AVOA techniques for fracture characterization

array:23 ["pii" => "S0016716914700789" "issn" => "00167169" "doi" => "10.1016/S0016-7169(14)70078-9" "estado" => "S300" "fechaPublicacion" => "2014-10-01" "aid" => "70078" "copyright" => "Universidad Nacional Autónoma de México" "copyrightAnyo" => "2014" "documento" => "article" "licencia" => "http://creativecommons.org/licenses/by-nc-nd/4.0/" "subdocumento" => "fla" "cita" => "Geofisica Internacional. 2014;53:457-71" "abierto" => array:3 [ "ES" => true "ES2" => true "LATM" => true ] "gratuito" => true "lecturas" => array:2 [ "total" => 1164 "formatos" => array:3 [ "EPUB" => 34 "HTML" => 620 "PDF" => 510 ] ] "itemSiguiente" => array:18 [ "pii" => "S0016716914700790" "issn" => "00167169" "doi" => "10.1016/S0016-7169(14)70079-0" "estado" => "S300" "fechaPublicacion" => "2014-10-01" "aid" => "70079" "copyright" => "Universidad Nacional Autónoma de México" "documento" => "article" "licencia" => "http://creativecommons.org/licenses/by-nc-nd/4.0/" "subdocumento" => "fla" "cita" => "Geofisica Internacional. 2014;53:473-90" "abierto" => array:3 [ "ES" => true "ES2" => true "LATM" => true ] "gratuito" => true "lecturas" => array:2 [ "total" => 2147 "formatos" => array:3 [ "EPUB" => 42 "HTML" => 1448 "PDF" => 657 ] ] "en" => array:11 [ "idiomaDefecto" => true "titulo" => "Thermomagnetic monitoring of lithic clasts burned under controlled temperature and field conditions. Implications for archaeomagnetism" "tienePdf" => "en" "tieneTextoCompleto" => "en" "tieneResumen" => array:2 [ 0 => "es" 1 => "en" ] "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "473" "paginaFinal" => "490" ] ] "contieneResumen" => array:2 [ "es" => true "en" => true ] "contieneTextoCompleto" => array:1 [ "en" => true ] "contienePdf" => array:1 [ "en" => true ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0055" "etiqueta" => "Figure 11" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr11.jpeg" "Alto" => 2460 "Ancho" => 1581 "Tamanyo" => 237811 ] ] "descripcion" => array:1 [ "en" => "Palaeointensity results obtained in obsidian and sandstone specimens, (a) and (b) respectively, with the multispecimen method. Lines connect sister-specimen of the different lithologies samples. Plots represent the relative percentage differences between the pTRMs acquired at different lab fields (40 and 50μT) and the total TRM (acquired during burning). Intersection of each line with the horizontal axis (zero difference) determines the intensity of the Earth's magnetic field during the burning. (c) Mean ambient field estimation calculated from both lithologies." ] ] ] "autores" => array:1 [ 0 => array:2 [ "autoresLista" => "Ángel Carrancho, Juan Morales, Avto Goguitchaichvili, Rodrigo Alonso, Marcos Terradillos" "autores" => array:5 [ 0 => array:2 [ "nombre" => "Ángel" "apellidos" => "Carrancho" ] 1 => array:2 [ "nombre" => "Juan" "apellidos" => "Morales" ] 2 => array:2 [ "nombre" => "Avto" "apellidos" => "Goguitchaichvili" ] 3 => array:2 [ "nombre" => "Rodrigo" "apellidos" => "Alonso" ] 4 => array:2 [ "nombre" => "Marcos" "apellidos" => "Terradillos" ] ] ] ] ] "idiomaDefecto" => "en" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716914700790?idApp=UINPBA00004N" "url" => "/00167169/0000005300000004/v1_201412221101/S0016716914700790/v1_201412221101/en/main.assets" ] "itemAnterior" => array:18 [ "pii" => "S0016716914700777" "issn" => "00167169" "doi" => "10.1016/S0016-7169(14)70077-7" "estado" => "S300" "fechaPublicacion" => "2014-10-01" "aid" => "70077" "copyright" => "Universidad Nacional Autónoma de México" "documento" => "article" "licencia" => "http://creativecommons.org/licenses/by-nc-nd/4.0/" "subdocumento" => "fla" "cita" => "Geofisica Internacional. 2014;53:435-56" "abierto" => array:3 [ "ES" => true "ES2" => true "LATM" => true ] "gratuito" => true "lecturas" => array:2 [ "total" => 2420 "formatos" => array:3 [ "EPUB" => 37 "HTML" => 1858 "PDF" => 525 ] ] "en" => array:11 [ "idiomaDefecto" => true "titulo" => "Ca. 13 Ma strike-slip deformation in coastal Sonora from a large-scale, en-echelon, brittle-ductile, dextral shear indicator: implications for the evolution of the California rift" "tienePdf" => "en" "tieneTextoCompleto" => "en" "tieneResumen" => array:2 [ 0 => "es" 1 => "en" ] "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "435" "paginaFinal" => "456" ] ] "contieneResumen" => array:2 [ "es" => true "en" => true ] "contieneTextoCompleto" => array:1 [ "en" => true ] "contienePdf" => array:1 [ "en" => true ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0040" "etiqueta" => "Figure 8" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr8.jpeg" "Alto" => 1292 "Ancho" => 1568 "Tamanyo" => 126722 ] ] "descripcion" => array:1 [ "en" => "Paleomagnetic data. (a-c) Orthogonal demagnetization diagrams of selected samples. Open (closed) symbols are projections on the vertical (horizontal) plane. (d) Stereographic projection of the characteristic magnetization (in situ), here open (closed) symbols are projections on the upper (lower) hemisphere." ] ] ] "autores" => array:2 [ 0 => array:2 [ "autoresLista" => "David García-Martínez, Jaime Roldán Quintana, Hector Mendívil-Quijada" "autores" => array:3 [ 0 => array:2 [ "nombre" => "David" "apellidos" => "García-Martínez" ] 1 => array:2 [ "nombre" => "Jaime Roldán" "apellidos" => "Quintana" ] 2 => array:2 [ "nombre" => "Hector" "apellidos" => "Mendívil-Quijada" ] ] ] 1 => array:2 [ "autoresLista" => "Roberto Stanley Molina Garza" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Roberto Stanley Molina" "apellidos" => "Garza" ] ] ] ] ] "idiomaDefecto" => "en" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716914700777?idApp=UINPBA00004N" "url" => "/00167169/0000005300000004/v1_201412221101/S0016716914700777/v1_201412221101/en/main.assets" ] "en" => array:18 [ "idiomaDefecto" => true "titulo" => "AVOA techniques for fracture characterization" "tieneTextoCompleto" => true "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "457" "paginaFinal" => "471" ] ] "autores" => array:1 [ 0 => array:4 [ "autoresLista" => "Vladimir Sabinin" "autores" => array:1 [ 0 => array:4 [ "nombre" => "Vladimir" "apellidos" => "Sabinin" "email" => array:1 [ 0 => "vsabinin@yahoo.com" ] "referencia" => array:1 [ 0 => array:2 [ "etiqueta" => "*" "identificador" => "cor0005" ] ] ] ] "afiliaciones" => array:1 [ 0 => array:2 [ "entidad" => "Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, Col. San Bartolo Atepehuacan, C.P. 07730, México D.F., México." "identificador" => "aff0005" ] ] "correspondencia" => array:1 [ 0 => array:3 [ "identificador" => "cor0005" "etiqueta" => "*" "correspondencia" => "Corresponding author:" ] ] ] ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0070" "etiqueta" => "Figure 14" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr14.jpeg" "Alto" => 852 "Ancho" => 1127 "Tamanyo" => 142222 ] ] "descripcion" => array:1 [ "en" => "Errors for the asymmetric set of azimuths; the noise, the upper boundary, VP3=3200, and fixed θmin=0." ] ] ] "textoCompleto" => "IntroductionThe analysis of azimuthal variation in reflection coefficients, or AVOA analysis (Amplitude Versus Offset and Azimuth), is widely applied for detecting and mapping highly fractured zones with azimuthally-oriented vertical cracks (<a class="elsevierStyleCrossRef" href="#bib0035">Mallik et al., 1998</a>; <a class="elsevierStyleCrossRef" href="#bib0025">Jenner, 2002</a>; <a class="elsevierStyleCrossRef" href="#bib0060">Sabinin & Chichinina, 2008</a>). The AVOA techniques are based on the <a class="elsevierStyleCrossRef" href="#bib0045">Rüger (1998)</a> approximation for the reflection coefficients in HTI medium, and give principal symmetry directions of HTI medium.Here, the computational aspects of AVOA techniques are considered, namely, applying amplitudes instead of reflection coefficients, smoothing the amplitudes, an incidence angle estimation, methods for obtaining the symmetry-axis angle, synthetic data for testing techniques, and a numerical experiment for investigating properties of the techniques. A new computational method for obtaining the symmetry-axis angle and a new filter for smoothing are suggested. All considered techniques are compared in synthetic anisotropic seismic data with noise, and without noise. The suggested new technique proved better than the others.BackgroundThe methodology of AVOA analysis is based on the concept of azimuthal anisotropy caused for the most part by parallel vertical fractures. It leads to the azimuthal anisotropy of amplitudes, in particular, to azimuthal variation in reflection coefficients. Let a fractured reservoir be represented by a model of a transversely isotropic medium with horizontal symmetry axis (HTI medium). The PP-wave reflection coefficient R at the interface (or at the reflecting boundary) between weakly anisotropic HTI media (or between an isotropic medium and an anisotropic HTI medium) is defined by the approximate formula (<a class="elsevierStyleCrossRef" href="#bib0045">Rüger, 1998</a>):<elsevierMultimedia ident="eq0005"></elsevierMultimedia>where θ is the incidence angle, and ϕ is the source-receiver-line azimuth with respect to the coordinate axis x. The term A is the normal-incidence reflection coefficient<elsevierMultimedia ident="eq0010"></elsevierMultimedia>where Z≡ρVP‖ is the vertical P-wave impedance, VP‖ is the vertical velocity (or velocity in the isotropy plane) of the P-wave, ρ i s density, Δ denotes the difference between the values of a parameter below and above the reflecting boundary, and the bar … indicates average of these values. VP‖=VP in the isotropic media.The coefficient B(ϕ) is a so-called AVO gradient, which can be written (<a class="elsevierStyleCrossRef" href="#bib0045">Rüger, 1998</a>) as<elsevierMultimedia ident="eq0015"></elsevierMultimedia>where ϕ0 is the angle of the symmetry axis with the x−-axis. The term Biso is the AVO-gradient isotropic part (equal to the AVO gradient for isotropic media), and Bani is the anisotropic part of the AVO gradient.The coefficient C(ϕ) can be written (<a class="elsevierStyleCrossRef" href="#bib0045">Rüger, 1998</a>) as,<elsevierMultimedia ident="eq0020"></elsevierMultimedia>where α≡ΔVP‖/2V¯P‖,β=12ΔεV, and γ=12ΔδV.Above, Thomsen-style anisotropy parameters ε(V), and δ(V) are negative for HTI media, and they are equal to zero for isotropic media.The main problem of AVOA analysis is to estimate the symmetry-axis angle ϕ0 from surface seismic data of amplitudes using numerical techniques.The techniques of AVOA are based on <a class="elsevierStyleCrossRef" href="#eq0005">equations (1)</a> – <a class="elsevierStyleCrossRef" href="#eq0020">(4)</a>. Note that <a class="elsevierStyleCrossRef" href="#eq0005">equation (1)</a> is intended for calculation of reflection coefficients, while in real data, one operates with amplitudes of reflected waves, not with reflection coefficients. This brings some problems which are discussed in the next section.Using amplitudes instead of reflection coefficientsWhile the background of AVOA analysis is based on Rüger's equation for the reflection coefficient <a class="elsevierStyleCrossRef" href="#eq0005">(1)</a>, in real data, AVOA analysis should use signal amplitudes. It is true that the amplitude is not equal to the reflection coefficient. Moreover, no picked instantaneous amplitude (sample) in the signal can be used instead of the reflection coefficient because the signal changes its form during propagation for many reasons. It seems that one should use an integral amplitude characteristic of the signal which adequately corresponds to the reflection coefficient. Let's call this characteristic simply by amplitude and denote it as P.<a name="p459"></a>The estimated value of ϕ0 is very sensitive to the definition of P, especially for data with noise. I suggest the following procedure for definition of P which gives good and stable results. The procedure calculates an average value of a signal envelope in a time window. In calculating the envelope, the Fourier transform of this signal is used: F = F+ + F−, where F is spectrum, F+ is the part of spectrum corresponding to positive frequencies (ω≥0), and F− is the part of negative frequencies. The envelope of the signal is given by the absolute value of inverse Fourier transform of the spectrum with double F+, and F−≡0 (<a class="elsevierStyleCrossRef" href="#bib0065">Sheriff & Geldart, 1983</a>).The sign of envelope is positive; therefore this approach is applicable only to seismograms with the constant sign of reflection coefficient in dependence on offset.For data with noise, the envelope is noisy, too (see <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>). Therefore, smoothing is necessary.<elsevierMultimedia ident="fig0005"></elsevierMultimedia>For smoothing, an algorithm of discrete transformations of wavelet by filters is applied. Four symmetric filters are constructed for it: the low-pass (h0) end high-pass (h1) analysis filters, and the low-pass (h2) and high-pass (h3) synthesis filters. The right-hand part of h0 consists from the filter derived by Abdelnour & Selesnick (2004). The left-hand part of h0 is symmetric to it. That is<elsevierMultimedia ident="eq0025"></elsevierMultimedia>where a=M/32,M=2/2,b=4a, and n=8. The central value is c=1−M.The high-pass analysis filter is constructed by formula h1(i) = (− 1)ih0 (n − i + 1) for i ≠ 0, and hl (0) = 0. The synthesis filters are calculated by formula h2(i) = (− 1)ih1 (i),h3 (i) = (− 1)ih0 (i), see (<a class="elsevierStyleCrossRef" href="#bib0010">Abdelnour & Selesnick, 2005</a>). The central values are h2 (0)=c and h3(0)=0.The smoothed signal is obtained by the decomposition algorithm; see <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a> (<a class="elsevierStyleCrossRef" href="#bib0075">WSBP, 2012</a>).<elsevierMultimedia ident="fig0010"></elsevierMultimedia>The input signal is x(j), j=1,…,m, m>>2n. It is decomposed into low and high components lo1(j) and hi1(j) in the first stage:<elsevierMultimedia ident="eq0030"></elsevierMultimedia>In the next stages (s=2,…, S), the each low component los-1(j) is decomposed by the same analysis filters.After all stages of decomposition finishing, the stages of applying the synthesis filters are fulfilled in reverse order (s = S, S - 1,…,1):<elsevierMultimedia ident="eq0035"></elsevierMultimedia>The output signal y (j) is obtained finally:<elsevierMultimedia ident="eq0040"></elsevierMultimedia>where the fitting amplitude coefficient p can be approximately estimated by the formula p=1+0.057S, where S is the number of stages.The advantage of this variant of discrete transformation algorithm in comparison with (<a class="elsevierStyleCrossRef" href="#bib0075">WSBP, 2012</a>) is the absence of shift functions in it due to applying the fully symmetric filters (i=-n,…,n).<a name="p460"></a>It must be noted that the algorithm gives unsatisfactory results at the edges of the signal because of truncating the filters in 2n edge points. Therefore, it is necessary m>>2n.The result of smoothing the signal of <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a> by the 3-stage algorithm is presented in <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>. The smoothness of resulting curve increases with increasing S. Also with increasing S, the algorithm slightly deforms the resulting impulse in comparison with the parent impulse without noise. Optimum in the smoothness and in the conservation of form is observed at the value S=3.<elsevierMultimedia ident="fig0015"></elsevierMultimedia>The limits of time window for calculating P with the help of envelope can be chosen by different ways. I use the following way. From the envelope of signal e(t), the maximum em and nearest local minimums, left el and right er, are calculated. The left limit of the time window is set in the point where e = el + 0.15(em−el), and the right limit – where e = er + 0.15(em−el), see vertical lines in <a class="elsevierStyleCrossRef" href="#fig0005">Figures 1</a>, <a class="elsevierStyleCrossRef" href="#fig0015">3</a>. Obviously, this algorithm correctly works only with smoothed signals.<a class="elsevierStyleCrossRef" href="#eq0005">Equation (1)</a> should be rewritten for using the amplitudes. If the source and the receivers are at the earth surface, then the amplitude of reflected PP-wave can be expressed as<elsevierMultimedia ident="eq0045"></elsevierMultimedia>where cg is the coefficient of geometrical spreading (divergence) from source to reflection point for this wave, cg = cg (θ, ϕ), Pini is the amplitude of the source (the initial amplitude), and R is the reflection coefficient, R = R(θ, ϕ) in the <a class="elsevierStyleCrossRef" href="#eq0005">equation (1)</a>.The amplitude for the normal-incidence wave can be written as<elsevierMultimedia ident="eq0050"></elsevierMultimedia>where cg0 is the normal-incidence coefficient of geometrical spreading, which does not depend on (θ, ϕ), and A is the normal-incidence reflection coefficient, A = const, see <a class="elsevierStyleCrossRef" href="#eq0005">equations (1)</a> - <a class="elsevierStyleCrossRef" href="#eq0010">(2)</a>. Then the reflection coefficient can be expressed as<elsevierMultimedia ident="eq0055"></elsevierMultimedia>Therefore the <a class="elsevierStyleCrossRef" href="#eq0005">equation (1)</a> for the reflection coefficient R transforms into the following equation for the amplitude P:<elsevierMultimedia ident="eq0060"></elsevierMultimedia>where m=P0/A, and rg≡(cg0 / cg)2. This equation should be used in the AVOA techniques instead of <a class="elsevierStyleCrossRef" href="#eq0005">(1)</a>.Note that cg can be expressed as cg = c(θ, ϕ)/r in 3D space, where r is a half of travel path from source to receiver, and c depends on the direction of wave propagation (for isotropic media, c=const). In assuming a weak anisotropy, one may assume a weak dependence of geometrical spreading on incidence angle: c≈const for a given source-to-receiver line with azimuth ϕ. Then, for a homogeneous medium above the reflecting boundary,<elsevierMultimedia ident="eq0065"></elsevierMultimedia>where z is the normal-incidence ray path, and cg0 ≡ c / z. It is the approximate formula for divergent correction.Also for multilayered media, the expressions for divergence correction can be found from <a class="elsevierStyleCrossRef" href="#bib0040">Newman (1973)</a>. A practical methodology for the P-wave geometrical-spreading correction in layered azimuthally anisotropic media can be found from <a class="elsevierStyleCrossRef" href="#bib0070">Xu & Tsvankin (2004)</a>.The incidence angle estimationIn the case of n isotropic layers above the reflecting boundary, one can obtain the incidence angle θ = θn from a solution of the following non-linear equation for a:<a name="p461"></a><elsevierMultimedia ident="eq0070"></elsevierMultimedia>where x0 is the half of offset, zi is the thickness of i-th layer, Vi is the velocity in i-th layer, and sin (θn)=aVn. For calculating the geometrical spreading, the travel path r=∑i=1nzi1−a2Vi2.In the case of the reflecting boundary being the lower boundary of anisotropic layer, the problem is more complicated because the last layer is anisotropic, and the velocity Vn depends on θn in it and is not known beforehand.The problem can be solved by <a class="elsevierStyleCrossRef" href="#bib0050">Sabinin (2012)</a>. An advantage of his method is that the value Vn in the anisotropic layer is unnecessary for calculating angle θn and path r. However, it uses additionally the impulse from the upper boundary of anisotropic layer what complicates the technique. It gives results not sufficiently better than the method <a class="elsevierStyleCrossRef" href="#eq0070">(8)</a>. Therefore, I use the simple method <a class="elsevierStyleCrossRef" href="#eq0070">(8)</a> here with setting an approximate value of Vn.The methods for estimation of symmetry axis angle by AVOAUsually, 3D seismic data used in AVOA analysis are received from a system of receivers and sources spacing in nodes of a rectangle grid at the surface. The symmetry axis angle is calculated for a small square (for a bin) including a node of the grid, by using seismic traces which have the Common Middle Point (CMP) located in this bin. If such traces are few, then neighbor bins are combined into a superbin, and calculations are made for it. Therefore, a preliminary stage of the estimation is an extraction of seismic traces of the superbin from the seismic data for taking them into consideration.For numerical methods of estimation of symmetry axis angle, one can use <a class="elsevierStyleCrossRef" href="#eq0060">equation (6)</a> as in Rüger's form:<elsevierMultimedia ident="eq0075"></elsevierMultimedia>as in the power form:<elsevierMultimedia ident="eq0080"></elsevierMultimedia>where T* = rgP (θ, ϕ), T = (1−s)T*, s = sin2θ, t = cos2 (ϕ−ϕ0), a = P0, b* = mBiso, b = b*−a, c = mBani, d* = mα, d = d*−b*, e* = mΔδ(V)/2, e = e*−c, f=mΔε(V)/2−e* and m = P0/A.The methods vary by simplifying ways applied, and can be separated into Sectored methods (S and SR), Linear methods (L and LR), and General method (G), where the letter ‘R’ denotes that the Ruger's form <a class="elsevierStyleCrossRef" href="#eq0075">(9a)</a> is used instead of <a class="elsevierStyleCrossRef" href="#eq0080">(9b)</a>.Sectored methodsThe method SR was suggested by <a class="elsevierStyleCrossRef" href="#bib0035">Mallik et al. (1998)</a> for the case of three azimuths with using <a class="elsevierStyleCrossRef" href="#eq0005">equations (1)</a>, and <a class="elsevierStyleCrossRef" href="#eq0015">(3)</a>. It took its perfect form in the work by <a class="elsevierStyleCrossRef" href="#bib0060">Sabinin & Chichinina (2008)</a> who used <a class="elsevierStyleCrossRef" href="#eq0060">equations (6)</a>, <a class="elsevierStyleCrossRef" href="#eq0015">(3)</a>, and <a class="elsevierStyleCrossRef" href="#eq0020">(4)</a>. For this method, the traces of superbin are sorted by n azimuthal sectors. It is adopted that all traces of the sector have the same value of azimuth equal to the middle azimuth of the sector. Because of this, sectored methods introduce in ϕ0 an own error no more than a half of the sector size.Here, the method S applied to <a class="elsevierStyleCrossRef" href="#eq0080">equation (9b)</a> is presented. If in the sector of azimuth ϕ (j=1, …, n), there are kj traces with incidence angles θj (i=1,…, kj) in the last layer above the target boundary, then one can write from <a class="elsevierStyleCrossRef" href="#eq0080">(9b)</a> for this sector j:<elsevierMultimedia ident="eq0085"></elsevierMultimedia>where Tij is the value T calculated from the trace i in the sector j. In each sector, Bj1=mjBj,Cj1=mjCj where mj=Pj/A, and Pj,Bj1,Cj1 are the fitting constants.Having Tij and θi for all i in the sector j (kj≥3), one can calculate si=sin2θi, and then Pj, Bj1, and Cj1 from <a class="elsevierStyleCrossRef" href="#eq0085">(10)</a> by the least-squares method. For this, it is minimized the functional of error for each sector j:<elsevierMultimedia ident="eq0090"></elsevierMultimedia>For minimizing Fj, it is necessary to solve the system of three equations:<elsevierMultimedia ident="eq0095"></elsevierMultimedia>It gives: Cj1=b1f1−a1g1b12−a1c1,Bj1=f1−Cj1b1/a1, and Pj=u0BCj1−ABj1/kj, where a1=A2−Bkj,b1=AB−Ckj,c1=B2−Dkj,f1=Au0−u1kj,g1=Bu0−u2kj,<elsevierMultimedia ident="eq0100"></elsevierMultimedia><a name="p462"></a>These calculations should be made for all sectors.Then, the unknown value φ0 can be obtained from the system of equations of type <a class="elsevierStyleCrossRef" href="#eq0015">(3)</a>, see <a class="elsevierStyleCrossRef" href="#eq0080">(9b)</a>:<elsevierMultimedia ident="eq0105"></elsevierMultimedia>where j=1, …, n, and tj=cos2(φ−φ0). The unknown constants b1, and c1 have a sense: b1A=Biso−A, and c1A=Bani.<a class="elsevierStyleCrossRef" href="#eq0105">Equation (11)</a> is transformed into more convenient form:<elsevierMultimedia ident="eq0110"></elsevierMultimedia>where Uj=Bj1/Pj, b0 = b1 + 0.5c1, c0 = 0.5c1, g = cos(2φ0), h = sin(2φ0), gj = cos(2φj) and hj = sin(2φj).The system <a class="elsevierStyleCrossRef" href="#eq0110">(12)</a> has three unknowns (b0, c0, and φ0), therefore it should be n≥3 for obtaining solution. The system <a class="elsevierStyleCrossRef" href="#eq0110">(12)</a> has two solutions (two φ0 differing in π/2, and two c0 of opposite signs, correspondently), and is solved by the least-squares method, too. It is minimized the functional of error:<elsevierMultimedia ident="eq0115"></elsevierMultimedia>The following system of three equations should be solved:<elsevierMultimedia ident="eq0120"></elsevierMultimedia>It gives: tan2ϕ0≡hg=b1f1−a1f2b1f2−c1f1, c0=f1/(ga1+hb1), and b0=[u0−c0(Ag+Bh)]/n, where a1=A2−Cn, b1=AB−Dn, c1=B2−En, f1=Au0−u1n, f2=Bu0−u2n, A=∑j=1ngj,B=∑j=1nhj,C=∑j=1ngj2,<elsevierMultimedia ident="eq0125"></elsevierMultimedia>The condition for distinguishing symmetry-axis from fracture-strike directions is derived by <a class="elsevierStyleCrossRef" href="#bib0060">Sabinin & Chichinina (2008)</a>, and uses <a class="elsevierStyleCrossRef" href="#eq0020">equation (4)</a>. Here it is presented in more general form.In terms of <a class="elsevierStyleCrossRef" href="#eq0080">equations (9b)</a>, <a class="elsevierStyleCrossRef" href="#eq0085">(10)</a>, and <a class="elsevierStyleCrossRef" href="#eq0105">(11)</a>, <a class="elsevierStyleCrossRef" href="#eq0020">equation (4)</a> can be written as<elsevierMultimedia ident="eq0130"></elsevierMultimedia>where j = 1, …, n, d1 = (α − Biso)/A, e1 = (Δδ(V)/2 − Bani)/A, and f1=(Δε(V)−Δδ(V))/(2A).When substituting the value φ0±π2 instead of φ0 into <a class="elsevierStyleCrossRef" href="#eq0130">equation (14)</a>, the sign of the second term e1tj switches to the opposite sign, because <a class="elsevierStyleCrossRef" href="#eq0130">equation (14)</a> takes the form<elsevierMultimedia ident="eq0135"></elsevierMultimedia>The last term of <a class="elsevierStyleCrossRef" href="#eq0105">equation (11)</a>c1tj switches its sign, too. One can combine Δδ(V) = 2A(c1 + e1) from definitions to <a class="elsevierStyleCrossRef" href="#eq0105">equations (11)</a>, <a class="elsevierStyleCrossRef" href="#eq0130">(14)</a>, and conclude that the sign of Δδ(V) is switched, too. For calculating Δδ(V), it should be solved system <a class="elsevierStyleCrossRef" href="#eq0130">(14)</a> which is similar to <a class="elsevierStyleCrossRef" href="#eq0085">(10)</a> by the method of solution.If the HTI layer is situated between isotropic layers then Δδ(V) must be negative for upper reflecting boundary of the HTI layer, and positive for lower boundary. If the calculated value of Δδ(V) has this sign then φ0 is the symmetry-axis angle. In opposite case, it is the fracture-strike direction.It must be noted that Δε(V) = 2A(c1 + e1 + f1), and also can be used for distinguishing solutions because ε(V) and δ(V) have the same sign.The formal condition that the second derivative of functional <a class="elsevierStyleCrossRef" href="#eq0115">(13)</a> must be positive in the minimum of functional can also be applied. Because of errors in data, it should be used as an additional condition to previous ones, and should have a form ∂2F/∂ϕ02> a small value.Linear methodsThe method LR was suggested by <a class="elsevierStyleCrossRef" href="#bib0025">Jenner (2002)</a> for <a class="elsevierStyleCrossRef" href="#eq0005">equation (1)</a>. It is not needed in sectoring data. All traces of superbin are taken into consideration together.Here, it is applied to <a class="elsevierStyleCrossRef" href="#eq0080">equation (9b)</a>, the method L. <a class="elsevierStyleCrossRef" href="#eq0080">Equation (9b)</a> is truncated after a line part respecting s. If the superbin has n traces(i=1, …, n), with incidence angles θi at the target boundary, and with azimuthal angles <a name="p463"></a>φi, then one can write the result of truncation in the form:<elsevierMultimedia ident="eq0140"></elsevierMultimedia>where Ti is the value T calculated from the trace i, si = sin2θi, b0 = b + 0.5c, c0 = 0.5c, g = cos(2φ0), h = sin(2φ0), gi = cos(2φi), and hi = sin(2φi).The values si, gi, and hi are known because they can be calculated from headers of seismograms and parameters of medium. Let us consider the functional of error:<elsevierMultimedia ident="eq0145"></elsevierMultimedia>Functional F must be minimized over parameters a, b0, c0, and φ0. For this, it is necessary to solve the system of four equations:<elsevierMultimedia ident="eq0150"></elsevierMultimedia>Solution of system <a class="elsevierStyleCrossRef" href="#eq0150">(17)</a> gives the equation for obtaining φ0:<elsevierMultimedia ident="eq0155"></elsevierMultimedia>where A1=a1b1−a22,B1=a1c1−a32, A2 = a1b2 − a2a3, H1 = F2a1 − F1a2, and H2 = F3a1 − F1a3 in which a1 = nB − A2, b1 = nI − D2, c1 = nJ − E2, a2 = nG − AD, b2 = nK − ED, a3 = nH − AE, F1 = nf1 − Af0, F2 = nj2 − Df0, and F3 = nf3 − Ef0, and finally:<elsevierMultimedia ident="eq0160"></elsevierMultimedia>The other parameters are:<elsevierMultimedia ident="eq0165"></elsevierMultimedia>and a=(f0−b0A−c0gD−c0hE)/n.From <a class="elsevierStyleCrossRef" href="#eq0155">(18)</a>, one can see that the solution φ0 has a period of π2. This value of the period means that we must use an additional condition for understanding what value φ0 is the symmetry-axis azimuth. This condition may be Bani>0 if VS / VP>0.56 (<a class="elsevierStyleCrossRef" href="#bib0015">Chichinina et al., 2003</a>). In general case, it can be the condition ∂2F/∂ϕ02> a small positive value, where F is the functional of error <a class="elsevierStyleCrossRef" href="#eq0145">(16)</a>.General method (G)The method is constructed by analogy with the GM method by <a class="elsevierStyleCrossRef" href="#bib0055">Sabinin (2013)</a>. It is not needed in sectoring, too. All traces of superbin are taken into consideration together. If the superbin has n traces (i=1, …, n), with incidence angles θi at the target boundary, and with azimuthal angles φi, then <a class="elsevierStyleCrossRef" href="#eq0080">equation (9b)</a> can be written as:<elsevierMultimedia ident="eq0170"></elsevierMultimedia>where Ti is the value T calculated from the trace i, si=sin2θi, and ti=cos2(φi−φ0).Let us consider the functional of error:<elsevierMultimedia ident="eq0175"></elsevierMultimedia>Functional F must be minimized over parameters a, b, c, d, e, f, and φ0. For this, it is necessary to solve the system of seven equations:<elsevierMultimedia ident="eq0180"></elsevierMultimedia>The six first equations of system <a class="elsevierStyleCrossRef" href="#eq0180">(20)</a> give a line system for deriving expressions for the parameters a, b, c, d, e, and f (for details, see <a class="elsevierStyleCrossRef" href="#sec055">Appendix</a>).The last equation of <a class="elsevierStyleCrossRef" href="#eq0180">(20)</a> can be transformed into a non-linear equation for obtaining φ0 (for details, see <a class="elsevierStyleCrossRef" href="#sec055">Appendix</a>).Thus, system <a class="elsevierStyleCrossRef" href="#eq0180">(20)</a> is non-linear on φ0, and is solved by the method of bisecting. It has more than one solution usually. From these local solutions, one chooses that one which gives a minimum for functional <a class="elsevierStyleCrossRef" href="#eq0175">(19)</a>.As was observed from calculations, the solutions of system <a class="elsevierStyleCrossRef" href="#eq0180">(20)</a> near the symmetry axis angle, and near the fracture strike angle give close values of functional <a class="elsevierStyleCrossRef" href="#eq0175">(19)</a>. It means that additional criterions are practically needed for separating these directions. For the case <a name="p464"></a>of HTI layer situated between isotropic layers, it can be the condition of negative values for calculated ε(V) and δ(V) in the anisotropic layer, as above. For this, from definitions to <a class="elsevierStyleCrossRef" href="#eq0080">equations (9b)</a> and <a class="elsevierStyleCrossRef" href="#eq0010">(2)</a>, one can calculate from the solution of <a class="elsevierStyleCrossRef" href="#eq0180">(20)</a> at the interface:<elsevierMultimedia ident="eq0185"></elsevierMultimedia><elsevierMultimedia ident="eq0190"></elsevierMultimedia>In the case of interface between anisotropic layers, it is needed additionally to know the predefined signs of Δε(V), and Δδ(V) for comparison.The additional criterion can also be the maximum of second derivative of functional <a class="elsevierStyleCrossRef" href="#eq0175">(19)</a>, ∂F/∂φ0.Comparing the AVOA techniquesThe techniques using the methods above for estimation of symmetry axis angle were compared in ability to give the most precise value of φ0 for HTI medium. At present, reliable field methods of obtaining φ0 do not exist. Therefore, I generated synthetic seismograms for an artificial three-layer medium with the anisotropic layer in the middle by applying the technique by <a class="elsevierStyleCrossRef" href="#bib0050">Sabinin (2012)</a> of 2D wave modeling. I set φ0=60°, and derived models of the anisotropic layer for different values of φj by rotating the stiffness tensor for anisotropic layer (<a class="elsevierStyleCrossRef" href="#bib0030">MacBeth, 1999</a>) around z axis relatively to φ0. Anisotropic parameters en=0.35, and et=0.2 (see <a class="elsevierStyleCrossRef" href="#bib0030">MacBeth, 1999</a>) were used in the stiffness tensor.Host rock velocity VP in three layers from above had the values 3200, 4000, and 4800 (the other variant was 3200), m/s, VS was twice less, densities were equal, and thicknesses of two first layers were 1600, and 400m. A source of explosion type generated one Ricker impulse of frequency 30Hz. Receivers were spaced over every 100m beginning from the source, and they measured z-component of velocity. There were 50 offsets, and 50 traces in each seismogram.There were three goals: to investigate how the techniques behave on different sets of incidence angles, how the techniques are influenced by non-symmetry in φj relatively to φ0, and how the techniques are influenced by noise.Therefore, for the first goal, I made calculations of φ0 for different intervals of offsets: from a minimum offset till a maximum offset, provided the minimum offset was fixed at the number one, and the number of maximum offset was changed from number 50 down to 3 in one set of the intervals; and the maximum offset was fixed at the 50-th, and the minimum offset was changed from number 1 to 48 in the other set of the intervals. Naturally, the maximum incidence angle θmax corresponding to the maximum offset, and the minimum incidence angle θmin corresponding to the minimum offset was also correspondently changed in these sets of offsets.For the second goal, I obtained different sets of the synthetic seismograms corresponding to different azimuths, one seismogram for each azimuth. The sets of azimuths were uniform, and differed by symmetry. I did not aim to find the best or the worst set from them. I only supposed that a symmetric set can be better than an asymmetric one. I kept for testing the symmetric set of azimuths φj={−150°, −120°, −90°, −60°, −30°, 0°, 30°, 60°, 90°, 120°, 150°, 180°}, and the asymmetric set φj={85°, 95°, 105°, 115°, 125°, 135°, 145°, 155°, 165°}.For the third goal, I took the best variant for the symmetric set of seismograms to eliminate the errors as due to the non-symmetry, as due to a finite-difference simulation when applying the artificial noise. The FD simulation by <a class="elsevierStyleCrossRef" href="#bib0050">Sabinin (2012)</a> uses PML boundary conditions which give non-visible (see <a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a>) but nonzero waves reflected from the boundaries of area. This slightly distorts the form of some synthetic impulses.<elsevierMultimedia ident="fig0020"></elsevierMultimedia>For the synthetic seismic data being quasireal, I added a random Gauss normal noise to the seismograms generated, different for each seismogram. Maximum amplitude of the noise was chosen as 10% of the maximum amplitude of the wave reflected from the top boundary of the anisotropic layer in the first trace of seismogram.Finally, I added the noise to the seismograms of the asymmetric set.All seismograms were smoothed by filters <a class="elsevierStyleCrossRef" href="#eq0025">(5)</a> in the techniques. High-frequency components of the noise are eliminated well after smoothing, as shown in <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>. It is principally impossible to eliminate low frequencies compared with the frequency of signal. Therefore, the signal after smoothing remains slightly deformed. I suppose that just these deformations affect the estimated value of φ0 in the case of noise.<a name="p465"></a>The same sets of the time windows were used for all the techniques, and for all intervals of offsets.As illustration, in <a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a>, the seismogram without noise for azimuth 5° is presented for the variant of VP3=4800 m/s; and in <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>, the seismogram with noise for azimuth 30° is presented for the variant of VP3=3200 m/s.<elsevierMultimedia ident="fig0025"></elsevierMultimedia>As one can see from <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>, the amplitudes of noise reach really up to 50% of the maximum wave amplitudes in the middle traces, and up to 100% in the far traces.The techniques were applied as to upper (1050 ms), as to down boundary (1250 ms) of the anisotropic layer.In <a class="elsevierStyleCrossRef" href="#fig0030">Figures 6</a>, <a class="elsevierStyleCrossRef" href="#fig0035">7</a>, the error of estimated φ0 in degrees (difference with the correct value 60°) is presented for the symmetric set of azimuths and the upper boundary, variant VP3=3200. <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a> is for fixed θmin=0°, and <a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a> is for fixed θmax=56.853°. The sectored methods show some instability for small values of θmax−θmin in comparison with the others. All methods increase the error in the case of small θmax (<a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>).<elsevierMultimedia ident="fig0030"></elsevierMultimedia><elsevierMultimedia ident="fig0035"></elsevierMultimedia>For the lower boundary and in the variant VP3=4800, the general and linear methods also show increasing errors for small θmax, and small θmax−θmin, see <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a>, and <a class="elsevierStyleCrossRef" href="#fig0045">Figure 9</a>. However, the errors of these methods are sufficiently less than of the sectored methods.<elsevierMultimedia ident="fig0040"></elsevierMultimedia><elsevierMultimedia ident="fig0045"></elsevierMultimedia>In <a class="elsevierStyleCrossRef" href="#fig0050">Figures 10</a>, <a class="elsevierStyleCrossRef" href="#fig0055">11</a>, the variant of <a class="elsevierStyleCrossRef" href="#fig0030">Figs. 6</a>, <a class="elsevierStyleCrossRef" href="#fig0035">7</a> with the added noise is presented. The sectored methods demonstrate so great errors and instability that can not be recommended for applying. The other methods show large errors only for small θmax (less than 30°).<elsevierMultimedia ident="fig0050"></elsevierMultimedia><elsevierMultimedia ident="fig0055"></elsevierMultimedia>The asymmetric set of azimuths is presented by results in <a class="elsevierStyleCrossRef" href="#fig0060">Figures 12</a>–<a class="elsevierStyleCrossRef" href="#fig0075">15</a>. The variant of upper boundary and VP3=3200 without noise is presented in <a class="elsevierStyleCrossRef" href="#fig0060">Figures 12</a>, <a class="elsevierStyleCrossRef" href="#fig0065">13</a>, and the same with the noise – in <a class="elsevierStyleCrossRef" href="#fig0070">Figures 14</a>, <a class="elsevierStyleCrossRef" href="#fig0075">15</a>.<elsevierMultimedia ident="fig0060"></elsevierMultimedia><elsevierMultimedia ident="fig0065"></elsevierMultimedia><elsevierMultimedia ident="fig0070"></elsevierMultimedia><elsevierMultimedia ident="fig0075"></elsevierMultimedia>Typical peculiarities of the asymmetric set are: great errors of the sectored methods with instability in noised data, and stable large errors of the linear methods (up to 7°). The general method remains of small errors. The noise causes instability of all methods in the interval of θmax<36°, provided even the general method (G) gives large errors in this interval.<a name="p466"></a><a name="p467"></a><a name="p468"></a><a name="p469"></a><a name="p470"></a>Discussion and conclusionSome unexpected results were obtained. The first is that the sectored S and SR methods are failed. They can be used only in seismic data without noise, and for mainly symmetric distributions of azimuths φ in the 3D data (<a class="elsevierStyleCrossRef" href="#fig0030">Figures 6</a>–<a class="elsevierStyleCrossRef" href="#fig0045">9</a>). This is too ideal conditions.The second is that the linear L and LR methods have an additional nearly constant error in mainly asymmetric distributions of azimuths φ in the data (<a class="elsevierStyleCrossRef" href="#fig0060">Figures 12</a>–<a class="elsevierStyleCrossRef" href="#fig0075">15</a>). This error is probably connected with the truncation of high terms in <a class="elsevierStyleCrossRef" href="#eq0005">equation (1)</a> of Ruger, because the general method G has not such error. Therefore, the linear methods should be applied to azimuthally symmetric data.The third is that the smoothing data with noise by simple filters <a class="elsevierStyleCrossRef" href="#eq0025">(5)</a> gives relatively stable estimated values of φ0 in a wide interval of incidence angles θ for the methods L, LR, and G (<a class="elsevierStyleCrossRef" href="#fig0050">Figures 10</a>, <a class="elsevierStyleCrossRef" href="#fig0055">11</a>, <a class="elsevierStyleCrossRef" href="#fig0070">14</a>, <a class="elsevierStyleCrossRef" href="#fig0075">15</a>). The interval of instability is near the normal incidence, and has a width of θmax<40°, different in different variants (<a class="elsevierStyleCrossRef" href="#fig0050">Figures 10</a>, <a class="elsevierStyleCrossRef" href="#fig0070">14</a>). For data without noise, this interval is θmax<10° (<a class="elsevierStyleCrossRef" href="#fig0030">Figures 6</a>, <a class="elsevierStyleCrossRef" href="#fig0040">8</a>). Presence of the interval of instability is an intrinsic property of the formula <a class="elsevierStyleCrossRef" href="#eq0005">(1)</a> in connection with the least-squares method. Errors in amplitudes become relatively more with decreasing θ in definition of φ0 by <a class="elsevierStyleCrossRef" href="#eq0005">equation (1)</a>.The results show a superior of the general method (G). On the whole, its errors are less than of the others. Unfortunately, it has an intrinsic problem of choosing the right solution from the local solutions of non-linear system <a class="elsevierStyleCrossRef" href="#eq0180">(20)</a>. All criterions described above do not guarantee the correct choosing. It is especially difficult in the interval of instability. All the methods have such problem of distinguishing solutions. The best in this sense is the method L. Its criterions are failed very rarely. Therefore, I recommend applying the method G in a coupling with the method L: after estimation of φ0 by L, the value φ0 is defined more precisely by G with expertly taking into consideration the local solutions of <a class="elsevierStyleCrossRef" href="#eq0180">(20)</a>. The other recommendation is to avoid the interval of instability.In applying to field data, the techniques can give worse results. The real data have much more interferences of waves than the synthetic data. It is practically impossible to clear each interfered wave of the other by filters. Distorted by this way impulses can lead to unpredictable results." "textoCompletoSecciones" => array:1 [ "secciones" => array:12 [ 0 => array:2 [ "identificador" => "xres400573" "titulo" => "Resumen" ] 1 => array:2 [ "identificador" => "xpalclavsec378193" "titulo" => "Palabras clave" ] 2 => array:2 [ "identificador" => "xres400574" "titulo" => "Abstract" ] 3 => array:2 [ "identificador" => "xpalclavsec378194" "titulo" => "Key words" ] 4 => array:2 [ "identificador" => "sec0005" "titulo" => "Introduction" ] 5 => array:2 [ "identificador" => "sec0010" "titulo" => "Background" ] 6 => array:2 [ "identificador" => "sec0015" "titulo" => "Using amplitudes instead of reflection coefficients" ] 7 => array:2 [ "identificador" => "sec0020" "titulo" => "The incidence angle estimation" ] 8 => array:3 [ "identificador" => "sec0025" "titulo" => "The methods for estimation of symmetry axis angle by AVOA" "secciones" => array:3 [ 0 => array:2 [ "identificador" => "sec0030" "titulo" => "Sectored methods" ] 1 => array:2 [ "identificador" => "sec0035" "titulo" => "Linear methods" ] 2 => array:2 [ "identificador" => "sec0040" "titulo" => "General method (G)" ] ] ] 9 => array:2 [ "identificador" => "sec0045" "titulo" => "Comparing the AVOA techniques" ] 10 => array:2 [ "identificador" => "sec0050" "titulo" => "Discussion and conclusion" ] 11 => array:1 [ "titulo" => "References" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "fechaRecibido" => "2013-09-02" "fechaAceptado" => "2013-12-05" "PalabrasClave" => array:2 [ "es" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Palabras clave" "identificador" => "xpalclavsec378193" "palabras" => array:4 [ 0 => "AVOA" 1 => "medio HTI" 2 => "anisotropía sísmica" 3 => "caracterización de yacimientos fracturados" ] ] ] "en" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Key words" "identificador" => "xpalclavsec378194" "palabras" => array:4 [ 0 => "AVOA" 1 => "HTI medium" 2 => "seismic anisotropy" 3 => "fracture-reservoir characterization" ] ] ] ] "tieneResumen" => true "resumen" => array:2 [ "es" => array:2 [ "titulo" => "Resumen" "resumen" => "Se tomaron en consideración distintos aspectos de algunas técnicas computacionales para el análisis AVOA (Amplitud Versus Offset y Azimut), para la composición de fracturas, en particular: utilizando amplitudes en lugar de coeficientes de refección, suavizando los datos sísmicos y el método de la estimación numérica para calcular la dirección. Se estimó un nuevo método de cálculo y se indica un nuevo método suavizado. Se compararan distintos métodos de cálculo en los datos sintéticos de superficie de reflección, con y sin ruido. Se obtuvieron propiedades de los métodos numéricos, dependientes de conjuntos distintos de los azimut y los offset. Se muestra una superioridad del nuevo método." ] "en" => array:2 [ "titulo" => "Abstract" "resumen" => "Different aspects of computational techniques for AVOA analysis (Amplitude Versus Offset and Azimuth) for fracture characterization are considered, in particular: using amplitudes instead of reflection coefficients, smoothing seismic data, and numerical methods for estimation of fracture directions. A new computational method and a new filter for smoothing are suggested. The different computational methods are compared in synthetic reflection surface data with noise, and without noise. Properties of the numerical methods in dependence on different sets of azimuths and offsets are obtained. It is shown a superiority of the new method.<a name="p458"></a>" ] ] "apendice" => array:1 [ 0 => array:1 [ "seccion" => array:1 [ 0 => array:4 [ "apendice" => "Let's define:<elsevierMultimedia ident="eq0195"></elsevierMultimedia> Then, from the first six equations of system <a class="elsevierStyleCrossRef" href="#eq0180">(20)</a>, one can derive the formulas for unknown parameters:<elsevierMultimedia ident="eq0200"></elsevierMultimedia>where a01 = A2−Cn, a02 = AB−Dn, a02 = AC−Fn, a04 = AD−Gn, a05 = AE−Hn, b01 = B2−En, a02 = BC− Gn, b03 = BD−Hn, b04 = BE−Kn, c01 = C2−Ln, c02 = CD−Mn, c03 = CE−Nn, d01 = D2−Nn, d02 = DE−On, e01 = E2−Pn, k1 = AU0−U1n, k2 = BU0−U2n, k3 = CU0−U3n, k4 = DU0−U4n, k5 = EU0−U5n,<elsevierMultimedia ident="eq0205"></elsevierMultimedia><a name="p471"></a> The seventh equation of system <a class="elsevierStyleCrossRef" href="#eq0180">(20)</a> takes a form:<elsevierMultimedia ident="eq0210"></elsevierMultimedia>where<elsevierMultimedia ident="eq0215"></elsevierMultimedia>" "etiqueta" => "Appendix" "titulo" => "Solution of system <a class="elsevierStyleCrossRef" href="#eq0180">(20)</a>" "identificador" => "sec055" ] ] ] ] "multimedia" => array:58 [ 0 => array:7 [ "identificador" => "fig0005" "etiqueta" => "Figure 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 613 "Ancho" => 882 "Tamanyo" => 79972 ] ] "descripcion" => array:1 [ "en" => "A signal with noise (thin line) in time, and its envelope." ] ] 1 => array:7 [ "identificador" => "fig0010" "etiqueta" => "Figure 2" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr2.jpeg" "Alto" => 188 "Ancho" => 882 "Tamanyo" => 22724 ] ] "descripcion" => array:1 [ "en" => "The 3-stage decomposition algorithm." ] ] 2 => array:7 [ "identificador" => "fig0015" "etiqueta" => "Figure 3" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr3.jpeg" "Alto" => 610 "Ancho" => 882 "Tamanyo" => 67468 ] ] "descripcion" => array:1 [ "en" => "The signal with noise (thin line), the smoothed signal (thick line), and the envelope of smoothed signal." ] ] 3 => array:7 [ "identificador" => "fig0020" "etiqueta" => "Figure 4" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr4.jpeg" "Alto" => 1134 "Ancho" => 1881 "Tamanyo" => 220774 ] ] "descripcion" => array:1 [ "en" => "Synthetic seismogram without noise. Azimuth 5°, VP3=4800. Axis x – time in ms, axis y – numbers of traces. Zero time is origin of the source impulse." ] ] 4 => array:7 [ "identificador" => "fig0025" "etiqueta" => "Figure 5" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr5.jpeg" "Alto" => 1136 "Ancho" => 1882 "Tamanyo" => 632005 ] ] "descripcion" => array:1 [ "en" => "Synthetic seismogram with added 10% noise. Azimuth 30°, VP3=3200. Axis x – time in ms, axis y – numbers of traces." ] ] 5 => array:7 [ "identificador" => "fig0030" "etiqueta" => "Figure 6" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr6.jpeg" "Alto" => 959 "Ancho" => 1253 "Tamanyo" => 103550 ] ] "descripcion" => array:1 [ "en" => "Errors for the symmetric set of azimuths; the upper boundary, and fixed θmin = 0." ] ] 6 => array:7 [ "identificador" => "fig0035" "etiqueta" => "Figure 7" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr7.jpeg" "Alto" => 850 "Ancho" => 1133 "Tamanyo" => 95915 ] ] "descripcion" => array:1 [ "en" => "Errors for the symmetric set of oe azimuths; the upper boundary, and fixed θmax = 56.853°." ] ] 7 => array:7 [ "identificador" => "fig0040" "etiqueta" => "Figure 8" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr8.jpeg" "Alto" => 864 "Ancho" => 1130 "Tamanyo" => 98451 ] ] "descripcion" => array:1 [ "en" => "Errors for the symmetric set of azimuths; the lower boundary, variant VP3=4800, and fixed θmin=0." ] ] 8 => array:7 [ "identificador" => "fig0045" "etiqueta" => "Figure 9" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr9.jpeg" "Alto" => 848 "Ancho" => 1133 "Tamanyo" => 108309 ] ] "descripcion" => array:1 [ "en" => "Errors for the symmetric set of azimuths; the lower boundary, variant VP3=4800, and fixed θmax=62.6°." ] ] 9 => array:7 [ "identificador" => "fig0050" "etiqueta" => "Figure 10" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr10.jpeg" "Alto" => 850 "Ancho" => 1122 "Tamanyo" => 130932 ] ] "descripcion" => array:1 [ "en" => "Errors for the symmetric set of azimuths; the noise, the upper boundary, VP3=3200, and fixed θmin=0." ] ] 10 => array:7 [ "identificador" => "fig0055" "etiqueta" => "Figure 11" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr11.jpeg" "Alto" => 850 "Ancho" => 1127 "Tamanyo" => 136925 ] ] "descripcion" => array:1 [ "en" => "Errors for the symmetric set of azimuths; the noise, the upper boundary, VP3=3200, and fixed θmax=56.853°." ] ] 11 => array:7 [ "identificador" => "fig0060" "etiqueta" => "Figure 12" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr12.jpeg" "Alto" => 852 "Ancho" => 1122 "Tamanyo" => 114490 ] ] "descripcion" => array:1 [ "en" => "Errors for the asymmetric set of azimuths; the upper boundary, variant VP3=3200, and fixed θmin=0." ] ] 12 => array:7 [ "identificador" => "fig0065" "etiqueta" => "Figure 13" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr13.jpeg" "Alto" => 847 "Ancho" => 1123 "Tamanyo" => 108102 ] ] "descripcion" => array:1 [ "en" => "Errors for the asymmetric set of azimuths; the upper boundary, variant VP3=3200, and fixed θmax=56.853°." ] ] 13 => array:7 [ "identificador" => "fig0070" "etiqueta" => "Figure 14" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr14.jpeg" "Alto" => 852 "Ancho" => 1127 "Tamanyo" => 142222 ] ] "descripcion" => array:1 [ "en" => "Errors for the asymmetric set of azimuths; the noise, the upper boundary, VP3=3200, and fixed θmin=0." ] ] 14 => array:7 [ "identificador" => "fig0075" "etiqueta" => "Figure 15" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr15.jpeg" "Alto" => 849 "Ancho" => 1124 "Tamanyo" => 129412 ] ] "descripcion" => array:1 [ "en" => "Errors for the asymmetric set of azimuths; the noise, the upper boundary, variant VP3=3200, and fixed θmax=56.853°." ] ] 15 => array:6 [ "identificador" => "eq0005" "etiqueta" => "(1)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Rθ,ϕ=A+Bϕsin2θ+Cϕsin2θtan2θ," "Fichero" => "si1.jpeg" "Tamanyo" => 2929 "Alto" => 19 "Ancho" => 314 ] ] 16 => array:6 [ "identificador" => "eq0010" "etiqueta" => "(2)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "A=ΔZ/2Z¯" "Fichero" => "si2.jpeg" "Tamanyo" => 1075 "Alto" => 20 "Ancho" => 100 ] ] 17 => array:6 [ "identificador" => "eq0015" "etiqueta" => "(3)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Bϕ=Biso+Banicos2ϕ−ϕ0," "Fichero" => "si6.jpeg" "Tamanyo" => 2082 "Alto" => 18 "Ancho" => 228 ] ] 18 => array:6 [ "identificador" => "eq0020" "etiqueta" => "(4)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Cϕ=α+βcos4ϕ−ϕ0+γsin2ϕ−ϕ0cos2ϕ−ϕ0," "Fichero" => "si7.jpeg" "Tamanyo" => 3662 "Alto" => 19 "Ancho" => 400 ] ] 19 => array:6 [ "identificador" => "eq0025" "etiqueta" => "(5)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "h0−n,…n=b,b,−a,a,b,b,a,−a,c,                −a,a,b,b,a,−a,b,b," "Fichero" => "si10.jpeg" "Tamanyo" => 3569 "Alto" => 39 "Ancho" => 295 ] ] 20 => array:5 [ "identificador" => "eq0030" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "lo1j=∑i=−nnh0ixi+j,hi1j=∑i=−nnh1ixi+j,j=1,…,m." "Fichero" => "si12.jpeg" "Tamanyo" => 5010 "Alto" => 125 "Ancho" => 192 ] ] 21 => array:5 [ "identificador" => "eq0035" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "los−1j=∑i=−nnh2ilosi+j    +∑i=−nnh3ihisi+j,    j=1,…,m." "Fichero" => "si13.jpeg" "Tamanyo" => 4893 "Alto" => 125 "Ancho" => 215 ] ] 22 => array:5 [ "identificador" => "eq0040" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "yj=lo0jp," "Fichero" => "si14.jpeg" "Tamanyo" => 1107 "Alto" => 16 "Ancho" => 109 ] ] 23 => array:5 [ "identificador" => "eq0045" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "P=cg2RPini," "Fichero" => "si15.jpeg" "Tamanyo" => 904 "Alto" => 21 "Ancho" => 88 ] ] 24 => array:5 [ "identificador" => "eq0050" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "P0=cg02APini," "Fichero" => "si16.jpeg" "Tamanyo" => 1039 "Alto" => 24 "Ancho" => 100 ] ] 25 => array:5 [ "identificador" => "eq0055" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "R=Acg02Pcg2P0." "Fichero" => "si17.jpeg" "Tamanyo" => 1165 "Alto" => 38 "Ancho" => 79 ] ] 26 => array:6 [ "identificador" => "eq0060" "etiqueta" => "(6)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "rgPθ,ϕ=P0+mBϕsin2θ+mCϕsin2θtan2θ," "Fichero" => "si18.jpeg" "Tamanyo" => 3307 "Alto" => 20 "Ancho" => 359 ] ] 27 => array:6 [ "identificador" => "eq0065" "etiqueta" => "(7)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "rg=cg02cg2=r2z2=1cos2θ," "Fichero" => "si19.jpeg" "Tamanyo" => 1589 "Alto" => 38 "Ancho" => 153 ] ] 28 => array:6 [ "identificador" => "eq0070" "etiqueta" => "(8)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "x0=a∑i=1nziVi1−a2Vi2," "Fichero" => "si20.jpeg" "Tamanyo" => 1895 "Alto" => 46 "Ancho" => 130 ] ] 29 => array:6 [ "identificador" => "eq0075" "etiqueta" => "(9a)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "T*=a+b*+cts+d*+e*t+ft2s2/1−s," "Fichero" => "si22.jpeg" "Tamanyo" => 2732 "Alto" => 20 "Ancho" => 340 ] ] 30 => array:6 [ "identificador" => "eq0080" "etiqueta" => "(9b)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "T=a+sb+ct+s2d+et+ft2," "Fichero" => "si23.jpeg" "Tamanyo" => 2103 "Alto" => 20 "Ancho" => 259 ] ] 31 => array:6 [ "identificador" => "eq0085" "etiqueta" => "(10)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Tij=Pj+Bj1si+Cj1si2," "Fichero" => "si24.jpeg" "Tamanyo" => 1554 "Alto" => 22 "Ancho" => 161 ] ] 32 => array:5 [ "identificador" => "eq0090" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Fj=∑i=1kjPj+Bj1si+Cj1si2−Tij2." "Fichero" => "si29.jpeg" "Tamanyo" => 2807 "Alto" => 52 "Ancho" => 245 ] ] 33 => array:5 [ "identificador" => "eq0095" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∂Fj/∂Pj=0,∂Fj/∂Bj1=0,∂Fj/∂Cj1=0." "Fichero" => "si30.jpeg" "Tamanyo" => 2652 "Alto" => 22 "Ancho" => 281 ] ] 34 => array:5 [ "identificador" => "eq0100" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "A=∑i=1kjsi,B=∑i=1kjsi2,C=∑i=1kjsi3,D=∑i=1kjsi4,u0=∑i=1kjTij,u1=∑i=1kjsiTij,andu2=∑i=1kjsi2Tij." "Fichero" => "si34.jpeg" "Tamanyo" => 7824 "Alto" => 111 "Ancho" => 317 ] ] 35 => array:6 [ "identificador" => "eq0105" "etiqueta" => "(11)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Bj1/Pj=b1+c1tj′" "Fichero" => "si35.jpeg" "Tamanyo" => 1215 "Alto" => 22 "Ancho" => 127 ] ] 36 => array:6 [ "identificador" => "eq0110" "etiqueta" => "(12)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Uj=b0+c0ggj+c0hhj′" "Fichero" => "si36.jpeg" "Tamanyo" => 1550 "Alto" => 17 "Ancho" => 173 ] ] 37 => array:6 [ "identificador" => "eq0115" "etiqueta" => "(13)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "F=∑j=1nb0+c0ggj+c0hhj−Uj2." "Fichero" => "si38.jpeg" "Tamanyo" => 2552 "Alto" => 54 "Ancho" => 252 ] ] 38 => array:5 [ "identificador" => "eq0120" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∂F/∂b0=0,∂F/∂c0=0,∂F/∂ϕ0=0." "Fichero" => "si39.jpeg" "Tamanyo" => 2228 "Alto" => 14 "Ancho" => 252 ] ] 39 => array:5 [ "identificador" => "eq0125" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "D=∑j=1ngjhj,E=∑j=1nhj2,u0=∑j=1nUj,u1=∑j=1ngjUj,andu2=∑j=1nhjUj." "Fichero" => "si42.jpeg" "Tamanyo" => 6147 "Alto" => 106 "Ancho" => 273 ] ] 40 => array:6 [ "identificador" => "eq0130" "etiqueta" => "(14)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Cj1/Pj=d1+e1tj+f1tj2," "Fichero" => "si43.jpeg" "Tamanyo" => 1623 "Alto" => 22 "Ancho" => 180 ] ] 41 => array:5 [ "identificador" => "eq0135" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Cj1/Pj=d1+e1−e1tj+f1tj2." "Fichero" => "si45.jpeg" "Tamanyo" => 1970 "Alto" => 22 "Ancho" => 227 ] ] 42 => array:6 [ "identificador" => "eq0140" "etiqueta" => "(15)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ti=a+sib0+c0ggi+c0hhi." "Fichero" => "si47.jpeg" "Tamanyo" => 1897 "Alto" => 16 "Ancho" => 221 ] ] 43 => array:6 [ "identificador" => "eq0145" "etiqueta" => "(16)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "F=∑i=1na+b0si+c0gsigi+c0hsihi−Ti2." "Fichero" => "si48.jpeg" "Tamanyo" => 3084 "Alto" => 46 "Ancho" => 299 ] ] 44 => array:6 [ "identificador" => "eq0150" "etiqueta" => "(17)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∂F/∂a=0,∂F/∂b0=0,∂F/∂c0=0,∂F/∂ϕ0=0." "Fichero" => "si49.jpeg" "Tamanyo" => 2822 "Alto" => 14 "Ancho" => 330 ] ] 45 => array:6 [ "identificador" => "eq0155" "etiqueta" => "(18)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "tan2φ0=hg=A2H1−A1H2A2H2−B1H1," "Fichero" => "si50.jpeg" "Tamanyo" => 1907 "Alto" => 24 "Ancho" => 190 ] ] 46 => array:5 [ "identificador" => "eq0160" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "A=∑i=1nsi,B=∑i=1nsi2,D=∑i=1ngisi,E=∑i=1nhisi,G=∑i=1ngisi2,H=∑i=1nhisi2,I=∑i=1ngi2si2,J=∑i=1nhi2si2,K=∑i=1ngihisi2,f0=∑i=1nTi,f1=∑i=1nsiTi,f2=∑i=1ngisiTi,andf3=∑i=1nhisiTi." "Fichero" => "si52.jpeg" "Tamanyo" => 14890 "Alto" => 264 "Ancho" => 285 ] ] 47 => array:5 [ "identificador" => "eq0165" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "c0=A2H1−A1H2hA22−A1B1,b0=F1−c0ga2−c0ha3/a1," "Fichero" => "si53.jpeg" "Tamanyo" => 3174 "Alto" => 28 "Ancho" => 335 ] ] 48 => array:5 [ "identificador" => "eq0170" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Ti=a+bsi+csiti+dsi2+esi2ti+fsi2ti2," "Fichero" => "si56.jpeg" "Tamanyo" => 2364 "Alto" => 19 "Ancho" => 278 ] ] 49 => array:6 [ "identificador" => "eq0175" "etiqueta" => "(19)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "F=∑i=1na+bsi+csiti+dsi2+esi2ti+fsi2ti2−Ti2." "Fichero" => "si57.jpeg" "Tamanyo" => 3769 "Alto" => 76 "Ancho" => 203 ] ] 50 => array:6 [ "identificador" => "eq0180" "etiqueta" => "(20)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∂F/∂a=0,∂F/∂b=0,∂F/∂c=0,∂F/∂d=0,∂F/∂e=0,∂F/∂f=0,∂F/∂ϕ0=0." "Fichero" => "si58.jpeg" "Tamanyo" => 4610 "Alto" => 38 "Ancho" => 309 ] ] 51 => array:6 [ "identificador" => "eq0185" "etiqueta" => "(21)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "ΔεV=2Ac+e+f/a," "Fichero" => "si59.jpeg" "Tamanyo" => 1635 "Alto" => 19 "Ancho" => 181 ] ] 52 => array:6 [ "identificador" => "eq0190" "etiqueta" => "(22)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "ΔδV=2Ac+e/a." "Fichero" => "si60.jpeg" "Tamanyo" => 1346 "Alto" => 19 "Ancho" => 150 ] ] 53 => array:5 [ "identificador" => "eq0195" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "A=∑i=1nsi,B=∑i=1nsiti,C=∑i=1nsi2,D=∑i=1nsi2ti,E=∑i=1nsi2ti2,F=∑i=1nsi3,G=∑i=1nsi3ti,H=∑i=1nsi3ti2, K=∑i=1nsi3ti3, L=∑i=1nsi4,M=∑i=1nsi4ti, N=∑i=1nsi4ti2, O=∑i=1nsi4ti3,P=∑i=1nsi4ti4, U0=∑i=1nTi, U1=∑i=1nTisi,U2=∑i=1nTisiti, U3=∑i=1nTisi2, U4=∑i=1nTisi2ti,and U5=∑i=1nTisi2ti2." "Fichero" => "si61.jpeg" "Tamanyo" => 22845 "Alto" => 333 "Ancho" => 268 ] ] 54 => array:5 [ "identificador" => "eq0200" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "f=a2f1−a1f2a22−a1b1,e=f1−fa2a1,d=g1−fa13−ea12a11,c=h1−fa24−ea23−da22a21,b=k1−fa05−ea04−da03−ca02a01,a=U0−fE−eD−dC−cB−bA/n," "Fichero" => "si62.jpeg" "Tamanyo" => 10452 "Alto" => 187 "Ancho" => 282 ] ] 55 => array:5 [ "identificador" => "eq0205" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "a21=a022−a01b01,a22=a02a03−a01b02,a23=a02a04−a01b03,a24=a02a05−a01b04,b21=a032−a01c01,b22=a03a04−a01c02,b23=a03a05−a01c03,c21=a042−a01d01,c22=a04a05−a01d02,d21=a052−a01e01,h1=a02k1−a01k2,h2=a03k1−a01k3,h3=a04k1−a01k4,h4=a05k1−a01k5,a11=a222−a21b21,a12=a22a23−a21b22a13=a22a24−a21b23,b11=a232−a21c21,b12=a23a24−a21c22,c11=a242−a21d21,g1=a22h1−a21h2,g2=a23h1−a21h3,g3=a24h1−a21h4,a1=a122−a11b11a2=a12a13−a11b12, b1=a132−a11c11, f1=a12g1−a11g2,and f2=a13g1−a11g3." "Fichero" => "si63.jpeg" "Tamanyo" => 30743 "Alto" => 289 "Ancho" => 349 ] ] 56 => array:5 [ "identificador" => "eq0210" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "caA1+bB1+cC1+dD1+eE1+fF1−U6+eaB1+bD1+cE1+dG1+eH1+fK1−U7+2faC1+bE1+cF1+dH1+eK1+fL1−U8=0" "Fichero" => "si64.jpeg" "Tamanyo" => 8239 "Alto" => 64 "Ancho" => 359 ] ] 57 => array:5 [ "identificador" => "eq0215" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "A1=∑i=1nyisi,B1=∑i=1nyisi2,C1=∑i=1nyitisi2D1=∑i=1nyisi3,E1=∑i=1nyitisi3,F1=∑i=1nyisi3ti2,G1=∑i=1nyisi4,H1=∑i=1nyitisi4,K1=∑i=1nyiti2si4,L1=∑i=1nyiti3si4,U6=∑i=1nyiTisi,U7=∑i=1nyiTisi2U8=∑i=1nyiTitisi2,andyi=sin2ϕi−ϕ0." "Fichero" => "si65.jpeg" "Tamanyo" => 17915 "Alto" => 264 "Ancho" => 333 ] ] ] "bibliografia" => array:2 [ "titulo" => "References" "seccion" => array:1 [ 0 => array:2 [ "identificador" => "bibs0005" "bibliografiaReferencia" => array:15 [ 0 => array:3 [ "identificador" => "bib0005" "etiqueta" => "Abdelnour and Selesnick, 2000" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Symmetric nearly orthogonal and orthogonal nearly symmetric wavelets" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:2 [ 0 => "A.F. Abdelnour" 1 => "I.W. Selesnick" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:6 [ "tituloSerie" => "Arabian J. for Science and Engineering" "fecha" => "2004" "volumen" => "29" "numero" => "2C" "paginaInicial" => "3" "paginaFinal" => "16" ] ] ] ] ] ] 1 => array:3 [ "identificador" => "bib0010" "etiqueta" => "Abdelnour and Selesnick, 2005" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Symmetric nearly shift-invariant tight frame wavelets" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:2 [ 0 => "A.F. Abdelnour" 1 => "I.W. Selesnick" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:6 [ "tituloSerie" => "IEEE Trans. Signal Process" "fecha" => "2005" "volumen" => "53" "numero" => "1" "paginaInicial" => "231" "paginaFinal" => "239" ] ] ] ] ] ] 2 => array:3 [ "identificador" => "bib0015" "etiqueta" => "Chichinina et al., 2003" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Numerical modeling of P-wave AVOA in media containing vertical fractures / Expanded abstract" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:3 [ 0 => "T. Chichinina" 1 => "V. Sabinin" 2 => "G. Ronquillo Jarrillo" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Libro" => array:6 [ "titulo" => "The Sixth International Conference on Mathematical and Numerical Aspects of Wave Propagation “Waves’ 2003”" "fecha" => "2003" "paginaInicial" => "897" "paginaFinal" => "902" "editorial" => "Finland" "editorialLocalizacion" => "Springer" ] ] ] ] ] ] 3 => array:3 [ "identificador" => "bib0020" "etiqueta" => "Dasgupta and Clark, 1998" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Estimation of Q from surface seismic reflection data" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:2 [ 0 => "R. Dasgupta" 1 => "R.A. Clark" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:5 [ "tituloSerie" => "Geophysics" "fecha" => "1998" "volumen" => "63" "paginaInicial" => "2120" "paginaFinal" => "2128" ] ] ] ] ] ] 4 => array:3 [ "identificador" => "bib0025" "etiqueta" => "Jenner, 2002" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Azimuthal AVO: Methodology and data examples" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "E. Jenner" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:6 [ "tituloSerie" => "The Leading Edge" "fecha" => "2002" "volumen" => "21" "numero" => "8" "paginaInicial" => "782" "paginaFinal" => "786" ] ] ] ] ] ] 5 => array:3 [ "identificador" => "bib0030" "etiqueta" => "MacBeth, 1999" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Azimuthal variation in P-wave signatures due to flow" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "C. MacBeth" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:5 [ "tituloSerie" => "Geophysics" "fecha" => "1999" "volumen" => "64" "paginaInicial" => "1181" "paginaFinal" => "1192" ] ] ] ] ] ] 6 => array:3 [ "identificador" => "bib0035" "etiqueta" => "Mallik et al., 1998" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Determination of the principal directions of azimuthal anisotropy from P-wave seismic data" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:4 [ 0 => "S. Mallik" 1 => "K.L Craft" 2 => "L.J. Meister" 3 => "R.E. Chambers" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:5 [ "tituloSerie" => "Geophysics" "fecha" => "1998" "volumen" => "63" "paginaInicial" => "692" "paginaFinal" => "706" ] ] ] ] ] ] 7 => array:3 [ "identificador" => "bib0040" "etiqueta" => "Newman, 1973" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Divergence effects in a layered earth" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "P. Newman" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:5 [ "tituloSerie" => "Geophysics" "fecha" => "1973" "volumen" => "38" "paginaInicial" => "481" "paginaFinal" => "488" ] ] ] ] ] ] 8 => array:3 [ "identificador" => "bib0045" "etiqueta" => "Rüger, 1998" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Variation of P-wave reflectivity with offset and azimuth in anisotropic media" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "A. Rüger" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:5 [ "tituloSerie" => "Geophysics" "fecha" => "1998" "volumen" => "63" "paginaInicial" => "935" "paginaFinal" => "947" ] ] ] ] ] ] 9 => array:3 [ "identificador" => "bib0050" "etiqueta" => "Sabinin, 2012" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Viscoelastic modeling and factor Q for reflection data" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "V. Sabinin" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:6 [ "tituloSerie" => "Geofisica Internacional" "fecha" => "2012" "volumen" => "51" "numero" => "4" "paginaInicial" => "377" "paginaFinal" => "391" ] ] ] ] ] ] 10 => array:3 [ "identificador" => "bib0055" "etiqueta" => "Sabinin, 2013" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "QVOA techniques for fracture characterization" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "V. Sabinin" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:6 [ "tituloSerie" => "Geofisica Internacional" "fecha" => "2013" "volumen" => "52" "numero" => "4" "paginaInicial" => "311" "paginaFinal" => "320" ] ] ] ] ] ] 11 => array:3 [ "identificador" => "bib0060" "etiqueta" => "Sabinin and Chichinina, 2008" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "AVOA technique for fracture characterization: resolving ambiguity" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:2 [ 0 => "V. Sabinin" 1 => "T. Chichinina" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:6 [ "tituloSerie" => "Geofisica Internacional" "fecha" => "2008" "volumen" => "47" "numero" => "1" "paginaInicial" => "3" "paginaFinal" => "11" ] ] ] ] ] ] 12 => array:3 [ "identificador" => "bib0065" "etiqueta" => "Sheriff and Geldart, 1983" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:1 [ "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:2 [ 0 => "R.E. Sheriff" 1 => "L.P. Geldart" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Libro" => array:5 [ "titulo" => "Exploration Seismology, Volume 2, Data-processing and interpretation" "fecha" => "1983" "paginaInicial" => "400" "editorial" => "Cambridge University Press" "editorialLocalizacion" => "Cambridge" ] ] ] ] ] ] 13 => array:3 [ "identificador" => "bib0070" "etiqueta" => "Xu and Tsvankin, 2004" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "Geometrical-spreading correction for P-waves in layered azimuthally anisotropic media" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:2 [ 0 => "X. Xu" 1 => "Tsvankin" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:4 [ "tituloSerie" => "74th Ann. Internat. Mtg.: Soc. of Expl. Geophys." "fecha" => "2004" "paginaInicial" => "111" "paginaFinal" => "114" ] ] ] ] ] ] 14 => array:3 [ "identificador" => "bib0075" "etiqueta" => "WSBP, 2012" "referencia" => array:1 [ 0 => array:1 [ "referenciaCompleta" => "WSBP, 2012, (Wavelet Software at Brooklyn Poly), Matlab Implementation of Wavelet Transforms. <a id="intr0005" class="elsevierStyleInterRef" href="http://eeweb.poly.edu/iselesni/WaveletSoftware">http://eeweb.poly.edu/iselesni/WaveletSoftware</a>." ] ] ] ] ] ] ] ] "idiomaDefecto" => "en" "url" => "/00167169/0000005300000004/v1_201412221101/S0016716914700789/v1_201412221101/en/main.assets" "Apartado" => array:4 [ "identificador" => "32621" "tipo" => "SECCION" "en" => array:2 [ "titulo" => "Original paper" "idiomaDefecto" => true ] "idiomaDefecto" => "en" ] "PDF" => "https://static.elsevier.es/multimedia/00167169/0000005300000004/v1_201412221101/S0016716914700789/v1_201412221101/en/main.pdf?idApp=UINPBA00004N&text.app=https://www.elsevier.es/" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716914700789?idApp=UINPBA00004N"]

Statistics

Vladimir Sabinin

Corresponding author

vsabinin@yahoo.com

Corresponding author:

Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas 152, Col. San Bartolo Atepehuacan, C.P. 07730, México D.F., México.

Read

2124

Times
was read the article

836

Total PDF

1288

Total HTML

Share statistics

 array:23 [
  "pii" => "S0016716914700789"
  "issn" => "00167169"
  "doi" => "10.1016/S0016-7169(14)70078-9"
  "estado" => "S300"
  "fechaPublicacion" => "2014-10-01"
  "aid" => "70078"
  "copyright" => "Universidad Nacional Aut&#243;noma de M&#233;xico"
  "copyrightAnyo" => "2014"
  "documento" => "article"
  "licencia" => "http://creativecommons.org/licenses/by-nc-nd/4.0/"
  "subdocumento" => "fla"
  "cita" => "Geofisica Internacional. 2014;53:457-71"
  "abierto" => array:3 [
    "ES" => true
    "ES2" => true
    "LATM" => true
  ]
  "gratuito" => true
  "lecturas" => array:2 [
    "total" => 1164
    "formatos" => array:3 [
      "EPUB" => 34
      "HTML" => 620
      "PDF" => 510
    ]
  ]
  "itemSiguiente" => array:18 [
    "pii" => "S0016716914700790"
    "issn" => "00167169"
    "doi" => "10.1016/S0016-7169(14)70079-0"
    "estado" => "S300"
    "fechaPublicacion" => "2014-10-01"
    "aid" => "70079"
    "copyright" => "Universidad Nacional Aut&#243;noma de M&#233;xico"
    "documento" => "article"
    "licencia" => "http://creativecommons.org/licenses/by-nc-nd/4.0/"
    "subdocumento" => "fla"
    "cita" => "Geofisica Internacional. 2014;53:473-90"
    "abierto" => array:3 [
      "ES" => true
      "ES2" => true
      "LATM" => true
    ]
    "gratuito" => true
    "lecturas" => array:2 [
      "total" => 2147
      "formatos" => array:3 [
        "EPUB" => 42
        "HTML" => 1448
        "PDF" => 657
      ]
    ]
    "en" => array:11 [
      "idiomaDefecto" => true
      "titulo" => "Thermomagnetic monitoring of lithic clasts burned under controlled temperature and field conditions&#46; Implications for archaeomagnetism"
      "tienePdf" => "en"
      "tieneTextoCompleto" => "en"
      "tieneResumen" => array:2 [
        0 => "es"
        1 => "en"
      ]
      "paginas" => array:1 [
        0 => array:2 [
          "paginaInicial" => "473"
          "paginaFinal" => "490"
        ]
      ]
      "contieneResumen" => array:2 [
        "es" => true
        "en" => true
      ]
      "contieneTextoCompleto" => array:1 [
        "en" => true
      ]
      "contienePdf" => array:1 [
        "en" => true
      ]
      "resumenGrafico" => array:2 [
        "original" => 0
        "multimedia" => array:7 [
          "identificador" => "fig0055"
          "etiqueta" => "Figure 11"
          "tipo" => "MULTIMEDIAFIGURA"
          "mostrarFloat" => true
          "mostrarDisplay" => false
          "figura" => array:1 [
            0 => array:4 [
              "imagen" => "gr11.jpeg"
              "Alto" => 2460
              "Ancho" => 1581
              "Tamanyo" => 237811
            ]
          ]
          "descripcion" => array:1 [
            "en" => "<p id="spar0060" class="elsevierStyleSimplePara elsevierViewall">Palaeointensity results obtained in obsidian and sandstone specimens&#44; &#40;a&#41; and &#40;b&#41; respectively&#44; with the multispecimen method&#46; Lines connect sister-specimen of the different lithologies samples&#46; Plots represent the relative percentage differences between the pTRMs acquired at different lab fields &#40;40 and 50<span class="elsevierStyleHsp" style=""></span>&#956;T&#41; and the total TRM &#40;acquired during burning&#41;&#46; Intersection of each line with the horizontal axis &#40;zero difference&#41; determines the intensity of the Earth&#39;s magnetic field during the burning&#46; &#40;c&#41; Mean ambient field estimation calculated from both lithologies&#46;</p>"
          ]
        ]
      ]
      "autores" => array:1 [
        0 => array:2 [
          "autoresLista" => "&#193;ngel Carrancho, Juan Morales, Avto Goguitchaichvili, Rodrigo Alonso, Marcos Terradillos"
          "autores" => array:5 [
            0 => array:2 [
              "nombre" => "&#193;ngel"
              "apellidos" => "Carrancho"
            ]
            1 => array:2 [
              "nombre" => "Juan"
              "apellidos" => "Morales"
            ]
            2 => array:2 [
              "nombre" => "Avto"
              "apellidos" => "Goguitchaichvili"
            ]
            3 => array:2 [
              "nombre" => "Rodrigo"
              "apellidos" => "Alonso"
            ]
            4 => array:2 [
              "nombre" => "Marcos"
              "apellidos" => "Terradillos"
            ]
          ]
        ]
      ]
    ]
    "idiomaDefecto" => "en"
    "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716914700790?idApp=UINPBA00004N"
    "url" => "/00167169/0000005300000004/v1_201412221101/S0016716914700790/v1_201412221101/en/main.assets"
  ]
  "itemAnterior" => array:18 [
    "pii" => "S0016716914700777"
    "issn" => "00167169"
    "doi" => "10.1016/S0016-7169(14)70077-7"
    "estado" => "S300"
    "fechaPublicacion" => "2014-10-01"
    "aid" => "70077"
    "copyright" => "Universidad Nacional Aut&#243;noma de M&#233;xico"
    "documento" => "article"
    "licencia" => "http://creativecommons.org/licenses/by-nc-nd/4.0/"
    "subdocumento" => "fla"
    "cita" => "Geofisica Internacional. 2014;53:435-56"
    "abierto" => array:3 [
      "ES" => true
      "ES2" => true
      "LATM" => true
    ]
    "gratuito" => true
    "lecturas" => array:2 [
      "total" => 2420
      "formatos" => array:3 [
        "EPUB" => 37
        "HTML" => 1858
        "PDF" => 525
      ]
    ]
    "en" => array:11 [
      "idiomaDefecto" => true
      "titulo" => "<span class="elsevierStyleItalic">Ca&#46;</span> 13 Ma strike-slip deformation in coastal Sonora from a large-scale&#44; <span class="elsevierStyleItalic">en-echelon&#44;</span> brittle-ductile&#44; dextral shear indicator&#58; implications for the evolution of the California rift"
      "tienePdf" => "en"
      "tieneTextoCompleto" => "en"
      "tieneResumen" => array:2 [
        0 => "es"
        1 => "en"
      ]
      "paginas" => array:1 [
        0 => array:2 [
          "paginaInicial" => "435"
          "paginaFinal" => "456"
        ]
      ]
      "contieneResumen" => array:2 [
        "es" => true
        "en" => true
      ]
      "contieneTextoCompleto" => array:1 [
        "en" => true
      ]
      "contienePdf" => array:1 [
        "en" => true
      ]
      "resumenGrafico" => array:2 [
        "original" => 0
        "multimedia" => array:7 [
          "identificador" => "fig0040"
          "etiqueta" => "Figure 8"
          "tipo" => "MULTIMEDIAFIGURA"
          "mostrarFloat" => true
          "mostrarDisplay" => false
          "figura" => array:1 [
            0 => array:4 [
              "imagen" => "gr8.jpeg"
              "Alto" => 1292
              "Ancho" => 1568
              "Tamanyo" => 126722
            ]
          ]
          "descripcion" => array:1 [
            "en" => "<p id="spar0040" class="elsevierStyleSimplePara elsevierViewall">Paleomagnetic data&#46; &#40;a-c&#41; Orthogonal demagnetization diagrams of selected samples&#46; Open &#40;closed&#41; symbols are projections on the vertical &#40;horizontal&#41; plane&#46; &#40;d&#41; Stereographic projection of the characteristic magnetization &#40;in situ&#41;&#44; here open &#40;closed&#41; symbols are projections on the upper &#40;lower&#41; hemisphere&#46;</p>"
          ]
        ]
      ]
      "autores" => array:2 [
        0 => array:2 [
          "autoresLista" => "David Garc&#237;a-Mart&#237;nez, Jaime Rold&#225;n Quintana, Hector Mend&#237;vil-Quijada"
          "autores" => array:3 [
            0 => array:2 [
              "nombre" => "David"
              "apellidos" => "Garc&#237;a-Mart&#237;nez"
            ]
            1 => array:2 [
              "nombre" => "Jaime Rold&#225;n"
              "apellidos" => "Quintana"
            ]
            2 => array:2 [
              "nombre" => "Hector"
              "apellidos" => "Mend&#237;vil-Quijada"
            ]
          ]
        ]
        1 => array:2 [
          "autoresLista" => "Roberto Stanley Molina Garza"
          "autores" => array:1 [
            0 => array:2 [
              "nombre" => "Roberto Stanley Molina"
              "apellidos" => "Garza"
            ]
          ]
        ]
      ]
    ]
    "idiomaDefecto" => "en"
    "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716914700777?idApp=UINPBA00004N"
    "url" => "/00167169/0000005300000004/v1_201412221101/S0016716914700777/v1_201412221101/en/main.assets"
  ]
  "en" => array:18 [
    "idiomaDefecto" => true
    "titulo" => "AVOA techniques for fracture characterization"
    "tieneTextoCompleto" => true
    "paginas" => array:1 [
      0 => array:2 [
        "paginaInicial" => "457"
        "paginaFinal" => "471"
      ]
    ]
    "autores" => array:1 [
      0 => array:4 [
        "autoresLista" => "Vladimir Sabinin"
        "autores" => array:1 [
          0 => array:4 [
            "nombre" => "Vladimir"
            "apellidos" => "Sabinin"
            "email" => array:1 [
              0 => "vsabinin&#64;yahoo&#46;com"
            ]
            "referencia" => array:1 [
              0 => array:2 [
                "etiqueta" => "<span class="elsevierStyleSup">&#42;</span>"
                "identificador" => "cor0005"
              ]
            ]
          ]
        ]
        "afiliaciones" => array:1 [
          0 => array:2 [
            "entidad" => "Instituto Mexicano del Petr&#243;leo&#44; Eje Central L&#225;zaro C&#225;rdenas 152&#44; Col&#46; San Bartolo Atepehuacan&#44; C&#46;P&#46; 07730&#44; M&#233;xico D&#46;F&#46;&#44; M&#233;xico&#46;"
            "identificador" => "aff0005"
          ]
        ]
        "correspondencia" => array:1 [
          0 => array:3 [
            "identificador" => "cor0005"
            "etiqueta" => "&#42;"
            "correspondencia" => "Corresponding author&#58;"
          ]
        ]
      ]
    ]
    "resumenGrafico" => array:2 [
      "original" => 0
      "multimedia" => array:7 [
        "identificador" => "fig0070"
        "etiqueta" => "Figure 14"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr14.jpeg"
            "Alto" => 852
            "Ancho" => 1127
            "Tamanyo" => 142222
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0070" class="elsevierStyleSimplePara elsevierViewall">Errors for the asymmetric set of azimuths&#59; the noise&#44; the upper boundary&#44; <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3200&#44; and fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">min</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;</p>"
        ]
      ]
    ]
    "textoCompleto" => "<span class="elsevierStyleSections"><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0025">Introduction</span><p id="par0005" class="elsevierStylePara elsevierViewall">The analysis of azimuthal variation in reflection coefficients&#44; or AVOA analysis &#40;Amplitude Versus Offset and Azimuth&#41;&#44; is widely applied for detecting and mapping highly fractured zones with azimuthally-oriented vertical cracks &#40;<a class="elsevierStyleCrossRef" href="#bib0035">Mallik <span class="elsevierStyleItalic">et al&#46;&#44;</span> 1998</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0025">Jenner&#44; 2002</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0060">Sabinin &#38; Chichinina&#44; 2008</a>&#41;&#46; The AVOA techniques are based on the <a class="elsevierStyleCrossRef" href="#bib0045">R&#252;ger &#40;1998&#41;</a> approximation for the reflection coefficients in HTI medium&#44; and give principal symmetry directions of HTI medium&#46;</p><p id="par0010" class="elsevierStylePara elsevierViewall">Here&#44; the computational aspects of AVOA techniques are considered&#44; namely&#44; applying amplitudes instead of reflection coefficients&#44; smoothing the amplitudes&#44; an incidence angle estimation&#44; methods for obtaining the symmetry-axis angle&#44; synthetic data for testing techniques&#44; and a numerical experiment for investigating properties of the techniques&#46; A new computational method for obtaining the symmetry-axis angle and a new filter for smoothing are suggested&#46; All considered techniques are compared in synthetic anisotropic seismic data with noise&#44; and without noise&#46; The suggested new technique proved better than the others&#46;</p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0030">Background</span><p id="par0015" class="elsevierStylePara elsevierViewall">The methodology of AVOA analysis is based on the concept of azimuthal anisotropy caused for the most part by parallel vertical fractures&#46; It leads to the azimuthal anisotropy of amplitudes&#44; in particular&#44; to azimuthal variation in reflection coefficients&#46; Let a fractured reservoir be represented by a model of a transversely isotropic medium with horizontal symmetry axis &#40;HTI medium&#41;&#46; The PP-wave reflection coefficient <span class="elsevierStyleItalic">R</span> at the interface &#40;or at the reflecting boundary&#41; between weakly anisotropic HTI media &#40;or between an isotropic medium and an anisotropic HTI medium&#41; is defined by the approximate formula &#40;<a class="elsevierStyleCrossRef" href="#bib0045">R&#252;ger&#44; 1998</a>&#41;&#58;<elsevierMultimedia ident="eq0005"></elsevierMultimedia>where <span class="elsevierStyleItalic">&#952;</span> is the incidence angle&#44; and <span class="elsevierStyleItalic">&#981;</span> is the source-receiver-line azimuth with respect to the coordinate axis <span class="elsevierStyleItalic">x</span>&#46; The term <span class="elsevierStyleItalic">A</span> is the normal-incidence reflection coefficient<elsevierMultimedia ident="eq0010"></elsevierMultimedia>where Z&#8801;&#961;VP&#8214; is the vertical P-wave impedance&#44; VP&#8214; is the vertical velocity &#40;or velocity in the isotropy plane&#41; of the P-wave&#44; <span class="elsevierStyleItalic">&#961;</span> i s density&#44; <span class="elsevierStyleItalic">&#916;</span> denotes the difference between the values of a parameter below an<span class="elsevierStyleUnderline">d</span> above the reflecting boundary&#44; and the bar &#8230; indicates average of these values&#46; VP&#8214;&#61;VP in the isotropic media&#46;</p><p id="par0020" class="elsevierStylePara elsevierViewall">The coefficient <span class="elsevierStyleItalic">B</span>&#40;<span class="elsevierStyleItalic">&#981;</span>&#41; is a so-called AVO gradient&#44; which can be written &#40;<a class="elsevierStyleCrossRef" href="#bib0045">R&#252;ger&#44; 1998</a>&#41; as<elsevierMultimedia ident="eq0015"></elsevierMultimedia>where <span class="elsevierStyleItalic">&#981;</span><span class="elsevierStyleInf">0</span> is the angle of the symmetry axis with the <span class="elsevierStyleItalic">x&#8722;</span>-axis&#46; The term <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">iso</span></span> is the AVO-gradient isotropic part &#40;equal to the AVO gradient for isotropic media&#41;&#44; and <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ani</span></span> is the anisotropic part of the AVO gradient&#46;</p><p id="par0025" class="elsevierStylePara elsevierViewall">The coefficient <span class="elsevierStyleItalic">C</span>&#40;<span class="elsevierStyleItalic">&#981;</span>&#41; can be written &#40;<a class="elsevierStyleCrossRef" href="#bib0045">R&#252;ger&#44; 1998</a>&#41; as&#44;<elsevierMultimedia ident="eq0020"></elsevierMultimedia>where &#945;&#8801;&#916;VP&#8214;&#47;2V&#175;P&#8214;&#44;&#946;&#61;12&#916;&#949;V&#44; and &#947;&#61;12&#916;&#948;V&#46;</p><p id="par0030" class="elsevierStylePara elsevierViewall">Above&#44; Thomsen-style anisotropy parameters <span class="elsevierStyleItalic">&#949;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span>&#44; and <span class="elsevierStyleItalic">&#948;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span> are negative for HTI media&#44; and they are equal to zero for isotropic media&#46;</p><p id="par0035" class="elsevierStylePara elsevierViewall">The main problem of AVOA analysis is to estimate the symmetry-axis angle <span class="elsevierStyleItalic">&#981;</span><span class="elsevierStyleInf">0</span> from surface seismic data of amplitudes using numerical techniques&#46;</p><p id="par0040" class="elsevierStylePara elsevierViewall">The techniques of AVOA are based on <a class="elsevierStyleCrossRef" href="#eq0005">equations &#40;1&#41;</a> &#8211; <a class="elsevierStyleCrossRef" href="#eq0020">&#40;4&#41;</a>&#46; Note that <a class="elsevierStyleCrossRef" href="#eq0005">equation &#40;1&#41;</a> is intended for calculation of reflection coefficients&#44; while in real data&#44; one operates with amplitudes of reflected waves&#44; not with reflection coefficients&#46; This brings some problems which are discussed in the next section&#46;</p></span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0035">Using amplitudes instead of reflection coefficients</span><p id="par0045" class="elsevierStylePara elsevierViewall">While the background of AVOA analysis is based on R&#252;ger&#39;s equation for the reflection coefficient <a class="elsevierStyleCrossRef" href="#eq0005">&#40;1&#41;</a>&#44; in real data&#44; AVOA analysis should use signal amplitudes&#46; It is true that the amplitude is not equal to the reflection coefficient&#46; Moreover&#44; no picked instantaneous amplitude &#40;sample&#41; in the signal can be used instead of the reflection coefficient because the signal changes its form during propagation for many reasons&#46; It seems that one should use an integral amplitude characteristic of the signal which adequately corresponds to the reflection coefficient&#46; Let&#39;s call this characteristic simply by amplitude and denote it as <span class="elsevierStyleItalic">P</span>&#46;<a name="p459"></a></p><p id="par0050" class="elsevierStylePara elsevierViewall">The estimated value of <span class="elsevierStyleItalic">&#981;</span><span class="elsevierStyleInf">0</span> is very sensitive to the definition of <span class="elsevierStyleItalic">P</span>&#44; especially for data with noise&#46; I suggest the following procedure for definition of <span class="elsevierStyleItalic">P</span> which gives good and stable results&#46; The procedure calculates an average value of a signal envelope in a time window&#46; In calculating the envelope&#44; the Fourier transform of this signal is used&#58; <span class="elsevierStyleItalic">F</span> &#61; <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">&#43;</span> &#43; <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">&#8722;</span>&#44; where <span class="elsevierStyleItalic">F</span> is spectrum&#44; <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">&#43;</span> is the part of spectrum corresponding to positive frequencies &#40;&#969;<span class="elsevierStyleHsp" style=""></span>&#8805;<span class="elsevierStyleHsp" style=""></span>0&#41;&#44; and <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">&#8722;</span> is the part of negative frequencies&#46; The envelope of the signal is given by the absolute value of inverse Fourier transform of the spectrum with double <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">&#43;</span>&#44; and <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">&#8722;</span><span class="elsevierStyleHsp" style=""></span>&#8801;<span class="elsevierStyleHsp" style=""></span>0 &#40;<a class="elsevierStyleCrossRef" href="#bib0065">Sheriff &#38; Geldart&#44; 1983</a>&#41;&#46;</p><p id="par0055" class="elsevierStylePara elsevierViewall">The sign of envelope is positive&#59; therefore this approach is applicable only to seismograms with the constant sign of reflection coefficient in dependence on offset&#46;</p><p id="par0060" class="elsevierStylePara elsevierViewall">For data with noise&#44; the envelope is noisy&#44; too &#40;see <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>&#41;&#46; Therefore&#44; smoothing is necessary&#46;</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia><p id="par0065" class="elsevierStylePara elsevierViewall">For smoothing&#44; an algorithm of discrete transformations of wavelet by filters is applied&#46; Four symmetric filters are constructed for it&#58; the low-pass &#40;<span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">0</span>&#41; end high-pass &#40;<span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">1</span>&#41; analysis filters&#44; and the low-pass &#40;<span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">2</span>&#41; and high-pass &#40;<span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">3</span>&#41; synthesis filters&#46; The right-hand part of <span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">0</span> consists from the filter derived by Abdelnour &#38; Selesnick &#40;2004&#41;&#46; The left-hand part of <span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">0</span> is symmetric to it&#46; That is<elsevierMultimedia ident="eq0025"></elsevierMultimedia>where a&#61;M&#47;32&#44;M&#61;2&#47;2&#44;b&#61;4a&#44; and <span class="elsevierStyleItalic">n</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>8&#46; The central value is <span class="elsevierStyleItalic">c</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#8722;<span class="elsevierStyleItalic">M</span>&#46;</p><p id="par0070" class="elsevierStylePara elsevierViewall">The high-pass analysis filter is constructed by formula <span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">1</span>&#40;<span class="elsevierStyleItalic">i</span>&#41; &#61; &#40;&#8722; 1&#41;<span class="elsevierStyleSup"><span class="elsevierStyleItalic">i</span></span><span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">0</span> &#40;<span class="elsevierStyleItalic">n</span> &#8722; <span class="elsevierStyleItalic">i</span> &#43; 1&#41; for <span class="elsevierStyleItalic">i</span> &#8800; 0&#44; and <span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">l</span></span> &#40;0&#41; &#61; 0&#46; The synthesis filters are calculated by formula <span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">2</span>&#40;<span class="elsevierStyleItalic">i</span>&#41; &#61; &#40;&#8722; 1&#41;<span class="elsevierStyleSup"><span class="elsevierStyleItalic">i</span></span><span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">1</span></span> &#40;<span class="elsevierStyleItalic">i</span>&#41;&#44;<span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">3</span> &#40;<span class="elsevierStyleItalic">i</span>&#41; &#61; &#40;&#8722; 1&#41;<span class="elsevierStyleSup"><span class="elsevierStyleItalic">i</span></span><span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">0</span> &#40;<span class="elsevierStyleItalic">i</span>&#41;&#44; see &#40;<a class="elsevierStyleCrossRef" href="#bib0010">Abdelnour &#38; Selesnick&#44; 2005</a>&#41;&#46; The central values are <span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">2</span> &#40;0&#41;<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">c</span> and <span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf">3</span>&#40;0&#41;<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;</p><p id="par0075" class="elsevierStylePara elsevierViewall">The smoothed signal is obtained by the decomposition algorithm&#59; see <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a> &#40;<a class="elsevierStyleCrossRef" href="#bib0075">WSBP&#44; 2012</a>&#41;&#46;</p><elsevierMultimedia ident="fig0010"></elsevierMultimedia><p id="par0080" class="elsevierStylePara elsevierViewall">The input signal is <span class="elsevierStyleItalic">x</span>&#40;<span class="elsevierStyleItalic">j</span>&#41;&#44; <span class="elsevierStyleItalic">j</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#44;&#8230;&#44;<span class="elsevierStyleItalic">m</span>&#44; <span class="elsevierStyleItalic">m</span><span class="elsevierStyleHsp" style=""></span>&#62;<span class="elsevierStyleHsp" style=""></span>&#62;<span class="elsevierStyleHsp" style=""></span>2<span class="elsevierStyleItalic">n</span>&#46; It is decomposed into low and high components <span class="elsevierStyleItalic">lo</span><span class="elsevierStyleInf">1</span>&#40;<span class="elsevierStyleItalic">j</span>&#41; and <span class="elsevierStyleItalic">hi</span><span class="elsevierStyleInf">1</span>&#40;<span class="elsevierStyleItalic">j</span>&#41; in the first stage&#58;<elsevierMultimedia ident="eq0030"></elsevierMultimedia></p><p id="par0085" class="elsevierStylePara elsevierViewall">In the next stages &#40;<span class="elsevierStyleItalic">s</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>2&#44;&#8230;&#44; <span class="elsevierStyleItalic">S</span>&#41;&#44; the each low component <span class="elsevierStyleItalic">lo</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">s</span><span class="elsevierStyleHsp" style=""></span>-<span class="elsevierStyleHsp" style=""></span>1</span>&#40;<span class="elsevierStyleItalic">j</span>&#41; is decomposed by the same analysis filters&#46;</p><p id="par0090" class="elsevierStylePara elsevierViewall">After all stages of decomposition finishing&#44; the stages of applying the synthesis filters are fulfilled in reverse order &#40;<span class="elsevierStyleItalic">s</span> &#61; <span class="elsevierStyleItalic">S</span>&#44; <span class="elsevierStyleItalic">S</span> - 1&#44;&#8230;&#44;1&#41;&#58;<elsevierMultimedia ident="eq0035"></elsevierMultimedia></p><p id="par0095" class="elsevierStylePara elsevierViewall">The output signal <span class="elsevierStyleItalic">y</span> &#40;<span class="elsevierStyleItalic">j</span>&#41; is obtained finally&#58;<elsevierMultimedia ident="eq0040"></elsevierMultimedia>where the fitting amplitude coefficient <span class="elsevierStyleItalic">p</span> can be approximately estimated by the formula <span class="elsevierStyleItalic">p</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#43;0&#46;057<span class="elsevierStyleItalic">S</span>&#44; where <span class="elsevierStyleItalic">S</span> is the number of stages&#46;</p><p id="par0100" class="elsevierStylePara elsevierViewall">The advantage of this variant of discrete transformation algorithm in comparison with &#40;<a class="elsevierStyleCrossRef" href="#bib0075">WSBP&#44; 2012</a>&#41; is the absence of shift functions in it due to applying the fully symmetric filters &#40;<span class="elsevierStyleItalic">i</span>&#61;-<span class="elsevierStyleItalic">n</span>&#44;&#8230;&#44;<span class="elsevierStyleItalic">n</span>&#41;&#46;<a name="p460"></a></p><p id="par0105" class="elsevierStylePara elsevierViewall">It must be noted that the algorithm gives unsatisfactory results at the edges of the signal because of truncating the filters in 2<span class="elsevierStyleItalic">n</span> edge points&#46; Therefore&#44; it is necessary <span class="elsevierStyleItalic">m</span><span class="elsevierStyleHsp" style=""></span>&#62;&#62;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">2n</span>&#46;</p><p id="par0110" class="elsevierStylePara elsevierViewall">The result of smoothing the signal of <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a> by the 3-stage algorithm is presented in <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>&#46; The smoothness of resulting curve increases with increasing <span class="elsevierStyleItalic">S</span>&#46; Also with increasing <span class="elsevierStyleItalic">S</span>&#44; the algorithm slightly deforms the resulting impulse in comparison with the parent impulse without noise&#46; Optimum in the smoothness and in the conservation of form is observed at the value <span class="elsevierStyleItalic">S</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3&#46;</p><elsevierMultimedia ident="fig0015"></elsevierMultimedia><p id="par0115" class="elsevierStylePara elsevierViewall">The limits of time window for calculating <span class="elsevierStyleItalic">P</span> with the help of envelope can be chosen by different ways&#46; I use the following way&#46; From the envelope of signal <span class="elsevierStyleItalic">e</span>&#40;<span class="elsevierStyleItalic">t</span>&#41;&#44; the maximum <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span> and nearest local minimums&#44; left <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">l</span></span> and right <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">r</span></span>&#44; are calculated&#46; The left limit of the time window is set in the point where <span class="elsevierStyleItalic">e</span> &#61; <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">l</span></span> &#43; 0&#46;15&#40;<span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf">m</span>&#8722;<span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">l</span></span>&#41;&#44; and the right limit &#8211; where <span class="elsevierStyleItalic">e</span> &#61; <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">r</span></span> &#43; 0&#46;15&#40;<span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf">m</span>&#8722;<span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">l</span></span>&#41;&#44; see vertical lines in <a class="elsevierStyleCrossRef" href="#fig0005">Figures 1</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0015">3</a>&#46; Obviously&#44; this algorithm correctly works only with smoothed signals&#46;</p><p id="par0120" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#eq0005">Equation &#40;1&#41;</a> should be rewritten for using the amplitudes&#46; If the source and the receivers are at the earth surface&#44; then the amplitude of reflected PP-wave can be expressed as<elsevierMultimedia ident="eq0045"></elsevierMultimedia>where <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> is the coefficient of geometrical spreading &#40;divergence&#41; from source to reflection point for this wave&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> &#61; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> &#40;<span class="elsevierStyleItalic">&#952;</span>&#44; <span class="elsevierStyleItalic">&#981;</span>&#41;&#44; <span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ini</span></span> is the amplitude of the source &#40;the initial amplitude&#41;&#44; and <span class="elsevierStyleItalic">R</span> is the reflection coefficient&#44; <span class="elsevierStyleItalic">R</span> &#61; <span class="elsevierStyleItalic">R</span>&#40;<span class="elsevierStyleItalic">&#952;</span>&#44; <span class="elsevierStyleItalic">&#981;</span>&#41; in the <a class="elsevierStyleCrossRef" href="#eq0005">equation &#40;1&#41;</a>&#46;</p><p id="par0125" class="elsevierStylePara elsevierViewall">The amplitude for the normal-incidence wave can be written as<elsevierMultimedia ident="eq0050"></elsevierMultimedia>where <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span>0</span> is the normal-incidence coefficient of geometrical spreading&#44; which does not depend on &#40;<span class="elsevierStyleItalic">&#952;</span>&#44; <span class="elsevierStyleItalic">&#981;</span>&#41;&#44; and <span class="elsevierStyleItalic">A</span> is the normal-incidence reflection coefficient&#44; <span class="elsevierStyleItalic">A</span> &#61; <span class="elsevierStyleItalic">const&#44;</span> see <a class="elsevierStyleCrossRef" href="#eq0005">equations &#40;1&#41;</a> - <a class="elsevierStyleCrossRef" href="#eq0010">&#40;2&#41;</a>&#46; Then the reflection coefficient can be expressed as<elsevierMultimedia ident="eq0055"></elsevierMultimedia></p><p id="par0130" class="elsevierStylePara elsevierViewall">Therefore the <a class="elsevierStyleCrossRef" href="#eq0005">equation &#40;1&#41;</a> for the reflection coefficient <span class="elsevierStyleItalic">R</span> transforms into the following equation for the amplitude <span class="elsevierStyleItalic">P</span>&#58;<elsevierMultimedia ident="eq0060"></elsevierMultimedia>where <span class="elsevierStyleItalic">m</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">0</span>&#47;<span class="elsevierStyleItalic">A</span>&#44; and <span class="elsevierStyleItalic">r</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span><span class="elsevierStyleHsp" style=""></span>&#8801;<span class="elsevierStyleHsp" style=""></span>&#40;<span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span>0</span> &#47; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span>&#41;<span class="elsevierStyleSup">2</span>&#46; This equation should be used in the AVOA techniques instead of <a class="elsevierStyleCrossRef" href="#eq0005">&#40;1&#41;</a>&#46;</p><p id="par0135" class="elsevierStylePara elsevierViewall">Note that <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> can be expressed as <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span> &#61; <span class="elsevierStyleItalic">c</span>&#40;<span class="elsevierStyleItalic">&#952;</span>&#44; <span class="elsevierStyleItalic">&#981;</span>&#41;&#47;<span class="elsevierStyleItalic">r</span> in 3D space&#44; where <span class="elsevierStyleItalic">r</span> is a half of travel path from source to receiver&#44; and <span class="elsevierStyleItalic">c</span> depends on the direction of wave propagation &#40;for isotropic media&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">const</span>&#41;&#46; In assuming a weak anisotropy&#44; one may assume a weak dependence of geometrical spreading on incidence angle&#58; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleHsp" style=""></span>&#8776;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">const</span> for a given source-to-receiver line with azimuth <span class="elsevierStyleItalic">&#981;</span>&#46; Then&#44; for a homogeneous medium above the reflecting boundary&#44;<elsevierMultimedia ident="eq0065"></elsevierMultimedia>where <span class="elsevierStyleItalic">z</span> is the normal-incidence ray path&#44; and <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span>0</span> &#8801; <span class="elsevierStyleItalic">c</span> &#47; <span class="elsevierStyleItalic">z</span>&#46; It is the approximate formula for divergent correction&#46;</p><p id="par0140" class="elsevierStylePara elsevierViewall">Also for multilayered media&#44; the expressions for divergence correction can be found from <a class="elsevierStyleCrossRef" href="#bib0040">Newman &#40;1973&#41;</a>&#46; A practical methodology for the P-wave geometrical-spreading correction in layered azimuthally anisotropic media can be found from <a class="elsevierStyleCrossRef" href="#bib0070">Xu &#38; Tsvankin &#40;2004&#41;</a>&#46;</p></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0040">The incidence angle estimation</span><p id="par0145" class="elsevierStylePara elsevierViewall">In the case of <span class="elsevierStyleItalic">n</span> isotropic layers above the reflecting boundary&#44; one can obtain the incidence angle <span class="elsevierStyleItalic">&#952;</span> &#61; <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span> from a solution of the following non-linear equation for <span class="elsevierStyleItalic">a</span>&#58;<a name="p461"></a><elsevierMultimedia ident="eq0070"></elsevierMultimedia>where <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">0</span> is the half of offset&#44; <span class="elsevierStyleItalic">z</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> is the thickness of <span class="elsevierStyleItalic">i</span>-th layer&#44; <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> is the velocity in <span class="elsevierStyleItalic">i</span>-th layer&#44; and sin &#40;<span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span>&#41;<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">aV</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span>&#46; For calculating the geometrical spreading&#44; the travel path r&#61;&#8721;i&#61;1nzi1&#8722;a2Vi2&#46;</p><p id="par0150" class="elsevierStylePara elsevierViewall">In the case of the reflecting boundary being the lower boundary of anisotropic layer&#44; the problem is more complicated because the last layer is anisotropic&#44; and the velocity <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span> depends on <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span> in it and is not known beforehand&#46;</p><p id="par0155" class="elsevierStylePara elsevierViewall">The problem can be solved by <a class="elsevierStyleCrossRef" href="#bib0050">Sabinin &#40;2012&#41;</a>&#46; An advantage of his method is that the value <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span> in the anisotropic layer is unnecessary for calculating angle <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span> and path <span class="elsevierStyleItalic">r</span>&#46; However&#44; it uses additionally the impulse from the upper boundary of anisotropic layer what complicates the technique&#46; It gives results not sufficiently better than the method <a class="elsevierStyleCrossRef" href="#eq0070">&#40;8&#41;</a>&#46; Therefore&#44; I use the simple method <a class="elsevierStyleCrossRef" href="#eq0070">&#40;8&#41;</a> here with setting an approximate value of <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span>&#46;</p></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0045">The methods for estimation of symmetry axis angle by AVOA</span><p id="par0160" class="elsevierStylePara elsevierViewall">Usually&#44; 3D seismic data used in AVOA analysis are received from a system of receivers and sources spacing in nodes of a rectangle grid at the surface&#46; The symmetry axis angle is calculated for a small square &#40;for a bin&#41; including a node of the grid&#44; by using seismic traces which have the Common Middle Point &#40;CMP&#41; located in this bin&#46; If such traces are few&#44; then neighbor bins are combined into a superbin&#44; and calculations are made for it&#46; Therefore&#44; a preliminary stage of the estimation is an extraction of seismic traces of the superbin from the seismic data for taking them into consideration&#46;</p><p id="par0165" class="elsevierStylePara elsevierViewall">For numerical methods of estimation of symmetry axis angle&#44; one can use <a class="elsevierStyleCrossRef" href="#eq0060">equation &#40;6&#41;</a> as in R&#252;ger&#39;s form&#58;<elsevierMultimedia ident="eq0075"></elsevierMultimedia>as in the power form&#58;<elsevierMultimedia ident="eq0080"></elsevierMultimedia>where <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf">&#42;</span> &#61; <span class="elsevierStyleItalic">r</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">g</span></span><span class="elsevierStyleItalic">P</span> &#40;<span class="elsevierStyleItalic">&#952;</span>&#44; <span class="elsevierStyleItalic">&#981;</span>&#41;&#44; <span class="elsevierStyleItalic">T</span> &#61; &#40;1&#8722;<span class="elsevierStyleItalic">s</span>&#41;<span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf">&#42;</span>&#44; <span class="elsevierStyleItalic">s</span> &#61; <span class="elsevierStyleItalic">sin</span><span class="elsevierStyleSup">2</span><span class="elsevierStyleItalic">&#952;</span>&#44; <span class="elsevierStyleItalic">t</span> &#61; <span class="elsevierStyleItalic">cos</span><span class="elsevierStyleSup">2</span> &#40;<span class="elsevierStyleItalic">&#981;</span>&#8722;<span class="elsevierStyleItalic">&#981;</span><span class="elsevierStyleInf">0</span>&#41;&#44; <span class="elsevierStyleItalic">a</span> &#61; <span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">0</span>&#44; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">&#42;</span> &#61; <span class="elsevierStyleItalic">mB</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">iso</span></span>&#44; <span class="elsevierStyleItalic">b</span> &#61; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">&#42;</span>&#8722;<span class="elsevierStyleItalic">a</span>&#44; <span class="elsevierStyleItalic">c</span> &#61; <span class="elsevierStyleItalic">mB</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ani</span></span>&#44; <span class="elsevierStyleItalic">d</span><span class="elsevierStyleInf">&#42;</span> &#61; <span class="elsevierStyleItalic">m</span>&#945;&#44; <span class="elsevierStyleItalic">d</span> &#61; <span class="elsevierStyleItalic">d</span><span class="elsevierStyleInf">&#42;</span>&#8722;<span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">&#42;</span>&#44; <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf">&#42;</span> &#61; <span class="elsevierStyleItalic">m</span>&#916;&#948;<span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span>&#47;2&#44; <span class="elsevierStyleItalic">e</span> &#61; <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf">&#42;</span>&#8722;<span class="elsevierStyleItalic">c</span>&#44; <span class="elsevierStyleItalic">f</span>&#61;<span class="elsevierStyleItalic">m&#916;&#949;</span><span class="elsevierStyleSup"><span class="elsevierStyleItalic">&#40;V&#41;</span></span>&#47;2&#8722;<span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf">&#42;</span> and <span class="elsevierStyleItalic">m</span> &#61; <span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">0</span>&#47;A&#46;</p><p id="par0170" class="elsevierStylePara elsevierViewall">The methods vary by simplifying ways applied&#44; and can be separated into Sectored methods &#40;S and SR&#41;&#44; Linear methods &#40;L and LR&#41;&#44; and General method &#40;G&#41;&#44; where the letter &#8216;R&#8217; denotes that the Ruger&#39;s form <a class="elsevierStyleCrossRef" href="#eq0075">&#40;9a&#41;</a> is used instead of <a class="elsevierStyleCrossRef" href="#eq0080">&#40;9b&#41;</a>&#46;</p><span id="sec0030" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0050">Sectored methods</span><p id="par0175" class="elsevierStylePara elsevierViewall">The method <span class="elsevierStyleItalic">SR</span> was suggested by <a class="elsevierStyleCrossRef" href="#bib0035">Mallik <span class="elsevierStyleItalic">et al&#46;</span> &#40;1998&#41;</a> for the case of three azimuths with using <a class="elsevierStyleCrossRef" href="#eq0005">equations &#40;1&#41;</a>&#44; and <a class="elsevierStyleCrossRef" href="#eq0015">&#40;3&#41;</a>&#46; It took its perfect form in the work by <a class="elsevierStyleCrossRef" href="#bib0060">Sabinin &#38; Chichinina &#40;2008&#41;</a> who used <a class="elsevierStyleCrossRef" href="#eq0060">equations &#40;6&#41;</a>&#44; <a class="elsevierStyleCrossRef" href="#eq0015">&#40;3&#41;</a>&#44; and <a class="elsevierStyleCrossRef" href="#eq0020">&#40;4&#41;</a>&#46; For this method&#44; the traces of superbin are sorted by <span class="elsevierStyleItalic">n</span> azimuthal sectors&#46; It is adopted that all traces of the sector have the same value of azimuth equal to the middle azimuth of the sector&#46; Because of this&#44; sectored methods introduce in <span class="elsevierStyleItalic">&#981;</span><span class="elsevierStyleInf">0</span> an own error no more than a half of the sector size&#46;</p><p id="par0180" class="elsevierStylePara elsevierViewall">Here&#44; the method S applied to <a class="elsevierStyleCrossRef" href="#eq0080">equation &#40;9b&#41;</a> is presented&#46; If in the sector of azimuth <span class="elsevierStyleItalic">&#981;</span> &#40;<span class="elsevierStyleItalic">j</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#44; &#8230;&#44; <span class="elsevierStyleItalic">n</span>&#41;&#44; there are <span class="elsevierStyleItalic">k</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span> traces with incidence angles <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span> &#40;<span class="elsevierStyleItalic">i</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#44;&#8230;&#44; <span class="elsevierStyleItalic">k</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span>&#41; in the last layer above the target boundary&#44; then one can write from <a class="elsevierStyleCrossRef" href="#eq0080">&#40;9b&#41;</a> for this sector <span class="elsevierStyleItalic">j</span>&#58;<elsevierMultimedia ident="eq0085"></elsevierMultimedia>where <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span> is the value <span class="elsevierStyleItalic">T</span> calculated from the trace <span class="elsevierStyleItalic">i</span> in the sector <span class="elsevierStyleItalic">j</span>&#46; In each sector&#44; Bj1&#61;mjBj&#44;Cj1&#61;mjCj where <span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span>&#47;<span class="elsevierStyleItalic">A</span>&#44; and Pj&#44;Bj1&#44;Cj1 are the fitting constants&#46;</p><p id="par0185" class="elsevierStylePara elsevierViewall">Having <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span> and <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> for all <span class="elsevierStyleItalic">i</span> in the sector <span class="elsevierStyleItalic">j</span> &#40;<span class="elsevierStyleItalic">k</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span><span class="elsevierStyleHsp" style=""></span>&#8805;<span class="elsevierStyleHsp" style=""></span>3&#41;&#44; one can calculate <span class="elsevierStyleItalic">s</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>sin<span class="elsevierStyleSup">2</span><span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleItalic">i</span>&#44; and then <span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span>&#44; Bj1&#44; and Cj1 from <a class="elsevierStyleCrossRef" href="#eq0085">&#40;10&#41;</a> by the least-squares method&#46; For this&#44; it is minimized the functional of error for each sector <span class="elsevierStyleItalic">j</span>&#58;<elsevierMultimedia ident="eq0090"></elsevierMultimedia></p><p id="par0190" class="elsevierStylePara elsevierViewall">For minimizing <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span>&#44; it is necessary to solve the system of three equations&#58;<elsevierMultimedia ident="eq0095"></elsevierMultimedia></p><p id="par0195" class="elsevierStylePara elsevierViewall">It gives&#58; Cj1&#61;b1f1&#8722;a1g1b12&#8722;a1c1&#44;Bj1&#61;f1&#8722;Cj1b1&#47;a1&#44; and Pj&#61;u0BCj1&#8722;ABj1&#47;kj&#44; where a1&#61;A2&#8722;Bkj&#44;b1&#61;AB&#8722;Ckj&#44;c1&#61;B2&#8722;Dkj&#44;f1&#61;Au0&#8722;u1kj&#44;g1&#61;Bu0&#8722;u2kj&#44;<elsevierMultimedia ident="eq0100"></elsevierMultimedia><a name="p462"></a></p><p id="par0200" class="elsevierStylePara elsevierViewall">These calculations should be made for all sectors&#46;</p><p id="par0205" class="elsevierStylePara elsevierViewall">Then&#44; the unknown value <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> can be obtained from the system of equations of type <a class="elsevierStyleCrossRef" href="#eq0015">&#40;3&#41;</a>&#44; see <a class="elsevierStyleCrossRef" href="#eq0080">&#40;9b&#41;</a>&#58;<elsevierMultimedia ident="eq0105"></elsevierMultimedia>where <span class="elsevierStyleItalic">j</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#44; &#8230;&#44; <span class="elsevierStyleItalic">n</span>&#44; and <span class="elsevierStyleItalic">t</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>cos<span class="elsevierStyleSup">2</span>&#40;<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#41;&#46; The unknown constants <span class="elsevierStyleItalic">b</span><span class="elsevierStyleSup">1</span>&#44; and <span class="elsevierStyleItalic">c</span><span class="elsevierStyleSup">1</span> have a sense&#58; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleSup">1</span><span class="elsevierStyleItalic">A</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">iso</span></span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">A</span>&#44; and <span class="elsevierStyleItalic">c</span><span class="elsevierStyleSup">1</span><span class="elsevierStyleItalic">A</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ani</span></span>&#46;</p><p id="par0210" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#eq0105">Equation &#40;11&#41;</a> is transformed into more convenient form&#58;<elsevierMultimedia ident="eq0110"></elsevierMultimedia>where Uj&#61;Bj1&#47;Pj&#44; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">0</span> &#61; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleSup">1</span> &#43; 0&#46;5<span class="elsevierStyleItalic">c</span><span class="elsevierStyleSup">1</span>&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">0</span> &#61; 0&#46;5<span class="elsevierStyleItalic">c</span><span class="elsevierStyleSup">1</span>&#44; <span class="elsevierStyleItalic">g</span> &#61; cos&#40;2<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#41;&#44; <span class="elsevierStyleItalic">h</span> &#61; sin&#40;2<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#41;&#44; <span class="elsevierStyleItalic">g</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span> &#61; cos&#40;2<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span>&#41; and <span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span> &#61; sin&#40;2<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span>&#41;&#46;</p><p id="par0215" class="elsevierStylePara elsevierViewall">The system <a class="elsevierStyleCrossRef" href="#eq0110">&#40;12&#41;</a> has three unknowns &#40;<span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">0</span>&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">0</span>&#44; and <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#41;&#44; therefore it should be <span class="elsevierStyleItalic">n</span><span class="elsevierStyleHsp" style=""></span>&#8805;<span class="elsevierStyleHsp" style=""></span>3 for obtaining solution&#46; The system <a class="elsevierStyleCrossRef" href="#eq0110">&#40;12&#41;</a> has two solutions &#40;two <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> differing in <span class="elsevierStyleItalic">&#960;</span>&#47;2&#44; and two <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">0</span> of opposite signs&#44; correspondently&#41;&#44; and is solved by the least-squares method&#44; too&#46; It is minimized the functional of error&#58;<elsevierMultimedia ident="eq0115"></elsevierMultimedia></p><p id="par0220" class="elsevierStylePara elsevierViewall">The following system of three equations should be solved&#58;<elsevierMultimedia ident="eq0120"></elsevierMultimedia></p><p id="par0225" class="elsevierStylePara elsevierViewall">It gives&#58; tan2&#981;0&#8801;hg&#61;b1f1&#8722;a1f2b1f2&#8722;c1f1&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf">1</span>&#47;&#40;<span class="elsevierStyleItalic">ga</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">hb</span><span class="elsevierStyleInf">1</span>&#41;&#44; and <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#91;<span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">0</span>&#40;<span class="elsevierStyleItalic">Ag</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">Bh</span>&#41;&#93;&#47;<span class="elsevierStyleItalic">n</span>&#44; where <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">A</span><span class="elsevierStyleSup">2</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">Cn</span>&#44; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">AB</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">Dn</span>&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">B</span><span class="elsevierStyleSup">2</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">En</span>&#44; <span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">Au</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">n</span>&#44; <span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">Bu</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">n</span>&#44; A&#61;&#8721;j&#61;1ngj&#44;B&#61;&#8721;j&#61;1nhj&#44;C&#61;&#8721;j&#61;1ngj2&#44;<elsevierMultimedia ident="eq0125"></elsevierMultimedia></p><p id="par0230" class="elsevierStylePara elsevierViewall">The condition for distinguishing symmetry-axis from fracture-strike directions is derived by <a class="elsevierStyleCrossRef" href="#bib0060">Sabinin &#38; Chichinina &#40;2008&#41;</a>&#44; and uses <a class="elsevierStyleCrossRef" href="#eq0020">equation &#40;4&#41;</a>&#46; Here it is presented in more general form&#46;</p><p id="par0235" class="elsevierStylePara elsevierViewall">In terms of <a class="elsevierStyleCrossRef" href="#eq0080">equations &#40;9b&#41;</a>&#44; <a class="elsevierStyleCrossRef" href="#eq0085">&#40;10&#41;</a>&#44; and <a class="elsevierStyleCrossRef" href="#eq0105">&#40;11&#41;</a>&#44; <a class="elsevierStyleCrossRef" href="#eq0020">equation &#40;4&#41;</a> can be written as<elsevierMultimedia ident="eq0130"></elsevierMultimedia>where <span class="elsevierStyleItalic">j</span> &#61; 1&#44; &#8230;&#44; <span class="elsevierStyleItalic">n</span>&#44; <span class="elsevierStyleItalic">d</span><span class="elsevierStyleSup">1</span> &#61; &#40;<span class="elsevierStyleItalic">&#945;</span> &#8722; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">iso</span></span>&#41;&#47;<span class="elsevierStyleItalic">A</span>&#44; <span class="elsevierStyleItalic">e</span><span class="elsevierStyleSup">1</span> &#61; &#40;<span class="elsevierStyleItalic">&#916;&#948;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span>&#47;2 &#8722; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ani</span></span>&#41;&#47;<span class="elsevierStyleItalic">A</span>&#44; and <span class="elsevierStyleItalic">f</span><span class="elsevierStyleSup">1</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#40;<span class="elsevierStyleItalic">&#916;&#949;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#916;&#948;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span>&#41;&#47;&#40;2<span class="elsevierStyleItalic">A</span>&#41;&#46;</p><p id="par0240" class="elsevierStylePara elsevierViewall">When substituting the value &#966;0&#177;&#960;2 instead of <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> into <a class="elsevierStyleCrossRef" href="#eq0130">equation &#40;14&#41;</a>&#44; the sign of the second term <span class="elsevierStyleItalic">e</span><span class="elsevierStyleSup">1</span><span class="elsevierStyleItalic">t</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span> switches to the opposite sign&#44; because <a class="elsevierStyleCrossRef" href="#eq0130">equation &#40;14&#41;</a> takes the form<elsevierMultimedia ident="eq0135"></elsevierMultimedia></p><p id="par0245" class="elsevierStylePara elsevierViewall">The last term of <a class="elsevierStyleCrossRef" href="#eq0105">equation &#40;11&#41;</a><span class="elsevierStyleItalic">c</span><span class="elsevierStyleSup">1</span><span class="elsevierStyleItalic">t</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span> switches its sign&#44; too&#46; One can combine <span class="elsevierStyleItalic">&#916;&#948;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span> &#61; 2<span class="elsevierStyleItalic">A</span>&#40;<span class="elsevierStyleItalic">c</span>1 &#43; <span class="elsevierStyleItalic">e</span><span class="elsevierStyleSup">1</span>&#41; from definitions to <a class="elsevierStyleCrossRef" href="#eq0105">equations &#40;11&#41;</a>&#44; <a class="elsevierStyleCrossRef" href="#eq0130">&#40;14&#41;</a>&#44; and conclude that the sign of <span class="elsevierStyleItalic">&#916;&#948;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span> is switched&#44; too&#46; For calculating <span class="elsevierStyleItalic">&#916;&#948;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span>&#44; it should be solved system <a class="elsevierStyleCrossRef" href="#eq0130">&#40;14&#41;</a> which is similar to <a class="elsevierStyleCrossRef" href="#eq0085">&#40;10&#41;</a> by the method of solution&#46;</p><p id="par0250" class="elsevierStylePara elsevierViewall">If the HTI layer is situated between isotropic layers then <span class="elsevierStyleItalic">&#916;&#948;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span> must be negative <span class="elsevierStyleItalic">for upper reflecting boundary</span> of the HTI layer&#44; and positive <span class="elsevierStyleItalic">for lower boundary</span>&#46; If the calculated value of <span class="elsevierStyleItalic">&#916;&#948;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span> has this sign then <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> is the symmetry-axis angle&#46; In opposite case&#44; it is the fracture-strike direction&#46;</p><p id="par0255" class="elsevierStylePara elsevierViewall">It must be noted that <span class="elsevierStyleItalic">&#916;&#949;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span> &#61; 2<span class="elsevierStyleItalic">A</span>&#40;<span class="elsevierStyleItalic">c</span><span class="elsevierStyleSup">1</span> &#43; <span class="elsevierStyleItalic">e</span><span class="elsevierStyleSup">1</span> &#43; <span class="elsevierStyleItalic">f</span><span class="elsevierStyleSup">1</span>&#41;&#44; and also can be used for distinguishing solutions because <span class="elsevierStyleItalic">&#949;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span> and <span class="elsevierStyleItalic">&#948;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span> have the same sign&#46;</p><p id="par0260" class="elsevierStylePara elsevierViewall">The formal condition that the second derivative of functional <a class="elsevierStyleCrossRef" href="#eq0115">&#40;13&#41;</a> must be positive in the minimum of functional can also be applied&#46; Because of errors in data&#44; it should be used as an additional condition to previous ones&#44; and should have a form &#8706;2F&#47;&#8706;&#981;02&#62; a small value&#46;</p></span><span id="sec0035" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0055">Linear methods</span><p id="par0265" class="elsevierStylePara elsevierViewall">The method LR was suggested by <a class="elsevierStyleCrossRef" href="#bib0025">Jenner &#40;2002&#41;</a> for <a class="elsevierStyleCrossRef" href="#eq0005">equation &#40;1&#41;</a>&#46; It is not needed in sectoring data&#46; All traces of superbin are taken into consideration together&#46;</p><p id="par0270" class="elsevierStylePara elsevierViewall">Here&#44; it is applied to <a class="elsevierStyleCrossRef" href="#eq0080">equation &#40;9b&#41;</a>&#44; the method L&#46; <a class="elsevierStyleCrossRef" href="#eq0080">Equation &#40;9b&#41;</a> is truncated after a line part respecting <span class="elsevierStyleItalic">s</span>&#46; If the superbin has <span class="elsevierStyleItalic">n</span> traces&#40;<span class="elsevierStyleItalic">i</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#44; &#8230;&#44; <span class="elsevierStyleItalic">n</span>&#41;&#44; with incidence angles <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> at the target boundary&#44; and with azimuthal angles <a name="p463"></a><span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span>&#44; then one can write the result of truncation in the form&#58;<elsevierMultimedia ident="eq0140"></elsevierMultimedia>where <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> is the value <span class="elsevierStyleItalic">T</span> calculated from the trace <span class="elsevierStyleItalic">i</span>&#44; <span class="elsevierStyleItalic">s</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> &#61; sin<span class="elsevierStyleSup">2</span><span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">i</span>&#44; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">0</span> &#61; b &#43; 0&#46;5<span class="elsevierStyleItalic">c</span>&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">0</span> &#61; 0&#46;5<span class="elsevierStyleItalic">c</span>&#44; <span class="elsevierStyleItalic">g</span> &#61; cos&#40;2<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#41;&#44; <span class="elsevierStyleItalic">h</span> &#61; sin&#40;2<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#41;&#44; <span class="elsevierStyleItalic">g</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> &#61; cos&#40;2<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span>&#41;&#44; and <span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> &#61; sin&#40;2<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span>&#41;&#46;</p><p id="par0275" class="elsevierStylePara elsevierViewall">The values <span class="elsevierStyleItalic">s</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span>&#44; <span class="elsevierStyleItalic">g</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span>&#44; and <span class="elsevierStyleItalic">h</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> are known because they can be calculated from headers of seismograms and parameters of medium&#46; Let us consider the functional of error&#58;<elsevierMultimedia ident="eq0145"></elsevierMultimedia></p><p id="par0280" class="elsevierStylePara elsevierViewall">Functional <span class="elsevierStyleItalic">F</span> must be minimized over parameters <span class="elsevierStyleItalic">a</span>&#44; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">0</span>&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">0</span>&#44; and <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#46; For this&#44; it is necessary to solve the system of four equations&#58;<elsevierMultimedia ident="eq0150"></elsevierMultimedia></p><p id="par0285" class="elsevierStylePara elsevierViewall">Solution of system <a class="elsevierStyleCrossRef" href="#eq0150">&#40;17&#41;</a> gives the equation for obtaining <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#58;<elsevierMultimedia ident="eq0155"></elsevierMultimedia>where A1&#61;a1b1&#8722;a22&#44;B1&#61;a1c1&#8722;a32&#44; <span class="elsevierStyleItalic">A</span><span class="elsevierStyleInf">2</span> &#61; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">2</span> &#8722; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span>&#44; <span class="elsevierStyleItalic">H</span><span class="elsevierStyleInf">1</span> &#61; <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span> &#8722; <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span>&#44; and <span class="elsevierStyleItalic">H</span><span class="elsevierStyleInf">2</span> &#61; <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span> &#8722; <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span> in which <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span> &#61; <span class="elsevierStyleItalic">nB</span> &#8722; <span class="elsevierStyleItalic">A</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">1</span> &#61; <span class="elsevierStyleItalic">nI</span> &#8722; <span class="elsevierStyleItalic">D</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">1</span> &#61; <span class="elsevierStyleItalic">nJ</span> &#8722; <span class="elsevierStyleItalic">E</span><span class="elsevierStyleSup">2</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span> &#61; <span class="elsevierStyleItalic">nG</span> &#8722; <span class="elsevierStyleItalic">AD</span>&#44; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">2</span> &#61; <span class="elsevierStyleItalic">nK</span> &#8722; <span class="elsevierStyleItalic">ED</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span> &#61; <span class="elsevierStyleItalic">nH</span> &#8722; <span class="elsevierStyleItalic">AE</span>&#44; <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">1</span> &#61; <span class="elsevierStyleItalic">nf</span><span class="elsevierStyleInf">1</span> &#8722; <span class="elsevierStyleItalic">Af</span><span class="elsevierStyleInf">0</span>&#44; <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">2</span> &#61; <span class="elsevierStyleItalic">nj</span><span class="elsevierStyleInf">2</span> &#8722; <span class="elsevierStyleItalic">Df</span><span class="elsevierStyleInf">0</span>&#44; and <span class="elsevierStyleItalic">F</span><span class="elsevierStyleInf">3</span> &#61; <span class="elsevierStyleItalic">nf</span><span class="elsevierStyleInf">3</span> &#8722; <span class="elsevierStyleItalic">Ef</span><span class="elsevierStyleInf">0</span>&#44; and finally&#58;<elsevierMultimedia ident="eq0160"></elsevierMultimedia></p><p id="par0290" class="elsevierStylePara elsevierViewall">The other parameters are&#58;<elsevierMultimedia ident="eq0165"></elsevierMultimedia>and <span class="elsevierStyleItalic">a</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#40;<span class="elsevierStyleItalic">f</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleItalic">A</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleItalic">gD</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleItalic">hE</span>&#41;&#47;<span class="elsevierStyleItalic">n</span>&#46;</p><p id="par0295" class="elsevierStylePara elsevierViewall">From <a class="elsevierStyleCrossRef" href="#eq0155">&#40;18&#41;</a>&#44; one can see that the solution <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> has a period of &#960;2&#46; This value of the period means that we must use an additional condition for understanding what value <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> is the symmetry-axis azimuth&#46; This condition may be <span class="elsevierStyleItalic">B</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ani</span></span><span class="elsevierStyleHsp" style=""></span>&#62;<span class="elsevierStyleHsp" style=""></span>0 if <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">S</span></span> &#47; <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span></span><span class="elsevierStyleHsp" style=""></span>&#62;<span class="elsevierStyleHsp" style=""></span>0&#46;56 &#40;<a class="elsevierStyleCrossRef" href="#bib0015">Chichinina <span class="elsevierStyleItalic">et al&#46;</span>&#44; 2003</a>&#41;&#46; In general case&#44; it can be the condition &#8706;2F&#47;&#8706;&#981;02&#62; a small positive value&#44; where <span class="elsevierStyleItalic">F</span> is the functional of error <a class="elsevierStyleCrossRef" href="#eq0145">&#40;16&#41;</a>&#46;</p></span><span id="sec0040" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0060">General method &#40;G&#41;</span><p id="par0300" class="elsevierStylePara elsevierViewall">The method is constructed by analogy with the GM method by <a class="elsevierStyleCrossRef" href="#bib0055">Sabinin &#40;2013&#41;</a>&#46; It is not needed in sectoring&#44; too&#46; All traces of superbin are taken into consideration together&#46; If the superbin has <span class="elsevierStyleItalic">n</span> traces &#40;<span class="elsevierStyleItalic">i</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#44; &#8230;&#44; <span class="elsevierStyleItalic">n</span>&#41;&#44; with incidence angles <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> at the target boundary&#44; and with azimuthal angles <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span>&#44; then <a class="elsevierStyleCrossRef" href="#eq0080">equation &#40;9b&#41;</a> can be written as&#58;<elsevierMultimedia ident="eq0170"></elsevierMultimedia>where <span class="elsevierStyleItalic">T</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> is the value <span class="elsevierStyleItalic">T</span> calculated from the trace <span class="elsevierStyleItalic">i</span>&#44; <span class="elsevierStyleItalic">s</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>sin<span class="elsevierStyleSup">2</span><span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span>&#44; and <span class="elsevierStyleItalic">t</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>cos<span class="elsevierStyleSup">2</span>&#40;<span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#41;&#46;</p><p id="par0305" class="elsevierStylePara elsevierViewall">Let us consider the functional of error&#58;<elsevierMultimedia ident="eq0175"></elsevierMultimedia></p><p id="par0310" class="elsevierStylePara elsevierViewall">Functional <span class="elsevierStyleItalic">F</span> must be minimized over parameters <span class="elsevierStyleItalic">a</span>&#44; <span class="elsevierStyleItalic">b</span>&#44; <span class="elsevierStyleItalic">c</span>&#44; <span class="elsevierStyleItalic">d</span>&#44; <span class="elsevierStyleItalic">e</span>&#44; <span class="elsevierStyleItalic">f</span>&#44; and <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#46; For this&#44; it is necessary to solve the system of seven equations&#58;<elsevierMultimedia ident="eq0180"></elsevierMultimedia></p><p id="par0315" class="elsevierStylePara elsevierViewall">The six first equations of system <a class="elsevierStyleCrossRef" href="#eq0180">&#40;20&#41;</a> give a line system for deriving expressions for the parameters <span class="elsevierStyleItalic">a</span>&#44; <span class="elsevierStyleItalic">b</span>&#44; <span class="elsevierStyleItalic">c</span>&#44; <span class="elsevierStyleItalic">d</span>&#44; <span class="elsevierStyleItalic">e</span>&#44; and <span class="elsevierStyleItalic">f</span> &#40;for details&#44; see <a class="elsevierStyleCrossRef" href="#sec055">Appendix</a>&#41;&#46;</p><p id="par0320" class="elsevierStylePara elsevierViewall">The last equation of <a class="elsevierStyleCrossRef" href="#eq0180">&#40;20&#41;</a> can be transformed into a non-linear equation for obtaining <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> &#40;for details&#44; see <a class="elsevierStyleCrossRef" href="#sec055">Appendix</a>&#41;&#46;</p><p id="par0325" class="elsevierStylePara elsevierViewall">Thus&#44; system <a class="elsevierStyleCrossRef" href="#eq0180">&#40;20&#41;</a> is non-linear on <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#44; and is solved by the method of bisecting&#46; It has more than one solution usually&#46; From these local solutions&#44; one chooses that one which gives a minimum for functional <a class="elsevierStyleCrossRef" href="#eq0175">&#40;19&#41;</a>&#46;</p><p id="par0330" class="elsevierStylePara elsevierViewall">As was observed from calculations&#44; the solutions of system <a class="elsevierStyleCrossRef" href="#eq0180">&#40;20&#41;</a> near the symmetry axis angle&#44; and near the fracture strike angle give close values of functional <a class="elsevierStyleCrossRef" href="#eq0175">&#40;19&#41;</a>&#46; It means that additional criterions are practically needed for separating these directions&#46; For the case <a name="p464"></a>of HTI layer situated between isotropic layers&#44; it can be the condition of negative values for calculated <span class="elsevierStyleItalic">&#949;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span> and <span class="elsevierStyleItalic">&#948;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span> in the anisotropic layer&#44; as above&#46; For this&#44; from definitions to <a class="elsevierStyleCrossRef" href="#eq0080">equations &#40;9b&#41;</a> and <a class="elsevierStyleCrossRef" href="#eq0010">&#40;2&#41;</a>&#44; one can calculate from the solution of <a class="elsevierStyleCrossRef" href="#eq0180">&#40;20&#41;</a> at the interface&#58;<elsevierMultimedia ident="eq0185"></elsevierMultimedia><elsevierMultimedia ident="eq0190"></elsevierMultimedia></p><p id="par0335" class="elsevierStylePara elsevierViewall">In the case of interface between anisotropic layers&#44; it is needed additionally to know the predefined signs of <span class="elsevierStyleItalic">&#916;&#949;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span>&#44; and <span class="elsevierStyleItalic">&#916;&#948;</span><span class="elsevierStyleSup">&#40;<span class="elsevierStyleItalic">V</span>&#41;</span> for comparison&#46;</p><p id="par0340" class="elsevierStylePara elsevierViewall">The additional criterion can also be the maximum of second derivative of functional <a class="elsevierStyleCrossRef" href="#eq0175">&#40;19&#41;</a>&#44; <span class="elsevierStyleItalic">&#8706;F</span>&#47;<span class="elsevierStyleItalic">&#8706;&#966;</span><span class="elsevierStyleInf">0</span>&#46;</p></span></span><span id="sec0045" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0065">Comparing the AVOA techniques</span><p id="par0345" class="elsevierStylePara elsevierViewall">The techniques using the methods above for estimation of symmetry axis angle were compared in ability to give the most precise value of <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> for HTI medium&#46; At present&#44; reliable field methods of obtaining <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> do not exist&#46; Therefore&#44; I generated synthetic seismograms for an artificial three-layer medium with the anisotropic layer in the middle by applying the technique by <a class="elsevierStyleCrossRef" href="#bib0050">Sabinin &#40;2012&#41;</a> of 2D wave modeling&#46; I set <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>60&#176;&#44; and derived models of the anisotropic layer for different values of <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span> by rotating the stiffness tensor for anisotropic layer &#40;<a class="elsevierStyleCrossRef" href="#bib0030">MacBeth&#44; 1999</a>&#41; around <span class="elsevierStyleItalic">z</span> axis relatively to <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#46; Anisotropic parameters <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">n</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;35&#44; and <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">t</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;2 &#40;see <a class="elsevierStyleCrossRef" href="#bib0030">MacBeth&#44; 1999</a>&#41; were used in the stiffness tensor&#46;</p><p id="par0350" class="elsevierStylePara elsevierViewall">Host rock velocity <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span></span> in three layers from above had the values 3200&#44; 4000&#44; and 4800 &#40;the other variant was 3200&#41;&#44; <span class="elsevierStyleItalic">m</span>&#47;<span class="elsevierStyleItalic">s</span>&#44; <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">S</span></span> was twice less&#44; densities were equal&#44; and thicknesses of two first layers were 1600&#44; and 400<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">m</span>&#46; A source of explosion type generated one Ricker impulse of frequency 30<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">Hz</span>&#46; Receivers were spaced over every 100<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">m</span> beginning from the source&#44; and they measured z-component of velocity&#46; There were 50 offsets&#44; and 50 traces in each seismogram&#46;</p><p id="par0355" class="elsevierStylePara elsevierViewall">There were three goals&#58; to investigate how the techniques behave on different sets of incidence angles&#44; how the techniques are influenced by non-symmetry in <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span> relatively to <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span>&#44; and how the techniques are influenced by noise&#46;</p><p id="par0360" class="elsevierStylePara elsevierViewall">Therefore&#44; for the first goal&#44; I made calculations of <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> for different intervals of offsets&#58; from a minimum offset till a maximum offset&#44; provided the minimum offset was fixed at the number one&#44; and the number of maximum offset was changed from number 50 down to 3 in one set of the intervals&#59; and the maximum offset was fixed at the 50-th&#44; and the minimum offset was changed from number 1 to 48 in the other set of the intervals&#46; Naturally&#44; the maximum incidence angle <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span> corresponding to the maximum offset&#44; and the minimum incidence angle <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">min</span> corresponding to the minimum offset was also correspondently changed in these sets of offsets&#46;</p><p id="par0365" class="elsevierStylePara elsevierViewall">For the second goal&#44; I obtained different sets of the synthetic seismograms corresponding to different azimuths&#44; one seismogram for each azimuth&#46; The sets of azimuths were uniform&#44; and differed by symmetry&#46; I did not aim to find the best or the worst set from them&#46; I only supposed that a symmetric set can be better than an asymmetric one&#46; I kept for testing the symmetric set of azimuths <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;&#8722;150&#176;&#44; &#8722;120&#176;&#44; &#8722;90&#176;&#44; &#8722;60&#176;&#44; &#8722;30&#176;&#44; 0&#176;&#44; 30&#176;&#44; 60&#176;&#44; 90&#176;&#44; 120&#176;&#44; 150&#176;&#44; 180&#176;&#125;&#44; and the asymmetric set <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;85&#176;&#44; 95&#176;&#44; 105&#176;&#44; 115&#176;&#44; 125&#176;&#44; 135&#176;&#44; 145&#176;&#44; 155&#176;&#44; 165&#176;&#125;&#46;</p><p id="par0370" class="elsevierStylePara elsevierViewall">For the third goal&#44; I took the best variant for the symmetric set of seismograms to eliminate the errors as due to the non-symmetry&#44; as due to a finite-difference simulation when applying the artificial noise&#46; The FD simulation by <a class="elsevierStyleCrossRef" href="#bib0050">Sabinin &#40;2012&#41;</a> uses PML boundary conditions which give non-visible &#40;see <a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a>&#41; but nonzero waves reflected from the boundaries of area&#46; This slightly distorts the form of some synthetic impulses&#46;</p><elsevierMultimedia ident="fig0020"></elsevierMultimedia><p id="par0375" class="elsevierStylePara elsevierViewall">For the synthetic seismic data being quasireal&#44; I added a random Gauss normal noise to the seismograms generated&#44; different for each seismogram&#46; Maximum amplitude of the noise was chosen as 10&#37; of the maximum amplitude of the wave reflected from the top boundary of the anisotropic layer in the first trace of seismogram&#46;</p><p id="par0380" class="elsevierStylePara elsevierViewall">Finally&#44; I added the noise to the seismograms of the asymmetric set&#46;</p><p id="par0385" class="elsevierStylePara elsevierViewall">All seismograms were smoothed by filters <a class="elsevierStyleCrossRef" href="#eq0025">&#40;5&#41;</a> in the techniques&#46; High-frequency components of the noise are eliminated well after smoothing&#44; as shown in <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>&#46; It is principally impossible to eliminate low frequencies compared with the frequency of signal&#46; Therefore&#44; the signal after smoothing remains slightly deformed&#46; I suppose that just these deformations affect the estimated value of <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> in the case of noise&#46;<a name="p465"></a></p><p id="par0390" class="elsevierStylePara elsevierViewall">The same sets of the time windows were used for all the techniques&#44; and for all intervals of offsets&#46;</p><p id="par0395" class="elsevierStylePara elsevierViewall">As illustration&#44; in <a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a>&#44; the seismogram without noise for azimuth 5&#176; is presented for the variant of <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>4800 <span class="elsevierStyleItalic">m</span>&#47;<span class="elsevierStyleItalic">s</span>&#59; and in <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>&#44; the seismogram with noise for azimuth 30&#176; is presented for the variant of <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3200 <span class="elsevierStyleItalic">m</span>&#47;<span class="elsevierStyleItalic">s</span>&#46;</p><elsevierMultimedia ident="fig0025"></elsevierMultimedia><p id="par0400" class="elsevierStylePara elsevierViewall">As one can see from <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>&#44; the amplitudes of noise reach really up to 50&#37; of the maximum wave amplitudes in the middle traces&#44; and up to 100&#37; in the far traces&#46;</p><p id="par0405" class="elsevierStylePara elsevierViewall">The techniques were applied as to upper &#40;1050 ms&#41;&#44; as to down boundary &#40;1250 ms&#41; of the anisotropic layer&#46;</p><p id="par0410" class="elsevierStylePara elsevierViewall">In <a class="elsevierStyleCrossRef" href="#fig0030">Figures 6</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0035">7</a>&#44; the error of estimated <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> in degrees &#40;difference with the correct value 60&#176;&#41; is presented for the symmetric set of azimuths and the upper boundary&#44; variant <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3200&#46; <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a> is for fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">min</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#176;&#44; and <a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a> is for fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>56&#46;853&#176;&#46; The sectored methods show some instability for small values of <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">min</span> in comparison with the others&#46; All methods increase the error in the case of small <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span> &#40;<a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>&#41;&#46;</p><elsevierMultimedia ident="fig0030"></elsevierMultimedia><elsevierMultimedia ident="fig0035"></elsevierMultimedia><p id="par0415" class="elsevierStylePara elsevierViewall">For the lower boundary and in the variant <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>4800&#44; the general and linear methods also show increasing errors for small <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span>&#44; and small <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span><span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">min</span>&#44; see <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a>&#44; and <a class="elsevierStyleCrossRef" href="#fig0045">Figure 9</a>&#46; However&#44; the errors of these methods are sufficiently less than of the sectored methods&#46;</p><elsevierMultimedia ident="fig0040"></elsevierMultimedia><elsevierMultimedia ident="fig0045"></elsevierMultimedia><p id="par0420" class="elsevierStylePara elsevierViewall">In <a class="elsevierStyleCrossRef" href="#fig0050">Figures 10</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0055">11</a>&#44; the variant of <a class="elsevierStyleCrossRef" href="#fig0030">Figs&#46; 6</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0035">7</a> with the added noise is presented&#46; The sectored methods demonstrate so great errors and instability that can not be recommended for applying&#46; The other methods show large errors only for small <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span> &#40;less than 30&#176;&#41;&#46;</p><elsevierMultimedia ident="fig0050"></elsevierMultimedia><elsevierMultimedia ident="fig0055"></elsevierMultimedia><p id="par0425" class="elsevierStylePara elsevierViewall">The asymmetric set of azimuths is presented by results in <a class="elsevierStyleCrossRef" href="#fig0060">Figures 12</a>&#8211;<a class="elsevierStyleCrossRef" href="#fig0075">15</a>&#46; The variant of upper boundary and <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3200 without noise is presented in <a class="elsevierStyleCrossRef" href="#fig0060">Figures 12</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0065">13</a>&#44; and the same with the noise &#8211; in <a class="elsevierStyleCrossRef" href="#fig0070">Figures 14</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0075">15</a>&#46;</p><elsevierMultimedia ident="fig0060"></elsevierMultimedia><elsevierMultimedia ident="fig0065"></elsevierMultimedia><elsevierMultimedia ident="fig0070"></elsevierMultimedia><elsevierMultimedia ident="fig0075"></elsevierMultimedia><p id="par0430" class="elsevierStylePara elsevierViewall">Typical peculiarities of the asymmetric set are&#58; great errors of the sectored methods with instability in noised data&#44; and stable large errors of the linear methods &#40;up to 7&#176;&#41;&#46; The general method remains of small errors&#46; The noise causes instability of all methods in the interval of <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span><span class="elsevierStyleHsp" style=""></span>&#60;<span class="elsevierStyleHsp" style=""></span>36&#176;&#44; provided even the general method &#40;G&#41; gives large errors in this interval&#46;<a name="p466"></a><a name="p467"></a><a name="p468"></a><a name="p469"></a><a name="p470"></a></p></span><span id="sec0050" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0070">Discussion and conclusion</span><p id="par0435" class="elsevierStylePara elsevierViewall">Some unexpected results were obtained&#46; The first is that the sectored S and SR methods are failed&#46; They can be used only in seismic data without noise&#44; and for mainly symmetric distributions of azimuths <span class="elsevierStyleItalic">&#966;</span> in the 3D data &#40;<a class="elsevierStyleCrossRef" href="#fig0030">Figures 6</a>&#8211;<a class="elsevierStyleCrossRef" href="#fig0045">9</a>&#41;&#46; This is too ideal conditions&#46;</p><p id="par0440" class="elsevierStylePara elsevierViewall">The second is that the linear <span class="elsevierStyleItalic">L</span> and <span class="elsevierStyleItalic">LR</span> methods have an additional nearly constant error in mainly asymmetric distributions of azimuths <span class="elsevierStyleItalic">&#966;</span> in the data &#40;<a class="elsevierStyleCrossRef" href="#fig0060">Figures 12</a>&#8211;<a class="elsevierStyleCrossRef" href="#fig0075">15</a>&#41;&#46; This error is probably connected with the truncation of high terms in <a class="elsevierStyleCrossRef" href="#eq0005">equation &#40;1&#41;</a> of Ruger&#44; because the general method G has not such error&#46; Therefore&#44; the linear methods should be applied to azimuthally symmetric data&#46;</p><p id="par0445" class="elsevierStylePara elsevierViewall">The third is that the smoothing data with noise by simple filters <a class="elsevierStyleCrossRef" href="#eq0025">&#40;5&#41;</a> gives relatively stable estimated values of <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> in a wide interval of incidence angles <span class="elsevierStyleItalic">&#952;</span> for the methods L&#44; LR&#44; and G &#40;<a class="elsevierStyleCrossRef" href="#fig0050">Figures 10</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0055">11</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0070">14</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0075">15</a>&#41;&#46; The interval of instability is near the normal incidence&#44; and has a width of <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span><span class="elsevierStyleHsp" style=""></span>&#60;<span class="elsevierStyleHsp" style=""></span>40&#176;&#44; different in different variants &#40;<a class="elsevierStyleCrossRef" href="#fig0050">Figures 10</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0070">14</a>&#41;&#46; For data without noise&#44; this interval is <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span><span class="elsevierStyleHsp" style=""></span>&#60;<span class="elsevierStyleHsp" style=""></span>10&#176; &#40;<a class="elsevierStyleCrossRef" href="#fig0030">Figures 6</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0040">8</a>&#41;&#46; Presence of the interval of instability is an intrinsic property of the formula <a class="elsevierStyleCrossRef" href="#eq0005">&#40;1&#41;</a> in connection with the least-squares method&#46; Errors in amplitudes become relatively more with decreasing <span class="elsevierStyleItalic">&#952;</span> in definition of <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> by <a class="elsevierStyleCrossRef" href="#eq0005">equation &#40;1&#41;</a>&#46;</p><p id="par0450" class="elsevierStylePara elsevierViewall">The results show a superior of the general method &#40;G&#41;&#46; On the whole&#44; its errors are less than of the others&#46; Unfortunately&#44; it has an intrinsic problem of choosing the right solution from the local solutions of non-linear system <a class="elsevierStyleCrossRef" href="#eq0180">&#40;20&#41;</a>&#46; All criterions described above do not guarantee the correct choosing&#46; It is especially difficult in the interval of instability&#46; All the methods have such problem of distinguishing solutions&#46; The best in this sense is the method L&#46; Its criterions are failed very rarely&#46; Therefore&#44; I recommend applying the method G in a coupling with the method L&#58; after estimation of <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> by L&#44; the value <span class="elsevierStyleItalic">&#966;</span><span class="elsevierStyleInf">0</span> is defined more precisely by G with expertly taking into consideration the local solutions of <a class="elsevierStyleCrossRef" href="#eq0180">&#40;20&#41;</a>&#46; The other recommendation is to avoid the interval of instability&#46;</p><p id="par0455" class="elsevierStylePara elsevierViewall">In applying to field data&#44; the techniques can give worse results&#46; The real data have much more interferences of waves than the synthetic data&#46; It is practically impossible to clear each interfered wave of the other by filters&#46; Distorted by this way impulses can lead to unpredictable results&#46;</p></span></span>"
    "textoCompletoSecciones" => array:1 [
      "secciones" => array:12 [
        0 => array:2 [
          "identificador" => "xres400573"
          "titulo" => "Resumen"
        ]
        1 => array:2 [
          "identificador" => "xpalclavsec378193"
          "titulo" => "Palabras clave"
        ]
        2 => array:2 [
          "identificador" => "xres400574"
          "titulo" => "Abstract"
        ]
        3 => array:2 [
          "identificador" => "xpalclavsec378194"
          "titulo" => "Key words"
        ]
        4 => array:2 [
          "identificador" => "sec0005"
          "titulo" => "Introduction"
        ]
        5 => array:2 [
          "identificador" => "sec0010"
          "titulo" => "Background"
        ]
        6 => array:2 [
          "identificador" => "sec0015"
          "titulo" => "Using amplitudes instead of reflection coefficients"
        ]
        7 => array:2 [
          "identificador" => "sec0020"
          "titulo" => "The incidence angle estimation"
        ]
        8 => array:3 [
          "identificador" => "sec0025"
          "titulo" => "The methods for estimation of symmetry axis angle by AVOA"
          "secciones" => array:3 [
            0 => array:2 [
              "identificador" => "sec0030"
              "titulo" => "Sectored methods"
            ]
            1 => array:2 [
              "identificador" => "sec0035"
              "titulo" => "Linear methods"
            ]
            2 => array:2 [
              "identificador" => "sec0040"
              "titulo" => "General method &#40;G&#41;"
            ]
          ]
        ]
        9 => array:2 [
          "identificador" => "sec0045"
          "titulo" => "Comparing the AVOA techniques"
        ]
        10 => array:2 [
          "identificador" => "sec0050"
          "titulo" => "Discussion and conclusion"
        ]
        11 => array:1 [
          "titulo" => "References"
        ]
      ]
    ]
    "pdfFichero" => "main.pdf"
    "tienePdf" => true
    "fechaRecibido" => "2013-09-02"
    "fechaAceptado" => "2013-12-05"
    "PalabrasClave" => array:2 [
      "es" => array:1 [
        0 => array:4 [
          "clase" => "keyword"
          "titulo" => "Palabras clave"
          "identificador" => "xpalclavsec378193"
          "palabras" => array:4 [
            0 => "AVOA"
            1 => "medio HTI"
            2 => "anisotrop&#237;a s&#237;smica"
            3 => "caracterizaci&#243;n de yacimientos fracturados"
          ]
        ]
      ]
      "en" => array:1 [
        0 => array:4 [
          "clase" => "keyword"
          "titulo" => "Key words"
          "identificador" => "xpalclavsec378194"
          "palabras" => array:4 [
            0 => "AVOA"
            1 => "HTI medium"
            2 => "seismic anisotropy"
            3 => "fracture-reservoir characterization"
          ]
        ]
      ]
    ]
    "tieneResumen" => true
    "resumen" => array:2 [
      "es" => array:2 [
        "titulo" => "Resumen"
        "resumen" => "<p id="spar0080" class="elsevierStyleSimplePara elsevierViewall">Se tomaron en consideraci&#243;n distintos aspectos de algunas t&#233;cnicas computacionales para el an&#225;lisis AVOA &#40;Amplitud Versus Offset y Azimut&#41;&#44; para la composici&#243;n de fracturas&#44; en particular&#58; utilizando amplitudes en lugar de coeficientes de refecci&#243;n&#44; suavizando los datos s&#237;smicos y el m&#233;todo de la estimaci&#243;n num&#233;rica para calcular la direcci&#243;n&#46; Se estim&#243; un nuevo m&#233;todo de c&#225;lculo y se indica un nuevo m&#233;todo suavizado&#46; Se compararan distintos m&#233;todos de c&#225;lculo en los datos sint&#233;ticos de superficie de reflecci&#243;n&#44; con y sin ruido&#46; Se obtuvieron propiedades de los m&#233;todos num&#233;ricos&#44; dependientes de conjuntos distintos de los azimut y los offset&#46; Se muestra una superioridad del nuevo m&#233;todo&#46;</p>"
      ]
      "en" => array:2 [
        "titulo" => "Abstract"
        "resumen" => "<p id="spar0085" class="elsevierStyleSimplePara elsevierViewall">Different aspects of computational techniques for AVOA analysis &#40;Amplitude Versus Offset and Azimuth&#41; for fracture characterization are considered&#44; in particular&#58; using amplitudes instead of reflection coefficients&#44; smoothing seismic data&#44; and numerical methods for estimation of fracture directions&#46; A new computational method and a new filter for smoothing are suggested&#46; The different computational methods are compared in synthetic reflection surface data with noise&#44; and without noise&#46; Properties of the numerical methods in dependence on different sets of azimuths and offsets are obtained&#46; It is shown a superiority of the new method&#46;<a name="p458"></a></p>"
      ]
    ]
    "apendice" => array:1 [
      0 => array:1 [
        "seccion" => array:1 [
          0 => array:4 [
            "apendice" => "<p id="par0460" class="elsevierStylePara elsevierViewall">Let&#39;s define&#58;<elsevierMultimedia ident="eq0195"></elsevierMultimedia></p> <p id="par0465" class="elsevierStylePara elsevierViewall">Then&#44; from the first six equations of system <a class="elsevierStyleCrossRef" href="#eq0180">&#40;20&#41;</a>&#44; one can derive the formulas for unknown parameters&#58;<elsevierMultimedia ident="eq0200"></elsevierMultimedia>where <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">01</span> &#61; <span class="elsevierStyleItalic">A</span><span class="elsevierStyleSup">2</span>&#8722;<span class="elsevierStyleItalic">Cn</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">02</span> &#61; <span class="elsevierStyleItalic">AB</span>&#8722;<span class="elsevierStyleItalic">Dn</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">02</span> &#61; <span class="elsevierStyleItalic">AC</span>&#8722;<span class="elsevierStyleItalic">Fn</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">04</span> &#61; <span class="elsevierStyleItalic">AD</span>&#8722;<span class="elsevierStyleItalic">Gn</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">05</span> &#61; <span class="elsevierStyleItalic">AE</span>&#8722;<span class="elsevierStyleItalic">Hn</span>&#44; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">01</span> &#61; <span class="elsevierStyleItalic">B</span><span class="elsevierStyleSup">2</span>&#8722;<span class="elsevierStyleItalic">En</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">02</span> &#61; <span class="elsevierStyleItalic">BC</span>&#8722; <span class="elsevierStyleItalic">Gn</span>&#44; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">03</span> &#61; <span class="elsevierStyleItalic">BD</span>&#8722;<span class="elsevierStyleItalic">Hn</span>&#44; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">04</span> &#61; <span class="elsevierStyleItalic">BE</span>&#8722;<span class="elsevierStyleItalic">Kn</span>&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">01</span> &#61; <span class="elsevierStyleItalic">C</span><span class="elsevierStyleSup">2</span>&#8722;<span class="elsevierStyleItalic">Ln</span>&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">02</span> &#61; <span class="elsevierStyleItalic">CD</span>&#8722;<span class="elsevierStyleItalic">Mn</span>&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">03</span> &#61; <span class="elsevierStyleItalic">CE</span>&#8722;<span class="elsevierStyleItalic">Nn</span>&#44; <span class="elsevierStyleItalic">d</span><span class="elsevierStyleInf">01</span> &#61; <span class="elsevierStyleItalic">D</span><span class="elsevierStyleSup">2</span>&#8722;<span class="elsevierStyleItalic">Nn</span>&#44; <span class="elsevierStyleItalic">d</span><span class="elsevierStyleInf">02</span> &#61; <span class="elsevierStyleItalic">DE</span>&#8722;<span class="elsevierStyleItalic">On</span>&#44; <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf">01</span> &#61; <span class="elsevierStyleItalic">E</span><span class="elsevierStyleSup">2</span>&#8722;<span class="elsevierStyleItalic">Pn</span>&#44; <span class="elsevierStyleItalic">k</span><span class="elsevierStyleInf">1</span> &#61; <span class="elsevierStyleItalic">AU</span><span class="elsevierStyleInf">0</span>&#8722;<span class="elsevierStyleItalic">U</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">n</span>&#44; <span class="elsevierStyleItalic">k</span><span class="elsevierStyleInf">2</span> &#61; <span class="elsevierStyleItalic">BU</span><span class="elsevierStyleInf">0</span>&#8722;<span class="elsevierStyleItalic">U</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">n</span>&#44; <span class="elsevierStyleItalic">k</span><span class="elsevierStyleInf">3</span> &#61; <span class="elsevierStyleItalic">CU</span><span class="elsevierStyleInf">0</span>&#8722;<span class="elsevierStyleItalic">U</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">n</span>&#44; <span class="elsevierStyleItalic">k</span><span class="elsevierStyleInf">4</span> &#61; <span class="elsevierStyleItalic">DU</span><span class="elsevierStyleInf">0</span>&#8722;<span class="elsevierStyleItalic">U</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleItalic">n</span>&#44; <span class="elsevierStyleItalic">k</span><span class="elsevierStyleInf">5</span> &#61; <span class="elsevierStyleItalic">EU</span><span class="elsevierStyleInf">0</span>&#8722;<span class="elsevierStyleItalic">U</span><span class="elsevierStyleInf">5</span><span class="elsevierStyleItalic">n</span>&#44;<elsevierMultimedia ident="eq0205"></elsevierMultimedia><a name="p471"></a></p> <p id="par0470" class="elsevierStylePara elsevierViewall">The seventh equation of system <a class="elsevierStyleCrossRef" href="#eq0180">&#40;20&#41;</a> takes a form&#58;<elsevierMultimedia ident="eq0210"></elsevierMultimedia>where<elsevierMultimedia ident="eq0215"></elsevierMultimedia></p>"
            "etiqueta" => "Appendix"
            "titulo" => "Solution of system <a class="elsevierStyleCrossRef" href="#eq0180">&#40;20&#41;</a>"
            "identificador" => "sec055"
          ]
        ]
      ]
    ]
    "multimedia" => array:58 [
      0 => array:7 [
        "identificador" => "fig0005"
        "etiqueta" => "Figure 1"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr1.jpeg"
            "Alto" => 613
            "Ancho" => 882
            "Tamanyo" => 79972
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">A signal with noise &#40;thin line&#41; in time&#44; and its envelope&#46;</p>"
        ]
      ]
      1 => array:7 [
        "identificador" => "fig0010"
        "etiqueta" => "Figure 2"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr2.jpeg"
            "Alto" => 188
            "Ancho" => 882
            "Tamanyo" => 22724
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0010" class="elsevierStyleSimplePara elsevierViewall">The 3-stage decomposition algorithm&#46;</p>"
        ]
      ]
      2 => array:7 [
        "identificador" => "fig0015"
        "etiqueta" => "Figure 3"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr3.jpeg"
            "Alto" => 610
            "Ancho" => 882
            "Tamanyo" => 67468
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0015" class="elsevierStyleSimplePara elsevierViewall">The signal with noise &#40;thin line&#41;&#44; the smoothed signal &#40;thick line&#41;&#44; and the envelope of smoothed signal&#46;</p>"
        ]
      ]
      3 => array:7 [
        "identificador" => "fig0020"
        "etiqueta" => "Figure 4"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr4.jpeg"
            "Alto" => 1134
            "Ancho" => 1881
            "Tamanyo" => 220774
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0020" class="elsevierStyleSimplePara elsevierViewall">Synthetic seismogram without noise&#46; Azimuth 5&#176;&#44; <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>4800&#46; Axis x &#8211; time in ms&#44; axis y &#8211; numbers of traces&#46; Zero time is origin of the source impulse&#46;</p>"
        ]
      ]
      4 => array:7 [
        "identificador" => "fig0025"
        "etiqueta" => "Figure 5"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr5.jpeg"
            "Alto" => 1136
            "Ancho" => 1882
            "Tamanyo" => 632005
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0025" class="elsevierStyleSimplePara elsevierViewall">Synthetic seismogram with added 10&#37; noise&#46; Azimuth 30&#176;&#44; <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3200&#46; Axis x &#8211; time in ms&#44; axis y &#8211; numbers of traces&#46;</p>"
        ]
      ]
      5 => array:7 [
        "identificador" => "fig0030"
        "etiqueta" => "Figure 6"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr6.jpeg"
            "Alto" => 959
            "Ancho" => 1253
            "Tamanyo" => 103550
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0030" class="elsevierStyleSimplePara elsevierViewall">Errors for the symmetric set of azimuths&#59; the upper boundary&#44; and fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">min</span> &#61; 0&#46;</p>"
        ]
      ]
      6 => array:7 [
        "identificador" => "fig0035"
        "etiqueta" => "Figure 7"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr7.jpeg"
            "Alto" => 850
            "Ancho" => 1133
            "Tamanyo" => 95915
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0035" class="elsevierStyleSimplePara elsevierViewall">Errors for the symmetric set of oe azimuths&#59; the upper boundary&#44; and fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span> &#61; 56&#46;853&#176;&#46;</p>"
        ]
      ]
      7 => array:7 [
        "identificador" => "fig0040"
        "etiqueta" => "Figure 8"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr8.jpeg"
            "Alto" => 864
            "Ancho" => 1130
            "Tamanyo" => 98451
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0040" class="elsevierStyleSimplePara elsevierViewall">Errors for the symmetric set of azimuths&#59; the lower boundary&#44; variant <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>4800&#44; and fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">min</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;</p>"
        ]
      ]
      8 => array:7 [
        "identificador" => "fig0045"
        "etiqueta" => "Figure 9"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr9.jpeg"
            "Alto" => 848
            "Ancho" => 1133
            "Tamanyo" => 108309
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0045" class="elsevierStyleSimplePara elsevierViewall">Errors for the symmetric set of azimuths&#59; the lower boundary&#44; variant <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>4800&#44; and fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>62&#46;6&#176;&#46;</p>"
        ]
      ]
      9 => array:7 [
        "identificador" => "fig0050"
        "etiqueta" => "Figure 10"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr10.jpeg"
            "Alto" => 850
            "Ancho" => 1122
            "Tamanyo" => 130932
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0050" class="elsevierStyleSimplePara elsevierViewall">Errors for the symmetric set of azimuths&#59; the noise&#44; the upper boundary&#44; <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3200&#44; and fixed <span class="elsevierStyleItalic">&#952;<span class="elsevierStyleInf">min</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;</span></p>"
        ]
      ]
      10 => array:7 [
        "identificador" => "fig0055"
        "etiqueta" => "Figure 11"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr11.jpeg"
            "Alto" => 850
            "Ancho" => 1127
            "Tamanyo" => 136925
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0055" class="elsevierStyleSimplePara elsevierViewall">Errors for the symmetric set of azimuths&#59; the noise&#44; the upper boundary&#44; <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3200&#44; and fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>56&#46;853&#176;&#46;</p>"
        ]
      ]
      11 => array:7 [
        "identificador" => "fig0060"
        "etiqueta" => "Figure 12"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr12.jpeg"
            "Alto" => 852
            "Ancho" => 1122
            "Tamanyo" => 114490
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0060" class="elsevierStyleSimplePara elsevierViewall">Errors for the asymmetric set of azimuths&#59; the upper boundary&#44; variant <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3200&#44; and fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">min</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;</p>"
        ]
      ]
      12 => array:7 [
        "identificador" => "fig0065"
        "etiqueta" => "Figure 13"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr13.jpeg"
            "Alto" => 847
            "Ancho" => 1123
            "Tamanyo" => 108102
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0065" class="elsevierStyleSimplePara elsevierViewall">Errors for the asymmetric set of azimuths&#59; the upper boundary&#44; variant <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3200&#44; and fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>56&#46;853&#176;&#46;</p>"
        ]
      ]
      13 => array:7 [
        "identificador" => "fig0070"
        "etiqueta" => "Figure 14"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr14.jpeg"
            "Alto" => 852
            "Ancho" => 1127
            "Tamanyo" => 142222
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0070" class="elsevierStyleSimplePara elsevierViewall">Errors for the asymmetric set of azimuths&#59; the noise&#44; the upper boundary&#44; <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3200&#44; and fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">min</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#46;</p>"
        ]
      ]
      14 => array:7 [
        "identificador" => "fig0075"
        "etiqueta" => "Figure 15"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr15.jpeg"
            "Alto" => 849
            "Ancho" => 1124
            "Tamanyo" => 129412
          ]
        ]
        "descripcion" => array:1 [
          "en" => "<p id="spar0075" class="elsevierStyleSimplePara elsevierViewall">Errors for the asymmetric set of azimuths&#59; the noise&#44; the upper boundary&#44; variant <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">P</span>3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>3200&#44; and fixed <span class="elsevierStyleItalic">&#952;</span><span class="elsevierStyleInf">max</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>56&#46;853&#176;&#46;</p>"
        ]
      ]
      15 => array:6 [
        "identificador" => "eq0005"
        "etiqueta" => "&#40;1&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "R&#952;&#44;&#981;&#61;A&#43;B&#981;sin2&#952;&#43;C&#981;sin2&#952;tan2&#952;&#44;"
          "Fichero" => "si1.jpeg"
          "Tamanyo" => 2929
          "Alto" => 19
          "Ancho" => 314
        ]
      ]
      16 => array:6 [
        "identificador" => "eq0010"
        "etiqueta" => "&#40;2&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "A&#61;&#916;Z&#47;2Z&#175;"
          "Fichero" => "si2.jpeg"
          "Tamanyo" => 1075
          "Alto" => 20
          "Ancho" => 100
        ]
      ]
      17 => array:6 [
        "identificador" => "eq0015"
        "etiqueta" => "&#40;3&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "B&#981;&#61;Biso&#43;Banicos2&#981;&#8722;&#981;0&#44;"
          "Fichero" => "si6.jpeg"
          "Tamanyo" => 2082
          "Alto" => 18
          "Ancho" => 228
        ]
      ]
      18 => array:6 [
        "identificador" => "eq0020"
        "etiqueta" => "&#40;4&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "C&#981;&#61;&#945;&#43;&#946;cos4&#981;&#8722;&#981;0&#43;&#947;sin2&#981;&#8722;&#981;0cos2&#981;&#8722;&#981;0&#44;"
          "Fichero" => "si7.jpeg"
          "Tamanyo" => 3662
          "Alto" => 19
          "Ancho" => 400
        ]
      ]
      19 => array:6 [
        "identificador" => "eq0025"
        "etiqueta" => "&#40;5&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "h0&#8722;n&#44;&#8230;n&#61;b&#44;b&#44;&#8722;a&#44;a&#44;b&#44;b&#44;a&#44;&#8722;a&#44;c&#44;&#8201;&#8201;&#8201;&#8201;&#8201;&#8201;&#8201;&#8201;&#8201;&#8201;&#8201;&#8201;&#8201;&#8201;&#8201;&#8201;&#8722;a&#44;a&#44;b&#44;b&#44;a&#44;&#8722;a&#44;b&#44;b&#44;"
          "Fichero" => "si10.jpeg"
          "Tamanyo" => 3569
          "Alto" => 39
          "Ancho" => 295
        ]
      ]
      20 => array:5 [
        "identificador" => "eq0030"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "lo1j&#61;&#8721;i&#61;&#8722;nnh0ixi&#43;j&#44;hi1j&#61;&#8721;i&#61;&#8722;nnh1ixi&#43;j&#44;j&#61;1&#44;&#8230;&#44;m&#46;"
          "Fichero" => "si12.jpeg"
          "Tamanyo" => 5010
          "Alto" => 125
          "Ancho" => 192
        ]
      ]
      21 => array:5 [
        "identificador" => "eq0035"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "los&#8722;1j&#61;&#8721;i&#61;&#8722;nnh2ilosi&#43;j&#8195;&#8195;&#8195;&#8201;&#43;&#8721;i&#61;&#8722;nnh3ihisi&#43;j&#44;&#8195;&#8195;&#8201;&#8201;j&#61;1&#44;&#8230;&#44;m&#46;"
          "Fichero" => "si13.jpeg"
          "Tamanyo" => 4893
          "Alto" => 125
          "Ancho" => 215
        ]
      ]
      22 => array:5 [
        "identificador" => "eq0040"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "yj&#61;lo0jp&#44;"
          "Fichero" => "si14.jpeg"
          "Tamanyo" => 1107
          "Alto" => 16
          "Ancho" => 109
        ]
      ]
      23 => array:5 [
        "identificador" => "eq0045"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "P&#61;cg2RPini&#44;"
          "Fichero" => "si15.jpeg"
          "Tamanyo" => 904
          "Alto" => 21
          "Ancho" => 88
        ]
      ]
      24 => array:5 [
        "identificador" => "eq0050"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "P0&#61;cg02APini&#44;"
          "Fichero" => "si16.jpeg"
          "Tamanyo" => 1039
          "Alto" => 24
          "Ancho" => 100
        ]
      ]
      25 => array:5 [
        "identificador" => "eq0055"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "R&#61;Acg02Pcg2P0&#46;"
          "Fichero" => "si17.jpeg"
          "Tamanyo" => 1165
          "Alto" => 38
          "Ancho" => 79
        ]
      ]
      26 => array:6 [
        "identificador" => "eq0060"
        "etiqueta" => "&#40;6&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "rgP&#952;&#44;&#981;&#61;P0&#43;mB&#981;sin2&#952;&#43;mC&#981;sin2&#952;tan2&#952;&#44;"
          "Fichero" => "si18.jpeg"
          "Tamanyo" => 3307
          "Alto" => 20
          "Ancho" => 359
        ]
      ]
      27 => array:6 [
        "identificador" => "eq0065"
        "etiqueta" => "&#40;7&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "rg&#61;cg02cg2&#61;r2z2&#61;1cos2&#952;&#44;"
          "Fichero" => "si19.jpeg"
          "Tamanyo" => 1589
          "Alto" => 38
          "Ancho" => 153
        ]
      ]
      28 => array:6 [
        "identificador" => "eq0070"
        "etiqueta" => "&#40;8&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "x0&#61;a&#8721;i&#61;1nziVi1&#8722;a2Vi2&#44;"
          "Fichero" => "si20.jpeg"
          "Tamanyo" => 1895
          "Alto" => 46
          "Ancho" => 130
        ]
      ]
      29 => array:6 [
        "identificador" => "eq0075"
        "etiqueta" => "&#40;9a&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "T&#42;&#61;a&#43;b&#42;&#43;cts&#43;d&#42;&#43;e&#42;t&#43;ft2s2&#47;1&#8722;s&#44;"
          "Fichero" => "si22.jpeg"
          "Tamanyo" => 2732
          "Alto" => 20
          "Ancho" => 340
        ]
      ]
      30 => array:6 [
        "identificador" => "eq0080"
        "etiqueta" => "&#40;9b&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "T&#61;a&#43;sb&#43;ct&#43;s2d&#43;et&#43;ft2&#44;"
          "Fichero" => "si23.jpeg"
          "Tamanyo" => 2103
          "Alto" => 20
          "Ancho" => 259
        ]
      ]
      31 => array:6 [
        "identificador" => "eq0085"
        "etiqueta" => "&#40;10&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "Tij&#61;Pj&#43;Bj1si&#43;Cj1si2&#44;"
          "Fichero" => "si24.jpeg"
          "Tamanyo" => 1554
          "Alto" => 22
          "Ancho" => 161
        ]
      ]
      32 => array:5 [
        "identificador" => "eq0090"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "Fj&#61;&#8721;i&#61;1kjPj&#43;Bj1si&#43;Cj1si2&#8722;Tij2&#46;"
          "Fichero" => "si29.jpeg"
          "Tamanyo" => 2807
          "Alto" => 52
          "Ancho" => 245
        ]
      ]
      33 => array:5 [
        "identificador" => "eq0095"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "&#8706;Fj&#47;&#8706;Pj&#61;0&#44;&#8706;Fj&#47;&#8706;Bj1&#61;0&#44;&#8706;Fj&#47;&#8706;Cj1&#61;0&#46;"
          "Fichero" => "si30.jpeg"
          "Tamanyo" => 2652
          "Alto" => 22
          "Ancho" => 281
        ]
      ]
      34 => array:5 [
        "identificador" => "eq0100"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "A&#61;&#8721;i&#61;1kjsi&#44;B&#61;&#8721;i&#61;1kjsi2&#44;C&#61;&#8721;i&#61;1kjsi3&#44;D&#61;&#8721;i&#61;1kjsi4&#44;u0&#61;&#8721;i&#61;1kjTij&#44;u1&#61;&#8721;i&#61;1kjsiTij&#44;andu2&#61;&#8721;i&#61;1kjsi2Tij&#46;"
          "Fichero" => "si34.jpeg"
          "Tamanyo" => 7824
          "Alto" => 111
          "Ancho" => 317
        ]
      ]
      35 => array:6 [
        "identificador" => "eq0105"
        "etiqueta" => "&#40;11&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "Bj1&#47;Pj&#61;b1&#43;c1tj&#8242;"
          "Fichero" => "si35.jpeg"
          "Tamanyo" => 1215
          "Alto" => 22
          "Ancho" => 127
        ]
      ]
      36 => array:6 [
        "identificador" => "eq0110"
        "etiqueta" => "&#40;12&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "Uj&#61;b0&#43;c0ggj&#43;c0hhj&#8242;"
          "Fichero" => "si36.jpeg"
          "Tamanyo" => 1550
          "Alto" => 17
          "Ancho" => 173
        ]
      ]
      37 => array:6 [
        "identificador" => "eq0115"
        "etiqueta" => "&#40;13&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "F&#61;&#8721;j&#61;1nb0&#43;c0ggj&#43;c0hhj&#8722;Uj2&#46;"
          "Fichero" => "si38.jpeg"
          "Tamanyo" => 2552
          "Alto" => 54
          "Ancho" => 252
        ]
      ]
      38 => array:5 [
        "identificador" => "eq0120"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "&#8706;F&#47;&#8706;b0&#61;0&#44;&#8706;F&#47;&#8706;c0&#61;0&#44;&#8706;F&#47;&#8706;&#981;0&#61;0&#46;"
          "Fichero" => "si39.jpeg"
          "Tamanyo" => 2228
          "Alto" => 14
          "Ancho" => 252
        ]
      ]
      39 => array:5 [
        "identificador" => "eq0125"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "D&#61;&#8721;j&#61;1ngjhj&#44;E&#61;&#8721;j&#61;1nhj2&#44;u0&#61;&#8721;j&#61;1nUj&#44;u1&#61;&#8721;j&#61;1ngjUj&#44;andu2&#61;&#8721;j&#61;1nhjUj&#46;"
          "Fichero" => "si42.jpeg"
          "Tamanyo" => 6147
          "Alto" => 106
          "Ancho" => 273
        ]
      ]
      40 => array:6 [
        "identificador" => "eq0130"
        "etiqueta" => "&#40;14&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "Cj1&#47;Pj&#61;d1&#43;e1tj&#43;f1tj2&#44;"
          "Fichero" => "si43.jpeg"
          "Tamanyo" => 1623
          "Alto" => 22
          "Ancho" => 180
        ]
      ]
      41 => array:5 [
        "identificador" => "eq0135"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "Cj1&#47;Pj&#61;d1&#43;e1&#8722;e1tj&#43;f1tj2&#46;"
          "Fichero" => "si45.jpeg"
          "Tamanyo" => 1970
          "Alto" => 22
          "Ancho" => 227
        ]
      ]
      42 => array:6 [
        "identificador" => "eq0140"
        "etiqueta" => "&#40;15&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "Ti&#61;a&#43;sib0&#43;c0ggi&#43;c0hhi&#46;"
          "Fichero" => "si47.jpeg"
          "Tamanyo" => 1897
          "Alto" => 16
          "Ancho" => 221
        ]
      ]
      43 => array:6 [
        "identificador" => "eq0145"
        "etiqueta" => "&#40;16&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "F&#61;&#8721;i&#61;1na&#43;b0si&#43;c0gsigi&#43;c0hsihi&#8722;Ti2&#46;"
          "Fichero" => "si48.jpeg"
          "Tamanyo" => 3084
          "Alto" => 46
          "Ancho" => 299
        ]
      ]
      44 => array:6 [
        "identificador" => "eq0150"
        "etiqueta" => "&#40;17&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "&#8706;F&#47;&#8706;a&#61;0&#44;&#8706;F&#47;&#8706;b0&#61;0&#44;&#8706;F&#47;&#8706;c0&#61;0&#44;&#8706;F&#47;&#8706;&#981;0&#61;0&#46;"
          "Fichero" => "si49.jpeg"
          "Tamanyo" => 2822
          "Alto" => 14
          "Ancho" => 330
        ]
      ]
      45 => array:6 [
        "identificador" => "eq0155"
        "etiqueta" => "&#40;18&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "tan2&#966;0&#61;hg&#61;A2H1&#8722;A1H2A2H2&#8722;B1H1&#44;"
          "Fichero" => "si50.jpeg"
          "Tamanyo" => 1907
          "Alto" => 24
          "Ancho" => 190
        ]
      ]
      46 => array:5 [
        "identificador" => "eq0160"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "A&#61;&#8721;i&#61;1nsi&#44;B&#61;&#8721;i&#61;1nsi2&#44;D&#61;&#8721;i&#61;1ngisi&#44;E&#61;&#8721;i&#61;1nhisi&#44;G&#61;&#8721;i&#61;1ngisi2&#44;H&#61;&#8721;i&#61;1nhisi2&#44;I&#61;&#8721;i&#61;1ngi2si2&#44;J&#61;&#8721;i&#61;1nhi2si2&#44;K&#61;&#8721;i&#61;1ngihisi2&#44;f0&#61;&#8721;i&#61;1nTi&#44;f1&#61;&#8721;i&#61;1nsiTi&#44;f2&#61;&#8721;i&#61;1ngisiTi&#44;andf3&#61;&#8721;i&#61;1nhisiTi&#46;"
          "Fichero" => "si52.jpeg"
          "Tamanyo" => 14890
          "Alto" => 264
          "Ancho" => 285
        ]
      ]
      47 => array:5 [
        "identificador" => "eq0165"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "c0&#61;A2H1&#8722;A1H2hA22&#8722;A1B1&#44;b0&#61;F1&#8722;c0ga2&#8722;c0ha3&#47;a1&#44;"
          "Fichero" => "si53.jpeg"
          "Tamanyo" => 3174
          "Alto" => 28
          "Ancho" => 335
        ]
      ]
      48 => array:5 [
        "identificador" => "eq0170"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "Ti&#61;a&#43;bsi&#43;csiti&#43;dsi2&#43;esi2ti&#43;fsi2ti2&#44;"
          "Fichero" => "si56.jpeg"
          "Tamanyo" => 2364
          "Alto" => 19
          "Ancho" => 278
        ]
      ]
      49 => array:6 [
        "identificador" => "eq0175"
        "etiqueta" => "&#40;19&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "F&#61;&#8721;i&#61;1na&#43;bsi&#43;csiti&#43;dsi2&#43;esi2ti&#43;fsi2ti2&#8722;Ti2&#46;"
          "Fichero" => "si57.jpeg"
          "Tamanyo" => 3769
          "Alto" => 76
          "Ancho" => 203
        ]
      ]
      50 => array:6 [
        "identificador" => "eq0180"
        "etiqueta" => "&#40;20&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "&#8706;F&#47;&#8706;a&#61;0&#44;&#8706;F&#47;&#8706;b&#61;0&#44;&#8706;F&#47;&#8706;c&#61;0&#44;&#8706;F&#47;&#8706;d&#61;0&#44;&#8706;F&#47;&#8706;e&#61;0&#44;&#8706;F&#47;&#8706;f&#61;0&#44;&#8706;F&#47;&#8706;&#981;0&#61;0&#46;"
          "Fichero" => "si58.jpeg"
          "Tamanyo" => 4610
          "Alto" => 38
          "Ancho" => 309
        ]
      ]
      51 => array:6 [
        "identificador" => "eq0185"
        "etiqueta" => "&#40;21&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "&#916;&#949;V&#61;2Ac&#43;e&#43;f&#47;a&#44;"
          "Fichero" => "si59.jpeg"
          "Tamanyo" => 1635
          "Alto" => 19
          "Ancho" => 181
        ]
      ]
      52 => array:6 [
        "identificador" => "eq0190"
        "etiqueta" => "&#40;22&#41;"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "&#916;&#948;V&#61;2Ac&#43;e&#47;a&#46;"
          "Fichero" => "si60.jpeg"
          "Tamanyo" => 1346
          "Alto" => 19
          "Ancho" => 150
        ]
      ]
      53 => array:5 [
        "identificador" => "eq0195"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "A&#61;&#8721;i&#61;1nsi&#44;B&#61;&#8721;i&#61;1nsiti&#44;C&#61;&#8721;i&#61;1nsi2&#44;D&#61;&#8721;i&#61;1nsi2ti&#44;E&#61;&#8721;i&#61;1nsi2ti2&#44;F&#61;&#8721;i&#61;1nsi3&#44;G&#61;&#8721;i&#61;1nsi3ti&#44;H&#61;&#8721;i&#61;1nsi3ti2&#44;&#8201;K&#61;&#8721;i&#61;1nsi3ti3&#44;&#8201;L&#61;&#8721;i&#61;1nsi4&#44;M&#61;&#8721;i&#61;1nsi4ti&#44;&#8201;N&#61;&#8721;i&#61;1nsi4ti2&#44;&#8201;O&#61;&#8721;i&#61;1nsi4ti3&#44;P&#61;&#8721;i&#61;1nsi4ti4&#44;&#8201;U0&#61;&#8721;i&#61;1nTi&#44;&#8201;U1&#61;&#8721;i&#61;1nTisi&#44;U2&#61;&#8721;i&#61;1nTisiti&#44;&#8201;U3&#61;&#8721;i&#61;1nTisi2&#44;&#8201;U4&#61;&#8721;i&#61;1nTisi2ti&#44;and&#8201;U5&#61;&#8721;i&#61;1nTisi2ti2&#46;"
          "Fichero" => "si61.jpeg"
          "Tamanyo" => 22845
          "Alto" => 333
          "Ancho" => 268
        ]
      ]
      54 => array:5 [
        "identificador" => "eq0200"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "f&#61;a2f1&#8722;a1f2a22&#8722;a1b1&#44;e&#61;f1&#8722;fa2a1&#44;d&#61;g1&#8722;fa13&#8722;ea12a11&#44;c&#61;h1&#8722;fa24&#8722;ea23&#8722;da22a21&#44;b&#61;k1&#8722;fa05&#8722;ea04&#8722;da03&#8722;ca02a01&#44;a&#61;U0&#8722;fE&#8722;eD&#8722;dC&#8722;cB&#8722;bA&#47;n&#44;"
          "Fichero" => "si62.jpeg"
          "Tamanyo" => 10452
          "Alto" => 187
          "Ancho" => 282
        ]
      ]
      55 => array:5 [
        "identificador" => "eq0205"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "a21&#61;a022&#8722;a01b01&#44;a22&#61;a02a03&#8722;a01b02&#44;a23&#61;a02a04&#8722;a01b03&#44;a24&#61;a02a05&#8722;a01b04&#44;b21&#61;a032&#8722;a01c01&#44;b22&#61;a03a04&#8722;a01c02&#44;b23&#61;a03a05&#8722;a01c03&#44;c21&#61;a042&#8722;a01d01&#44;c22&#61;a04a05&#8722;a01d02&#44;d21&#61;a052&#8722;a01e01&#44;h1&#61;a02k1&#8722;a01k2&#44;h2&#61;a03k1&#8722;a01k3&#44;h3&#61;a04k1&#8722;a01k4&#44;h4&#61;a05k1&#8722;a01k5&#44;a11&#61;a222&#8722;a21b21&#44;a12&#61;a22a23&#8722;a21b22a13&#61;a22a24&#8722;a21b23&#44;b11&#61;a232&#8722;a21c21&#44;b12&#61;a23a24&#8722;a21c22&#44;c11&#61;a242&#8722;a21d21&#44;g1&#61;a22h1&#8722;a21h2&#44;g2&#61;a23h1&#8722;a21h3&#44;g3&#61;a24h1&#8722;a21h4&#44;a1&#61;a122&#8722;a11b11a2&#61;a12a13&#8722;a11b12&#44;&#8201;b1&#61;a132&#8722;a11c11&#44;&#8201;f1&#61;a12g1&#8722;a11g2&#44;and&#8201;f2&#61;a13g1&#8722;a11g3&#46;"
          "Fichero" => "si63.jpeg"
          "Tamanyo" => 30743
          "Alto" => 289
          "Ancho" => 349
        ]
      ]
      56 => array:5 [
        "identificador" => "eq0210"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "caA1&#43;bB1&#43;cC1&#43;dD1&#43;eE1&#43;fF1&#8722;U6&#43;eaB1&#43;bD1&#43;cE1&#43;dG1&#43;eH1&#43;fK1&#8722;U7&#43;2faC1&#43;bE1&#43;cF1&#43;dH1&#43;eK1&#43;fL1&#8722;U8&#61;0"
          "Fichero" => "si64.jpeg"
          "Tamanyo" => 8239
          "Alto" => 64
          "Ancho" => 359
        ]
      ]
      57 => array:5 [
        "identificador" => "eq0215"
        "tipo" => "MULTIMEDIAFORMULA"
        "mostrarFloat" => false
        "mostrarDisplay" => true
        "Formula" => array:5 [
          "Matematica" => "A1&#61;&#8721;i&#61;1nyisi&#44;B1&#61;&#8721;i&#61;1nyisi2&#44;C1&#61;&#8721;i&#61;1nyitisi2D1&#61;&#8721;i&#61;1nyisi3&#44;E1&#61;&#8721;i&#61;1nyitisi3&#44;F1&#61;&#8721;i&#61;1nyisi3ti2&#44;G1&#61;&#8721;i&#61;1nyisi4&#44;H1&#61;&#8721;i&#61;1nyitisi4&#44;K1&#61;&#8721;i&#61;1nyiti2si4&#44;L1&#61;&#8721;i&#61;1nyiti3si4&#44;U6&#61;&#8721;i&#61;1nyiTisi&#44;U7&#61;&#8721;i&#61;1nyiTisi2U8&#61;&#8721;i&#61;1nyiTitisi2&#44;andyi&#61;sin2&#981;i&#8722;&#981;0&#46;"
          "Fichero" => "si65.jpeg"
          "Tamanyo" => 17915
          "Alto" => 264
          "Ancho" => 333
        ]
      ]
    ]
    "bibliografia" => array:2 [
      "titulo" => "References"
      "seccion" => array:1 [
        0 => array:2 [
          "identificador" => "bibs0005"
          "bibliografiaReferencia" => array:15 [
            0 => array:3 [
              "identificador" => "bib0005"
              "etiqueta" => "Abdelnour and Selesnick&#44; 2000"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "Symmetric nearly orthogonal and orthogonal nearly symmetric wavelets"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:2 [
                            0 => "A&#46;F&#46; Abdelnour"
                            1 => "I&#46;W&#46; Selesnick"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:6 [
                        "tituloSerie" => "Arabian J&#46; for Science and Engineering"
                        "fecha" => "2004"
                        "volumen" => "29"
                        "numero" => "2C"
                        "paginaInicial" => "3"
                        "paginaFinal" => "16"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            1 => array:3 [
              "identificador" => "bib0010"
              "etiqueta" => "Abdelnour and Selesnick&#44; 2005"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "Symmetric nearly shift-invariant tight frame wavelets"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:2 [
                            0 => "A&#46;F&#46; Abdelnour"
                            1 => "I&#46;W&#46; Selesnick"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:6 [
                        "tituloSerie" => "IEEE Trans&#46; Signal Process"
                        "fecha" => "2005"
                        "volumen" => "53"
                        "numero" => "1"
                        "paginaInicial" => "231"
                        "paginaFinal" => "239"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            2 => array:3 [
              "identificador" => "bib0015"
              "etiqueta" => "Chichinina et al&#46;&#44; 2003"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "Numerical modeling of P-wave AVOA in media containing vertical fractures &#47; Expanded abstract"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:3 [
                            0 => "T&#46; Chichinina"
                            1 => "V&#46; Sabinin"
                            2 => "G&#46; Ronquillo Jarrillo"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Libro" => array:6 [
                        "titulo" => "The Sixth International Conference on Mathematical and Numerical Aspects of Wave Propagation &#8220;Waves&#8217; 2003&#8221;"
                        "fecha" => "2003"
                        "paginaInicial" => "897"
                        "paginaFinal" => "902"
                        "editorial" => "Finland"
                        "editorialLocalizacion" => "Springer"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            3 => array:3 [
              "identificador" => "bib0020"
              "etiqueta" => "Dasgupta and Clark&#44; 1998"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "Estimation of Q from surface seismic reflection data"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:2 [
                            0 => "R&#46; Dasgupta"
                            1 => "R&#46;A&#46; Clark"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:5 [
                        "tituloSerie" => "Geophysics"
                        "fecha" => "1998"
                        "volumen" => "63"
                        "paginaInicial" => "2120"
                        "paginaFinal" => "2128"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            4 => array:3 [
              "identificador" => "bib0025"
              "etiqueta" => "Jenner&#44; 2002"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "Azimuthal AVO&#58; Methodology and data examples"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:1 [
                            0 => "E&#46; Jenner"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:6 [
                        "tituloSerie" => "The Leading Edge"
                        "fecha" => "2002"
                        "volumen" => "21"
                        "numero" => "8"
                        "paginaInicial" => "782"
                        "paginaFinal" => "786"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            5 => array:3 [
              "identificador" => "bib0030"
              "etiqueta" => "MacBeth&#44; 1999"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "Azimuthal variation in P-wave signatures due to flow"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:1 [
                            0 => "C&#46; MacBeth"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:5 [
                        "tituloSerie" => "Geophysics"
                        "fecha" => "1999"
                        "volumen" => "64"
                        "paginaInicial" => "1181"
                        "paginaFinal" => "1192"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            6 => array:3 [
              "identificador" => "bib0035"
              "etiqueta" => "Mallik et al&#46;&#44; 1998"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "Determination of the principal directions of azimuthal anisotropy from P-wave seismic data"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:4 [
                            0 => "S&#46; Mallik"
                            1 => "K&#46;L Craft"
                            2 => "L&#46;J&#46; Meister"
                            3 => "R&#46;E&#46; Chambers"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:5 [
                        "tituloSerie" => "Geophysics"
                        "fecha" => "1998"
                        "volumen" => "63"
                        "paginaInicial" => "692"
                        "paginaFinal" => "706"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            7 => array:3 [
              "identificador" => "bib0040"
              "etiqueta" => "Newman&#44; 1973"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "Divergence effects in a layered earth"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:1 [
                            0 => "P&#46; Newman"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:5 [
                        "tituloSerie" => "Geophysics"
                        "fecha" => "1973"
                        "volumen" => "38"
                        "paginaInicial" => "481"
                        "paginaFinal" => "488"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            8 => array:3 [
              "identificador" => "bib0045"
              "etiqueta" => "R&#252;ger&#44; 1998"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "Variation of P-wave reflectivity with offset and azimuth in anisotropic media"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:1 [
                            0 => "A&#46; R&#252;ger"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:5 [
                        "tituloSerie" => "Geophysics"
                        "fecha" => "1998"
                        "volumen" => "63"
                        "paginaInicial" => "935"
                        "paginaFinal" => "947"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            9 => array:3 [
              "identificador" => "bib0050"
              "etiqueta" => "Sabinin&#44; 2012"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "Viscoelastic modeling and factor Q for reflection data"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:1 [
                            0 => "V&#46; Sabinin"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:6 [
                        "tituloSerie" => "Geofisica Internacional"
                        "fecha" => "2012"
                        "volumen" => "51"
                        "numero" => "4"
                        "paginaInicial" => "377"
                        "paginaFinal" => "391"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            10 => array:3 [
              "identificador" => "bib0055"
              "etiqueta" => "Sabinin&#44; 2013"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "QVOA techniques for fracture characterization"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:1 [
                            0 => "V&#46; Sabinin"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:6 [
                        "tituloSerie" => "Geofisica Internacional"
                        "fecha" => "2013"
                        "volumen" => "52"
                        "numero" => "4"
                        "paginaInicial" => "311"
                        "paginaFinal" => "320"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            11 => array:3 [
              "identificador" => "bib0060"
              "etiqueta" => "Sabinin and Chichinina&#44; 2008"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "AVOA technique for fracture characterization&#58; resolving ambiguity"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:2 [
                            0 => "V&#46; Sabinin"
                            1 => "T&#46; Chichinina"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:6 [
                        "tituloSerie" => "Geofisica Internacional"
                        "fecha" => "2008"
                        "volumen" => "47"
                        "numero" => "1"
                        "paginaInicial" => "3"
                        "paginaFinal" => "11"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            12 => array:3 [
              "identificador" => "bib0065"
              "etiqueta" => "Sheriff and Geldart&#44; 1983"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:1 [
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:2 [
                            0 => "R&#46;E&#46; Sheriff"
                            1 => "L&#46;P&#46; Geldart"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Libro" => array:5 [
                        "titulo" => "Exploration Seismology&#44; Volume 2&#44; Data-processing and interpretation"
                        "fecha" => "1983"
                        "paginaInicial" => "400"
                        "editorial" => "Cambridge University Press"
                        "editorialLocalizacion" => "Cambridge"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            13 => array:3 [
              "identificador" => "bib0070"
              "etiqueta" => "Xu and Tsvankin&#44; 2004"
              "referencia" => array:1 [
                0 => array:2 [
                  "contribucion" => array:1 [
                    0 => array:2 [
                      "titulo" => "Geometrical-spreading correction for P-waves in layered azimuthally anisotropic media"
                      "autores" => array:1 [
                        0 => array:2 [
                          "etal" => false
                          "autores" => array:2 [
                            0 => "X&#46; Xu"
                            1 => "Tsvankin"
                          ]
                        ]
                      ]
                    ]
                  ]
                  "host" => array:1 [
                    0 => array:1 [
                      "Revista" => array:4 [
                        "tituloSerie" => "74th Ann&#46; Internat&#46; Mtg&#46;&#58; Soc&#46; of Expl&#46; Geophys&#46;"
                        "fecha" => "2004"
                        "paginaInicial" => "111"
                        "paginaFinal" => "114"
                      ]
                    ]
                  ]
                ]
              ]
            ]
            14 => array:3 [
              "identificador" => "bib0075"
              "etiqueta" => "WSBP&#44; 2012"
              "referencia" => array:1 [
                0 => array:1 [
                  "referenciaCompleta" => "WSBP&#44; 2012&#44; &#40;Wavelet Software at Brooklyn Poly&#41;&#44; Matlab Implementation of Wavelet Transforms&#46; <a id="intr0005" class="elsevierStyleInterRef" href="http://eeweb.poly.edu/iselesni/WaveletSoftware">http&#58;&#47;&#47;eeweb&#46;poly&#46;edu&#47;iselesni&#47;WaveletSoftware</a>&#46;"
                ]
              ]
            ]
          ]
        ]
      ]
    ]
  ]
  "idiomaDefecto" => "en"
  "url" => "/00167169/0000005300000004/v1_201412221101/S0016716914700789/v1_201412221101/en/main.assets"
  "Apartado" => array:4 [
    "identificador" => "32621"
    "tipo" => "SECCION"
    "en" => array:2 [
      "titulo" => "Original paper"
      "idiomaDefecto" => true
    ]
    "idiomaDefecto" => "en"
  ]
  "PDF" => "https://static.elsevier.es/multimedia/00167169/0000005300000004/v1_201412221101/S0016716914700789/v1_201412221101/en/main.pdf?idApp=UINPBA00004N&text.app=https://www.elsevier.es/"
  "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716914700789?idApp=UINPBA00004N"
]

Article information

ISSN: 00167169
Original language: English

DOI:10.1016/S0016-7169(14)70078-9

The statistics are updated each day

Year/Month	Html	Pdf	Total

2024 November	2	0	2
2024 October	9	0	9
2024 September	12	1	13
2024 August	18	3	21
2024 July	21	8	29
2024 June	15	3	18
2024 May	11	3	14
2024 April	10	2	12
2024 March	19	11	30
2024 February	7	3	10
2024 January	13	8	21
2023 December	9	2	11
2023 November	12	5	17
2023 October	19	8	27
2023 September	7	1	8
2023 August	10	4	14
2023 July	9	4	13
2023 June	10	3	13
2023 May	17	3	20
2023 April	7	0	7
2023 March	15	7	22
2023 February	11	8	19
2023 January	15	4	19
2022 December	11	2	13
2022 November	14	9	23
2022 October	22	5	27
2022 September	13	5	18
2022 August	7	6	13
2022 July	9	9	18
2022 June	6	7	13
2022 May	12	11	23
2022 April	9	6	15
2022 March	9	6	15
2022 February	6	3	9
2022 January	19	6	25
2021 December	9	9	18
2021 November	13	5	18
2021 October	13	9	22
2021 September	11	13	24
2021 August	23	9	32
2021 July	14	7	21
2021 June	18	6	24
2021 May	11	5	16
2021 April	26	14	40
2021 March	17	12	29
2021 February	5	3	8
2021 January	10	9	19
2020 December	11	6	17
2020 November	7	7	14
2020 October	10	13	23
2020 September	8	9	17
2020 August	8	4	12
2020 July	10	4	14
2020 June	7	2	9
2020 May	8	4	12
2020 April	6	3	9
2020 March	5	5	10
2020 February	6	3	9
2020 January	5	2	7
2019 December	11	6	17
2019 November	10	2	12
2019 October	11	2	13
2019 September	14	3	17
2019 August	7	0	7
2019 July	6	6	12
2019 June	15	20	35
2019 May	53	47	100
2019 April	13	25	38
2019 March	1	4	5
2019 February	6	8	14
2019 January	2	9	11
2018 December	1	4	5
2018 November	1	3	4
2018 October	6	4	10
2018 September	11	7	18
2018 August	5	10	15
2018 July	1	1	2
2018 June	6	3	9
2018 May	9	4	13
2018 April	1	0	1
2018 March	4	0	4
2018 February	1	0	1
2018 January	10	1	11
2017 December	1	0	1
2017 November	10	0	10
2017 October	9	3	12
2017 September	12	2	14
2017 August	7	2	9
2017 July	3	1	4
2017 June	13	10	23
2017 May	9	24	33
2017 April	11	13	24
2017 March	16	16	32
2017 February	9	8	17
2017 January	10	4	14
2016 December	16	7	23
2016 November	11	4	15
2016 October	7	6	13
2016 September	9	4	13
2016 August	9	3	12
2016 July	12	3	15
2016 June	11	8	19
2016 May	12	14	26
2016 April	7	26	33
2016 March	6	12	18
2016 February	18	17	35
2016 January	11	18	29
2015 December	15	16	31
2015 November	12	13	25
2015 October	18	16	34
2015 September	18	10	28
2015 August	9	5	14
2015 July	7	3	10
2015 June	3	1	4
2015 May	11	6	17
2015 April	22	18	40
2015 March	19	9	28
2015 February	19	24	43
2015 January	15	12	27

Show all

Indexed in:

Follow us:

Statistics

Indexed in:

Follow us:

Statistics

Subscribe to our newsletter