Two algorithms to compute the electric resistivity response using Green's functions for 3D structures

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The numbers of the curves correspond to the buoys 1 and 2. Daily average wind velocity from January to May 2002 in Lake Alchichica is presented in the rectangle in the superior right corner." ] ] ] "autores" => array:1 [ 0 => array:2 [ "autoresLista" => "Anatoliy Filonov, Iryna Tereshchenko, Javier Alcocer, Cesar Monzón" "autores" => array:4 [ 0 => array:2 [ "nombre" => "Anatoliy" "apellidos" => "Filonov" ] 1 => array:2 [ "nombre" => "Iryna" "apellidos" => "Tereshchenko" ] 2 => array:2 [ "nombre" => "Javier" "apellidos" => "Alcocer" ] 3 => array:2 [ "nombre" => "Cesar" "apellidos" => "Monzón" ] ] ] ] ] "idiomaDefecto" => "en" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716915000021?idApp=UINPBA00004N" "url" => "/00167169/0000005400000001/v1_201505130244/S0016716915000021/v1_201505130244/en/main.assets" ] "en" => array:19 [ "idiomaDefecto" => true "titulo" => "Two algorithms to compute the electric resistivity response using Green's functions for 3D structures" "tieneTextoCompleto" => true "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "7" "paginaFinal" => "20" ] ] "autores" => array:1 [ 0 => array:4 [ "autoresLista" => "E. Leticia Flores-Márquez, Andrés Tejero-Andrade, Adrián León-Sánchez, Claudia Arango-Galván, René Chávez-Segura" "autores" => array:5 [ 0 => array:4 [ "nombre" => "E. Leticia" "apellidos" => "Flores-Márquez" "email" => array:1 [ 0 => "leticia@geofisica.unam.mx" ] "referencia" => array:2 [ 0 => array:2 [ "etiqueta" => "a" "identificador" => "aff0005" ] 1 => array:2 [ "etiqueta" => "*" "identificador" => "cor0005" ] ] ] 1 => array:3 [ "nombre" => "Andrés" "apellidos" => "Tejero-Andrade" "referencia" => array:1 [ 0 => array:2 [ "etiqueta" => "b" "identificador" => "aff0010" ] ] ] 2 => array:3 [ "nombre" => "Adrián" "apellidos" => "León-Sánchez" "referencia" => array:1 [ 0 => array:2 [ "etiqueta" => "b" "identificador" => "aff0010" ] ] ] 3 => array:3 [ "nombre" => "Claudia" "apellidos" => "Arango-Galván" "referencia" => array:1 [ 0 => array:2 [ "etiqueta" => "a" "identificador" => "aff0005" ] ] ] 4 => array:3 [ "nombre" => "René" "apellidos" => "Chávez-Segura" "referencia" => array:1 [ 0 => array:2 [ "etiqueta" => "a" "identificador" => "aff0005" ] ] ] ] "afiliaciones" => array:2 [ 0 => array:3 [ "entidad" => "Instituto de Geofísica, Universidad Nacional Autónoma de México Circuito Exterior, Cd. Universitaria, 04510 México D.F., México" "etiqueta" => "a" "identificador" => "aff0005" ] 1 => array:3 [ "entidad" => "División de Ciencias de la Tierra Facultad de Ingeniería, Universidad Nacional Autónoma de México Circuito Interior, Cd. Universitaria, 04510 México D.F., México" "etiqueta" => "b" "identificador" => "aff0010" ] ] "correspondencia" => array:1 [ 0 => array:3 [ "identificador" => "cor0005" "etiqueta" => "⁎" "correspondencia" => "Corresponding author." ] ] ] ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0030" "etiqueta" => "Figure 6" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr6.jpeg" "Alto" => 4230 "Ancho" => 2714 "Tamanyo" => 1305903 ] ] "descripcion" => array:1 [ "en" => "Third synthetic example constituted by (a) 4 immersed 3D bodies ρc = 20 Ωm in a homogeneous half space, b) The results of the SIM model, c) the results published by <a class="elsevierStyleCrossRef" href="#bib0165">Flores et al. (2001)</a>." ] ] ] "textoCompleto" => "IntroductionThe last three decades have been characterized by an increased use of computerized methods in the interpretation of geoelectrical data, due to the evolution of the computer systems. Most reconstructive algorithms are iterative and need a forward solution, i.e., to compute the electrical response for a given resistivity distribution and a given set array of current injection electrodes. Thus, the electrical potential needs to be calculated at a set of measured points. This forward problem consists on solving an elliptic partial differential equation (PDE): the Poisson equation, with boundary conditions. The formulation leads to solve a system with two kinds of unknown quantities: the electrical potential and a current-related quantity.The PDE problem is usually solved with finite-difference schemes that specially has been helpful to compute the apparent electrical resistivity in a two-dimensional medium (e.g. <a class="elsevierStyleCrossRefs" href="#bib0270">Forsythe and Wasow, 1960; Mufti, 1976; Dey and Morrison, 1979; Marchuk, 1989; Thomée, 1989; Spitzer, 1995; Zhang et al., 1995; Loke and Barker, 1996</a>). Another scheme extensively used in solving this PDE problem has been finite-element scheme (e.g. <a class="elsevierStyleCrossRefs" href="#bib0035">Coggon, 1971; Strang and Fix, 1973; Wait, 1977; Fox et al., 1980; Pridmore et al., 1980; Johnson, 1987; Ciarlet, 1991; Sasaki, 1994</a>; Tsourlous and Ogilvy; 1999; <a class="elsevierStyleCrossRefs" href="#bib0110">Li and Spitzer, 2002, 2005; Marescot et al., 2008; Ren and Tang, 2010</a>). Finite volume schemes have also produced excellent results in computing electrical resistivity (e.g. <a class="elsevierStyleCrossRefs" href="#bib0220">Snyder, 1976; Baliga and Patankar, 1980; Cai et al., 1991; Eskola, 1992; Perez-Flores, 1995; Perez-Flores et al., 2001; León-Sánchez, 2004; Pidlisecky et al., 2007</a>). The methods based on a finite-element scheme have been widely studied in the past 40 years and give rise to very high-performing techniques as mixed methods (<a class="elsevierStyleCrossRef" href="#bib0105">Lesur et al., 1999</a>), or h-p methods (<a class="elsevierStyleCrossRef" href="#bib0015">Babuska and Suri, 1994</a>). Nevertheless, the already mentioned methods lead to very large systems of linear equations, which are very demanding even for the supercomputers.One limitation in integral methods is the heterogeneity of the medium and the geometrical complexity of the bodies immersed in the modeled medium. An alternative to reduce this limitation is to propose a linearization procedure or some hypothesis about the interaction between bodies, as the weak scattering problem (<a class="elsevierStyleCrossRefs" href="#bib0045">Eskola, 1992; Hvozdara and Kaikkonen, 1998</a>). Such alternatives make integral equation method a good option to solve PDE, since this method does not need linearization, even in the case of bodies with complex geometry.The boundary-element methods (BEM) (<a class="elsevierStyleCrossRefs" href="#bib0155">Okabe, 1981; Nedelec, 1985, 1994; Wendland, 1987</a>) can be thought as a particular version among the finite-element methods. An example of the application of this method to 3-D electrical modeling can be found in <a class="elsevierStyleCrossRef" href="#bib0185">Poirmeur and Vasseur (1988)</a>. In this methodology, only the boundaries between media, of constant resistivity, need to be discretized and integrated. Therefore, unbounded homogeneous media are easily treated, and 3-D problems are solved using only 2-D integrals. Moreover, the boundary- element method can be coupled with standard finite element methods. The modification of the integral equations method with BEM, introduced by <a class="elsevierStyleCrossRef" href="#bib0085">Hvozdara and Kaikkonen (1998)</a>, is physically more meaningful and not so much demanding on computer resources, which made the method more accessible for routine prospecting work.This work follows the integral solution of the forward DC geoelectrical problem introduced by Hvozdara and Kaikkonen (1998; <a class="elsevierStyleCrossRef" href="#bib0080">Hvozdara, 1982</a>), which consists of interpreting the electric response of three-dimensional disturbing body of non-uniform conductivity, immersed in a planar homogeneous half-space, under the assumption of weak scattering (<a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>). In this research two algorithms are proposed to solve this forward problem, by introducing the resistivity contrast between bodies and the homogeneous half-space and the concepts of:additive potential sources for immersed bodies and density surface charges, which result in two types of solutions: volume (VIM) and surface integral methods (SIM). SIM and BEM use the same theoretical background but the boundary surfaces in SIM are not discretized and therefore no finite element is employed. SIM and VIM are used to solve the geoelectrical problem, with mixed boundary conditions, by considering a dipole-dipole electrode array to reproduce an electric tomography profile. The results of some synthetic examples are compared with those obtained by alternative methods in solving PDE already published by other authors (e.g. <a class="elsevierStyleCrossRefs" href="#bib0240">Tsourlos and Ogilvy, 1999; Pridmore, 1978; Hvozdara and Kaikkonen, 1998; Perez-Flores et al., 2001</a>).<elsevierMultimedia ident="fig0005"></elsevierMultimedia>Theoretical SettingFor a 3D heterogeneous half-space with a resistivity ρ (r¯), the total electric potential for a point source at the surface z = 0, is expressed by:<elsevierMultimedia ident="eq0005"></elsevierMultimedia>This PDE problem with boundary conditions can be rewritten as:<elsevierMultimedia ident="eq0010"></elsevierMultimedia>One solution for this equation can be expressed for the potential U(r) using the Green's theorems and Green's function method:<elsevierMultimedia ident="eq0015"></elsevierMultimedia>where r→ ′=(x ′, y ′, z ′) is related to local coordinates system, r→=(x, y, z) related to global coordinates system, and ∮s  denotes the integral over the boundaries s. In particular the integral over all boundaries can be written as:<elsevierMultimedia ident="eq0020"></elsevierMultimedia>Where ∮s U (r→)▿′ G(r→,r→) dS→0 if, r→→∞ due to the boundary conditions (<a class="elsevierStyleCrossRef" href="#bib0085">Hvozdara and Kaikkonen, 1998</a>), and G is the Green's function. Green's function, G, is is defined for a half-space problem (eq. 3) for Neumann condition, where ∂G∂Z z=0=0.∇′U(r→ ′) −E→(r→ ′) and by using E→(r→′) ρ(r→′)J→ (r→′), the expression (3) can be rewritten as<elsevierMultimedia ident="eq0025"></elsevierMultimedia>The Volume Integral Method (VIM) evaluates U(r) from equation (4), so it is necessary to know the current density function J→(r→) in half-space. The computation of J→(r→) is not an easy task, since there are several types of currents involved, particularly those present in the heterogeneous half-space. The “weak scattering problem” assumes that the primary conduction current is more significant that the secondary, that is J→ 2r→<<J→ pr→ (<a class="elsevierStyleCrossRef" href="#bib0045">Eskola, 1992</a>). Due to the interaction between bodies, we can express J→ p(r→) as:<elsevierMultimedia ident="eq0030"></elsevierMultimedia>where sub-index s represents the location of source electrodes.The Neumann Green function for half space can be defined, as was done by Kaufman (1992) as:<elsevierMultimedia ident="eq0035"></elsevierMultimedia>where rrg2= (x − x′)2 + (y − y′)2. Introducing this definition into eq. 4, and evaluating eq. 5 in z= 0, it becomes:<elsevierMultimedia ident="eq0040"></elsevierMultimedia><a class="elsevierStyleCrossRef" href="#bib0060">Gómez-Treviño (1987)</a>, Pérez-Flores et al. (2001) and <a class="elsevierStyleCrossRef" href="#bib0100">León-Sánchez (2004)</a> used a similar relation to estimate the apparent resistivity ρ(r→) in a heterogeneous half-space.Also U(r→) can be expressed as a surface integral, leading to the Surface Integral Method (SIM).If ρ (r→) J→ (r→) E→ p(r→)=E→ p(r→)+E→ 2(r→) then eq. (4) can be rewritten as<elsevierMultimedia ident="eq0045"></elsevierMultimedia>Here E→p (r→) is the primary electrical field due to the point source and E→2 (r→) the secondary electric field due to the heterogeneities of the medium.The first term of the right hand of equation (7) is equal to the primary source's potential Up (r→). The second term implies the whole halfspace volume. This integral could be separated in volumes for each heterogeneous body, for instance if we define:<elsevierMultimedia ident="eq0050"></elsevierMultimedia>Using the next vector property for each Vi,<elsevierMultimedia ident="eq0055"></elsevierMultimedia>and assuming that the resistivity of each body within the half-space (<a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>), is constant, then ▿′ .E→2 (r→) 0. Thus, we can use the divergence theorem for each Vi and eq. 8 becomes:<elsevierMultimedia ident="eq0060"></elsevierMultimedia>This equation should be applied to the whole surface delimitating each immersed body; in our case, we assume the body as a regular prism. Then, the corresponding integral for the case of two contiguous prismatic bodies (b1,b2) with a common surface, is:<elsevierMultimedia ident="eq0065"></elsevierMultimedia>Where n→12 is the unit normal vector of the surfaces (1, 2) between the two bodies.The boundary conditions allow to define:<elsevierMultimedia ident="eq0070"></elsevierMultimedia>where σS(r→′) is the density surface charges and ¿0 is the free-space electrical permittivity.Taking into account the equations (10 to 13), the electric potential (eq. 7) is rewritten as:<elsevierMultimedia ident="eq0075"></elsevierMultimedia>Where number 6 denotes the total number of surfaces of one prismatic body and M the number of bodies within the half-space, this eq. constitutes the SIM.<a class="elsevierStyleCrossRef" href="#bib0045">Eskola (1992)</a> has obtained an expression similar to equation (14) using different analytical approach, under the same type of hypothesis.However, a problem to solve is to know σ (r→ ′) (the density surface charges) for each surface of the each prismatic body. Kaufman (1992) expressed σ (r→ ′) for two contiguous surfaces as:<elsevierMultimedia ident="eq0080"></elsevierMultimedia>where<elsevierMultimedia ident="eq0085"></elsevierMultimedia>If we neglect the normal electrical secondary field ld in expression (16), i.e.E→n→2,b1 (r→) and, E→n→2,b2 (r→) then the equation (15) becomes:<elsevierMultimedia ident="eq0090"></elsevierMultimedia>Where, R12, is the reflectivity coefficient between surfaces. This expression is the approximation of the induced surface electric charge and it is equivalent to the so-called “weak scattering problem”.Numerical ApproachEquations (6 and 14) are expressed in arbitrary coordinate systems with a fixed origin. However, to solve the corresponding integrals we redefine the origin of the coordinate system at the middle point of the prismatic body; that is the “local coordinate system”. The transformation between both coordinate systems will be defined as follows: Assuming P an arbitrary point in the space, its position vector in terms of the global coordinates system is r→ and r→ ′ is its position vector in terms of the local coordinate system. Consequently, the relationship between the origins for both systems is defined by r→a (see <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>), that is:<elsevierMultimedia ident="eq0095"></elsevierMultimedia>where r→′ = (x′, y′,z′), r→a = (xa, yb,zc), and r→=(x, y, z.)<elsevierMultimedia ident="fig0010"></elsevierMultimedia>Assuming isolated heterogeneous bodies immersed in a half space, let us introduce the resistivity contrast as ρ = ρc-ρm, where ρm is the resistivity of the half-space and ρc is the resistivity of the immersed body.Then, by applying equation (6) for a quadrupole array, that is to a vertical electric sounding (VES) (where electrodes are usually named: A, B, N, M, A and B indicate current electrodes and M and N reception electrodes), the potential UsNM associated with a point source electrode can be expressed as eq. (19). This equation also assumes the concept of additive potential sources (Orellana, 1972):<elsevierMultimedia ident="eq0100"></elsevierMultimedia>This equation allows us to compute the secondary electric potential by the volume integral method, VIM.In the eq. 14, SIM, the density surface charges expressed in terms of the local coordinates system is:<elsevierMultimedia ident="eq0105"></elsevierMultimedia>By substituting equation (20) in equation (14), the contribution of each surface of the immersed body, to the secondary potential field for the same quadrupole array, is expressed as:<elsevierMultimedia ident="eq0110"></elsevierMultimedia>Then the apparent resistivity ρa can be expressed as:<elsevierMultimedia ident="eq0115"></elsevierMultimedia>To solve the integrals involved in equations (19 and 21) (VIM and SIM, respectively) we use the Gauss-Legendre Quadrature, by using the subroutines QGAUS and DQDAGI, that are in-cluded in the IMLS Fortran numerical libraries (<a class="elsevierStyleCrossRefs" href="#bib0135">Meissner, 1995; Press et al., 1992</a>). DQDAGI subroutine makes use of Gauss-Kronrod approximation with 21 points, and by using an e-algorithm (<a class="elsevierStyleCrossRef" href="#bib0180">Piessens et al., 1983</a>), these integrals can be estimated even when the ending interval is a singularity.The computational program developed in this work computes the apparent resistivity profile for 3D inmersed bodies, by entering the data listed in <a class="elsevierStyleCrossRef" href="#tbl0005">table 1</a>. The output data are the apparent resistivity values in an array that corresponds to a resistivity pseudo-section cutting the half-space in the input direction.<elsevierMultimedia ident="tbl0005"></elsevierMultimedia>Synthetic examplesIn order to illustrate the validity of the VIM and SIM developed in this work, we studied some synthetic examples and compared them to results obtained by others authors.A stratified media, with three layers of different resistivities, constitutes the first example (<a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>a). One case considers a middle conductor layer: 100, 10, 100 ohm-m (<a class="elsevierStyleCrossRef" href="#fig0015">Figures 3</a>b); and the other case considers a middle resistive layer: 10, 100, 10 ohm-m (<a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>c). The results of the SIM model for a dipole-dipole array are compared (<a class="elsevierStyleCrossRef" href="#fig0015">Figures 3</a>b and <a class="elsevierStyleCrossRef" href="#fig0015">3</a>c) to those results obtained by applying the algorithm based on the adaptative digital filtering proposed by <a class="elsevierStyleCrossRef" href="#bib0005">Anderson (1979)</a>, which uses Hankel transforms. This comparison shows coincidences in the computed resistivity values at the subsurface assignation points corresponding to an electrodic separation of a = 1m, and until the level n = 14; however, after level n = 15 the results show differences between values (each level n corresponds to 0.5 m), because the computed induced charge by SIM is a poor approximation. It is important to point out that SIM is one method that needs to model closed bodies and the middle layer was considered as a body of 400 by 400 m and thickness of T = 2.5 m, the depth D = 5 m this assumption involves numerical errors that could explain the enlargement of the differences between both methods at depth (for levels n > 15 and depth > 10.5 m). But also it is important to point out the assumption of weak scattering concerns the use of Born approximation (Guozhong and Torres-Verdín, 2006) and this is also a contribution in those discrepancies, as it was signaled by <a class="elsevierStyleCrossRef" href="#bib0255">Zhdanov and Fang (1996)</a>, the Born approximation produces curves of the correct shape but incorrect magnitude. In summary, we can conclude the approximation with SIM is good enough.<elsevierMultimedia ident="fig0015"></elsevierMultimedia>The second example consists of a 3D homogeneous half-space, with ρm = 100 ohm-m, and one conductor immersed prismatic body, of ρc = 20 ohm-m (<a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a>). The results of VIM method (<a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>a) shows differences between 3 and 21 ohm-m in the lower values region compared to those computed by <a class="elsevierStyleCrossRef" href="#bib0195">Pridmore (1978)</a>; while the SIM modeling of a dipoledipole array over the prism are compared (<a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>b) to those obtained by <a class="elsevierStyleCrossRef" href="#bib0240">Tsourlos and Ogilvy (1999)</a>. As it is observed, the differences between values are within 1 and 10 ohm-m. In contrast (<a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>d), and only rise up to 14 ohm-m compared to those obtained by <a class="elsevierStyleCrossRef" href="#bib0240">Tsourlos and Ogilvy (1999)</a>, <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>c. In spite of the differences depicted between the results of SIM and VIM, the results are good enough since the computed resistivity values do not exceed 15 ohm-m (<a class="elsevierStyleCrossRef" href="#fig0025">Figures 5</a>a and <a class="elsevierStyleCrossRef" href="#fig0025">5</a>b). That is about 18% of the resistivity contrast between body and half-space.<elsevierMultimedia ident="fig0020"></elsevierMultimedia><elsevierMultimedia ident="fig0025"></elsevierMultimedia>The third example showed in <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>a, is constituted by the synthetic example published by <a class="elsevierStyleCrossRef" href="#bib0165">Perez-Flores et al. (2001)</a> with 4 immersed bodies of constant resistivity, ρc = 20 ohm-m. This model is based on a volume integral scheme (<a class="elsevierStyleCrossRef" href="#bib0165">Perez-Flores et al., 2001</a>) and it is similar to the hypothesis of the VIM proposed here. The comparison between SIM and model shows similar results (<a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>b). Also, the VIM shows quite the same data for this particular case (not showed in figure); however, for general cases, we would expect bigger differences from VIM results. A possible explanation is that the electrode separation is smaller than the dimensions of the bodies.<elsevierMultimedia ident="fig0030"></elsevierMultimedia>The fourth example presented consists of two conductive bodies (<a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a>) of ρc = 20 ohm-m, immersed in a homogeneous half-space of ρm = 100 ohm-m. The bodies have the same dimensions, 10 m thick (T, in the z direction), 10 m long (L, in the x direction) and 10 m width (W, in the y direction) and both are located at 2.5 m depth (D). This example is proposed just to show the interaction between bodies by changing the separation between them, with two possibilities: closer and distant (far) bodies, with S equal to 6 m and 40 m respectively. We assume a dipole-dipole array consisting of 31 electrodes, with a 5 m distance between them. <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a> shows the results obtained with SIM and VIM for the case with S = 6 m. The apparent resistivity values with SIM are those expected for the bodies. In contrast, VIM's resistivity values are bigger than those expected. It is important to point out that we obtain two minimum resistivities in the location corresponding to the bodies, as we expect, those anomalies in resistivities correspond to the bodies. However, it is also observed a third anomaly at the center of the resistivity image that corresponds to a numerical feature, of a lower resistivity value. <a class="elsevierStyleCrossRef" href="#fig0045">Figure 9</a> shows results for the case S =40 m, they are similar to those obtained for isolate bodies (<a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>). As well as previous case, it is also observed a third anomaly at the center of the resistivity image that corresponds to a numerical feature.<elsevierMultimedia ident="fig0035"></elsevierMultimedia><elsevierMultimedia ident="fig0040"></elsevierMultimedia><elsevierMultimedia ident="fig0045"></elsevierMultimedia>In all the studies cases, we can observe, SIM produces better approach than VIM in computing the electrical potential.ConclusionsThis paper introduces two algorithms for the integral solution of the forward DC geoelectrical problem introduced by <a class="elsevierStyleCrossRef" href="#bib0085">Hvozdara and Kaikkonen (1998)</a> with mixed boundary conditions using Green's function. The two types of solutions: volume (VIM) and surface integral methods (SIM) make use of the resistivity contrast between immersed bodies and the homogeneous half- space. These methods also use the concepts of: additive potential sources for immersed bodies, and density surface charges. Both algorithms are not so much demanding on computer time and memory because they do not produce to very large systems of linear equations. This made the methods more accessible for personal computers, quotidian prospecting work and also makes it attractive for educational purposes. In particular could be useful to easily validate the field measurements interpretation.The algorithms developed here can help in the interpretation of the field data obtained from resistivity profile methods, in two and three dimensions. The advantage of using the integral equation technique is that it is performed for each immersed body in the half space, in contrast to the usual procedure in finite-element and finite-difference methods. In order to find the induced charge, we do not need to define a grid on the surface of the body, due to the fact that we use the density surface charges on each surface.The conducted tests with synthetic data indicated that both algorithms (SIM and VIM) produced reasonably good results compared to already published results for similar problems, obtained by other algorithms. The synthetic examples allow us to conclude that SIM produces a better approximation of the apparent resistivity values than those based on the volume integral (VIM).These results are particularly attractive for computation in parallel, because they provide the mode to obtain the forward response for each body in simultaneous way." "textoCompletoSecciones" => array:1 [ "secciones" => array:12 [ 0 => array:3 [ "identificador" => "xres502922" "titulo" => "Resumen" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0005" ] ] ] 1 => array:2 [ "identificador" => "xpalclavsec524093" "titulo" => "Palabras clave" ] 2 => array:3 [ "identificador" => "xres502923" "titulo" => "Abstract" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0010" ] ] ] 3 => array:2 [ "identificador" => "xpalclavsec524092" "titulo" => "Keywords" ] 4 => array:2 [ "identificador" => "sec0005" "titulo" => "Introduction" ] 5 => array:2 [ "identificador" => "sec0010" "titulo" => "Theoretical Setting" ] 6 => array:2 [ "identificador" => "sec0015" "titulo" => "Numerical Approach" ] 7 => array:2 [ "identificador" => "sec0020" "titulo" => "Synthetic examples" ] 8 => array:2 [ "identificador" => "sec0025" "titulo" => "Conclusions" ] 9 => array:2 [ "identificador" => "xack163407" "titulo" => "Acknowledgements" ] 10 => array:1 [ "titulo" => "Further reading" ] 11 => array:1 [ "titulo" => "References" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "fechaRecibido" => "2013-06-13" "fechaAceptado" => "2014-05-05" "PalabrasClave" => array:2 [ "es" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Palabras clave" "identificador" => "xpalclavsec524093" "palabras" => array:1 [ 0 => "Modelo eléctrico 3D, funciones de Green, método integral, teorema de Gauss, condiciones de frontera." ] ] ] "en" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Keywords" "identificador" => "xpalclavsec524092" "palabras" => array:1 [ 0 => "3D electrical model, Green's functions, integral method, Gauss theorem, Boundary conditions." ] ] ] ] "tieneResumen" => true "resumen" => array:2 [ "es" => array:2 [ "titulo" => "Resumen" "resumen" => "Se introduce una solución integral para el problema directo de la respuesta geoeléctrica DC para cuerpos tri-dimensionales en un semi- espacio, mediante las funciones de Green. El primer algoritmo que se presenta se basa en el método integral de volumen (MIV); aquí, únicamente la corriente eléctrica primaria se utiliza para calcular el potencial eléctrico. El segundo caso emplea el método integral de superficie (MIS), en donde se asume que la carga inducida es debida al campo eléctrico primario. Ambos algoritmos son una combinación de integrales de volumen y de condiciones de frontera. Este artículo muestra la aplicabilidad de estos algoritmos para generar imágenes de perfiles de resistividad que reproducen algunos arreglos de electrodos para ejemplos sintéticos tradicionales, y posteriormente estas imágenes se comparan con resultados ya publicados en la literatura. Finalmente, la comparación entre estos resultados muestra que el concepto de carga inducida utilizada en MIS produce una mejor aproximación, que el esquema MIV en el cálculo del potencial eléctrico." ] "en" => array:2 [ "titulo" => "Abstract" "resumen" => "An integral solution of the forward DC geoelectric response for three-dimensional target-bodies in a half-space, based on Green's functions, is introduced. The first algorithm presented is based on a volume integral method (VIM); here, only the primary electrical current is involved to compute the electric potential. The second one employs the surface integral method (SIM), and it is assumed the induced charge is due to the primary electrical field. Both algorithms are a combination of boundary and volume integrals. This paper shows the applicability of these algorithms to generate resistivity profile images reproducing some electrode arrays for traditional synthetic examples, and then these images were compared with already published results. Finally, the comparison between results shows the concept of induced charge used in SIM produces a better approach than VIM scheme in computing the electrical potential." ] ] "multimedia" => array:33 [ 0 => array:7 [ "identificador" => "fig0005" "etiqueta" => "Figure 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 1205 "Ancho" => 2138 "Tamanyo" => 145245 ] ] "descripcion" => array:1 [ "en" => "Conceptual model of a heterogeneous half-space formed by some bodies, with different but constant resistivity values, ρ1 ...ρ6, immersed in a homogeneous medium with a constant resistivity value ρ0." ] ] 1 => array:7 [ "identificador" => "fig0010" "etiqueta" => "Figure 2" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr2.jpeg" "Alto" => 1641 "Ancho" => 2282 "Tamanyo" => 243635 ] ] "descripcion" => array:1 [ "en" => "Relationship between two coordinate systems: global, refers to the external coordinates a n d local, that it is centered at the origin of the immersed resistive body r→=r→′+r→a." ] ] 2 => array:7 [ "identificador" => "fig0015" "etiqueta" => "Figure 3" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr3.jpeg" "Alto" => 2531 "Ancho" => 2107 "Tamanyo" => 150341 ] ] "descripcion" => array:1 [ "en" => "a) Synthetic example assuming a stratified half-space of three layers with D = 5 m and T = 2.5 m. The log-log plot shows the comparison between the SIM and the Anderson filter (<a class="elsevierStyleCrossRef" href="#bib0005">Anderson, 1979</a>) using two different contrasts of resistivities: b) a middle conductive layer with ρm = 100 Ωm, ρl=10 Ωm and c) a middle resistive stratum with ρm = 10 Ωm, ρl=100 Ωm." ] ] 3 => array:7 [ "identificador" => "fig0020" "etiqueta" => "Figure 4" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr4.jpeg" "Alto" => 1103 "Ancho" => 2496 "Tamanyo" => 96108 ] ] "descripcion" => array:1 [ "en" => "The schematic model shows a 3D body, ρc = 20 Ωm, immersed in a homogeneous half-space ρm = 100 Ωm. a/2 is the depth to the top of the body, 2a is the longitude of all sides of the cube, and a is the inter-electrode separation. Distance along the profile is x-coordinate (meters) and z-coordinate designates the positive depth (meters)." ] ] 4 => array:7 [ "identificador" => "fig0025" "etiqueta" => "Figure 5" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr5.jpeg" "Alto" => 2975 "Ancho" => 2322 "Tamanyo" => 653933 ] ] "descripcion" => array:1 [ "en" => "Comparison between results of the second example, constituted by the conductor immersed prismatic body shown in <a class="elsevierStyleCrossRef" href="#fig0020">fig. 4</a>, a) the results of the VIM model, b) the results of the SIM model, and those already published c) <a class="elsevierStyleCrossRef" href="#bib0240">Tsourlos and Ogilvy (1999)</a>, and d) <a class="elsevierStyleCrossRef" href="#bib0195">Pridmore (1978)</a>." ] ] 5 => array:7 [ "identificador" => "fig0030" "etiqueta" => "Figure 6" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr6.jpeg" "Alto" => 4230 "Ancho" => 2714 "Tamanyo" => 1305903 ] ] "descripcion" => array:1 [ "en" => "Third synthetic example constituted by (a) 4 immersed 3D bodies ρc = 20 Ωm in a homogeneous half space, b) The results of the SIM model, c) the results published by <a class="elsevierStyleCrossRef" href="#bib0165">Flores et al. (2001)</a>." ] ] 6 => array:7 [ "identificador" => "fig0035" "etiqueta" => "Figure 7" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr7.jpeg" "Alto" => 1603 "Ancho" => 2952 "Tamanyo" => 103702 ] ] "descripcion" => array:1 [ "en" => "The schematic model shows a homogeneous half-space (ρm) and two immersed bodies of constant resistivity ρc. D is depth from soil to the top of the bodies (a/2), S is the horizontal distance between bodies, (T) high of bodies, (W) wide in y direction and (L) large in x direction, and a is the inter-electrodic separation." ] ] 7 => array:7 [ "identificador" => "fig0040" "etiqueta" => "Figure 8" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr8.jpeg" "Alto" => 2241 "Ancho" => 2254 "Tamanyo" => 395324 ] ] "descripcion" => array:1 [ "en" => "Pseudo-section model obtained for the fourth example (<a class="elsevierStyleCrossRef" href="#fig0035">figure 7</a>), with ρc = 20 Ωm and ρm = 100 Ωm. Simulating a dipole-dipole array of 31 electrodes, with a = 5 m. The apparent resistivities were computed for separation between immerse bodies of S = 6 m. a) SIM and b) VIM." ] ] 8 => array:7 [ "identificador" => "fig0045" "etiqueta" => "Figure 9" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr9.jpeg" "Alto" => 2174 "Ancho" => 2100 "Tamanyo" => 393055 ] ] "descripcion" => array:1 [ "en" => "Pseudo-section model obtained for the same characteristic of the bodies of the fourth example (<a class="elsevierStyleCrossRef" href="#fig0035">fig. 7</a>), with ρc = 20 Ωm and ρm = 100 Ωm, but for separation between immerse bodies of S = 40 m, simulating a dipole-dipole array of 31 electrodes, with a = 5 m, a) SIM and b) VIM." ] ] 9 => array:7 [ "identificador" => "tbl0005" "etiqueta" => "Table 1" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table 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class="td" title="table-entry " align="left" valign="top">Resistivity of each body \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">Resistivity of the half-space \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="left" valign="top">Direction of the 2D output section \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab802339.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Required data in the computational program to computes the apparent resistivity." ] ] 10 => array:6 [ "identificador" => "eq0005" "etiqueta" => "(1)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:1 [ "imagen" => array:1 [ 0 => array:4 [ "Fichero" => "fx1.jpeg" "Tamanyo" => 63307 "Alto" => 473 "Ancho" => 1345 ] ] ] ] 11 => array:6 [ "identificador" => "eq0010" "etiqueta" => "(2)" "tipo" => "MULTIMEDIAFORMULA" 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Turburllo" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:5 [ "tituloSerie" => "First Break" "fecha" => "1985" "volumen" => "3" "paginaInicial" => "16" "paginaFinal" => "20" ] ] ] ] ] ] 4 => array:3 [ "identificador" => "bib0095" "etiqueta" => "Kauffman, 1992" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:1 [ "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "A.A. Kauffman" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:2 [ "tituloSerie" => "Geophysical field theory and method: Part A - Gravitational, electric and magnetic fields." "fecha" => "1992" ] ] ] ] ] ] 5 => array:3 [ "identificador" => "bib0160" "etiqueta" => "Oldenburg, 1978" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:2 [ "titulo" => "The interpretation of direct current resistivity measurements" "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "W.D. Oldenburg" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:5 [ "tituloSerie" => "Geophysics" "fecha" => "1978" "volumen" => "43" "paginaInicial" => "610" "paginaFinal" => "625" ] ] ] ] ] ] 6 => array:3 [ "identificador" => "bib0210" "etiqueta" => "Roach, 2004" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:1 [ "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "G.F. Roach" ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Revista" => array:2 [ "tituloSerie" => "Green's functions." "fecha" => "2004" ] ] ] ] ] ] ] ] ] ] ] "agradecimientos" => array:1 [ 0 => array:4 [ "identificador" => "xack163407" "titulo" => "Acknowledgements" "texto" => "This study was financed by the PAPIIT- DGAPA-UNAM research projects IN112906 and IN104006." 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E. Leticia Flores-Márqueza,

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leticia@geofisica.unam.mx

Corresponding author.

, Andrés Tejero-Andradeb, Adrián León-Sánchezb, Claudia Arango-Galvána, René Chávez-Seguraa

a Instituto de Geofísica, Universidad Nacional Autónoma de México Circuito Exterior, Cd. Universitaria, 04510 México D.F., México

b División de Ciencias de la Tierra Facultad de Ingeniería, Universidad Nacional Autónoma de México Circuito Interior, Cd. Universitaria, 04510 México D.F., México

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    "textoCompleto" => "<span class="elsevierStyleSections"><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0025">Introduction</span><p id="par0005" class="elsevierStylePara elsevierViewall">The last three decades have been characterized by an increased use of computerized methods in the interpretation of geoelectrical data&#44; due to the evolution of the computer systems&#46; Most reconstructive algorithms are iterative and need a forward solution&#44; i&#46;e&#46;&#44; to compute the electrical response for a given resistivity distribution and a given set array of current injection electrodes&#46; Thus&#44; the electrical potential needs to be calculated at a set of measured points&#46; This forward problem consists on solving an elliptic partial differential equation &#40;PDE&#41;&#58; the Poisson equation&#44; with boundary conditions&#46; The formulation leads to solve a system with two kinds of unknown quantities&#58; the electrical potential and a current-related quantity&#46;</p><p id="par0010" class="elsevierStylePara elsevierViewall">The PDE problem is usually solved with finite-difference schemes that specially has been helpful to compute the apparent electrical resistivity in a two-dimensional medium &#40;e&#46;g&#46; <a class="elsevierStyleCrossRefs" href="#bib0270">Forsythe and Wasow&#44; 1960&#59; Mufti&#44; 1976&#59; Dey and Morrison&#44; 1979&#59; Marchuk&#44; 1989&#59; Thom&#233;e&#44; 1989&#59; Spitzer&#44; 1995&#59; Zhang <span class="elsevierStyleItalic">et al&#46;</span>&#44; 1995&#59; Loke and Barker&#44; 1996</a>&#41;&#46; Another scheme extensively used in solving this PDE problem has been finite-element scheme &#40;e&#46;g&#46; <a class="elsevierStyleCrossRefs" href="#bib0035">Coggon&#44; 1971&#59; Strang and Fix&#44; 1973&#59; Wait&#44; 1977&#59; Fox <span class="elsevierStyleItalic">et al&#46;</span>&#44; 1980&#59; Pridmore <span class="elsevierStyleItalic">et al&#46;</span>&#44; 1980&#59; Johnson&#44; 1987&#59; Ciarlet&#44; 1991&#59; Sasaki&#44; 1994</a>&#59; Tsourlous and Ogilvy&#59; 1999&#59; <a class="elsevierStyleCrossRefs" href="#bib0110">Li and Spitzer&#44; 2002&#44; 2005&#59; Marescot et al&#46;&#44; 2008&#59; Ren and Tang&#44; 2010</a>&#41;&#46; Finite volume schemes have also produced excellent results in computing electrical resistivity &#40;e&#46;g&#46; <a class="elsevierStyleCrossRefs" href="#bib0220">Snyder&#44; 1976&#59; Baliga and Patankar&#44; 1980&#59; Cai <span class="elsevierStyleItalic">et al&#46;</span>&#44; 1991&#59; Eskola&#44; 1992&#59; Perez-Flores&#44; 1995&#59; Perez-Flores <span class="elsevierStyleItalic">et al&#46;</span>&#44; 2001&#59; Le&#243;n-S&#225;nchez&#44; 2004&#59; Pidlisecky <span class="elsevierStyleItalic">et al&#46;</span>&#44; 2007</a>&#41;&#46; The methods based on a finite-element scheme have been widely studied in the past 40 years and give rise to very high-performing techniques as mixed methods &#40;<a class="elsevierStyleCrossRef" href="#bib0105">Lesur <span class="elsevierStyleItalic">et al&#46;</span>&#44; 1999</a>&#41;&#44; or h-p methods &#40;<a class="elsevierStyleCrossRef" href="#bib0015">Babuska and Suri&#44; 1994</a>&#41;&#46; Nevertheless&#44; the already mentioned methods lead to very large systems of linear equations&#44; which are very demanding even for the supercomputers&#46;</p><p id="par0015" class="elsevierStylePara elsevierViewall">One limitation in integral methods is the heterogeneity of the medium and the geometrical complexity of the bodies immersed in the modeled medium&#46; An alternative to reduce this limitation is to propose a linearization procedure or some hypothesis about the interaction between bodies&#44; as the weak scattering problem &#40;<a class="elsevierStyleCrossRefs" href="#bib0045">Eskola&#44; 1992&#59; Hvozdara and Kaikkonen&#44; 1998</a>&#41;&#46; Such alternatives make integral equation method a good option to solve PDE&#44; since this method does not need linearization&#44; even in the case of bodies with complex geometry&#46;</p><p id="par0020" class="elsevierStylePara elsevierViewall">The boundary-element methods &#40;BEM&#41; &#40;<a class="elsevierStyleCrossRefs" href="#bib0155">Okabe&#44; 1981&#59; Nedelec&#44; 1985&#44; 1994&#59; Wendland&#44; 1987</a>&#41; can be thought as a particular version among the finite-element methods&#46; An example of the application of this method to 3-D electrical modeling can be found in <a class="elsevierStyleCrossRef" href="#bib0185">Poirmeur and Vasseur &#40;1988&#41;</a>&#46; In this methodology&#44; only the boundaries between media&#44; of constant resistivity&#44; need to be discretized and integrated&#46; Therefore&#44; unbounded homogeneous media are easily treated&#44; and 3-D problems are solved using only 2-D integrals&#46; Moreover&#44; the boundary- element method can be coupled with standard finite element methods&#46; The modification of the integral equations method with BEM&#44; introduced by <a class="elsevierStyleCrossRef" href="#bib0085">Hvozdara and Kaikkonen &#40;1998&#41;</a>&#44; is physically more meaningful and not so much demanding on computer resources&#44; which made the method more accessible for routine prospecting work&#46;</p><p id="par0025" class="elsevierStylePara elsevierViewall">This work follows the integral solution of the forward DC geoelectrical problem introduced by Hvozdara and Kaikkonen &#40;1998&#59; <a class="elsevierStyleCrossRef" href="#bib0080">Hvozdara&#44; 1982</a>&#41;&#44; which consists of interpreting the electric response of three-dimensional disturbing body of non-uniform conductivity&#44; immersed in a planar homogeneous half-space&#44; under the assumption of weak scattering &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>&#41;&#46; In this research two algorithms are proposed to solve this forward problem&#44; by introducing the resistivity contrast between bodies and the homogeneous half-space and the concepts of&#58;additive potential sources for immersed bodies and density surface charges&#44; which result in two types of solutions&#58; volume &#40;<span class="elsevierStyleItalic">VIM</span>&#41; and surface integral methods &#40;<span class="elsevierStyleItalic">SIM</span>&#41;&#46; <span class="elsevierStyleItalic">SIM</span> and BEM use the same theoretical background but the boundary surfaces in <span class="elsevierStyleItalic">SIM</span> are not discretized and therefore no finite element is employed&#46; <span class="elsevierStyleItalic">SIM</span> and <span class="elsevierStyleItalic">VIM</span> are used to solve the geoelectrical problem&#44; with mixed boundary conditions&#44; by considering a dipole-dipole electrode array to reproduce an electric tomography profile&#46; The results of some synthetic examples are compared with those obtained by alternative methods in solving PDE already published by other authors &#40;e&#46;g&#46; <a class="elsevierStyleCrossRefs" href="#bib0240">Tsourlos and Ogilvy&#44; 1999&#59; Pridmore&#44; 1978&#59; Hvozdara and Kaikkonen&#44; 1998&#59; Perez-Flores <span class="elsevierStyleItalic">et al&#46;</span>&#44; 2001</a>&#41;&#46;</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0030">Theoretical Setting</span><p id="par0030" class="elsevierStylePara elsevierViewall">For a 3D heterogeneous half-space with a resistivity &#961; &#40;r&#175;&#41;&#44; the total electric potential for a point source at the surface <span class="elsevierStyleItalic">z</span> &#61; 0&#44; is expressed by&#58;<elsevierMultimedia ident="eq0005"></elsevierMultimedia></p><p id="par0035" class="elsevierStylePara elsevierViewall">This PDE problem with boundary conditions can be rewritten as&#58;<elsevierMultimedia ident="eq0010"></elsevierMultimedia></p><p id="par0040" class="elsevierStylePara elsevierViewall">One solution for this equation can be expressed for the potential U&#40;r&#41; using the Green&#39;s theorems and Green&#39;s function method&#58;<elsevierMultimedia ident="eq0015"></elsevierMultimedia>where r&#8594; &#8242;&#61;&#40;<span class="elsevierStyleItalic">x</span> &#8242;&#44; <span class="elsevierStyleItalic">y</span> &#8242;&#44; <span class="elsevierStyleItalic">z</span> &#8242;&#41; is related to <span class="elsevierStyleItalic">local coordinates system</span>&#44; r&#8594;<span class="elsevierStyleSup"><span class="elsevierStyleItalic">&#61;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#44; <span class="elsevierStyleItalic">y</span>&#44; <span class="elsevierStyleItalic">z</span>&#41; related to <span class="elsevierStyleItalic">global coordinates system</span>&#44; and &#8750;s&#8201; denotes the integral over the boundaries <span class="elsevierStyleItalic">s</span>&#46; In particular the integral over all boundaries can be written as&#58;<elsevierMultimedia ident="eq0020"></elsevierMultimedia></p><p id="par0045" class="elsevierStylePara elsevierViewall">Where &#8750;s&#8201;<span class="elsevierStyleItalic">U</span> &#40;r&#8594;&#41;&#9663;&#8242; <span class="elsevierStyleItalic">G</span>&#40;r&#8594;&#44;r&#8594;&#41; <span class="elsevierStyleItalic">d</span>S&#8594;<span class="elsevierStyleGlyphqbnd"></span>0 if&#44; r&#8594;&#8594;&#8734; due to the boundary conditions &#40;<a class="elsevierStyleCrossRef" href="#bib0085">Hvozdara and Kaikkonen&#44; 1998</a>&#41;&#44; and G is the Green&#39;s function&#46; Green&#39;s function&#44; G&#44; is is defined for a half-space problem &#40;eq&#46; 3&#41; for Neumann condition&#44; where &#8706;G&#8706;Z&#8201;z&#61;0&#61;0&#46;</p><p id="par0050" class="elsevierStylePara elsevierViewall">&#8711;&#8242;<span class="elsevierStyleItalic">U</span>&#40;r&#8594; &#8242;&#41; <span class="elsevierStyleGlyphqbnd"></span> &#8722;E&#8594;&#40;r&#8594; &#8242;&#41; and by using E&#8594;&#40;r&#8594;&#8242;&#41; <span class="elsevierStyleGlyphqbnd"></span><span class="elsevierStyleItalic">&#961;</span>&#40;r&#8594;&#8242;&#41;J&#8594; &#40;r&#8594;&#8242;&#41;&#44; the expression &#40;3&#41; can be rewritten as<elsevierMultimedia ident="eq0025"></elsevierMultimedia></p><p id="par0055" class="elsevierStylePara elsevierViewall">The <span class="elsevierStyleItalic">Volume Integral Method <span class="elsevierStyleBold">&#40;VIM</span>&#41;</span> evaluates U&#40;r&#41; from equation &#40;4&#41;&#44; so it is necessary to know the current density function J&#8594;&#40;r&#8594;&#41; in half-space&#46; The computation of J&#8594;&#40;r&#8594;&#41; is not an easy task&#44; since there are several types of currents involved&#44; particularly those present in the heterogeneous half-space&#46; The &#8220;weak scattering problem&#8221; assumes that the primary conduction current is more significant that the secondary&#44; that is J&#8594;&#8201;2r&#8594;&#60;&#60;J&#8594;&#8201;pr&#8594; &#40;<a class="elsevierStyleCrossRef" href="#bib0045">Eskola&#44; 1992</a>&#41;&#46; Due to the interaction between bodies&#44; we can express J&#8594;&#8201;p&#40;r&#8594;&#41; as&#58;<elsevierMultimedia ident="eq0030"></elsevierMultimedia>where sub-index <span class="elsevierStyleItalic">s</span> represents the location of source electrodes&#46;</p><p id="par0060" class="elsevierStylePara elsevierViewall">The Neumann Green function for half space can be defined&#44; as was done by Kaufman &#40;1992&#41; as&#58;<elsevierMultimedia ident="eq0035"></elsevierMultimedia></p><p id="par0065" class="elsevierStylePara elsevierViewall">where <span class="elsevierStyleItalic">r</span>rg2&#61; &#40;<span class="elsevierStyleItalic">x</span> &#8722; <span class="elsevierStyleItalic">x</span>&#8242;&#41;2 &#43; &#40;<span class="elsevierStyleItalic">y</span> &#8722; <span class="elsevierStyleItalic">y</span>&#8242;&#41;2&#46; Introducing this definition into eq&#46; 4&#44; and evaluating eq&#46; 5 in <span class="elsevierStyleItalic">z</span>&#61; 0&#44; it becomes&#58;<elsevierMultimedia ident="eq0040"></elsevierMultimedia></p><p id="par0070" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#bib0060">G&#243;mez-Trevi&#241;o &#40;1987&#41;</a>&#44; P&#233;rez-Flores <span class="elsevierStyleItalic">et al&#46;</span> &#40;2001&#41; and <a class="elsevierStyleCrossRef" href="#bib0100">Le&#243;n-S&#225;nchez &#40;2004&#41;</a> used a similar relation to estimate the apparent resistivity &#961;&#40;r&#8594;&#41; in a heterogeneous half-space&#46;</p><p id="par0075" class="elsevierStylePara elsevierViewall">Also <span class="elsevierStyleItalic">U</span>&#40;r&#8594;&#41; can be expressed as a surface integral&#44; leading to the <span class="elsevierStyleItalic">Surface Integral Method <span class="elsevierStyleBold">&#40;SIM&#41;</span></span>&#46;If <span class="elsevierStyleItalic">&#961;</span> &#40;r&#8594;&#41; J&#8594; &#40;r&#8594;&#41; <span class="elsevierStyleGlyphqbnd"></span>E&#8594;&#8201;p&#40;r&#8594;&#41;&#61;E&#8594;&#8201;p&#40;r&#8594;&#41;&#43;E&#8594;&#8201;2&#40;r&#8594;&#41; then eq&#46; &#40;4&#41; can be rewritten as<elsevierMultimedia ident="eq0045"></elsevierMultimedia></p><p id="par0080" class="elsevierStylePara elsevierViewall">Here E&#8594;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span> &#40;r&#8594;&#41; is the primary electrical field due to the point source and E&#8594;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span> &#40;r&#8594;&#41; the secondary electric field due to the heterogeneities of the medium&#46;</p><p id="par0085" class="elsevierStylePara elsevierViewall">The first term of the right hand of equation &#40;7&#41; is equal to the primary source&#39;s potential <span class="elsevierStyleItalic">U</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">p</span></span> &#40;r&#8594;&#41;&#46; The second term implies the whole halfspace volume&#46; This integral could be separated in volumes for each heterogeneous body&#44; for instance if we define&#58;<elsevierMultimedia ident="eq0050"></elsevierMultimedia></p><p id="par0090" class="elsevierStylePara elsevierViewall">Using the next vector property for each <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span>&#44;<elsevierMultimedia ident="eq0055"></elsevierMultimedia>and assuming that the resistivity of each body within the half-space &#40;<a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>&#41;&#44; is constant&#44; then &#9663;&#8242; <span class="elsevierStyleSup">&#46;</span>E&#8594;<span class="elsevierStyleInf">2</span> &#40;r&#8594;&#41; <span class="elsevierStyleGlyphqbnd"></span> 0&#46; Thus&#44; we can use the divergence theorem for each <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span> and eq&#46; 8 becomes&#58;<elsevierMultimedia ident="eq0060"></elsevierMultimedia></p><p id="par0095" class="elsevierStylePara elsevierViewall">This equation should be applied to the whole surface delimitating each immersed body&#59; in our case&#44; we assume the body as a regular prism&#46; Then&#44; the corresponding integral for the case of two contiguous prismatic bodies &#40;b<span class="elsevierStyleInf">1</span>&#44;b<span class="elsevierStyleInf">2</span>&#41; with a common surface&#44; is&#58;<elsevierMultimedia ident="eq0065"></elsevierMultimedia></p><p id="par0100" class="elsevierStylePara elsevierViewall">Where n&#8594;12 is the unit normal vector of the surfaces &#40;1&#44; 2&#41; between the two bodies&#46;</p><p id="par0105" class="elsevierStylePara elsevierViewall">The boundary conditions allow to define&#58;<elsevierMultimedia ident="eq0070"></elsevierMultimedia>where &#963;<span class="elsevierStyleInf">S</span>&#40;r&#8594;&#8242;&#41; is the density surface charges and ¿<span class="elsevierStyleInf">0</span> is the free-space electrical permittivity&#46;</p><p id="par0110" class="elsevierStylePara elsevierViewall">Taking into account the equations &#40;10 to 13&#41;&#44; the electric potential &#40;eq&#46; 7&#41; is rewritten as&#58;<elsevierMultimedia ident="eq0075"></elsevierMultimedia></p><p id="par0115" class="elsevierStylePara elsevierViewall">Where number 6 denotes the total number of surfaces of one prismatic body and M the number of bodies within the half-space&#44; this eq&#46; constitutes the <span class="elsevierStyleItalic"><span class="elsevierStyleBold">SIM</span>&#46;</span></p><p id="par0120" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#bib0045">Eskola &#40;1992&#41;</a> has obtained an expression similar to equation &#40;14&#41; using different analytical approach&#44; under the same type of hypothesis&#46;</p><p id="par0125" class="elsevierStylePara elsevierViewall">However&#44; a problem to solve is to know &#963; &#40;r&#8594; &#8242;&#41; &#40;the density surface charges&#41; for each surface of the each <span class="elsevierStyleUnderline">p</span>rismatic body&#46; Kaufman &#40;1992&#41; expressed &#963; &#40;r&#8594; &#8242;&#41; for two contiguous surfaces as&#58;<elsevierMultimedia ident="eq0080"></elsevierMultimedia>where<elsevierMultimedia ident="eq0085"></elsevierMultimedia></p><p id="par0130" class="elsevierStylePara elsevierViewall">If we neglect the normal electrical secondary field ld in expression &#40;16&#41;&#44; <span class="elsevierStyleItalic">i&#46;e&#46;</span>E&#8594;n&#8594;<span class="elsevierStyleInf">2&#44;b1 &#40;</span>r&#8594;<span class="elsevierStyleInf">&#41;</span> and&#44; E&#8594;n&#8594;<span class="elsevierStyleInf">2&#44;b2 &#40;</span>r&#8594;<span class="elsevierStyleInf">&#41;</span> then the equation &#40;15&#41; becomes&#58;<elsevierMultimedia ident="eq0090"></elsevierMultimedia></p><p id="par0135" class="elsevierStylePara elsevierViewall">Where&#44; R<span class="elsevierStyleInf">12</span>&#44; is the reflectivity coefficient between surfaces&#46; This expression is the approximation of the induced surface electric charge and it is equivalent to the so-called &#8220;<span class="elsevierStyleItalic">weak scattering problem</span>&#8221;&#46;</p></span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0035">Numerical Approach</span><p id="par0140" class="elsevierStylePara elsevierViewall">Equations &#40;6 and 14&#41; are expressed in arbitrary coordinate systems with a fixed origin&#46; However&#44; to solve the corresponding integrals we redefine the origin of the coordinate system at the middle point of the prismatic body&#59; that is the &#8220;<span class="elsevierStyleItalic">local coordinate system</span>&#8221;&#46; The transformation between both coordinate systems will be defined as follows&#58; Assuming <span class="elsevierStyleItalic"><span class="elsevierStyleBold">P</span></span> an arbitrary point in the space&#44; its position vector in terms of the global coordinates system is r&#8594; and r&#8594; &#8242; is its position vector in terms of the <span class="elsevierStyleItalic">local coordinate system</span>&#46; Consequently&#44; the relationship between the origins for both systems is defined by r&#8594;<span class="elsevierStyleInf">a</span> &#40;see <a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a>&#41;&#44; that is&#58;<elsevierMultimedia ident="eq0095"></elsevierMultimedia>where r&#8594;&#8242; &#61; &#40;x&#8242;&#44; y&#8242;&#44;z&#8242;&#41;&#44; r&#8594;<span class="elsevierStyleInf">a</span> &#61; &#40;x<span class="elsevierStyleInf">a</span>&#44; y<span class="elsevierStyleInf">b</span>&#44;z<span class="elsevierStyleInf">c</span>&#41;&#44; and r&#8594;&#61;&#40;x&#44; y&#44; z&#46;&#41;</p><elsevierMultimedia ident="fig0010"></elsevierMultimedia><p id="par0145" class="elsevierStylePara elsevierViewall">Assuming isolated heterogeneous bodies immersed in a half space&#44; let us introduce the resistivity contrast as <span class="elsevierStyleItalic">&#961;</span> &#61; <span class="elsevierStyleItalic">&#961;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">c</span></span>-<span class="elsevierStyleItalic">&#961;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span>&#44; where <span class="elsevierStyleItalic">&#961;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span> is the resistivity of the half-space and <span class="elsevierStyleItalic">&#961;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">c</span></span> is the resistivity of the immersed body&#46;</p><p id="par0150" class="elsevierStylePara elsevierViewall">Then&#44; by applying equation &#40;6&#41; for a quadrupole array&#44; that is to a vertical electric sounding &#40;VES&#41; &#40;where electrodes are usually named&#58; A&#44; B&#44; N&#44; M&#44; A and B indicate current electrodes and M and N reception electrodes&#41;&#44; the potential UsNM associated with a point source electrode can be expressed as eq&#46; &#40;19&#41;&#46; This equation also assumes the concept of additive potential sources &#40;Orellana&#44; 1972&#41;&#58;<elsevierMultimedia ident="eq0100"></elsevierMultimedia></p><p id="par0155" class="elsevierStylePara elsevierViewall">This equation allows us to compute the secondary electric potential by the volume integral method&#44; <span class="elsevierStyleItalic"><span class="elsevierStyleBold">VIM</span></span>&#46;</p><p id="par0160" class="elsevierStylePara elsevierViewall">In the eq&#46; 14&#44; <span class="elsevierStyleItalic">SIM</span>&#44; the density surface charges expressed in terms of the <span class="elsevierStyleItalic">local coordinates system</span> is&#58;<elsevierMultimedia ident="eq0105"></elsevierMultimedia></p><p id="par0165" class="elsevierStylePara elsevierViewall">By substituting equation &#40;20&#41; in equation &#40;14&#41;&#44; the contribution of each surface of the immersed body&#44; to the secondary potential field for the same quadrupole array&#44; is expressed as&#58;<elsevierMultimedia ident="eq0110"></elsevierMultimedia></p><p id="par0170" class="elsevierStylePara elsevierViewall">Then the apparent resistivity <span class="elsevierStyleItalic">&#961;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">a</span></span> can be expressed as&#58;<elsevierMultimedia ident="eq0115"></elsevierMultimedia></p><p id="par0175" class="elsevierStylePara elsevierViewall">To solve the integrals involved in equations &#40;19 and 21&#41; &#40;<span class="elsevierStyleItalic">VIM</span> and <span class="elsevierStyleItalic">SIM</span>&#44; respectively&#41; we use the Gauss-Legendre Quadrature&#44; by using the subroutines QGAUS and DQDAGI&#44; that are in-cluded in the IMLS Fortran numerical libraries &#40;<a class="elsevierStyleCrossRefs" href="#bib0135">Meissner&#44; 1995&#59; Press <span class="elsevierStyleItalic">et al&#46;</span>&#44; 1992</a>&#41;&#46; DQDAGI subroutine makes use of Gauss-Kronrod approximation with 21 points&#44; and by using an e-algorithm &#40;<a class="elsevierStyleCrossRef" href="#bib0180">Piessens <span class="elsevierStyleItalic">et al&#46;</span>&#44; 1983</a>&#41;&#44; these integrals can be estimated even when the ending interval is a singularity&#46;</p><p id="par0180" class="elsevierStylePara elsevierViewall">The computational program developed in this work computes the apparent resistivity profile for 3D inmersed bodies&#44; by entering the data listed in <a class="elsevierStyleCrossRef" href="#tbl0005">table 1</a>&#46; The output data are the apparent resistivity values in an array that corresponds to a resistivity pseudo-section cutting the half-space in the input direction&#46;</p><elsevierMultimedia ident="tbl0005"></elsevierMultimedia></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0040">Synthetic examples</span><p id="par0185" class="elsevierStylePara elsevierViewall">In order to illustrate the validity of the <span class="elsevierStyleItalic">VIM</span> and <span class="elsevierStyleItalic">SIM</span> developed in this work&#44; we studied some synthetic examples and compared them to results obtained by others authors&#46;</p><p id="par0190" class="elsevierStylePara elsevierViewall">A stratified media&#44; with three layers of different resistivities&#44; constitutes the first example &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>a&#41;&#46; One case considers a middle conductor layer&#58; 100&#44; 10&#44; 100 ohm-m &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Figures 3</a>b&#41;&#59; and the other case considers a middle resistive layer&#58; 10&#44; 100&#44; 10 ohm-m &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>c&#41;&#46; The results of the SIM model for a dipole-dipole array are compared &#40;<a class="elsevierStyleCrossRef" href="#fig0015">Figures 3</a>b and <a class="elsevierStyleCrossRef" href="#fig0015">3</a>c&#41; to those results obtained by applying the algorithm based on the adaptative digital filtering proposed by <a class="elsevierStyleCrossRef" href="#bib0005">Anderson &#40;1979&#41;</a>&#44; which uses Hankel transforms&#46; This comparison shows coincidences in the computed resistivity values at the subsurface assignation points corresponding to an electrodic separation of a &#61; 1m&#44; and until the level <span class="elsevierStyleItalic">n</span> &#61; 14&#59; however&#44; after level <span class="elsevierStyleItalic">n</span> &#61; 15 the results show differences between values &#40;each level n corresponds to 0&#46;5 m&#41;&#44; because the computed induced charge by <span class="elsevierStyleItalic">SIM</span> is a poor approximation&#46; It is important to point out that SIM is one method that needs to model closed bodies and the middle layer was considered as a body of 400 by 400 m and thickness of <span class="elsevierStyleItalic">T</span> &#61; 2&#46;5 m&#44; the depth D &#61; 5 m this assumption involves numerical errors that could explain the enlargement of the differences between both methods at depth &#40;for levels n &#62; 15 and depth &#62; 10&#46;5 m&#41;&#46; But also it is important to point out the assumption of weak scattering concerns the use of Born approximation &#40;Guozhong and Torres-Verd&#237;n&#44; 2006&#41; and this is also a contribution in those discrepancies&#44; as it was signaled by <a class="elsevierStyleCrossRef" href="#bib0255">Zhdanov and Fang &#40;1996&#41;</a>&#44; the Born approximation produces curves of the correct shape but incorrect magnitude&#46; In summary&#44; we can conclude the approximation with SIM is good enough&#46;</p><elsevierMultimedia ident="fig0015"></elsevierMultimedia><p id="par0195" class="elsevierStylePara elsevierViewall">The second example consists of a 3D homogeneous half-space&#44; with <span class="elsevierStyleItalic">&#961;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span> &#61; 100 ohm-m&#44; and one conductor immersed prismatic body&#44; of <span class="elsevierStyleItalic">&#961;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">c</span></span> &#61; 20 ohm-m &#40;<a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a>&#41;&#46; The results of VIM method &#40;<a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>a&#41; shows differences between 3 and 21 ohm-m in the lower values region compared to those computed by <a class="elsevierStyleCrossRef" href="#bib0195">Pridmore &#40;1978&#41;</a>&#59; while the SIM modeling of a dipoledipole array over the prism are compared &#40;<a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>b&#41; to those obtained by <a class="elsevierStyleCrossRef" href="#bib0240">Tsourlos and Ogilvy &#40;1999&#41;</a>&#46; As it is observed&#44; the differences between values are within 1 and 10 ohm-m&#46; In contrast &#40;<a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>d&#41;&#44; and only rise up to 14 ohm-m compared to those obtained by <a class="elsevierStyleCrossRef" href="#bib0240">Tsourlos and Ogilvy &#40;1999&#41;</a>&#44; <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a>c&#46; In spite of the differences depicted between the results of SIM and VIM&#44; the results are good enough since the computed resistivity values do not exceed 15 ohm-m &#40;<a class="elsevierStyleCrossRef" href="#fig0025">Figures 5</a>a and <a class="elsevierStyleCrossRef" href="#fig0025">5</a>b&#41;&#46; That is about 18&#37; of the resistivity contrast between body and half-space&#46;</p><elsevierMultimedia ident="fig0020"></elsevierMultimedia><elsevierMultimedia ident="fig0025"></elsevierMultimedia><p id="par0200" class="elsevierStylePara elsevierViewall">The third example showed in <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>a&#44; is constituted by the synthetic example published by <a class="elsevierStyleCrossRef" href="#bib0165">Perez-Flores <span class="elsevierStyleItalic">et al&#46;</span> &#40;2001&#41;</a> with 4 immersed bodies of constant resistivity&#44; <span class="elsevierStyleItalic">&#961;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">c</span></span> &#61; 20 ohm-m&#46; This model is based on a volume integral scheme &#40;<a class="elsevierStyleCrossRef" href="#bib0165">Perez-Flores <span class="elsevierStyleItalic">et al&#46;</span>&#44; 2001</a>&#41; and it is similar to the hypothesis of the <span class="elsevierStyleItalic">VIM</span> proposed here&#46; The comparison between SIM and model shows similar results &#40;<a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>b&#41;&#46; Also&#44; the VIM shows quite the same data for this particular case &#40;not showed in figure&#41;&#59; however&#44; for general cases&#44; we would expect bigger differences from VIM results&#46; A possible explanation is that the electrode separation is smaller than the dimensions of the bodies&#46;</p><elsevierMultimedia ident="fig0030"></elsevierMultimedia><p id="par0205" class="elsevierStylePara elsevierViewall">The fourth example presented consists of two conductive bodies &#40;<a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a>&#41; of <span class="elsevierStyleItalic">&#961;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">c</span></span> &#61; 20 ohm-m&#44; immersed in a homogeneous half-space of <span class="elsevierStyleItalic">&#961;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">m</span></span> &#61; 100 ohm-m&#46; The bodies have the same dimensions&#44; 10 m thick &#40;T&#44; in the z direction&#41;&#44; 10 m long &#40;L&#44; in the x direction&#41; and 10 m width &#40;W&#44; in the y direction&#41; and both are located at 2&#46;5 m depth &#40;D&#41;&#46; This example is proposed just to show the interaction between bodies by changing the separation between them&#44; with two possibilities&#58; closer and distant &#40;far&#41; bodies&#44; with S equal to 6 m and 40 m respectively&#46; We assume a dipole-dipole array consisting of 31 electrodes&#44; with a 5 m distance between them&#46; <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a> shows the results obtained with <span class="elsevierStyleItalic">SIM</span> and <span class="elsevierStyleItalic">VIM</span> for the case with S &#61; 6 m&#46; The apparent resistivity values with <span class="elsevierStyleItalic">SIM</span> are those expected for the bodies&#46; In contrast&#44; <span class="elsevierStyleItalic">VIM&#39;s</span> resistivity values are bigger than those expected&#46; It is important to point out that we obtain two minimum resistivities in the location corresponding to the bodies&#44; as we expect&#44; those anomalies in resistivities correspond to the bodies&#46; However&#44; it is also observed a third anomaly at the center of the resistivity image that corresponds to a numerical feature&#44; of a lower resistivity value&#46; <a class="elsevierStyleCrossRef" href="#fig0045">Figure 9</a> shows results for the case S &#61;40 m&#44; they are similar to those obtained for isolate bodies &#40;<a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>&#41;&#46; As well as previous case&#44; it is also observed a third anomaly at the center of the resistivity image that corresponds to a numerical feature&#46;</p><elsevierMultimedia ident="fig0035"></elsevierMultimedia><elsevierMultimedia ident="fig0040"></elsevierMultimedia><elsevierMultimedia ident="fig0045"></elsevierMultimedia><p id="par0210" class="elsevierStylePara elsevierViewall">In all the studies cases&#44; we can observe&#44; <span class="elsevierStyleItalic">SIM</span> produces better approach than <span class="elsevierStyleItalic">VIM</span> in computing the electrical potential&#46;</p></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0045">Conclusions</span><p id="par0215" class="elsevierStylePara elsevierViewall">This paper introduces two algorithms for the integral solution of the forward DC geoelectrical problem introduced by <a class="elsevierStyleCrossRef" href="#bib0085">Hvozdara and Kaikkonen &#40;1998&#41;</a> with mixed boundary conditions using Green&#39;s function&#46; The two types of solutions&#58; volume &#40;VIM&#41; and surface integral methods &#40;SIM&#41; make use of the resistivity contrast between immersed bodies and the homogeneous half- space&#46; These methods also use the concepts of&#58; additive potential sources for immersed bodies&#44; and density surface charges&#46; Both algorithms are not so much demanding on computer time and memory because they do not produce to very large systems of linear equations&#46; This made the methods more accessible for personal computers&#44; quotidian prospecting work and also makes it attractive for educational purposes&#46; In particular could be useful to easily validate the field measurements interpretation&#46;</p><p id="par0220" class="elsevierStylePara elsevierViewall">The algorithms developed here can help in the interpretation of the field data obtained from resistivity profile methods&#44; in two and three dimensions&#46; The advantage of using the integral equation technique is that it is performed for each immersed body in the half space&#44; in contrast to the usual procedure in finite-element and finite-difference methods&#46; In order to find the induced charge&#44; we do not need to define a grid on the surface of the body&#44; due to the fact that we use the density surface charges on each surface&#46;</p><p id="par0225" class="elsevierStylePara elsevierViewall">The conducted tests with synthetic data indicated that both algorithms &#40;SIM and VIM&#41; produced reasonably good results compared to already published results for similar problems&#44; obtained by other algorithms&#46; The synthetic examples allow us to conclude that SIM produces a better approximation of the apparent resistivity values than those based on the volume integral &#40;VIM&#41;&#46;</p><p id="par0230" class="elsevierStylePara elsevierViewall">These results are particularly attractive for computation in parallel&#44; because they provide the mode to obtain the forward response for each body in simultaneous way&#46;</p></span></span>"
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          "titulo" => "<span class="elsevierStyleSectionTitle" id="sect0065">Further reading</span>"
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        "resumen" => "<span id="abst0005" class="elsevierStyleSection elsevierViewall"><p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">Se introduce una soluci&#243;n integral para el problema directo de la respuesta geoel&#233;ctrica DC para cuerpos tri-dimensionales en un semi- espacio&#44; mediante las funciones de Green&#46; El primer algoritmo que se presenta se basa en el m&#233;todo integral de volumen &#40;MIV&#41;&#59; aqu&#237;&#44; &#250;nicamente la corriente el&#233;ctrica primaria se utiliza para calcular el potencial el&#233;ctrico&#46; El segundo caso emplea el m&#233;todo integral de superficie &#40;MIS&#41;&#44; en donde se asume que la carga inducida es debida al campo el&#233;ctrico primario&#46; Ambos algoritmos son una combinaci&#243;n de integrales de volumen y de condiciones de frontera&#46; Este art&#237;culo muestra la aplicabilidad de estos algoritmos para generar im&#225;genes de perfiles de resistividad que reproducen algunos arreglos de electrodos para ejemplos sint&#233;ticos tradicionales&#44; y posteriormente estas im&#225;genes se comparan con resultados ya publicados en la literatura&#46; Finalmente&#44; la comparaci&#243;n entre estos resultados muestra que el concepto de carga inducida utilizada en MIS produce una mejor aproximaci&#243;n&#44; que el esquema MIV en el c&#225;lculo del potencial el&#233;ctrico&#46;</p></span>"
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        "resumen" => "<span id="abst0010" class="elsevierStyleSection elsevierViewall"><p id="spar0010" class="elsevierStyleSimplePara elsevierViewall">An integral solution of the forward DC geoelectric response for three-dimensional target-bodies in a half-space&#44; based on Green&#39;s functions&#44; is introduced&#46; The first algorithm presented is based on a volume integral method &#40;<span class="elsevierStyleItalic">VIM</span>&#41;&#59; here&#44; only the primary electrical current is involved to compute the electric potential&#46; The second one employs the surface integral method &#40;<span class="elsevierStyleItalic">SIM</span>&#41;&#44; and it is assumed the induced charge is due to the primary electrical field&#46; Both algorithms are a combination of boundary and volume integrals&#46; This paper shows the applicability of these algorithms to generate resistivity profile images reproducing some electrode arrays for traditional synthetic examples&#44; and then these images were compared with already published results&#46; Finally&#44; the comparison between results shows the concept of induced charge used in <span class="elsevierStyleItalic">SIM</span> produces a better approach than <span class="elsevierStyleItalic">VIM</span> scheme in computing the electrical potential&#46;</p></span>"
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          "en" => "<p id="spar0025" class="elsevierStyleSimplePara elsevierViewall">a&#41; Synthetic example assuming a stratified half-space of three layers with D &#61; 5 m and T &#61; 2&#46;5 m&#46; The log-log plot shows the comparison between the SIM and the Anderson filter &#40;<a class="elsevierStyleCrossRef" href="#bib0005">Anderson&#44; 1979</a>&#41; using two different contrasts of resistivities&#58; b&#41; a middle conductive layer with <span class="elsevierStyleItalic">&#961;<span class="elsevierStyleInf">m</span></span> &#61; 100 <span class="elsevierStyleItalic">&#937;<span class="elsevierStyleInf">m</span></span>&#44; <span class="elsevierStyleItalic">&#961;<span class="elsevierStyleInf">l</span></span>&#61;10 <span class="elsevierStyleItalic">&#937;<span class="elsevierStyleInf">m</span></span> and c&#41; a middle resistive stratum with <span class="elsevierStyleItalic">&#961;<span class="elsevierStyleInf">m</span></span> &#61; 10 <span class="elsevierStyleItalic">&#937;<span class="elsevierStyleInf">m</span></span>&#44; <span class="elsevierStyleItalic">&#961;<span class="elsevierStyleInf">l</span></span>&#61;100 <span class="elsevierStyleItalic">&#937;<span class="elsevierStyleInf">m</span></span>&#46;</p>"
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          "en" => "<p id="spar0035" class="elsevierStyleSimplePara elsevierViewall">Comparison between results of the second example&#44; constituted by the conductor immersed prismatic body shown in <a class="elsevierStyleCrossRef" href="#fig0020">fig&#46; 4</a>&#44; a&#41; the results of the <span class="elsevierStyleItalic">VIM</span> model&#44; b&#41; the results of the <span class="elsevierStyleItalic">SIM</span> model&#44; and those already published c&#41; <a class="elsevierStyleCrossRef" href="#bib0240">Tsourlos and Ogilvy &#40;1999&#41;</a>&#44; and d&#41; <a class="elsevierStyleCrossRef" href="#bib0195">Pridmore &#40;1978&#41;</a>&#46;</p>"
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                  \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head  " align="left" valign="top" scope="col" style="border-bottom: 2px solid black"><span class="elsevierStyleBold">Input data</span>&nbsp;\t\t\t\t\t\t\n
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Article information

ISSN: 00167169
Original language: English

DOI:10.1016/j.gi.2015.04.006

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