Effect of galvanic distortions on the series and parallel magnetotelluric impedances and comparison with other responses

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"autores" => array:5 [ 0 => array:2 [ "nombre" => "Bertha" "apellidos" => "Aguilar Reyes" ] 1 => array:2 [ "nombre" => "Francisco" "apellidos" => "Bautista" ] 2 => array:2 [ "nombre" => "Avto" "apellidos" => "Goguitchaichvili" ] 3 => array:2 [ "nombre" => "Juan Julio" "apellidos" => "Morales Contreras" ] 4 => array:2 [ "nombre" => "Julie" "apellidos" => "Battu" ] ] ] 1 => array:2 [ "autoresLista" => "Patricia Quintana Owen" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Patricia" "apellidos" => "Quintana Owen" ] ] ] 2 => array:2 [ "autoresLista" => "Claire Carvallo" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Claire" "apellidos" => "Carvallo" ] ] ] ] ] "idiomaDefecto" => "en" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716913714673?idApp=UINPBA00004N" "url" => "/00167169/0000005200000002/v2_201505081345/S0016716913714673/v2_201505081345/en/main.assets" ] "en" => array:18 [ "idiomaDefecto" => true "titulo" => "Effect of galvanic distortions on the series and parallel magnetotelluric impedances and comparison with other responses" "tieneTextoCompleto" => true "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "135" "paginaFinal" => "152" ] ] "autores" => array:1 [ 0 => array:4 [ "autoresLista" => "Enrique Gómez-Treviño, Francisco Javier Esparza Hernández, José Manuel Romo Jones" "autores" => array:3 [ 0 => array:2 [ "nombre" => "Enrique" "apellidos" => "Gómez-Treviño" ] 1 => array:4 [ "nombre" => "Francisco Javier" "apellidos" => "Esparza Hernández" "email" => array:1 [ 0 => "fesparz@cicese.mx" ] "referencia" => array:1 [ 0 => array:2 [ "etiqueta" => "*" "identificador" => "cor0005" ] ] ] 2 => array:2 [ "nombre" => "José Manuel" "apellidos" => "Romo Jones" ] ] "afiliaciones" => array:1 [ 0 => array:2 [ "entidad" => "División de Ciencia de la Tierra CICESE Km 107 carretera Tijuana-Ensenada, 22860 Ensenada, Baja California, México." "identificador" => "aff0005" ] ] "correspondencia" => array:1 [ 0 => array:3 [ "identificador" => "cor0005" "etiqueta" => "*" "correspondencia" => "Corresponding author" ] ] ] ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0005" "etiqueta" => "Figure 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 1080 "Ancho" => 1282 "Tamanyo" => 102650 ] ] "descripcion" => array:1 [ "en" => "A and B correspond to the te and tm modes of sounding # 11 of the synthetic data set coprod2s1 made available in mtnet.dias.ie by <a class="elsevierStyleCrossRef" href="#bib0080">Varentsov (1998)</a>. When θ ≠ 0 in <a class="elsevierStyleCrossRef" href="#eq0025">equation (5)</a> the elements of the resulting impedance all are mixtures of the phases of A and B. Calculations were made using c1= 1.97, c2 = −−0.77, c3 = −0.35 and c4 = 0.64. The graphs correspond to the phase of the impedance." ] ] ] "textoCompleto" => "Introduction<a name="p136"></a>The series impedance is defined in terms of the sum of squares of the elements of the impedance tensor, and the parallel impedance in terms of the sum of squares of the elements of the admittance tensor, the inverse of impedance. Both quantities, series and parallel, are scalar measures of impedance and unlike the individual elements of the tensor, are invariant under rotation of the system of coordinates. Other properties include that series is particularly sensitive to underground resistors and that parallel is particularly sensitive to underground conductors (<a class="elsevierStyleCrossRef" href="#bib0065">Romo et al, 2005</a>). Beside these properties very little is known about these invariants. Two-dimensional (2d) inversions of synthetic and field data throw very similar models to those obtained using the traditional te and tm sounding curves. In particular, we have noticed that using te and tm data that are corrected for galvanic effects produces results suspiciously similar to those obtained using uncorrected series and parallel responses (<a class="elsevierStyleCrossRef" href="#bib0005">Antonio-Carpio et al., 2011</a>). This could be due to a mere accident or it could be an indication of unknown properties of the two invariants. To clarify this issue it is necessary to make a rigorous appraisal of the effect of galvanic distortions on the series and parallel invariants.The impedance tensor of a regional 2-d conductivity structure can be severely distorted by local 3-d structures that are inductively small, where inductively small means that their size is smaller than the skin depth. The distortions are due to perturbations of the regional electric field by local 3-d charge distributions, and to perturbations of the regional magnetic field by also local current distributions. In principle, the local 3-d structures can be included in the interpretation process as part of the sought model of the Earth. However, this is seldom done. Rather, the process is divided into two steps: the distortion issue is first settled in some way and then the undistorted data is properly interpreted. It is possible to deal with distortions separately because they are not completely arbitrary but follow some rules. The most important fact is that the impedance tensor distorted by electric effects can be simulated by a real tensor multiplied by the undistorted impedance (Berdichevsky and Dmitriev, 1976a; <a class="elsevierStyleCrossRef" href="#bib0015">Bahr, 1988</a>). This means that the basic physics and its mathematical representation are well understood. However, what matters in practice is to recover the undistorted impedances. To this end, <a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey (1989)</a> proposed a factorization of the real distorting tensor that they call a physical decomposition. They introduced two concepts called twist and shear, which together with strike direction and static factors completely simulate distorted experimental impedances. Twist and shear are physically meaningful effects on electric fields that distort the impedance in distinctly mathematical terms. In fact, they can be distinctly identified in the numerical solution of the nonlinear inverse problem proposed by <a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey (1989)</a>. While it is true that the factorization is not unique, as noted by <a class="elsevierStyleCrossRef" href="#bib0025">Caldwell et al. (2004)</a>, many practical applications attest for the validity of the model to simulate distorted impedances (e.g. <a class="elsevierStyleCrossRef" href="#bib0050">Ledo and Jones, 2001</a>).When computing series and parallel impedances we do not know how the distortions propagate from the elements of the tensor to the invariants. Using Groom-Bailey’s factorization the procedure to find out is straightforward but the result is far from trivial. The paper is written in a tutorial-review manner including results for the determinant, the classical invariant, and for Egger’s (1983) invariant eigenvalues. This last application illustrates that the result for series and parallel modes is not due solely to their invariant character. A final discussion about the phase tensor of <a class="elsevierStyleCrossRef" href="#bib0025">Caldwell et al. (2004)</a> gives perspective to the other results, illustrating that invariance is not a necessary condition to avoid distortions.MethodologyThe algebra of distortionsAssume a two-dimensional (2-d) electrical resistivity distribution subject to plane electromagnetic waves from above. Placing the x-axis of a Cartesian coordinate system along strike, the components of the electric and magnetic fields are related as<elsevierMultimedia ident="eq0005"></elsevierMultimedia>The components Ex and Hy correspond to the TE or E-polarization mode and A is the respective impedance. Accordingly, Ey and Hx defines the TM or H-polarization mode with its impedance B. It has been established that the distortions of the electric field can be modeled using a real tensor C as<elsevierMultimedia ident="eq0010"></elsevierMultimedia>Behind the simplicity of this equation there are extensive studies that led to its development (e.g. <a class="elsevierStyleCrossRef" href="#bib0020">Berdichevsky and Dimitriv, 1976</a>; <a class="elsevierStyleCrossRef" href="#bib0015">Bahr, 1988</a>). Substituting (<a class="elsevierStyleCrossRef" href="#eq0005">1</a>) in (<a class="elsevierStyleCrossRef" href="#eq0010">2</a>) we obtain<a name="p137"></a><elsevierMultimedia ident="eq0015"></elsevierMultimedia>If we now rotate the system of coordinates by an angle θ the components of the distorted electric field in the rotated system are<elsevierMultimedia ident="eq0020"></elsevierMultimedia>The components of the magnetic field need to be rotated too. Solving for the unprimed coordinates and substituting the result along with <a class="elsevierStyleCrossRef" href="#eq0020">equation (4)</a> back into <a class="elsevierStyleCrossRef" href="#eq0015">equation (3)</a>, the resulting equation relates the electric and magnetic fields in the primed coordinates. The corresponding impedance is<elsevierMultimedia ident="eq0025"></elsevierMultimedia>Notice that in <a class="elsevierStyleCrossRef" href="#eq0025">equation (5)</a> the columns of the distorted impedance before rotation have the same phase because the distorting factors are real. After rotation the elements of Zm are all different combinations of A and B and their phases are mixtures of the phases of the two modes. This is illustrated in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a> using the impedances of sounding # 11 of the synthetic data set coprod2s1 made available in MTnet by <a class="elsevierStyleCrossRef" href="#bib0080">Varentsov (1998)</a>. The calculations were made using c1 = 1.97, c2 = −0.77, c3 = 0.35 and c3 = 0.64.<elsevierMultimedia ident="fig0005"></elsevierMultimedia>A useful factorization<a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey (1989)</a> proposed to factorize or decompose the tensor C such that<a name="p138"></a><elsevierMultimedia ident="eq0030"></elsevierMultimedia>where<elsevierMultimedia ident="eq0035"></elsevierMultimedia><elsevierMultimedia ident="eq0040"></elsevierMultimedia><elsevierMultimedia ident="eq0045"></elsevierMultimedia>and<elsevierMultimedia ident="eq0050"></elsevierMultimedia>Notation is changed a little as indicated in <a class="elsevierStyleCrossRef" href="#eq0035">equation (6a)</a>. The tensor A accounts for anisotropy and the scalar g for the site gain. All together they simulate static shifts on impedances A and B. We simply use a and b as the corresponding factors, instead of the more elegant arrangement with g and s. Tensor T twists the electric field so the parameter t is called twist. Tensor S produces a shear-like effect so the parameter e is called shear. R is simply a rotation matrix to account for the coordinate system to be off an angle θ from strike. Twist and shear are expressed in degrees such that t and e are the tangent of the corresponding angles.<a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a> illustrates what each of the operators does to an undistorted unitary electric field. The undistorted field varies from (0,1) to (0, −1) to cover different initial directions to allow a pattern to emerge. Each operator can be thought to be the distortion tensor in <a class="elsevierStyleCrossRef" href="#eq0010">equation (2)</a> as applied to an undistorted electric field. Twist is a simple rotation of the electric field similar to the rotation of coordinates, and it could be thought to be an overlap of the latter. However, this is not so because the rotation of coordinates involves both electric and magnetic fields or, equivalently, the whole impedance tensor and not only the electric field. Another important comment about this factorization or decomposition is that it does not reduce or increase the number of distorting parameters; it just spreads them in a useful way for their recovery from the elements of Zm.<elsevierMultimedia ident="fig0010"></elsevierMultimedia>The absorption of static factorsThe impedance tensor, distorted or undistorted, can be mathematically transformed into an exactly equivalent set of 8 real numbers. We consider in the next sections some of these transformations. <a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey (1989)</a> factorization, although<a name="p139"></a> mathematically expressed, is not a mathematical decomposition in the same sense that these others are. It is more a convenient proposal to allow for the distortions to be estimated from the elements of Zm through an inverse procedure. They do it by neutralizing the two distortion factors a and b due to static, simply by placing A next to Z2 The unknowns now become aA and bB.This reduction in the number of unknowns accomplishes two things. Zm on the left side of <a class="elsevierStyleCrossRef" href="#eq0030">equation (6)</a> provides with 8 real numbers as data, and on the right side we have 9 unknowns: θ, t, e, a, b and four real numbers to make Z2. This means that without this absorption the inversion cannot even be attempted. However, this is not the most important reason because the relative number of data can be increased by using many frequencies and by forcing θ, t and e to be the same for all frequencies. The second reason is more fundamental because it cannot be dealt with except by doing what they did. It is universally acknowledged that static distortions factors cannot be determined from the impedance tensor alone, and that independent information is needed to resolve them (e.g. <a class="elsevierStyleCrossRef" href="#bib0060">Pellerin and Hohmann, 1990</a>; <a class="elsevierStyleCrossRef" href="#bib0055">Ledo et al., 2002</a>). By absorbing the static factors into A and B the problem is solvable. We can mimic the recovery of static shifted curves by explicitly solving for them in <a class="elsevierStyleCrossRef" href="#eq0030">equation (6)</a>. The recovery equation is<elsevierMultimedia ident="eq0055"></elsevierMultimedia>First we compute a distorted Zm using <a class="elsevierStyleCrossRef" href="#eq0030">equation (6)</a>. The undistorted Z2 is the same sounding used in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>. The distortions are as follow: two different static factors a=2 and b=3 in one case and a=3 and b=2 in the other, and the same θ, t and e for both cases. The distorted, undistorted and recovered curves are shown in <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>. Notice that the recovered curves are scaled versions of each other, but the disturbed ones are not. Note also that the recovered curves are scaled versions of the original TE or A curve.<elsevierMultimedia ident="fig0015"></elsevierMultimedia>In general, Zm depends on θ, t, e, a and b, as indicated by <a class="elsevierStyleCrossRef" href="#eq0030">equation (6)</a>. In other words, all of them are needed to ft experimental impedances. The <a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey (1989)</a> approach does not use the static factors to fit Zm, but this does not<a name="p140"></a> mean that Zm is immune to them. In other words, the unprocessed Zm is immune to none of the distorting parameters: twist, shear, static and strike. This is a natural starting point to analyze other responses derived from Zm that are immune to one or more types of distortions.When dealing with invariants, we can dispense of the rotation matrix and its transpose and use the simplified expression<elsevierMultimedia ident="eq0060"></elsevierMultimedia>This equation follows from (<a class="elsevierStyleCrossRef" href="#eq0030">6</a>) by carrying out the multiplications. Again, we assume that a is absorbed in A and b in B.ResultsImmunity to twistThe first invariant to examine is the classical determinant of the impedance tensor (<a class="elsevierStyleCrossRef" href="#bib0020">Berdichevsky and Dimitriv, 1976</a>). In the absence of distortions, the determinant is simply given as<elsevierMultimedia ident="eq0065"></elsevierMultimedia>In general, when the four elements are present in the tensor, the determinant is given as<elsevierMultimedia ident="eq0070"></elsevierMultimedia>To determine the effect of distortions we must substitute <a class="elsevierStyleCrossRef" href="#eq0060">equation (8)</a> into <a class="elsevierStyleCrossRef" href="#eq0070">equation (10)</a>.The result is<elsevierMultimedia ident="eq0075"></elsevierMultimedia>The twist parameter t cancels in the algebra but not the shear parameter e. The effect on the determinant is equivalent to modify the static factors on either A or B, or both. With no static effects the amplitude of the determinant would still be smaller than the actual one if shear distortions are present. On average, we might expect the static factors of the determinant to be biased towards values less than unity.An invariant closely related to the determinant is the parallel impedance (<a class="elsevierStyleCrossRef" href="#bib0065">Romo et al. 2005</a>) which is given as<elsevierMultimedia ident="eq0080"></elsevierMultimedia>Making t=e=0 in <a class="elsevierStyleCrossRef" href="#eq0060">equation (8)</a> and using <a class="elsevierStyleCrossRef" href="#eq0080">equation (12)</a>, the undistorted parallel impedance is<elsevierMultimedia ident="eq0085"></elsevierMultimedia>On the other hand, the general case reduces to<elsevierMultimedia ident="eq0090"></elsevierMultimedia>Again, the distorted impedance is immune to twist, but not to shear and, of course neither to static, which is hidden in A and B. The effect of shear is most effective on the parallel impedance than on the determinant because of the square. This is illustrated in <a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a> for both invariants. The quantity plotted is the traditional apparent resistivity |Z|2/(ωµ0), where ω stands for angular frequency and µ0 is the permeability of free space.<elsevierMultimedia ident="fig0020"></elsevierMultimedia>Immunity to twist and shearIt is interesting how in the previous section twist separates quite neatly from shear, to leave only the latter as an active distorter. In this section we report an invariant that leaves no trace of both twist and shear. This invariant we call the series equivalent (<a class="elsevierStyleCrossRef" href="#bib0065">Romo et al., 2005</a>) corresponds to half the sum of squares of the elements of the impedance tensor (<a class="elsevierStyleCrossRef" href="#bib0075">Szarka and Menvielle, 1997</a>). This is<elsevierMultimedia ident="eq0095"></elsevierMultimedia>When t=e=0, this reduces to<elsevierMultimedia ident="eq0100"></elsevierMultimedia>In general, substituting <a class="elsevierStyleCrossRef" href="#eq0060">equation (8)</a> in (<a class="elsevierStyleCrossRef" href="#eq0095">15</a>) leads to<a name="p141"></a><elsevierMultimedia ident="eq0105"></elsevierMultimedia>It turns out that this reduces to the undistorted series impedance, which means that this type of invariant response is immune not only to strike, but also to both twist and shear. Of course, it is still affected by static because A and B have absorbed a and b, the two static factors.The distorted parallel impedance given by <a class="elsevierStyleCrossRef" href="#eq0090">equation (14)</a> differs from the undistorted impedance by a real factor that depends on shear. This implies that the corresponding phase is immune to shear, and because the complex impedance was already immune to strike and twist, this implies that the phase of the parallel impedance is immune to all three distorting parameters. The phase is given as<elsevierMultimedia ident="eq0110"></elsevierMultimedia>Immunity to strike, twist and shear isolates static effects. It can be appreciated in both <a class="elsevierStyleCrossRef" href="#eq0100">equation (16)</a> and (<a class="elsevierStyleCrossRef" href="#eq0110">18</a>) that static factors will produce phase mixing. This is illustrated in <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a> for the phase of the series impedance. Static effects, absent in the phase of the TE and TM modes, can severely distort the phase of the series impedance. Care must be taken before or while inverting the data. The same applies to the phase of the parallel impedance, for while the factors cancel in the product AB, they do not within the square root sign.<elsevierMultimedia ident="fig0025"></elsevierMultimedia>Immunity to twist, shear and statics<a class="elsevierStyleCrossRef" href="#eq0075">Equation (11)</a> indicates explicitly that the determinant is affected by shear through a multiplicative factor. On the other hand, implicitly it is affected by static though another factor (ab), which comes from the absorption of a and b into A and B. All together they compose a single factor that vanishes when determining the phase of the determinant. That is<elsevierMultimedia ident="eq0115"></elsevierMultimedia>The phase of the determinant is then immune to shear and static, along with twist and strike that shares with the complex determinant. We have arrived at a standstill, for there is nothing that we can vary that alters the phase of the determinant.Invariance is not sufficientThe twist tensor as defined in <a class="elsevierStyleCrossRef" href="#eq0040">equation 6(b)</a> is a rotation matrix that changes the direction of the electric field. This may seem to indicate that all invariants are immune to twist, as are the three invariants analyzed so far. However, this is not the case. Consider the two invariant eigenvalues of Eggers (1983), which are given by the quadratic formula<a name="p142"></a><elsevierMultimedia ident="eq0120"></elsevierMultimedia>where<elsevierMultimedia ident="eq0125"></elsevierMultimedia>To see how the distortions affect these invariants we need to use the general expression for Z given in <a class="elsevierStyleCrossRef" href="#eq0070">equation (10)</a>. Let us first substitute Z in (<a class="elsevierStyleCrossRef" href="#eq0120">20</a>) when t=e=0. The result is<elsevierMultimedia ident="eq0130"></elsevierMultimedia>In this case the eigenvalues reduce to either A or B and in the distorted case to<elsevierMultimedia ident="eq0135"></elsevierMultimedia>This is about the minimum expression one can obtain for the distorted eigenvalues. It can be observed that t and e are present in the expression and that there is no possible way that those in the numerator cancel out with those in the denominator. It is also clear that the effect of twist and shear cannot be absorbed as equivalent static factors on A and B, although they can in several places in the expression. The factor (1+te) multiplies B and (1-te) multiplies A, both outside and within the square root. However, the absorption cannot be completed for the product AB within the square root. Summarizing, the two eigenvalues are not immune to any of the distortions. <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a> illustrates this with the phase of the two eigenvalues using different factors for twist, shear and static. Notice that the curves show phase mixing of the original phases of A and B.<elsevierMultimedia ident="fig0030"></elsevierMultimedia>Invariance is not necessaryWe now come to a very interesting response that can be derived from the impedance tensor. It is called the phase tensor and was proposed by <a class="elsevierStyleCrossRef" href="#bib0025">Cadwell et al. (2004)</a>. This response is immune to twist, shear and static and it is not invariant. The measured impedance is separated into its real and imaginary parts as<a name="p143"></a><elsevierMultimedia ident="eq0140"></elsevierMultimedia>If Zm were a simple complex number, the tangent of its phase would be its imaginary part divided by its real part. In other words, the tangent of its phase would be the inverse of its real part multiplied by its imaginary part. But Zm is not a simple complex number, it is a tensor. However, nothing prevents to multiply the inverse of the real tensor by the imaginary tensor. That is<elsevierMultimedia ident="eq0145"></elsevierMultimedia>The product is a dimensionless tensor whose elements can be interpreted as the tangent of an angle. So far there is nothing impressive about this. However, let us now enquire about the effect of the different distortions on this dimensionless tensor. The undistorted tensor can be written as<elsevierMultimedia ident="eq0150"></elsevierMultimedia>Substituting this into (<a class="elsevierStyleCrossRef" href="#eq0030">6</a>) and doing the operation as in (<a class="elsevierStyleCrossRef" href="#eq0145">25</a>) we have that<elsevierMultimedia ident="eq0155"></elsevierMultimedia>Performing the operations there results that T, S and A cancel with their corresponding inverses. The expression reduces to<elsevierMultimedia ident="eq0160"></elsevierMultimedia>The phase tensor depends only on strike. This is a very important property because it allows obtaining the strike direction independently of the distorting parameters. <a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a> illustrates the dependence of diagonal elements of the phase tensor as strike is varied.<elsevierMultimedia ident="fig0035"></elsevierMultimedia>Series, parallel and determinant impedancesThe properties of the determinant in relation to distortions are well known (e.g. <a class="elsevierStyleCrossRef" href="#bib0010">Arango et al., 2009</a>) and derive from the fact that if Zm = CZ2, then det (Zm) = det (C) det (Z2). Given that C is real the phase of det (Zm) is the same as the phase of det (Z2). Stated in this way this result is more general than that of <a class="elsevierStyleCrossRef" href="#eq0075">equation (11)</a>, which assumes that the undistorted impedance is 2-D. The results for the series and parallel impedances derived above also assume a 2-D undistorted impedance. This means that for real data, seldom fully 2-D, we should expect the results to hold exactly for the determinant and only approximately for series and parallel. This issue is explored in the next three figures. First, we present a reference case<a name="p144"></a> using synthetic 2-D data. What we have done is to distort the original TE and TM data using <a class="elsevierStyleCrossRef" href="#eq0030">equation (6)</a> and then compute the three invariants of Zm. It can be observed in <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8c</a> and <a class="elsevierStyleCrossRef" href="#fig0040">8d</a> that both apparent resistivity and phase curves of the series impedance are immune to the distortions. The comparison is made for three shears, 0, 30 and 40 degrees. Strike direction and twist are not recorded because these and the other results presented below are exactly the same regardless of the value of these parameters. According to theory the only distortion left is due to shear and this is what <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a> shows. The determinant and the parallel apparent resistivities, as shown in <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8a</a> and <a class="elsevierStyleCrossRef" href="#fig0040">8c</a>, suffer a downward shift as predicted by <a class="elsevierStyleCrossRef" href="#eq0075">equations (11)</a> and (<a class="elsevierStyleCrossRef" href="#eq0090">14</a>), respectively. The shift being more pronounced the larger the shear, and the shift of the parallel resistivity being larger than that of the determinant, as it should be according to theory. It can also be observed in <a class="elsevierStyleCrossRef" href="#fig0040">Figures 8b</a> and <a class="elsevierStyleCrossRef" href="#fig0040">8c</a> that the corresponding phases are not affected by shear.<elsevierMultimedia ident="fig0040"></elsevierMultimedia>The results using field data are presented in <a class="elsevierStyleCrossRef" href="#fig0045">Figures 9</a> and <a class="elsevierStyleCrossRef" href="#fig0050">10</a>. We chose soundings from the coprod2 and bc87 data sets (<a class="elsevierStyleCrossRef" href="#bib0040">Jones, 1993a</a>, <a class="elsevierStyleCrossRef" href="#bib0045">1993b</a>). It can be observed that in both cases the corresponding curves behave very much like those using synthetic data. We used <a class="elsevierStyleCrossRef" href="#eq0030">equation (6)</a> to further distort both soundings with different strike directions, twists and shears and then computed the three invariants. It can be observed that the phases of the three invariants are immune to the changes of the distorting parameters, implying that they are free of distortions except for the implicit static factors. Notice that the phases of the determinant are exactly the same for the three shears (<a class="elsevierStyleCrossRef" href="#fig0045">Figures 9b</a> and <a class="elsevierStyleCrossRef" href="#fig0050">10c</a>) and that those of the series and parallel depart a little from each other (<a class="elsevierStyleCrossRef" href="#fig0045">Figures 9d</a> and <a class="elsevierStyleCrossRef" href="#fig0050">10d</a>). As mentioned before, this is because the determinant makes no assumption about dimensionality while the other two assume a 2-d undistorted impedance. Overall, the phases for the three shears are reasonably close considering that a shear of 40 degrees is close to the maximum of 45 degrees.<elsevierMultimedia ident="fig0045"></elsevierMultimedia><elsevierMultimedia ident="fig0050"></elsevierMultimedia>The example drawn from the bc87 data set illustrates an important point about what is removed and what remains to be done to completely clean the data. Notice that at short periods the series and parallel apparent resistivities (<a class="elsevierStyleCrossRef" href="#fig0050">Figure 10c</a>) are both horizontal for the null shear, and that the latter is depressed in relation to the former. This suggests that the parallel apparent resistivity curve has an intrinsic shear distortion that is depressing it from the undistorted series resistivity. This would lead us to shift the curve<a name="p145"></a><a name="p146"></a><a name="p147"></a><a name="p148"></a><a name="p149"></a><a name="p150"></a><a name="p151"></a> upwards and leave the phase unchanged to undo the effect of shear. However, the fact is that we do not know whether this shift is due to shear or static. The parallel resistivity curve could also be lifted by playing with the implicit static factors, in which case the phase curve would have to be modified according to <a class="elsevierStyleCrossRef" href="#eq0110">equation (18)</a>. Deciding which of the two is the proper way to proceed is outside the scope of this first communication. The present results are encouraging to continue the process of advancing our knowledge of the series and parallel invariants, whose properties are just beginning to be understood.ConclusionGalvanic distortions of magnetotelluric data can be neutralized by some invariants of the impedance tensor. Except for strike direction, invariants are devised without any thought about distortions. However, as it turns out for the case of 2-D data, the 3-D distorting effects can gradually be neutralized by an also gradual averaging process. Series and parallel impedances are averages of the tensor elements and, in turn, the determinant is an average of the series and parallel impedances. This gradual immunity is summarized in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>. The new results include: a) The amplitude of the series impedance is affected only by statics, being immune to twist, shear and strike, and so are the phases of series and parallel; b) the amplitude of the parallel impedance is immune to twist and strike but not to shear and statics. These results place the series and parallel impedances midway between invariants that are not immune to any of the distortions, like Egger’s eigenvalues, and the phase of the determinant which is immune to all of them.<elsevierMultimedia ident="tbl0005"></elsevierMultimedia>" "textoCompletoSecciones" => array:1 [ "secciones" => array:10 [ 0 => array:3 [ "identificador" => "xres497966" "titulo" => "Resumen" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0005" ] ] ] 1 => array:2 [ "identificador" => "xpalclavsec519512" "titulo" => "Palabras clave" ] 2 => array:3 [ "identificador" => "xres497965" "titulo" => "Abstract" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0010" ] ] ] 3 => array:2 [ "identificador" => "xpalclavsec519513" "titulo" => "Key words" ] 4 => array:2 [ "identificador" => "sec0005" "titulo" => "Introduction" ] 5 => array:3 [ "identificador" => "sec0010" "titulo" => "Methodology" "secciones" => array:3 [ 0 => array:2 [ "identificador" => "sec0015" "titulo" => "The algebra of distortions" ] 1 => array:2 [ "identificador" => "sec0020" "titulo" => "A useful factorization" ] 2 => array:2 [ "identificador" => "sec0025" "titulo" => "The absorption of static factors" ] ] ] 6 => array:3 [ "identificador" => "sec0030" "titulo" => "Results" "secciones" => array:6 [ 0 => array:2 [ "identificador" => "sec0035" "titulo" => "Immunity to twist" ] 1 => array:2 [ "identificador" => "sec0040" "titulo" => "Immunity to twist and shear" ] 2 => array:2 [ "identificador" => "sec0045" "titulo" => "Immunity to twist, shear and statics" ] 3 => array:2 [ "identificador" => "sec0050" "titulo" => "Invariance is not sufficient" ] 4 => array:2 [ "identificador" => "sec0055" "titulo" => "Invariance is not necessary" ] 5 => array:2 [ "identificador" => "sec0060" "titulo" => "Series, parallel and determinant impedances" ] ] ] 7 => array:2 [ "identificador" => "sec0065" "titulo" => "Conclusion" ] 8 => array:2 [ "identificador" => "xack161003" "titulo" => "Acknoledgments" ] 9 => array:1 [ "titulo" => "Bibliography" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "fechaRecibido" => "2012-02-09" "fechaAceptado" => "2013-01-17" "PalabrasClave" => array:2 [ "es" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Palabras clave" "identificador" => "xpalclavsec519512" "palabras" => array:3 [ 0 => "magnetotelurico" 1 => "distorsiones galvanicas" 2 => "invariantes" ] ] ] "en" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Key words" "identificador" => "xpalclavsec519513" "palabras" => array:3 [ 0 => "Magnetotelluric" 1 => "galvanic distortions" 2 => "invariants" ] ] ] ] "tieneResumen" => true "resumen" => array:2 [ "es" => array:2 [ "titulo" => "Resumen" "resumen" => "Las impedancias serie y paralelo del tensor magnetotelúrico se evalúan en relación con su relativa inmunidad a las distorsiones galvano-eléctricas. Las respuestas distorsionadas se modelan utilizando la descomposición del tensor en términos de giro, cizalla estática y rumbo de Groom y Bailey. Estos cuatro parámetros, junto con las impedancias sin distorsión, normalmente se consideran incógnitas y se obtienen de los datos mediante la solución de un problema inverso. En el presente trabajo utilizamos la descomposición como un modelo directo para simular sondeos distorsionados. Partiendo de respuestas 2-D sin distorsiones, el tensor se distorsiona suponiendo valores arbitrarios de giro, cizalla, estática y rumbo. Por definición las impedancias serie y paralelo son inmunes al rumbo porque son invariantes ante rotación. Adicionalmente, la impedancia serie es inmune a giros y a cizalla, y la impedancia paralelo sólo a giros. La impedancia paralelo depende de cizalla en la forma de un factor que desplaza hacia abajo las curvas de amplitud. Por otro lado, el efecto de la estática en ambas impedancias es más complicado que en el tensor mismo porque no se puede corregir con un simple desplazamiento de las curvas. En términos generales, hay un balance positivo por parte de las impedancias serie y paralelo sobre las respuestas te y tm porque los invariantes filtran varias distorsiones. Se muestra que la condición de invariante no es suficiente para tener inmunidad a cualquiera de las distorsiones. Se utiliza para esto los valores característicos de Eggers, los cuales son inmunes sólo al rumbo, como todos los invariantes. Se muestra además que la invariancia tampoco es una condición necesaria para ser inmune a las distorsiones, según lo atestigua el tensor de impedancia, el cual depende del rumbo pero está libre de las demás distorsiones. Los desarrollos se ilustran utilizando sondeos de los conjuntos de datos coprod2s1, coprod2 y bc87." ] "en" => array:2 [ "titulo" => "Abstract" "resumen" => "The series and parallel impedances of the magnetotelluric tensor are appraised in relation to their relative immunity to galvanic electric distortions. The distorted responses are modeled using the Groom-Bailey decomposition of the tensor in terms of twist, shear, statics and strike direction. These four parameters and the undistorted responses are normally considered as unknowns, and are obtained from field data through the solution of an inverse problem. In the present work we use the decomposition as a forward model to simulate distorted sounding curves. Starting with undistorted 2-d te and tm responses, the tensor is distorted by assuming arbitrary values of twist, shear, static and strike direction. By default, both series and parallel responses are immune to the strike direction because they are invariants under rotation. In addition, series responses are immune to twist and shear and parallel responses only to twist. The dependence of the latter on shear is in the form of a real factor that shifts downwards the amplitude curves. On the other hand, the effect of statics on both series and parallel responses is more complicated than that on the impedance tensor because it cannot be accounted for by a simple shift of the curves. On the whole, there is a positive balance on the part of the series and parallel impedances over the te and tm responses because some of the distortions are filtered out by the invariants. It is shown that invariance is not sufficient to be immune to any of the distortions. The example chosen is Eggers’ eigenvalues, which are immune only to the by-the-fault strike direction. Invariance is not necessary either, as evidenced by the phase tensor, whose elements depend on strike but are immune to all distortions. The derivations are illustrated using soundings from the synthetic coprod2s1 and field-recorded coprod2 and bc87 data sets." ] ] "multimedia" => array:43 [ 0 => array:7 [ "identificador" => "fig0005" "etiqueta" => "Figure 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 1080 "Ancho" => 1282 "Tamanyo" => 102650 ] ] "descripcion" => array:1 [ "en" => "A and B correspond to the te and tm modes of sounding # 11 of the synthetic data set coprod2s1 made available in mtnet.dias.ie by <a class="elsevierStyleCrossRef" href="#bib0080">Varentsov (1998)</a>. When θ ≠ 0 in <a class="elsevierStyleCrossRef" href="#eq0025">equation (5)</a> the elements of the resulting impedance all are mixtures of the phases of A and B. Calculations were made using c1= 1.97, c2 = −−0.77, c3 = −0.35 and c4 = 0.64. The graphs correspond to the phase of the impedance." ] ] 1 => array:7 [ "identificador" => "fig0010" "etiqueta" => "Figure 2" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr2.jpeg" "Alto" => 1314 "Ancho" => 1927 "Tamanyo" => 139523 ] ] "descripcion" => array:1 [ "en" => "The effect of the individual distortion tensor T, S and A are illustrated in a), b) and c), respectively. The figures illustrate what each of the operators does to an undistorted unitary electric field shown in a). The undistorted field varies from (0,1) to (0, −1) to cover different initial directions to allow a pattern to emerge. Each operator can be thought to be the distortion tensor in <a class="elsevierStyleCrossRef" href="#eq0010">equation (2)</a> as applied to a unitary and undistorted electric field. The figure is inspired in a similar one in <a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey (1989)</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" => 1196 "Ancho" => 1722 "Tamanyo" => 147316 ] ] "descripcion" => array:1 [ "en" => "The recovery of undistorted TE impedances using <a class="elsevierStyleCrossRef" href="#eq0055">equation (7)</a> is correct save for the static factors. The distorted Zm is computed using <a class="elsevierStyleCrossRef" href="#eq0030">equation (6)</a> and the indicated distorting parameters of twist, shear and strike. The undistorted Z2 is the same sounding used in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>. Notice that the recovered curves are scaled versions of each other, but the disturbed ones are not. Note also that the recovered curves are scaled versions of the original TE or A curve. In this instance it is pretended that we know the distorting parameters of twist, shear and strike in <a class="elsevierStyleCrossRef" href="#eq0055">equation (7)</a>." ] ] 3 => array:7 [ "identificador" => "fig0020" "etiqueta" => "Figure 4" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr4.jpeg" "Alto" => 994 "Ancho" => 1420 "Tamanyo" => 105958 ] ] "descripcion" => array:1 [ "en" => "Aside from statics, the amplitudes of the determinant and of the parallel impedance are distorted only by shear. This is illustrated in this figure where it can be observed that according to theory the effect is more effective on the parallel impedance." ] ] 4 => array:7 [ "identificador" => "fig0025" "etiqueta" => "Figure 5" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr5.jpeg" "Alto" => 1111 "Ancho" => 1567 "Tamanyo" => 103164 ] ] "descripcion" => array:1 [ "en" => "Immunity to strike, twist and shear isolates static effects as in the case of <a class="elsevierStyleCrossRef" href="#eq0100">equation (16)</a> for the series impedance. Unfortunately, if A and B are affected by statics (a and b different from unity) this produces a static-dependent phase mixing that is not present in the original impedances A and B. This is illustrated in this figure. Static effects, absent in the phase of the TE and TM modes, can severely distort the phase of the series impedance. The same applies to the phase of the parallel impedance, for while the factors cancel in the product AB, they do not within the square root sign in <a class="elsevierStyleCrossRef" href="#eq0110">equation (18)</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" => 1080 "Ancho" => 1572 "Tamanyo" => 110123 ] ] "descripcion" => array:1 [ "en" => "Eggers’ eigenvalues, the invariants defined by <a class="elsevierStyleCrossRef" href="#eq0135">equation (23)</a> are not immune to any of the distortion parameters of twist, shear and statics. This is illustrated here with the phase of the two eigenvalues. This case demonstrates that invariance under rotation is not a sufficient condition for immunity to any of the other distortions." ] ] 6 => array:7 [ "identificador" => "fig0035" "etiqueta" => "Figure 7" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr7.jpeg" "Alto" => 1086 "Ancho" => 1575 "Tamanyo" => 127806 ] ] "descripcion" => array:1 [ "en" => "Contrary to the invariants, the phase tensor depends on strike and at the same time it is immune to all other distortions. This is a very important property because it allows obtaining the strike direction independently of the distorting parameters. This figure illustrates the dependence of the diagonal elements of the phase tensor as strike is varied. The phase tensor is an example of a response that is not invariant but is immune to distortions. This result, together with that of <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>, imply that invariance is neither sufficient not necessary for immunity to distortions, and that each response must be evaluated on its own." ] ] 7 => array:7 [ "identificador" => "fig0040" "etiqueta" => "Figure 8" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:2 [ 0 => array:1 [ "imagen" => "gr8a.jpeg" ] 1 => array:1 [ "imagen" => "gr8b.jpeg" ] ] "descripcion" => array:1 [ "en" => "Synthetic te and tm data are distorted using <a class="elsevierStyleCrossRef" href="#eq0030">equation (6)</a> and then invariants of Zm are computed. It can be observed in b) and d) that the phases of the determinant (ψd), series (ψs), and parallel (ψp) impedances are immune to distortions, in agreement to theory. The corresponding amplitudes shown in a) and c), also in agreement to theory, are scaled versions of the originals (shear = 0 degrees). This includes the series impedance which is immune to this scaling due to shear. Strike direction and twist are not recorded because the results are exactly the same regardless of the values of these parameters." ] ] 8 => array:7 [ "identificador" => "fig0045" "etiqueta" => "Figure 9" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:2 [ 0 => array:1 [ "imagen" => "gr9a.jpeg" ] 1 => array:1 [ "imagen" => "gr9b.jpeg" ] ] "descripcion" => array:1 [ "en" => "Results for the determinant, series and parallel impedances using field data from the COPROD2 data set, show that the corresponding curves behave very much like those of <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a> using synthetic data. We used <a class="elsevierStyleCrossRef" href="#eq0030">equation (6)</a> to further distort both soundings with different strike directions, twists and shears and then computed the three invariants. It can be observed that the phases of the three invariants shown in b) and d) are immune to the changes of the distorting parameters, implying that they are free of distortions except for the implicit static factors. Notice that the phases of the determinant are exactly the same for the three shears and that those of the series and parallel depart a little from each other. As mentioned in the main text, this is because the determinant makes no assumption about dimensionality while series and parallel assume a 2-D undistorted impedance. The amplitudes shown in a) and c) are scaled versions of the originals (shear = 0 degrees) in agreement to theory. This includes the series invariant which is immune to this scaling due to shear. Overall, the phases for the three shears are reasonably close considering that a shear of 40 degrees is close to the maximum of 45 degrees." ] ] 9 => array:7 [ "identificador" => "fig0050" "etiqueta" => "Figure 10" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:2 [ 0 => array:1 [ "imagen" => "gr10a.jpeg" ] 1 => array:1 [ "imagen" => "gr10b.jpeg" ] ] "descripcion" => array:1 [ "en" => "Results for the determinant, series and parallel impedances using field data from the BC87 data set, show that the corresponding curves behave very much like those of <a class="elsevierStyleCrossRef" href="#fig0040">Figures 8</a> and <a class="elsevierStyleCrossRef" href="#fig0045">9</a>. It can be observed in c) that there is already a significant downwards shift of the parallel amplitude with respect to series that is not related to the added shear scaling. As discussed in the main text, this could be due to a preexisting shear distortion or to static effects. The rest of the curves in a), b) and d) follow the same patterns as in <a class="elsevierStyleCrossRef" href="#fig0040">Figures 8</a> and <a class="elsevierStyleCrossRef" href="#fig0045">9</a>. Notice that, as in the other tests, the series amplitude is immune to the scaling introduced by the shear distortion as predicted by <a class="elsevierStyleCrossRef" href="#eq0105">equation (17)</a>." ] ] 10 => 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 border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="" valign="top" scope="col" style="border-bottom: 2px solid black"> \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">Strike \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">Twist \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">Shear \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">Statics \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="left" valign="middle">Eggers’s eigenvalues \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">no \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">no \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">no \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="middle">Parallel amplitude \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">no \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">no \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="middle">Determinant amplitude \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">no \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">no \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="middle">Series amplitude \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">no \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="middle">Series phase \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">no \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="middle">Parallel phase \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">no \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="middle">Phase tensor \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">no \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \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="middle">Determinant phase \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="middle">yes \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab795037.png" ] ] ] ] "descripcion" => array:1 [ "en" => "The table summarizes the immunity of the different responses to the four distorting parameters, strike, twist, shear and statics." ] ] 11 => array:6 [ "identificador" => "eq0005" "etiqueta" => "(1)" "tipo" => "MULTIMEDIAFORMULA" 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"Fichero" => "si29.jpeg" "Tamanyo" => 2626 "Alto" => 49 "Ancho" => 205 ] ] 40 => array:6 [ "identificador" => "eq0150" "etiqueta" => "(26)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Z2=X2+iY2." "Fichero" => "si30.jpeg" "Tamanyo" => 980 "Alto" => 13 "Ancho" => 104 ] ] 41 => array:6 [ "identificador" => "eq0155" "etiqueta" => "(27)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Φ=(RTSAX2RT)−1(RTSAY2RT)." "Fichero" => "si31.jpeg" "Tamanyo" => 2624 "Alto" => 21 "Ancho" => 234 ] ] 42 => array:6 [ "identificador" => "eq0160" "etiqueta" => "(28)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Φ=RX2−1Y2RT." 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Enrique Gómez-Treviño, Francisco Javier Esparza Hernández

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fesparz@cicese.mx

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, José Manuel Romo Jones

División de Ciencia de la Tierra CICESE Km 107 carretera Tijuana-Ensenada, 22860 Ensenada, Baja California, México.

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          "en" => "<p id="spar0015" class="elsevierStyleSimplePara elsevierViewall"><span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span> correspond to the <span class="elsevierStyleSmallCaps">te</span> and <span class="elsevierStyleSmallCaps">tm</span> modes of sounding &#35; 11 of the synthetic data set <span class="elsevierStyleSmallCaps">coprod2s1</span> made available in <span class="elsevierStyleSmallCaps">mtn</span>et&#46;dias&#46;ie by <a class="elsevierStyleCrossRef" href="#bib0080">Varentsov &#40;1998&#41;</a>&#46; When <span class="elsevierStyleItalic">&#952;</span> &#8800; 0 in <a class="elsevierStyleCrossRef" href="#eq0025">equation &#40;5&#41;</a> the elements of the resulting impedance all are mixtures of the phases of <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span>&#46; Calculations were made using <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">1</span>&#61; 1&#46;97&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">2</span> &#61; &#8722;&#8722;0&#46;77&#44; c<span class="elsevierStyleInf">3</span> &#61; &#8722;0&#46;35 and <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">4</span> &#61; 0&#46;64&#46; The graphs correspond to the phase of the impedance&#46;</p>"
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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"><a name="p136"></a>The series impedance is defined in terms of the sum of squares of the elements of the impedance tensor&#44; and the parallel impedance in terms of the sum of squares of the elements of the admittance tensor&#44; the inverse of impedance&#46; Both quantities&#44; series and parallel&#44; are scalar measures of impedance and unlike the individual elements of the tensor&#44; are invariant under rotation of the system of coordinates&#46; Other properties include that series is particularly sensitive to underground resistors and that parallel is particularly sensitive to underground conductors &#40;<a class="elsevierStyleCrossRef" href="#bib0065">Romo <span class="elsevierStyleItalic">et al</span>&#44; 2005</a>&#41;&#46; Beside these properties very little is known about these invariants&#46; Two-dimensional &#40;<span class="elsevierStyleSmallCaps">2d</span>&#41; inversions of synthetic and field data throw very similar models to those obtained using the traditional <span class="elsevierStyleSmallCaps">te</span> and <span class="elsevierStyleSmallCaps">tm</span> sounding curves&#46; In particular&#44; we have noticed that using <span class="elsevierStyleSmallCaps">te</span> and <span class="elsevierStyleSmallCaps">tm</span> data that are corrected for galvanic effects produces results suspiciously similar to those obtained using uncorrected series and parallel responses &#40;<a class="elsevierStyleCrossRef" href="#bib0005">Antonio-Carpio <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2011</a>&#41;&#46; This could be due to a mere accident or it could be an indication of unknown properties of the two invariants&#46; To clarify this issue it is necessary to make a rigorous appraisal of the effect of galvanic distortions on the series and parallel invariants&#46;</p><p id="par0010" class="elsevierStylePara elsevierViewall">The impedance tensor of a regional 2-<span class="elsevierStyleSmallCaps">d</span> conductivity structure can be severely distorted by local 3-<span class="elsevierStyleSmallCaps">d</span> structures that are inductively small&#44; where inductively small means that their size is smaller than the skin depth&#46; The distortions are due to perturbations of the regional electric field by local 3-<span class="elsevierStyleSmallCaps">d</span> charge distributions&#44; and to perturbations of the regional magnetic field by also local current distributions&#46; In principle&#44; the local 3-<span class="elsevierStyleSmallCaps">d</span> structures can be included in the interpretation process as part of the sought model of the Earth&#46; However&#44; this is seldom done&#46; Rather&#44; the process is divided into two steps&#58; the distortion issue is first settled in some way and then the undistorted data is properly interpreted&#46; It is possible to deal with distortions separately because they are not completely arbitrary but follow some rules&#46; The most important fact is that the impedance tensor distorted by electric effects can be simulated by a real tensor multiplied by the undistorted impedance &#40;Berdichevsky and Dmitriev&#44; 1976a&#59; <a class="elsevierStyleCrossRef" href="#bib0015">Bahr&#44; 1988</a>&#41;&#46; This means that the basic physics and its mathematical representation are well understood&#46; However&#44; what matters in practice is to recover the undistorted impedances&#46; To this end&#44; <a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey &#40;1989&#41;</a> proposed a factorization of the real distorting tensor that they call a physical decomposition&#46; They introduced two concepts called twist and shear&#44; which together with strike direction and static factors completely simulate distorted experimental impedances&#46; Twist and shear are physically meaningful effects on electric fields that distort the impedance in distinctly mathematical terms&#46; In fact&#44; they can be distinctly identified in the numerical solution of the nonlinear inverse problem proposed by <a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey &#40;1989&#41;</a>&#46; While it is true that the factorization is not unique&#44; as noted by <a class="elsevierStyleCrossRef" href="#bib0025">Caldwell <span class="elsevierStyleItalic">et al</span>&#46; &#40;2004&#41;</a>&#44; many practical applications attest for the validity of the model to simulate distorted impedances &#40;<span class="elsevierStyleItalic">e&#46;g</span>&#46; <a class="elsevierStyleCrossRef" href="#bib0050">Ledo and Jones&#44; 2001</a>&#41;&#46;</p><p id="par0015" class="elsevierStylePara elsevierViewall">When computing series and parallel impedances we do not know how the distortions propagate from the elements of the tensor to the invariants&#46; Using Groom-Bailey&#8217;s factorization the procedure to find out is straightforward but the result is far from trivial&#46; The paper is written in a tutorial-review manner including results for the determinant&#44; the classical invariant&#44; and for Egger&#8217;s &#40;1983&#41; invariant eigenvalues&#46; This last application illustrates that the result for series and parallel modes is not due solely to their invariant character&#46; A final discussion about the phase tensor of <a class="elsevierStyleCrossRef" href="#bib0025">Caldwell <span class="elsevierStyleItalic">et al</span>&#46; &#40;2004&#41;</a> gives perspective to the other results&#44; illustrating that invariance is not a necessary condition to avoid distortions&#46;</p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0030">Methodology</span><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0035">The algebra of distortions</span><p id="par0020" class="elsevierStylePara elsevierViewall">Assume a two-dimensional &#40;2-<span class="elsevierStyleSmallCaps">d</span>&#41; electrical resistivity distribution subject to plane electromagnetic waves from above&#46; Placing the <span class="elsevierStyleItalic">x</span>-axis of a Cartesian coordinate system along strike&#44; the components of the electric and magnetic fields are related as<elsevierMultimedia ident="eq0005"></elsevierMultimedia></p><p id="par0025" class="elsevierStylePara elsevierViewall">The components <span class="elsevierStyleItalic">E<span class="elsevierStyleInf">x</span></span> and <span class="elsevierStyleItalic">H<span class="elsevierStyleInf">y</span></span> correspond to the <span class="elsevierStyleItalic">TE or E</span>-polarization mode and <span class="elsevierStyleItalic">A</span> is the respective impedance&#46; Accordingly&#44; <span class="elsevierStyleItalic">E<span class="elsevierStyleInf">y</span></span> and <span class="elsevierStyleItalic">H<span class="elsevierStyleInf">x</span></span> defines the <span class="elsevierStyleItalic">TM</span> or <span class="elsevierStyleItalic">H</span>-polarization mode with its impedance <span class="elsevierStyleItalic">B</span>&#46; It has been established that the distortions of the electric field can be modeled using a real tensor <span class="elsevierStyleBold"><span class="elsevierStyleItalic">C</span></span> as<elsevierMultimedia ident="eq0010"></elsevierMultimedia></p><p id="par0030" class="elsevierStylePara elsevierViewall">Behind the simplicity of this equation there are extensive studies that led to its development &#40;e&#46;g&#46; <a class="elsevierStyleCrossRef" href="#bib0020">Berdichevsky and Dimitriv&#44; 1976</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0015">Bahr&#44; 1988</a>&#41;&#46; Substituting &#40;<a class="elsevierStyleCrossRef" href="#eq0005">1</a>&#41; in &#40;<a class="elsevierStyleCrossRef" href="#eq0010">2</a>&#41; we obtain<a name="p137"></a><elsevierMultimedia ident="eq0015"></elsevierMultimedia></p><p id="par0035" class="elsevierStylePara elsevierViewall">If we now rotate the system of coordinates by an angle <span class="elsevierStyleItalic">&#952;</span> the components of the distorted electric field in the rotated system are<elsevierMultimedia ident="eq0020"></elsevierMultimedia></p><p id="par0040" class="elsevierStylePara elsevierViewall">The components of the magnetic field need to be rotated too&#46; Solving for the unprimed coordinates and substituting the result along with <a class="elsevierStyleCrossRef" href="#eq0020">equation &#40;4&#41;</a> back into <a class="elsevierStyleCrossRef" href="#eq0015">equation &#40;3&#41;</a>&#44; the resulting equation relates the electric and magnetic fields in the primed coordinates&#46; The corresponding impedance is<elsevierMultimedia ident="eq0025"></elsevierMultimedia></p><p id="par0045" class="elsevierStylePara elsevierViewall">Notice that in <a class="elsevierStyleCrossRef" href="#eq0025">equation &#40;5&#41;</a> the columns of the distorted impedance before rotation have the same phase because the distorting factors are real&#46; After rotation the elements of <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> are all different combinations of <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span> and their phases are mixtures of the phases of the two modes&#46; This is illustrated in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a> using the impedances of sounding &#35; 11 of the synthetic data set <span class="elsevierStyleSmallCaps">coprod2s1</span> made available in MTnet by <a class="elsevierStyleCrossRef" href="#bib0080">Varentsov &#40;1998&#41;</a>&#46; The calculations were made using <span class="elsevierStyleItalic">c<span class="elsevierStyleInf">1</span></span> &#61; 1&#46;97&#44; <span class="elsevierStyleItalic">c<span class="elsevierStyleInf">2</span></span> &#61; &#8722;0&#46;77&#44; <span class="elsevierStyleItalic">c<span class="elsevierStyleInf">3</span></span> &#61; 0&#46;35 and <span class="elsevierStyleItalic">c<span class="elsevierStyleInf">3</span> &#61;</span> 0&#46;64&#46;</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0040">A useful factorization</span><p id="par0050" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey &#40;1989&#41;</a> proposed to factorize or decompose the tensor <span class="elsevierStyleBold"><span class="elsevierStyleItalic">C</span></span> such that<a name="p138"></a><elsevierMultimedia ident="eq0030"></elsevierMultimedia></p><p id="par0055" class="elsevierStylePara elsevierViewall">where<elsevierMultimedia ident="eq0035"></elsevierMultimedia><elsevierMultimedia ident="eq0040"></elsevierMultimedia><elsevierMultimedia ident="eq0045"></elsevierMultimedia></p><p id="par0060" class="elsevierStylePara elsevierViewall">and<elsevierMultimedia ident="eq0050"></elsevierMultimedia></p><p id="par0065" class="elsevierStylePara elsevierViewall">Notation is changed a little as indicated in <a class="elsevierStyleCrossRef" href="#eq0035">equation &#40;6a&#41;</a>&#46; The tensor <span class="elsevierStyleBold"><span class="elsevierStyleItalic">A</span></span> accounts for anisotropy and the scalar <span class="elsevierStyleItalic">g</span> for the site gain&#46; All together they simulate static shifts on impedances <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span>&#46; We simply use <span class="elsevierStyleItalic">a</span> and <span class="elsevierStyleItalic">b</span> as the corresponding factors&#44; instead of the more elegant arrangement with <span class="elsevierStyleItalic">g</span> and <span class="elsevierStyleItalic">s</span>&#46; Tensor <span class="elsevierStyleBold"><span class="elsevierStyleItalic">T</span></span> twists the electric field so the parameter <span class="elsevierStyleItalic">t</span> is called twist&#46; Tensor <span class="elsevierStyleBold"><span class="elsevierStyleItalic">S</span></span> produces a shear-like effect so the parameter <span class="elsevierStyleItalic">e</span> is called shear&#46; <span class="elsevierStyleBold"><span class="elsevierStyleItalic">R</span></span> is simply a rotation matrix to account for the coordinate system to be off an angle <span class="elsevierStyleItalic">&#952;</span> from strike&#46; Twist and shear are expressed in degrees such that <span class="elsevierStyleItalic">t</span> and <span class="elsevierStyleItalic">e</span> are the tangent of the corresponding angles&#46;</p><p id="par0070" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#fig0010">Figure 2</a> illustrates what each of the operators does to an undistorted unitary electric field&#46; The undistorted field varies from &#40;0&#44;1&#41; to &#40;0&#44; &#8722;1&#41; to cover different initial directions to allow a pattern to emerge&#46; Each operator can be thought to be the distortion tensor in <a class="elsevierStyleCrossRef" href="#eq0010">equation &#40;2&#41;</a> as applied to an undistorted electric field&#46; Twist is a simple rotation of the electric field similar to the rotation of coordinates&#44; and it could be thought to be an overlap of the latter&#46; However&#44; this is not so because the rotation of coordinates involves both electric and magnetic fields or&#44; equivalently&#44; the whole impedance tensor and not only the electric field&#46; Another important comment about this factorization or decomposition is that it does not reduce or increase the number of distorting parameters&#59; it just spreads them in a useful way for their recovery from the elements of <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span>&#46;</p><elsevierMultimedia ident="fig0010"></elsevierMultimedia></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0045">The absorption of static factors</span><p id="par0075" class="elsevierStylePara elsevierViewall">The impedance tensor&#44; distorted or undistorted&#44; can be mathematically transformed into an exactly equivalent set of 8 real numbers&#46; We consider in the next sections some of these transformations&#46; <a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey &#40;1989&#41;</a> factorization&#44; although<a name="p139"></a> mathematically expressed&#44; is not a mathematical decomposition in the same sense that these others are&#46; It is more a convenient proposal to allow for the distortions to be estimated from the elements of <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> through an inverse procedure&#46; They do it by neutralizing the two distortion factors <span class="elsevierStyleItalic">a</span> and <span class="elsevierStyleItalic">b</span> due to static&#44; simply by placing <span class="elsevierStyleBold"><span class="elsevierStyleItalic">A</span></span> next to <span class="elsevierStyleItalic">Z</span><span class="elsevierStyleInf">2</span> The unknowns now become <span class="elsevierStyleItalic">aA</span> and <span class="elsevierStyleItalic">bB</span>&#46;</p><p id="par0080" class="elsevierStylePara elsevierViewall">This reduction in the number of unknowns accomplishes two things&#46; <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> on the left side of <a class="elsevierStyleCrossRef" href="#eq0030">equation &#40;6&#41;</a> provides with 8 real numbers as data&#44; and on the right side we have 9 unknowns&#58; <span class="elsevierStyleItalic">&#952;&#44; t&#44; e&#44; a&#44; b</span> and four real numbers to make <span class="elsevierStyleItalic">Z</span><span class="elsevierStyleInf">2</span>&#46; This means that without this absorption the inversion cannot even be attempted&#46; However&#44; this is not the most important reason because the relative number of data can be increased by using many frequencies and by forcing &#952;&#44; <span class="elsevierStyleItalic">t</span> and <span class="elsevierStyleItalic">e</span> to be the same for all frequencies&#46; The second reason is more fundamental because it cannot be dealt with except by doing what they did&#46; It is universally acknowledged that static distortions factors cannot be determined from the impedance tensor alone&#44; and that independent information is needed to resolve them &#40;e&#46;g&#46; <a class="elsevierStyleCrossRef" href="#bib0060">Pellerin and Hohmann&#44; 1990</a>&#59; <a class="elsevierStyleCrossRef" href="#bib0055">Ledo <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2002</a>&#41;&#46; By absorbing the static factors into <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span> the problem is solvable&#46; We can mimic the recovery of static shifted curves by explicitly solving for them in <a class="elsevierStyleCrossRef" href="#eq0030">equation &#40;6&#41;</a>&#46; The recovery equation is<elsevierMultimedia ident="eq0055"></elsevierMultimedia></p><p id="par0085" class="elsevierStylePara elsevierViewall">First we compute a distorted <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> using <a class="elsevierStyleCrossRef" href="#eq0030">equation &#40;6&#41;</a>&#46; The undistorted <span class="elsevierStyleItalic">Z</span><span class="elsevierStyleInf">2</span> is the same sounding used in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>&#46; The distortions are as follow&#58; two different static factors <span class="elsevierStyleItalic">a&#61;2</span> and <span class="elsevierStyleItalic">b&#61;3</span> in one case and <span class="elsevierStyleItalic">a&#61;3</span> and <span class="elsevierStyleItalic">b&#61;2</span> in the other&#44; and the same <span class="elsevierStyleItalic">&#952;&#44; t</span> and <span class="elsevierStyleItalic">e</span> for both cases&#46; The distorted&#44; undistorted and recovered curves are shown in <a class="elsevierStyleCrossRef" href="#fig0015">Figure 3</a>&#46; Notice that the recovered curves are scaled versions of each other&#44; but the disturbed ones are not&#46; Note also that the recovered curves are scaled versions of the original <span class="elsevierStyleItalic">TE</span> or <span class="elsevierStyleItalic">A</span> curve&#46;</p><elsevierMultimedia ident="fig0015"></elsevierMultimedia><p id="par0090" class="elsevierStylePara elsevierViewall">In general&#44; <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> depends on <span class="elsevierStyleItalic">&#952;&#44; t&#44; e&#44; a</span> and <span class="elsevierStyleItalic">b</span>&#44; as indicated by <a class="elsevierStyleCrossRef" href="#eq0030">equation &#40;6&#41;</a>&#46; In other words&#44; all of them are needed to ft experimental impedances&#46; The <a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey &#40;1989&#41;</a> approach does not use the static factors to fit <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span>&#44; but this does not<a name="p140"></a> mean that <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> is immune to them&#46; In other words&#44; the unprocessed <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> is immune to none of the distorting parameters&#58; twist&#44; shear&#44; static and strike&#46; This is a natural starting point to analyze other responses derived from <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> that are immune to one or more types of distortions&#46;</p><p id="par0095" class="elsevierStylePara elsevierViewall">When dealing with invariants&#44; we can dispense of the rotation matrix and its transpose and use the simplified expression<elsevierMultimedia ident="eq0060"></elsevierMultimedia></p><p id="par0100" class="elsevierStylePara elsevierViewall">This equation follows from &#40;<a class="elsevierStyleCrossRef" href="#eq0030">6</a>&#41; by carrying out the multiplications&#46; Again&#44; we assume that <span class="elsevierStyleItalic">a</span> is absorbed in <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">b</span> in <span class="elsevierStyleItalic">B</span>&#46;</p></span></span><span id="sec0030" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0050">Results</span><span id="sec0035" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0055">Immunity to twist</span><p id="par0105" class="elsevierStylePara elsevierViewall">The first invariant to examine is the classical determinant of the impedance tensor &#40;<a class="elsevierStyleCrossRef" href="#bib0020">Berdichevsky and Dimitriv&#44; 1976</a>&#41;&#46; In the absence of distortions&#44; the determinant is simply given as<elsevierMultimedia ident="eq0065"></elsevierMultimedia></p><p id="par0110" class="elsevierStylePara elsevierViewall">In general&#44; when the four elements are present in the tensor&#44; the determinant is given as<elsevierMultimedia ident="eq0070"></elsevierMultimedia></p><p id="par0115" class="elsevierStylePara elsevierViewall">To determine the effect of distortions we must substitute <a class="elsevierStyleCrossRef" href="#eq0060">equation &#40;8&#41;</a> into <a class="elsevierStyleCrossRef" href="#eq0070">equation &#40;10&#41;</a>&#46;</p><p id="par0120" class="elsevierStylePara elsevierViewall">The result is<elsevierMultimedia ident="eq0075"></elsevierMultimedia></p><p id="par0125" class="elsevierStylePara elsevierViewall">The twist parameter <span class="elsevierStyleItalic">t</span> cancels in the algebra but not the shear parameter <span class="elsevierStyleItalic">e</span>&#46; The effect on the determinant is equivalent to modify the static factors on either <span class="elsevierStyleItalic">A</span> or <span class="elsevierStyleItalic">B</span>&#44; or both&#46; With no static effects the amplitude of the determinant would still be smaller than the actual one if shear distortions are present&#46; On average&#44; we might expect the static factors of the determinant to be biased towards values less than unity&#46;</p><p id="par0130" class="elsevierStylePara elsevierViewall">An invariant closely related to the determinant is the parallel impedance &#40;<a class="elsevierStyleCrossRef" href="#bib0065">Romo <span class="elsevierStyleItalic">et al</span>&#46; 2005</a>&#41; which is given as<elsevierMultimedia ident="eq0080"></elsevierMultimedia></p><p id="par0135" class="elsevierStylePara elsevierViewall">Making <span class="elsevierStyleItalic">t&#61;e&#61;0</span> in <a class="elsevierStyleCrossRef" href="#eq0060">equation &#40;8&#41;</a> and using <a class="elsevierStyleCrossRef" href="#eq0080">equation &#40;12&#41;</a>&#44; the undistorted parallel impedance is<elsevierMultimedia ident="eq0085"></elsevierMultimedia></p><p id="par0140" class="elsevierStylePara elsevierViewall">On the other hand&#44; the general case reduces to<elsevierMultimedia ident="eq0090"></elsevierMultimedia></p><p id="par0145" class="elsevierStylePara elsevierViewall">Again&#44; the distorted impedance is immune to twist&#44; but not to shear and&#44; of course neither to static&#44; which is hidden in <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span>&#46; The effect of shear is most effective on the parallel impedance than on the determinant because of the square&#46; This is illustrated in <a class="elsevierStyleCrossRef" href="#fig0020">Figure 4</a> for both invariants&#46; The quantity plotted is the traditional apparent resistivity &#124;Z&#124;<span class="elsevierStyleSup">2</span>&#47;&#40;<span class="elsevierStyleItalic">&#969;&#181;</span><span class="elsevierStyleInf">0</span>&#41;&#44; where <span class="elsevierStyleItalic">&#969;</span> stands for angular frequency and <span class="elsevierStyleItalic">&#181;</span><span class="elsevierStyleInf">0</span> is the permeability of free space&#46;</p><elsevierMultimedia ident="fig0020"></elsevierMultimedia></span><span id="sec0040" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0060">Immunity to twist and shear</span><p id="par0150" class="elsevierStylePara elsevierViewall">It is interesting how in the previous section twist separates quite neatly from shear&#44; to leave only the latter as an active distorter&#46; In this section we report an invariant that leaves no trace of both twist and shear&#46; This invariant we call the series equivalent &#40;<a class="elsevierStyleCrossRef" href="#bib0065">Romo <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2005</a>&#41; corresponds to half the sum of squares of the elements of the impedance tensor &#40;<a class="elsevierStyleCrossRef" href="#bib0075">Szarka and Menvielle&#44; 1997</a>&#41;&#46; This is<elsevierMultimedia ident="eq0095"></elsevierMultimedia></p><p id="par0155" class="elsevierStylePara elsevierViewall">When <span class="elsevierStyleItalic">t&#61;e&#61;0</span>&#44; this reduces to<elsevierMultimedia ident="eq0100"></elsevierMultimedia></p><p id="par0160" class="elsevierStylePara elsevierViewall">In general&#44; substituting <a class="elsevierStyleCrossRef" href="#eq0060">equation &#40;8&#41;</a> in &#40;<a class="elsevierStyleCrossRef" href="#eq0095">15</a>&#41; leads to<a name="p141"></a><elsevierMultimedia ident="eq0105"></elsevierMultimedia></p><p id="par0165" class="elsevierStylePara elsevierViewall">It turns out that this reduces to the undistorted series impedance&#44; which means that this type of invariant response is immune not only to strike&#44; but also to both twist and shear&#46; Of course&#44; it is still affected by static because <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span> have absorbed <span class="elsevierStyleItalic">a</span> and <span class="elsevierStyleItalic">b</span>&#44; the two static factors&#46;</p><p id="par0170" class="elsevierStylePara elsevierViewall">The distorted parallel impedance given by <a class="elsevierStyleCrossRef" href="#eq0090">equation &#40;14&#41;</a> differs from the undistorted impedance by a real factor that depends on shear&#46; This implies that the corresponding phase is immune to shear&#44; and because the complex impedance was already immune to strike and twist&#44; this implies that the phase of the parallel impedance is immune to all three distorting parameters&#46; The phase is given as<elsevierMultimedia ident="eq0110"></elsevierMultimedia></p><p id="par0175" class="elsevierStylePara elsevierViewall">Immunity to strike&#44; twist and shear isolates static effects&#46; It can be appreciated in both <a class="elsevierStyleCrossRef" href="#eq0100">equation &#40;16&#41;</a> and &#40;<a class="elsevierStyleCrossRef" href="#eq0110">18</a>&#41; that static factors will produce phase mixing&#46; This is illustrated in <a class="elsevierStyleCrossRef" href="#fig0025">Figure 5</a> for the phase of the series impedance&#46; Static effects&#44; absent in the phase of the TE and TM modes&#44; can severely distort the phase of the series impedance&#46; Care must be taken before or while inverting the data&#46; The same applies to the phase of the parallel impedance&#44; for while the factors cancel in the product <span class="elsevierStyleItalic">AB</span>&#44; they do not within the square root sign&#46;</p><elsevierMultimedia ident="fig0025"></elsevierMultimedia></span><span id="sec0045" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0065">Immunity to twist&#44; shear and statics</span><p id="par0180" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#eq0075">Equation &#40;11&#41;</a> indicates explicitly that the determinant is affected by shear through a multiplicative factor&#46; On the other hand&#44; implicitly it is affected by static though another factor <span class="elsevierStyleItalic">&#40;ab&#41;</span>&#44; which comes from the absorption of <span class="elsevierStyleItalic">a</span> and <span class="elsevierStyleItalic">b</span> into <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span>&#46; All together they compose a single factor that vanishes when determining the phase of the determinant&#46; That is<elsevierMultimedia ident="eq0115"></elsevierMultimedia></p><p id="par0185" class="elsevierStylePara elsevierViewall">The phase of the determinant is then immune to shear and static&#44; along with twist and strike that shares with the complex determinant&#46; We have arrived at a standstill&#44; for there is nothing that we can vary that alters the phase of the determinant&#46;</p></span><span id="sec0050" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0070">Invariance is not sufficient</span><p id="par0190" class="elsevierStylePara elsevierViewall">The twist tensor as defined in <a class="elsevierStyleCrossRef" href="#eq0040">equation 6&#40;b&#41;</a> is a rotation matrix that changes the direction of the electric field&#46; This may seem to indicate that all invariants are immune to twist&#44; as are the three invariants analyzed so far&#46; However&#44; this is not the case&#46; Consider the two invariant eigenvalues of Eggers &#40;1983&#41;&#44; which are given by the quadratic formula<a name="p142"></a><elsevierMultimedia ident="eq0120"></elsevierMultimedia></p><p id="par0195" class="elsevierStylePara elsevierViewall">where<elsevierMultimedia ident="eq0125"></elsevierMultimedia></p><p id="par0200" class="elsevierStylePara elsevierViewall">To see how the distortions affect these invariants we need to use the general expression for <span class="elsevierStyleItalic">Z</span> given in <a class="elsevierStyleCrossRef" href="#eq0070">equation &#40;10&#41;</a>&#46; Let us first substitute <span class="elsevierStyleItalic">Z</span> in &#40;<a class="elsevierStyleCrossRef" href="#eq0120">20</a>&#41; when t&#61;e&#61;0&#46; The result is<elsevierMultimedia ident="eq0130"></elsevierMultimedia></p><p id="par0205" class="elsevierStylePara elsevierViewall">In this case the eigenvalues reduce to either <span class="elsevierStyleItalic">A</span> or <span class="elsevierStyleItalic">B</span> and in the distorted case to<elsevierMultimedia ident="eq0135"></elsevierMultimedia></p><p id="par0210" class="elsevierStylePara elsevierViewall">This is about the minimum expression one can obtain for the distorted eigenvalues&#46; It can be observed that <span class="elsevierStyleItalic">t</span> and <span class="elsevierStyleItalic">e</span> are present in the expression and that there is no possible way that those in the numerator cancel out with those in the denominator&#46; It is also clear that the effect of twist and shear cannot be absorbed as equivalent static factors on <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span>&#44; although they can in several places in the expression&#46; The factor <span class="elsevierStyleItalic">&#40;1&#43;te&#41;</span> multiplies <span class="elsevierStyleItalic">B</span> and <span class="elsevierStyleItalic">&#40;1-te&#41;</span> multiplies <span class="elsevierStyleItalic">A</span>&#44; both outside and within the square root&#46; However&#44; the absorption cannot be completed for the product <span class="elsevierStyleItalic">AB</span> within the square root&#46; Summarizing&#44; the two eigenvalues are not immune to any of the distortions&#46; <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a> illustrates this with the phase of the two eigenvalues using different factors for twist&#44; shear and static&#46; Notice that the curves show phase mixing of the original phases of <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span>&#46;</p><elsevierMultimedia ident="fig0030"></elsevierMultimedia></span><span id="sec0055" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0075">Invariance is not necessary</span><p id="par0215" class="elsevierStylePara elsevierViewall">We now come to a very interesting response that can be derived from the impedance tensor&#46; It is called the phase tensor and was proposed by <a class="elsevierStyleCrossRef" href="#bib0025">Cadwell <span class="elsevierStyleItalic">et al</span>&#46; &#40;2004&#41;</a>&#46; This response is immune to twist&#44; shear and static and it is not invariant&#46; The measured impedance is separated into its real and imaginary parts as<a name="p143"></a><elsevierMultimedia ident="eq0140"></elsevierMultimedia></p><p id="par0220" class="elsevierStylePara elsevierViewall">If <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> were a simple complex number&#44; the tangent of its phase would be its imaginary part divided by its real part&#46; In other words&#44; the tangent of its phase would be the inverse of its real part multiplied by its imaginary part&#46; But <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> is not a simple complex number&#44; it is a tensor&#46; However&#44; nothing prevents to multiply the inverse of the real tensor by the imaginary tensor&#46; That is<elsevierMultimedia ident="eq0145"></elsevierMultimedia></p><p id="par0225" class="elsevierStylePara elsevierViewall">The product is a dimensionless tensor whose elements can be interpreted as the tangent of an angle&#46; So far there is nothing impressive about this&#46; However&#44; let us now enquire about the effect of the different distortions on this dimensionless tensor&#46; The undistorted tensor can be written as<elsevierMultimedia ident="eq0150"></elsevierMultimedia></p><p id="par0230" class="elsevierStylePara elsevierViewall">Substituting this into &#40;<a class="elsevierStyleCrossRef" href="#eq0030">6</a>&#41; and doing the operation as in &#40;<a class="elsevierStyleCrossRef" href="#eq0145">25</a>&#41; we have that<elsevierMultimedia ident="eq0155"></elsevierMultimedia></p><p id="par0235" class="elsevierStylePara elsevierViewall">Performing the operations there results that <span class="elsevierStyleBold"><span class="elsevierStyleItalic">T&#44; S</span></span> and <span class="elsevierStyleBold"><span class="elsevierStyleItalic">A</span></span> cancel with their corresponding inverses&#46; The expression reduces to<elsevierMultimedia ident="eq0160"></elsevierMultimedia></p><p id="par0240" class="elsevierStylePara elsevierViewall">The phase tensor depends only on strike&#46; This is a very important property because it allows obtaining the strike direction independently of the distorting parameters&#46; <a class="elsevierStyleCrossRef" href="#fig0035">Figure 7</a> illustrates the dependence of diagonal elements of the phase tensor as strike is varied&#46;</p><elsevierMultimedia ident="fig0035"></elsevierMultimedia></span><span id="sec0060" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0080">Series&#44; parallel and determinant impedances</span><p id="par0245" class="elsevierStylePara elsevierViewall">The properties of the determinant in relation to distortions are well known &#40;e&#46;g&#46; <a class="elsevierStyleCrossRef" href="#bib0010">Arango <span class="elsevierStyleItalic">et al</span>&#46;&#44; 2009</a>&#41; and derive from the fact that if <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> &#61; <span class="elsevierStyleItalic">CZ</span><span class="elsevierStyleInf">2</span>&#44; then <span class="elsevierStyleItalic">det</span> &#40;<span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span>&#41; &#61; <span class="elsevierStyleItalic">det</span> &#40;<span class="elsevierStyleItalic">C</span>&#41; <span class="elsevierStyleItalic">det</span> &#40;<span class="elsevierStyleItalic">Z</span><span class="elsevierStyleInf">2</span>&#41;&#46; Given that <span class="elsevierStyleItalic">C</span> is real the phase of <span class="elsevierStyleItalic">det</span> &#40;<span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span>&#41; is the same as the phase of <span class="elsevierStyleItalic">det</span> &#40;<span class="elsevierStyleItalic">Z</span><span class="elsevierStyleInf">2</span>&#41;&#46; Stated in this way this result is more general than that of <a class="elsevierStyleCrossRef" href="#eq0075">equation &#40;11&#41;</a>&#44; which assumes that the undistorted impedance is 2-D&#46; The results for the series and parallel impedances derived above also assume a 2-D undistorted impedance&#46; This means that for real data&#44; seldom fully 2-D&#44; we should expect the results to hold exactly for the determinant and only approximately for series and parallel&#46; This issue is explored in the next three figures&#46; First&#44; we present a reference case<a name="p144"></a> using synthetic 2-D data&#46; What we have done is to distort the original TE and TM data using <a class="elsevierStyleCrossRef" href="#eq0030">equation &#40;6&#41;</a> and then compute the three invariants of <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span>&#46; It can be observed in <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8c</a> and <a class="elsevierStyleCrossRef" href="#fig0040">8d</a> that both apparent resistivity and phase curves of the series impedance are immune to the distortions&#46; The comparison is made for three shears&#44; 0&#44; 30 and 40 degrees&#46; Strike direction and twist are not recorded because these and the other results presented below are exactly the same regardless of the value of these parameters&#46; According to theory the only distortion left is due to shear and this is what <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a> shows&#46; The determinant and the parallel apparent resistivities&#44; as shown in <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8a</a> and <a class="elsevierStyleCrossRef" href="#fig0040">8c</a>&#44; suffer a downward shift as predicted by <a class="elsevierStyleCrossRef" href="#eq0075">equations &#40;11&#41;</a> and &#40;<a class="elsevierStyleCrossRef" href="#eq0090">14</a>&#41;&#44; respectively&#46; The shift being more pronounced the larger the shear&#44; and the shift of the parallel resistivity being larger than that of the determinant&#44; as it should be according to theory&#46; It can also be observed in <a class="elsevierStyleCrossRef" href="#fig0040">Figures 8b</a> and <a class="elsevierStyleCrossRef" href="#fig0040">8c</a> that the corresponding phases are not affected by shear&#46;</p><elsevierMultimedia ident="fig0040"></elsevierMultimedia><p id="par0250" class="elsevierStylePara elsevierViewall">The results using field data are presented in <a class="elsevierStyleCrossRef" href="#fig0045">Figures 9</a> and <a class="elsevierStyleCrossRef" href="#fig0050">10</a>&#46; We chose soundings from the <span class="elsevierStyleSmallCaps">coprod2</span> and <span class="elsevierStyleSmallCaps">bc</span>87 data sets &#40;<a class="elsevierStyleCrossRef" href="#bib0040">Jones&#44; 1993a</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0045">1993b</a>&#41;&#46; It can be observed that in both cases the corresponding curves behave very much like those using synthetic data&#46; We used <a class="elsevierStyleCrossRef" href="#eq0030">equation &#40;6&#41;</a> to further distort both soundings with different strike directions&#44; twists and shears and then computed the three invariants&#46; It can be observed that the phases of the three invariants are immune to the changes of the distorting parameters&#44; implying that they are free of distortions except for the implicit static factors&#46; Notice that the phases of the determinant are exactly the same for the three shears &#40;<a class="elsevierStyleCrossRef" href="#fig0045">Figures 9b</a> and <a class="elsevierStyleCrossRef" href="#fig0050">10c</a>&#41; and that those of the series and parallel depart a little from each other &#40;<a class="elsevierStyleCrossRef" href="#fig0045">Figures 9d</a> and <a class="elsevierStyleCrossRef" href="#fig0050">10d</a>&#41;&#46; As mentioned before&#44; this is because the determinant makes no assumption about dimensionality while the other two assume a 2-<span class="elsevierStyleSmallCaps">d</span> undistorted impedance&#46; Overall&#44; the phases for the three shears are reasonably close considering that a shear of 40 degrees is close to the maximum of 45 degrees&#46;</p><elsevierMultimedia ident="fig0045"></elsevierMultimedia><elsevierMultimedia ident="fig0050"></elsevierMultimedia><p id="par0255" class="elsevierStylePara elsevierViewall">The example drawn from the <span class="elsevierStyleSmallCaps">bc</span>87 data set illustrates an important point about what is removed and what remains to be done to completely clean the data&#46; Notice that at short periods the series and parallel apparent resistivities &#40;<a class="elsevierStyleCrossRef" href="#fig0050">Figure 10c</a>&#41; are both horizontal for the null shear&#44; and that the latter is depressed in relation to the former&#46; This suggests that the parallel apparent resistivity curve has an intrinsic shear distortion that is depressing it from the undistorted series resistivity&#46; This would lead us to shift the curve<a name="p145"></a><a name="p146"></a><a name="p147"></a><a name="p148"></a><a name="p149"></a><a name="p150"></a><a name="p151"></a> upwards and leave the phase unchanged to undo the effect of shear&#46; However&#44; the fact is that we do not know whether this shift is due to shear or static&#46; The parallel resistivity curve could also be lifted by playing with the implicit static factors&#44; in which case the phase curve would have to be modified according to <a class="elsevierStyleCrossRef" href="#eq0110">equation &#40;18&#41;</a>&#46; Deciding which of the two is the proper way to proceed is outside the scope of this first communication&#46; The present results are encouraging to continue the process of advancing our knowledge of the series and parallel invariants&#44; whose properties are just beginning to be understood&#46;</p></span></span><span id="sec0065" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0085">Conclusion</span><p id="par0260" class="elsevierStylePara elsevierViewall">Galvanic distortions of magnetotelluric data can be neutralized by some invariants of the impedance tensor&#46; Except for strike direction&#44; invariants are devised without any thought about distortions&#46; However&#44; as it turns out for the case of 2-D data&#44; the 3-D distorting effects can gradually be neutralized by an also gradual averaging process&#46; Series and parallel impedances are averages of the tensor elements and&#44; in turn&#44; the determinant is an average of the series and parallel impedances&#46; This gradual immunity is summarized in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>&#46; The new results include&#58; a&#41; The amplitude of the series impedance is affected only by statics&#44; being immune to twist&#44; shear and strike&#44; and so are the phases of series and parallel&#59; b&#41; the amplitude of the parallel impedance is immune to twist and strike but not to shear and statics&#46; These results place the series and parallel impedances midway between invariants that are not immune to any of the distortions&#44; like Egger&#8217;s eigenvalues&#44; and the phase of the determinant which is immune to all of them&#46;</p><elsevierMultimedia ident="tbl0005"></elsevierMultimedia></span></span>"
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              "titulo" => "The algebra of distortions"
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              "titulo" => "A useful factorization"
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              "titulo" => "Immunity to twist"
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            1 => array:2 [
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              "titulo" => "Immunity to twist and shear"
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              "titulo" => "Immunity to twist&#44; shear and statics"
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              "titulo" => "Invariance is not sufficient"
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              "titulo" => "Invariance is not necessary"
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              "titulo" => "Series&#44; parallel and determinant impedances"
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        "resumen" => "<span id="abst0005" class="elsevierStyleSection elsevierViewall"><p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">Las impedancias serie y paralelo del tensor magnetotel&#250;rico se eval&#250;an en relaci&#243;n con su relativa inmunidad a las distorsiones galvano-el&#233;ctricas&#46; Las respuestas distorsionadas se modelan utilizando la descomposici&#243;n del tensor en t&#233;rminos de giro&#44; cizalla est&#225;tica y rumbo de Groom y Bailey&#46; Estos cuatro par&#225;metros&#44; junto con las impedancias sin distorsi&#243;n&#44; normalmente se consideran inc&#243;gnitas y se obtienen de los datos mediante la soluci&#243;n de un problema inverso&#46; En el presente trabajo utilizamos la descomposici&#243;n como un modelo directo para simular sondeos distorsionados&#46; Partiendo de respuestas 2-D sin distorsiones&#44; el tensor se distorsiona suponiendo valores arbitrarios de giro&#44; cizalla&#44; est&#225;tica y rumbo&#46; Por definici&#243;n las impedancias serie y paralelo son inmunes al rumbo porque son invariantes ante rotaci&#243;n&#46; Adicionalmente&#44; la impedancia serie es inmune a giros y a cizalla&#44; y la impedancia paralelo s&#243;lo a giros&#46; La impedancia paralelo depende de cizalla en la forma de un factor que desplaza hacia abajo las curvas de amplitud&#46; Por otro lado&#44; el efecto de la est&#225;tica en ambas impedancias es m&#225;s complicado que en el tensor mismo porque no se puede corregir con un simple desplazamiento de las curvas&#46; En t&#233;rminos generales&#44; hay un balance positivo por parte de las impedancias serie y paralelo sobre las respuestas <span class="elsevierStyleSmallCaps">te</span> y <span class="elsevierStyleSmallCaps">tm</span> porque los invariantes filtran varias distorsiones&#46; Se muestra que la condici&#243;n de invariante no es suficiente para tener inmunidad a cualquiera de las distorsiones&#46; Se utiliza para esto los valores caracter&#237;sticos de Eggers&#44; los cuales son inmunes s&#243;lo al rumbo&#44; como todos los invariantes&#46; Se muestra adem&#225;s que la invariancia tampoco es una condici&#243;n necesaria para ser inmune a las distorsiones&#44; seg&#250;n lo atestigua el tensor de impedancia&#44; el cual depende del rumbo pero est&#225; libre de las dem&#225;s distorsiones&#46; Los desarrollos se ilustran utilizando sondeos de los conjuntos de datos <span class="elsevierStyleSmallCaps">coprod2s1&#44; coprod2</span> y <span class="elsevierStyleSmallCaps">bc</span>87&#46;</p></span>"
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        "resumen" => "<span id="abst0010" class="elsevierStyleSection elsevierViewall"><p id="spar0010" class="elsevierStyleSimplePara elsevierViewall">The series and parallel impedances of the magnetotelluric tensor are appraised in relation to their relative immunity to galvanic electric distortions&#46; The distorted responses are modeled using the Groom-Bailey decomposition of the tensor in terms of twist&#44; shear&#44; statics and strike direction&#46; These four parameters and the undistorted responses are normally considered as unknowns&#44; and are obtained from field data through the solution of an inverse problem&#46; In the present work we use the decomposition as a forward model to simulate distorted sounding curves&#46; Starting with undistorted 2-<span class="elsevierStyleSmallCaps">d te</span> and <span class="elsevierStyleSmallCaps">tm</span> responses&#44; the tensor is distorted by assuming arbitrary values of twist&#44; shear&#44; static and strike direction&#46; By default&#44; both series and parallel responses are immune to the strike direction because they are invariants under rotation&#46; In addition&#44; series responses are immune to twist and shear and parallel responses only to twist&#46; The dependence of the latter on shear is in the form of a real factor that shifts downwards the amplitude curves&#46; On the other hand&#44; the effect of statics on both series and parallel responses is more complicated than that on the impedance tensor because it cannot be accounted for by a simple shift of the curves&#46; On the whole&#44; there is a positive balance on the part of the series and parallel impedances over the <span class="elsevierStyleSmallCaps">te</span> and <span class="elsevierStyleSmallCaps">tm</span> responses because some of the distortions are filtered out by the invariants&#46; It is shown that invariance is not sufficient to be immune to any of the distortions&#46; The example chosen is Eggers&#8217; eigenvalues&#44; which are immune only to the by-the-fault strike direction&#46; Invariance is not necessary either&#44; as evidenced by the phase tensor&#44; whose elements depend on strike but are immune to all distortions&#46; The derivations are illustrated using soundings from the synthetic <span class="elsevierStyleSmallCaps">coprod2s1</span> and field-recorded <span class="elsevierStyleSmallCaps">coprod2</span> and <span class="elsevierStyleSmallCaps">bc87</span> data sets&#46;</p></span>"
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          "en" => "<p id="spar0015" class="elsevierStyleSimplePara elsevierViewall"><span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span> correspond to the <span class="elsevierStyleSmallCaps">te</span> and <span class="elsevierStyleSmallCaps">tm</span> modes of sounding &#35; 11 of the synthetic data set <span class="elsevierStyleSmallCaps">coprod2s1</span> made available in <span class="elsevierStyleSmallCaps">mtn</span>et&#46;dias&#46;ie by <a class="elsevierStyleCrossRef" href="#bib0080">Varentsov &#40;1998&#41;</a>&#46; When <span class="elsevierStyleItalic">&#952;</span> &#8800; 0 in <a class="elsevierStyleCrossRef" href="#eq0025">equation &#40;5&#41;</a> the elements of the resulting impedance all are mixtures of the phases of <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span>&#46; Calculations were made using <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">1</span>&#61; 1&#46;97&#44; <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">2</span> &#61; &#8722;&#8722;0&#46;77&#44; c<span class="elsevierStyleInf">3</span> &#61; &#8722;0&#46;35 and <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">4</span> &#61; 0&#46;64&#46; The graphs correspond to the phase of the impedance&#46;</p>"
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          "en" => "<p id="spar0020" class="elsevierStyleSimplePara elsevierViewall">The effect of the individual distortion tensor T&#44; S and <span class="elsevierStyleBold"><span class="elsevierStyleItalic">A</span></span> are illustrated in a&#41;&#44; b&#41; and c&#41;&#44; respectively&#46; The figures illustrate what each of the operators does to an undistorted unitary electric field shown in a&#41;&#46; The undistorted field varies from &#40;0&#44;1&#41; to &#40;0&#44; &#8722;1&#41; to cover different initial directions to allow a pattern to emerge&#46; Each operator can be thought to be the distortion tensor in <a class="elsevierStyleCrossRef" href="#eq0010">equation &#40;2&#41;</a> as applied to a unitary and undistorted electric field&#46; The figure is inspired in a similar one in <a class="elsevierStyleCrossRef" href="#bib0035">Groom and Bailey &#40;1989&#41;</a>&#46;</p>"
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          "en" => "<p id="spar0025" class="elsevierStyleSimplePara elsevierViewall">The recovery of undistorted TE impedances using <a class="elsevierStyleCrossRef" href="#eq0055">equation &#40;7&#41;</a> is correct save for the static factors&#46; The distorted <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> is computed using <a class="elsevierStyleCrossRef" href="#eq0030">equation &#40;6&#41;</a> and the indicated distorting parameters of twist&#44; shear and strike&#46; The undistorted <span class="elsevierStyleItalic">Z</span><span class="elsevierStyleInf">2</span> is the same sounding used in <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a>&#46; Notice that the recovered curves are scaled versions of each other&#44; but the disturbed ones are not&#46; Note also that the recovered curves are scaled versions of the original TE or <span class="elsevierStyleItalic">A</span> curve&#46; In this instance it is pretended that we know the distorting parameters of twist&#44; shear and strike in <a class="elsevierStyleCrossRef" href="#eq0055">equation &#40;7&#41;</a>&#46;</p>"
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        "identificador" => "fig0020"
        "etiqueta" => "Figure 4"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
          0 => array:4 [
            "imagen" => "gr4.jpeg"
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            "Ancho" => 1420
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          "en" => "<p id="spar0030" class="elsevierStyleSimplePara elsevierViewall">Aside from statics&#44; the amplitudes of the determinant and of the parallel impedance are distorted only by shear&#46; This is illustrated in this figure where it can be observed that according to theory the effect is more effective on the parallel impedance&#46;</p>"
        ]
      ]
      4 => array:7 [
        "identificador" => "fig0025"
        "etiqueta" => "Figure 5"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
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            "imagen" => "gr5.jpeg"
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          "en" => "<p id="spar0035" class="elsevierStyleSimplePara elsevierViewall">Immunity to strike&#44; twist and shear isolates static effects as in the case of <a class="elsevierStyleCrossRef" href="#eq0100">equation &#40;16&#41;</a> for the series impedance&#46; Unfortunately&#44; if <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span> are affected by statics &#40;<span class="elsevierStyleItalic">a</span> and <span class="elsevierStyleItalic">b</span> different from unity&#41; this produces a static-dependent phase mixing that is not present in the original impedances <span class="elsevierStyleItalic">A</span> and B&#46; This is illustrated in this figure&#46; Static effects&#44; absent in the phase of the TE and TM modes&#44; can severely distort the phase of the series impedance&#46; The same applies to the phase of the parallel impedance&#44; for while the factors cancel in the product <span class="elsevierStyleItalic">AB</span>&#44; they do not within the square root sign in <a class="elsevierStyleCrossRef" href="#eq0110">equation &#40;18&#41;</a>&#46;</p>"
        ]
      ]
      5 => array:7 [
        "identificador" => "fig0030"
        "etiqueta" => "Figure 6"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:1 [
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        "descripcion" => array:1 [
          "en" => "<p id="spar0040" class="elsevierStyleSimplePara elsevierViewall">Eggers&#8217; eigenvalues&#44; the invariants defined by <a class="elsevierStyleCrossRef" href="#eq0135">equation &#40;23&#41;</a> are not immune to any of the distortion parameters of twist&#44; shear and statics&#46; This is illustrated here with the phase of the two eigenvalues&#46; This case demonstrates that invariance under rotation is not a sufficient condition for immunity to any of the other distortions&#46;</p>"
        ]
      ]
      6 => array:7 [
        "identificador" => "fig0035"
        "etiqueta" => "Figure 7"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
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        "descripcion" => array:1 [
          "en" => "<p id="spar0045" class="elsevierStyleSimplePara elsevierViewall">Contrary to the invariants&#44; the phase tensor depends on strike and at the same time it is immune to all other distortions&#46; This is a very important property because it allows obtaining the strike direction independently of the distorting parameters&#46; This figure illustrates the dependence of the diagonal elements of the phase tensor as strike is varied&#46; The phase tensor is an example of a response that is not invariant but is immune to distortions&#46; This result&#44; together with that of <a class="elsevierStyleCrossRef" href="#fig0030">Figure 6</a>&#44; imply that invariance is neither sufficient not necessary for immunity to distortions&#44; and that each response must be evaluated on its own&#46;</p>"
        ]
      ]
      7 => array:7 [
        "identificador" => "fig0040"
        "etiqueta" => "Figure 8"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:2 [
          0 => array:1 [
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          1 => array:1 [
            "imagen" => "gr8b.jpeg"
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        "descripcion" => array:1 [
          "en" => "<p id="spar0050" class="elsevierStyleSimplePara elsevierViewall">Synthetic <span class="elsevierStyleSmallCaps">te</span> and <span class="elsevierStyleSmallCaps">tm</span> data are distorted using <a class="elsevierStyleCrossRef" href="#eq0030">equation &#40;6&#41;</a> and then invariants of <span class="elsevierStyleItalic">Z<span class="elsevierStyleInf">m</span></span> are computed&#46; It can be observed in b&#41; and d&#41; that the phases of the determinant &#40;<span class="elsevierStyleItalic">&#968;<span class="elsevierStyleInf">d</span></span>&#41;&#44; series &#40;<span class="elsevierStyleItalic">&#968;<span class="elsevierStyleInf">s</span></span>&#41;&#44; and parallel &#40;&#968;<span class="elsevierStyleInf">p</span>&#41; impedances are immune to distortions&#44; in agreement to theory&#46; The corresponding amplitudes shown in a&#41; and c&#41;&#44; also in agreement to theory&#44; are scaled versions of the originals &#40;shear &#61; 0 degrees&#41;&#46; This includes the series impedance which is immune to this scaling due to shear&#46; Strike direction and twist are not recorded because the results are exactly the same regardless of the values of these parameters&#46;</p>"
        ]
      ]
      8 => array:7 [
        "identificador" => "fig0045"
        "etiqueta" => "Figure 9"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:2 [
          0 => array:1 [
            "imagen" => "gr9a.jpeg"
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            "imagen" => "gr9b.jpeg"
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        "descripcion" => array:1 [
          "en" => "<p id="spar0055" class="elsevierStyleSimplePara elsevierViewall">Results for the determinant&#44; series and parallel impedances using field data from the COPROD2 data set&#44; show that the corresponding curves behave very much like those of <a class="elsevierStyleCrossRef" href="#fig0040">Figure 8</a> using synthetic data&#46; We used <a class="elsevierStyleCrossRef" href="#eq0030">equation &#40;6&#41;</a> to further distort both soundings with different strike directions&#44; twists and shears and then computed the three invariants&#46; It can be observed that the phases of the three invariants shown in b&#41; and d&#41; are immune to the changes of the distorting parameters&#44; implying that they are free of distortions except for the implicit static factors&#46; Notice that the phases of the determinant are exactly the same for the three shears and that those of the series and parallel depart a little from each other&#46; As mentioned in the main text&#44; this is because the determinant makes no assumption about dimensionality while series and parallel assume a 2-D undistorted impedance&#46; The amplitudes shown in a&#41; and c&#41; are scaled versions of the originals &#40;shear &#61; 0 degrees&#41; in agreement to theory&#46; This includes the series invariant which is immune to this scaling due to shear&#46; Overall&#44; the phases for the three shears are reasonably close considering that a shear of 40 degrees is close to the maximum of 45 degrees&#46;</p>"
        ]
      ]
      9 => array:7 [
        "identificador" => "fig0050"
        "etiqueta" => "Figure 10"
        "tipo" => "MULTIMEDIAFIGURA"
        "mostrarFloat" => true
        "mostrarDisplay" => false
        "figura" => array:2 [
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            "imagen" => "gr10b.jpeg"
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          "en" => "<p id="spar0060" class="elsevierStyleSimplePara elsevierViewall">Results for the determinant&#44; series and parallel impedances using field data from the BC87 data set&#44; show that the corresponding curves behave very much like those of <a class="elsevierStyleCrossRef" href="#fig0040">Figures 8</a> and <a class="elsevierStyleCrossRef" href="#fig0045">9</a>&#46; It can be observed in c&#41; that there is already a significant downwards shift of the parallel amplitude with respect to series that is not related to the added shear scaling&#46; As discussed in the main text&#44; this could be due to a preexisting shear distortion or to static effects&#46; The rest of the curves in a&#41;&#44; b&#41; and d&#41; follow the same patterns as in <a class="elsevierStyleCrossRef" href="#fig0040">Figures 8</a> and <a class="elsevierStyleCrossRef" href="#fig0045">9</a>&#46; Notice that&#44; as in the other tests&#44; the series amplitude is immune to the scaling introduced by the shear distortion as predicted by <a class="elsevierStyleCrossRef" href="#eq0105">equation &#40;17&#41;</a>&#46;</p>"
        ]
      ]
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        "identificador" => "tbl0005"
        "etiqueta" => "Table 1"
        "tipo" => "MULTIMEDIATABLA"
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                  <table border="0" frame="\n
                  \t\t\t\t\tvoid\n
                  \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head  " align="" valign="top" scope="col" style="border-bottom: 2px solid black">&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">Strike&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">Twist&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">Shear&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th><th class="td" title="table-head  " align="center" valign="middle" scope="col" style="border-bottom: 2px solid black">Statics&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry  " align="left" valign="middle">Eggers&#8217;s eigenvalues&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">no&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">no&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">no&nbsp;\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="middle">Parallel amplitude&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">no&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">no&nbsp;\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="middle">Determinant amplitude&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">no&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">no&nbsp;\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="middle">Series amplitude&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">no&nbsp;\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="middle">Series phase&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">no&nbsp;\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="middle">Parallel phase&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">no&nbsp;\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="middle">Phase tensor&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">no&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\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="middle">Determinant phase&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td><td class="td" title="table-entry  " align="center" valign="middle">yes&nbsp;\t\t\t\t\t\t\n
                  \t\t\t\t</td></tr></tbody></table>
                  """
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          "en" => "<p id="spar0065" class="elsevierStyleSimplePara elsevierViewall">The table summarizes the immunity of the different responses to the four distorting parameters&#44; strike&#44; twist&#44; shear and statics&#46;</p>"
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Article information

ISSN: 00167169
Original language: English

DOI:10.1016/S0016-7169(13)71468-5

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Year/Month	Html	Pdf	Total

2024 November	2	0	2
2024 October	19	5	24
2024 September	22	1	23
2024 August	25	1	26
2024 July	34	2	36
2024 June	31	4	35
2024 May	19	5	24
2024 April	20	5	25
2024 March	17	3	20
2024 February	9	4	13
2024 January	13	5	18
2023 December	13	4	17
2023 November	10	7	17
2023 October	17	10	27
2023 September	8	1	9
2023 August	11	4	15
2023 July	15	3	18
2023 June	14	2	16
2023 May	40	11	51
2023 April	14	3	17
2023 March	18	3	21
2023 February	21	8	29
2023 January	9	4	13
2022 December	10	5	15
2022 November	21	7	28
2022 October	12	11	23
2022 September	8	16	24
2022 August	18	5	23
2022 July	24	11	35
2022 June	10	7	17
2022 May	24	5	29
2022 April	25	9	34
2022 March	31	4	35
2022 February	34	8	42
2022 January	53	11	64
2021 December	30	8	38
2021 November	23	12	35
2021 October	18	8	26
2021 September	18	12	30
2021 August	21	3	24
2021 July	15	7	22
2021 June	18	7	25
2021 May	11	5	16
2021 April	39	26	65
2021 March	15	5	20
2021 February	5	3	8
2021 January	11	9	20
2020 December	7	8	15
2020 November	14	6	20
2020 October	12	6	18
2020 September	9	9	18
2020 August	19	7	26
2020 July	5	1	6
2020 June	8	4	12
2020 May	9	7	16
2020 April	6	4	10
2020 March	9	2	11
2020 February	9	1	10
2020 January	17	9	26
2019 December	10	16	26
2019 November	10	4	14
2019 October	7	1	8
2019 September	10	1	11
2019 August	6	0	6
2019 July	21	8	29
2019 June	37	15	52
2019 May	63	40	103
2019 April	39	11	50
2019 March	7	1	8
2019 February	6	2	8
2019 January	3	2	5
2018 December	3	3	6
2018 November	4	5	9
2018 October	7	10	17
2018 September	22	7	29
2018 August	3	1	4
2018 July	5	0	5
2018 June	4	4	8
2018 May	2	3	5
2018 April	3	8	11
2018 March	4	1	5
2018 February	3	0	3
2018 January	2	0	2
2017 December	5	0	5
2017 November	8	3	11
2017 October	6	0	6
2017 September	6	0	6
2017 August	12	1	13
2017 July	16	1	17
2017 June	25	3	28
2017 May	8	1	9
2017 April	12	3	15
2017 March	13	4	17
2017 February	8	4	12
2017 January	18	0	18
2016 December	19	2	21
2016 November	26	4	30
2016 October	22	7	29
2016 September	20	1	21
2016 August	24	2	26
2016 July	20	3	23
2016 June	8	3	11
2016 May	12	9	21
2016 April	8	6	14
2016 March	10	3	13
2016 February	9	5	14
2016 January	10	5	15
2015 December	7	3	10
2015 November	16	4	20
2015 October	25	5	30
2015 September	12	5	17
2015 August	9	1	10
2015 July	7	3	10
2015 June	1	0	1
2015 May	1	1	2
2015 April	1	1	2

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