 Express Letter
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3D inversion of MT impedances and intersite tensors, individually and jointly. New lessons learnt
Earth, Planets and Space volume 71, Article number: 4 (2019)
Abstract
A conventional magnetotelluric (MT) survey layout implies measurements of horizontal electric and magnetic fields at every site with subsequent estimation and interpretation of impedance tensors \({Z}\) or dependent responses, such as apparent resistivities and phases. In this work, we assess advantages and disadvantages of complementing or substituting conventional MT with intersite transfer functions such as intersite impedance tensor, \({Q}\), horizontal magnetic, \({M}\), and horizontal electric, \({T}\), tensors. Our analysis is based on a 3D inversion of synthetic responses calculated for a 3D model which consists of two buried adjacent (resistive and conductive) blocks and thin resistor above them. The (regularized) 3D inversion is performed using scalable 3D MT inverse solver with forward modelling engine based on a contracting integral equation approach. The inversion exploits gradienttype (quasiNewton) optimization algorithm and invokes adjoint sources approach to compute misfits’ gradients. From our model study, we conclude that: (1) 3D inversion of either \({Z}\) or \({Q}\) tensors recovers the “true” structures equally well. This, in particular, raises the question whether we need magnetic field measurements at every survey site in the course of 3D MT studies; (2) recovery of true structures is slightly worse if \({T}\) tensor is inverted, and significantly worse if \({M}\) tensor is inverted; (3) simultaneous inversion of \({Z}\) and \({M}\) (or \({Z}\) and \({T}\)) does not improve the recovery of true structures compared to individual inversion of \({Z}\) or \({Q}\); (4) location of reference site, which is required for calculating intersite \({Q}\), \({T}\) and \({M}\) tensors, has also marginal effect on the inversion results.
Introduction
One of the wellestablished geophysical techniques to explore the Earth’s interiors is magnetotelluric (MT) method (Chave and Jones 2012). During an MT survey, natural variations of the horizontal electric, \({\mathbf {E}}_\tau \), and magnetic, \({\mathbf {H}}_\tau \), fields are measured with further translating them into subsurface’s electrical conductivity distribution. Conductivity is ultimately interpreted in terms of Earth’s composition, temperature, volatile content and partial melt. It is assumed in MT that the source of timevarying fields \({\mathbf {E}}_\tau \) and \({\mathbf {H}}_\tau \) is vertically incident plane wave of variable polarization. The plane wave assumption allows one to write an equation which relates frequency domain \({\mathbf {E}}_\tau \) and \({\mathbf {H}}_\tau \) at a survey site \( \mathbf{r}_s\) via complexvalued impedance tensor \({Z}\) (Berdichevsky and Dmitriev 2008)
where \(\omega \) is angular frequency, \({\mathbf {E}}_\tau =[E_x,E_y]^{\mathrm {T}}\), \({\mathbf {H}}_\tau =[H_x,H_y]^{\mathrm {T}}\), and the superscript (\({\mathrm{T}}\)) denotes the transpose of a vector. Estimation of \({Z}\) at multiple survey sites and frequencies, with subsequent inversion of the estimated \({Z}\) in terms of subsurface conductivity constitute an essence of MT method. Starting with 1D MT inversions, and thereafter 2D, nowadays 3D MT inversions has become common practice due to the availability of efficient and scalable 3D forward modelling solvers and highperformance clusters. However, a proper 3D inversion requires 2D—preferably regular—grid of observations over the region of interest. Moreover, if one is interested in laterally detailed conductivity images the observation grid should be rather dense. This, in particular, means that MT survey would become logistically and instrumentally demanding, if conventional MT survey setup—which requires measurements of both electric and magnetic fields at each survey site—is invoked. It is relevant to notice that MT was originally introduced by Tikhonov (1950) and Cagniard (1953) in assumptions that the measurements are performed at a single site, and the conductivity distribution beneath this site is 1D, which inevitably leads to necessity of measuring both fields at the site. In case of a non1D problem setup and multiplesite survey layout, the requirement to measure both fields at each site seems could be relaxed. One of the options, which is already in use (e.g., Comeau et al. 2018), is a measurement of both fields on a subset of survey sites (say, on a coarser grid) and measurement of electric field only on an entire (dense) grid. In this scenario, one needs to invert instead of singlesite \({Z}\) the socalled intersite impedances \({Q}\) (Hermance and Thayer 1975)
where \( \mathbf{r}_r\) stands for the reference site(s) where both, electric and magnetic, fields are measured. Surprisingly enough no model study is reported yet with the discussion of advantages and disadvantages of complementing or substituting during 3D MT inversion the conventional (singlesite) impedances with intersite impedances. In this paper, we fill this gap and compare results of 3D inversion of \({Z}\) and \({Q}\). Our analysis is based on a 3D inversion of synthetic responses calculated for a 3D model which consists of two buried adjacent (resistive and conductive) blocks and thin resistor above them. We consider an ultimate case when both fields are available at one reference site and explore the dependence of the inversion results on the location of this site. For completeness, we also investigate the “performance” of a 3D inversion if two alternative intersite tensors, namely horizontal electric (telluric) tensor \({T}\) (Berdichevsky 1965; Yungul 1966)
and horizontal magnetic tensor \({M}\) (Berdichevsky and Dmitriev 2008)
are inverted. Here under “performance” we understand an ability of inversions to recover “true model”; note that the computational resources (memory, CPU time), needed to perform inversions of different tensors, are essentially the same. Final experiment we perform and discuss is a simultaneous inversion of \({Z}\) and \({M}\), and \({Z}\) and \({T}\).
We conclude the introduction with two remarks. The first refers to notations. We employ the notation \({T}\) for telluric tensor, following nomenclature which is routinely used in the literature where this type of responses is discussed (e.g., Berdichevsky 1965; Hermance and Thayer 1975; Yungul 1966). We know that many authors (e.g., Araya and Ritter 2016; Campanya et al. 2016) reserve \({T}\) for tippers—the singlesite response functions that relate vertical and horizontal magnetic field components
Here we use for tippers the notation \({W}\), following the original terminology [cf. Chapter 4 of Berdichevsky and Dmitriev (2008)]. The second remark concerns our decision not to discuss in the paper the 3D inversion of \({W}\), and \({Z}\) and \({W}\). This is done by purpose, since we believe that this topic is already well covered in the literature (e.g Meqbel et al. 2014; Rao et al. 2014; Siripunvaraporn and Egbert 2009; Tietze and Ritter 2013; Yang et al. 2015).
Modelling synthetic responses
To assess performance of 3D inversion of different MT responses, we use the model from Grayver (2015). The model consists of three rectangular blocks embedded in homogeneous background halfspace of conductivity 0.1 S/m ( cf. upper plots in Fig. 1). The shallow thin \(3\times 4\times 0.3\,{\text {km}}^3\) block of conductivity 0.002 S/m is located at the depth of 0.2 km and two deeper adjacent blocks of conductivities 1 S/m and 0.002 S/m, each of size \(7\times 4\times 4.5\,{\text {km}}^3\) are located at the depth of 1.5 km. Upper left and middle plots in Fig. 1 show plane view of the model at depths 300 and 2500 km, respectively. Upper right plot presents side view of the model taken along central, ydirected, profile, where x and yaxes are pointed up and right, respectively, at “plane view” plots.
Synthetic responses (data) were generated at a regular 2D \(13\times 13\) grid (black dots in upper left and middle plots of Fig. 1; spacing between grid points—1 km) for 16 frequencies evenly spaced on the logarithmic scale in the range of 0.001–100 Hz. Generation of the responses was performed using forward modelling code by Kruglyakov and Bloshanskaya (2017) which is based on contracting integral equation (CIE) approach (Pankratov et al. 1995; Singer 1995). Since CIEbased code was used for responses’ generation, the forward modelling domain was confined to anomalous regions (three blocks) only. The blocks were discretized by cubic cells of equal size of \(25\times 25\times 25\,{\text {m}}^3\). Two percent random Gaussian noise was added to each element of the generated responses.
Remarks on 3D MT inverse solution
We exploit for (regularized) inversion our own, scalable, 3D MT inverse solver which allows us to: (1) utilize different hardware from laptops to supercomputers; (2) deal with highly detailed and contrasting models, and (3) invert (separately or jointly) any type of single or/and intersite MT responses. Forward modelling code by Kruglyakov et al. (2016), also based on CIE approach, is called by inversion. To minimize target functional, the inverse solver uses gradienttype (quasiNewton) optimization algorithm, namely BFGS (Nocedal and Wright 2006). This (conventional) functional consists of the corresponding misfit term and a stabilizer. Stabilizer is weighted with parameter \(\lambda \) which regulates the smoothness of the model. In our implementation, a stabilizer approximates a gradientlike operator. The computation of misfit gradients is performed using adjoint sources formalism (Pankratov and Kuvshinov 2010).
During an inversion, one has to parametrize an inversion domain. We apply “mask parametrization” approach which allows us to merge any subset of forward modelling cells in order to account for (usually) irregular distribution of the observation sites and different resolution of MT data with respect to the depth. The sketch of such parametrization is demonstrated in Fig. 2.
Inversion setup
The inverse and forward modelling domains during inversion coincided and were set as \(16 \times 16 \times 10\,{\text {km}}^3\), thus including part of the volume occupied by the background medium. To diminish an “inverse crime,” the forward modelling grid was taken different from that used for the responses’ generation. Note that the inverse crime occurs when the same (or very nearly the same) ingredients are employed to synthesize as well as to invert data in an inverse problem; the first time the term is found in print seems to be in the book of Colton and Kress (1992). In lateral directions, the cell’s size was set 4 times larger, i.e., \(100\times 100\,\text {m}^2\), and in vertical direction the cell’s size increased geometrically from 0.2 to 600 m with overall vertical discretization of \(N_z=120\). Thus forward modelling domain was discretized by \(160\times 160 \times 120\) cells. The inverse modelling domain was discretized by \(20\times 20\times 40\) cells, i.e., one inverse modelling cell was a combination of \(8 \times 8 \times 3\) forward modelling cells. 1D section needed to calculate Green’s tensors in the course of forward modelling was chosen to coincide with background 1D section. Error floors of 0.02\(\sqrt{e_{xx}^2+e_{xy}^2+e_{yx}^2+e_{yy}^2}\) were adopted, where \(e_{xx},\ldots ,e_{yy}\) stand for elements of corresponding tensor at specific location. It is worth noting here that in case of \({Z}\) and \({Q}\) tensors the offdiagonal elements are dominant, whereas in case of \({M}\) and \({T}\) tensors—the diagonal elements are dominant.
Results of inversion
Inversion of (singlesite) \({Z}\)
Bottom plots in Fig. 1 demonstrates results of \({Z}\) inversion. Hereinafter the results of inversions are shown for the same cross sections as in upper plots of Fig. 1. Also, hereinafter the results are shown for the regularization parameter \(\lambda \) which corresponds to the knee of the Lcurve (Hansen 1992). One can observe from the figure that the shallow thin resistive block, as well as the buried conductive block, is recovered well, both in shapes and conductivity values. The recovery of buried resistive block is distinctly and expectedly worse; the image is blurred, and the block becomes less resolvable with depth.
Left upper plot in Fig. 3 shows cumulative (all periods, all sites, all tensor components) histograms of relative errors defined as
where \(\alpha ,\beta =\{x,y\}\), \(e_{\alpha ,\beta }\) denote synthetic responses and \(e^{\mathrm{{mod}}}_{\alpha ,\beta }\)—responses obtained either from starting or recovered models. The plot illustrates the fact that inversion gets to an appropriate target misfit given the number of data and noise in the data. Indeed, it is seen that substantially all normalized errors for the recovered model lie within target 2% error floor. Remarkably, the similar behaviour is observed for the histograms (cf. other plots in the figure) resulting from the inversions of intersite responses to be discussed below.
Inversion of intersite tensors
The results of inversion of intersite impedance tensor \({Q}\) are shown in Fig. 4. From top to bottom are the results for different location of the reference sites. These sites were placed, respectively, in an 1D environment, above the centre of conductive and above the centre of deep resistive blocks. Their locations are depicted in upper left and middle plots of Fig. 1 by yellow dots A, B and C. It is seen from Fig. 4 that location of reference site has marginal effect on the inversion results. Remarkably, inversion of \({Q}\) recovers the “true” structures as good as inversion of \({Z}\) does (cf. bottom plots of Fig. 1).
Next we inverted horizontal electric tensor \({T}\). The results are shown in Fig. 5a for the case when the reference site was placed in an 1D environment. We observe that results of inversion of \({T}\) are slightly worse than those obtained by inversion of either \({Z}\) or \({Q}\). The inversion results for the cases with (two) alternative locations of the reference site are not shown since they—as in the case of inversion of intersite impedances—differ insignificantly.
Further model experiment was the inversion of horizontal magnetic tensor \({M}\) (Fig. 5b) which was performed, again, for the case when the reference site was placed in an 1D environment. The results of \({M}\) inversion turned out to be notably worse than those from inversion of \({Z}\), \({Q}\) or \({T}\), irrespective of reference site location (the results for \({M}\) inversion for other locations of reference site are not shown, since they were very similar, as it was in cases of \({Q}\), or \({T}\) inversions).
Final experiment we performed was simultaneous inversion of \({Z}\) and \({M}\), and \({Z}\) and \({T}\) (see Fig. 5c, d). Note that Campanya et al. (2016) also discussed inversion of \({Z}\) and \({M}\) (however using different 3D model) and concluded that simultaneous inversion improves the recovery of true 3D structures. Our inversion does not support this inference. Contrary, joint inversion of \({Z}\) and \({M}\) recovers true structures slightly worse than inversion of \({Z}\) (cf. bottom plots of Fig. 1). Simultaneous inversion of \({Z}\) and \({T}\) likewise does not reveal the improvement in the recovery of true structures compared with individual inversion of \({Z}.\)
Conclusions
We assessed in the paper the advantages and disadvantages of complementing or substituting conventional MT responses (impedances, \({Z}\)) with intersite transfer functions such as intersite impedance tensor, \({Q}\), horizontal magnetic, \({M}\), and horizontal electric, \({T}\), tensors. Our analysis was based on a 3D inversion of synthetic responses calculated for a 3D conductivity model which included buried conductive and resistive blocks. From our model study, we conclude that 3D inversion of either \({Z}\) or \({Q}\) recovers the “true” structure equally well. This, in particular, further promotes MT survey layout where measurements of both (electrical and magnetic) fields are conducted on a subset of survey sites (say, on a coarser grid, or even at a single reference site), and measurements of only electric field on an entire (dense) grid.
We also observed that recovery of the true structures is slightly worse if \({T}\) tensor is inverted, and considerably worse if \({M}\) tensor is inverted.
We attempted to improve the recovery of true 3D structures by performing simultaneous inversion of \({Z}\) and \({M}\), or \({Z}\) and \({T}\); however, we found that such inversions did not do better job, compared to individual inversions of \({Z}\) or \({Q}\).
Finally, we note that the location of reference site, which is required for calculating intersite \({Q}\), \({T}\) and \({M}\) tensors, had marginal effect on the inversion results.
Abbreviations
 MT:

magnetotelluric
 BFGS:

Broyden–Fletcher–Goldfarb–Shanno optimization algorithm
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Authors’ contributions
AK initiated the study. MK performed forward and inverse modellings. MK and AK analysed and discussed the results, MK drafted the manuscript. Both authors read and approved the final manuscript.
Acknowlegements
The authors thank Colin Farquharson and Yosuo Ogawa for constructive comments on the manuscript. This work is partly supported by the Swiss National Science Foundation Grant No. ZK0Z2 163494, and the Swiss National Supercomputing Centre (CSCS) Grants (Projects IDs s660 and s828). The authors also acknowledge the team of HPC CMC Lomonosov MSU for access to Bluegene/P HPC.
Competing interests
The authors declare that they have no competing interests.
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Keywords
 Magnetotellurics
 3D inversion
 Intersite tensors