# 3-D inversion of MT impedances and inter-site tensors, individually and jointly. New lessons learnt

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## 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 inter-site transfer functions such as inter-site impedance tensor, \({Q}\), horizontal magnetic, \({M}\), and horizontal electric, \({T}\), tensors. Our analysis is based on a 3-D inversion of synthetic responses calculated for a 3-D model which consists of two buried adjacent (resistive and conductive) blocks and thin resistor above them. The (regularized) 3-D inversion is performed using scalable 3-D MT inverse solver with forward modelling engine based on a contracting integral equation approach. The inversion exploits gradient-type (quasi-Newton) optimization algorithm and invokes adjoint sources approach to compute misfits’ gradients. From our model study, we conclude that: (1) 3-D 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 3-D 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 inter-site \({Q}\), \({T}\) and \({M}\) tensors, has also marginal effect on the inversion results.

## Keywords

Magnetotellurics 3-D inversion Inter-site tensors## Abbreviations

- MT
magnetotelluric

- BFGS
Broyden–Fletcher–Goldfarb–Shanno optimization algorithm

## Introduction

## Modelling synthetic responses

*y*-directed, profile, where

*x*- and

*y*-axes are pointed up and right, respectively, at “plane view” plots.

Synthetic responses (data) were generated at a regular 2-D \(13\times 13\) 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 CIE-based 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 3-D MT inverse solution

We exploit for (regularized) inversion our own, scalable, 3-D 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 inter-site 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 gradient-type (quasi-Newton) 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 gradient-like operator. The computation of misfit gradients is performed using adjoint sources formalism (Pankratov and Kuvshinov 2010).

## 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. 1-D section needed to calculate Green’s tensors in the course of forward modelling was chosen to coincide with background 1-D 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 off-diagonal elements are dominant, whereas in case of \({M}\) and \({T}\) tensors—the diagonal elements are dominant.

## Results of inversion

### Inversion of (single-site) \({Z}\)

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 *L*-curve (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.

### Inversion of inter-site tensors

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 1-D 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 3-D model) and concluded that simultaneous inversion improves the recovery of true 3-D 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 inter-site transfer functions such as inter-site impedance tensor, \({Q}\), horizontal magnetic, \({M}\), and horizontal electric, \({T}\), tensors. Our analysis was based on a 3-D inversion of synthetic responses calculated for a 3-D conductivity model which included buried conductive and resistive blocks. From our model study, we conclude that 3-D 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 3-D 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 inter-site \({Q}\), \({T}\) and \({M}\) tensors, had marginal effect on the inversion results.

## Notes

### 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.

### Availability of data and materials

Not applicable.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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