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# Use of ssq rotational invariant of magnetotelluric impedances for estimating informative properties for galvanic distortion

- T. Rung-Arunwan
^{1, 2, 3, 4}Email authorView ORCID ID profile, - W. Siripunvaraporn
^{1, 2}and - H. Utada
^{3}

**Received:**17 November 2016**Accepted:**31 May 2017**Published:**14 June 2017

## Abstract

Several useful properties and parameters—a model of the regional mean one-dimensional (1D) conductivity profile, local and regional distortion indicators, and apparent gains—were defined in our recent paper using two rotational invariants (det: determinant and ssq: sum of squared elements) from a set of magnetotelluric (MT) data obtained by an array of observation sites. In this paper, we demonstrate their characteristics and benefits through synthetic examples using 1D and three-dimensional (3D) models. First, a model of the regional mean 1D conductivity profile is obtained using the average ssq impedance with different levels of galvanic distortion. In contrast to the Berdichevsky average using the average det impedance, the average ssq impedance is shown to yield a reliable estimate of the model of the regional mean 1D conductivity profile, even when severe galvanic distortion is contained in the data. Second, the local and regional distortion indicators were found to indicate the galvanic distortion as expressed by the splitting and shear parameters and to quantify their strengths in individual MT data and in the dataset as a whole. Third, the apparent gain was also shown to be a good approximation of the site gain, which is generally claimed to be undeterminable without external information. The model of the regional mean 1D profile could be used as an initial or a priori model in higher-dimensional inversions. The local and regional distortion indicators and apparent gains could be used to examine the existence and to guess the strength of the galvanic distortion. Although these conclusions were derived from synthetic tests using the Groom–Bailey distortion model, additional tests with different distortion models indicated that these conclusions are not strongly dependent on the choice of distortion model. These galvanic-distortion-related parameters would also assist in judging if a proper treatment is needed for the galvanic distortion when an MT dataset is given. Hence, this information derived from the dataset would be useful in MT data analysis and inversion.

## Keywords

- Magnetotellurics
- Rotational invariant
- Galvanic distortion

## Introduction

To obtain reliable three-dimensional (3D) inversion from magnetotelluric (MT) data, either distorted or undistorted, the choice of an initial or a priori model is crucial. The benefit of having a good model of the regional mean one-dimensional (1D) profile as an initial or a priori model has been reported in previous studies. For example, the optimal model of the mean 1D conductivity profile would minimize the lateral conductivity contrast, which could yield a better-conditioned system of equations (Avdeev 2005). Furthermore, the use of the mean 1D profile as an a priori model would result in the rapid and stable convergence of higher-dimensional inversion problems (Tada et al. 2012).

*z*, and \({\text {d}}S\) is a spherical surface element. Once the global mean 1D conductivity profile is obtained in this way (Eq. 1), the 3D conductivity distribution at any position within the Earth can be expressed as a combination of the global mean 1D model and the azimuthal conductivity contrast as

The definition of the global mean 1D conductivity profile and the azimuthal contrast is clear in theory, but the estimation of them is not easy in practice. Although it is possible to perform global induction studies using geomagnetic observatory data, there are significant differences among existing inverted models (e.g., Kelbert et al. 2009; Kuvshinov and Semenov 2012; Semenov and Kuvshinov 2012). Most likely, such attempts may include biases due to the nonuniformity of their site distributions or false images resulting from spatial aliasing (Utada and Munekane 2000) because the distributions of existing geomagnetic observatories and MT observation sites are spatially nonuniform. More importantly, the EM induction method is generally sensitive to the conductivity beneath each observation site.

The regional mean 1D profile \(\sigma _{\text {R}}(z)\) is practically unknown beforehand, although it can be estimated from \(\sigma (x,y,z)\) inverted from the observed data. However, in this study, we use an alternate method in which \(\sigma _{\text {R}}(z)\) is first estimated from the observed data in the area of interest. The conductivity model, either \(\sigma (x,y,z)\) or \(\Delta \sigma (x,y,z)\), can then be estimated by 3D inversion using \(\sigma _{\text {R}}(z)\) as a priori information or a starting model.

This poses the problem of how to reliably estimate a model of the regional mean 1D profile from an array of MT observations in a general 3D situation. Baba et al. (2010) already presented a solution to such a problem in the case of a seafloor MT study. Here, we consider the case in which MT data are obtained from an array of observations on land. The solution of this problem is not straightforward because MT data on land are usually affected by galvanic distortion, i.e., an alteration in the MT impedance due to near-surface small-scale heterogeneity that is smaller than a typical site spacing and confined to be shallower than the inductive scale length of interest (Ledo et al. 1998; Utada and Munekane 2000; Bibby et al. 2005). In other words, the physical dimensions of the distorting bodies are smaller than their inductive scale length and also that of the host.

Berdichevsky et al. (1980) proposed a scheme to estimate a model of the mean 1D profile from distorted data by averaging the effective resistivity, which is equivalent to the apparent resistivity derived from the determinant of the impedance tensor (hereafter denoted as the det impedance, \(Z_{\rm{det}}= \sqrt{Z_{xx}Z_{yy}-Z_{xy}Z_{yx}}\) ). This is a statistical approach to smooth the effect of galvanic distortion that is supposed to be a random phenomenon, and the average in this method is referred to as the Berdichevsky average (Rung-Arunwan et al. 2016). When it was first introduced, the det impedance was generally used in regional studies (e.g., Berdichevsky et al. 1980; Jones 1988; Berdichevsky et al. 1989). However, it was later adopted in two-dimensional (2D) MT applications (e.g., Oldenburg and Ellis 1993; Pedersen and Engels 2005). It is also applied as the current channeling indicator (Lezaeta and Haak 2003) and used in environmental applications (Falgàs et al. 2009). The det impedance has also been used in recent works. For example, Seama et al. (2007) inverted the det impedances from marine MT data at each observation point to obtain 1D conductivity profiles beneath the Philippine Sea, Arango et al. (2009) used the det impedance to interpret 3D MT data, and Baba et al. (2010) and Avdeeva et al. (2015) used the average det impedance in the same way as Berdichevsky et al. (1980).

However, when the det impedance was re-examined on the basis of the present knowledge of galvanic distortion by applying the Groom–Bailey model (Groom and Bailey 1989), it was found that the magnitude of the det impedance is always biased downward by the geometric distortion expressed by the shear and splitting parameters (see Gómez-Treviño et al. 2013; Rung-Arunwan et al. 2016). Note that Gómez-Treviño et al. (2013) studied the effect of galvanic distortion on the rotational invariants in the case of 2D regional structures. Even in the absence of a site gain, the Berdichevsky average causes downward bias in the apparent resistivity as compared with those from the regional mean 1D conductivity profile (\(\sigma _{\text {R}}(z)\)) defined by Eqs. (3) or (4) (Rung-Arunwan et al. 2016).

Rung-Arunwan et al. (2016) proposed another method for estimating the model of the mean 1D profile that was similar to the method of using the Berdichevsky average but redefined it with another rotational invariant: the sum of the squared elements of the impedance tensor (ssq impedance) \(Z_{\text {ssq}}=\sqrt{ \left( Z_{xx}^2 + Z_{xy}^2 + Z_{yx}^2 + Z_{yy}^2\right) /2}\) (Szarka and Menvielle 1997). Note that the ssq and det impedances are identical in the case of 1D earth, but for 2D and 3D earth, the induction sensed by the ssq and det impedances is different (see also Szarka and Menvielle 1997) by \(Z_{\text {ssq}}^2 - Z_{\text {det}}^2 = \frac{1}{2}(Z_{xx}-Z_{yy})^2 + \frac{1}{2}(Z_{xy}+Z_{yx})^2\). In comparison with the det impedance, the ssq impedance has been proven to be less biased by the distortion parameters (see Gómez-Treviño et al. 2013; Rung-Arunwan et al. 2016). An example of the field data from the western part of Thailand is shown in Fig. 1. The field example is consistent with the theoretical prediction presented in Rung-Arunwan et al. (2016) that the det impedance will have a smaller magnitude than the ssq impedance. According to the prediction, the downward bias for the det impedances is supposed to be caused by a geometric (shear and splitting) effect because the phase characteristics are almost identical. Consequently, the use of average ssq impedance is expected to more reliably estimate the model of the regional mean 1D profile than the use of the det impedance. Thus, this field example motivated us to present a systematic investigation of the approaches proposed by Rung-Arunwan et al. (2016) in this paper.

Identification and removal methods for galvanic distortion remain undetermined (Chp. 6 in Chave and Jones 2012), although several attempts to solve the problem of galvanic distortion have been presented. Some studies assumed a 2D Earth (e.g., Bahr 1988; Groom and Bailey 1989), whereas others confronted the nonuniqueness of the obtained solution (e.g., Bibby et al. 2005). Moreover, Gómez-Treviño et al. (2014) presented an approach to estimate the 2D regional impedance and distortion parameters, i.e., twist and shear in the Groom–Bailey model, using the det and ssq impedances. Inversion based on the phase tensor (Caldwell et al. 2004), which yields a well-defined distortion-free solution, is also a promising strategy. However, the phase tensor is only a partial solution; thus, the inverted model strongly depends on the initial model (Patro et al. 2013; Tietze et al. 2015). In addition to decomposition approaches, inversion schemes that simultaneously solve the static shift (e.g., Sasaki and Meju 2006) have become feasible, but the geometric distortion is not controlled. Avdeeva et al. (2015) proposed 3D inversion with the solution of the full distortion matrix, but this approach does not allow this static shift to be a free parameter.

Although a number of approaches for handling galvanic distortion have been developed, an approach for determining the presence of galvanic distortion in the observed data has not been presented, except the concept of galvanic distortion indicators by Rung-Arunwan et al. (2016). The ability to identify the presence of galvanic distortion—either geometric or scaling—contained in the observed data and to quantify their intensity is undoubtedly important because the application of the galvanic distortion treatment to the observed data without knowing the presence of galvanic distortion and doing so may either improve or deteriorate the reliability of MT data interpretation.

Rung-Arunwan et al. (2016) proposed two types of galvanic distortion indicators. First, the local and regional distortion indicators are used to determine the strength of the geometric distortion as expressed by the shear and splitting effects on the basis of the fact that the geometric distortion (shear and splitting) has different effects on the det and ssq impedances. Second, the apparent gain is defined to be an approximation of the site gain (scaling in the impedance magnitude), which has been presumed to be indeterminable without other independent information (Groom et al. 1993; Bibby et al. 2005). These parameters may help quantitatively indicate the strength of the galvanic distortion posed in MT data. In addition, the employment of these two types of properties allows the effect of the site gain to be separated from the effects of the twist, shear, and splitting parameters. Most importantly, we can use these parameters to determine the necessary treatment of galvanic distortion for a given dataset, such as whether or not a removal scheme should be applied in the inversion (e.g., Sasaki and Meju 2006; Avdeeva et al. 2015).

## Theoretical background

This section briefly summarizes the method for estimating a model of the regional mean 1D profile and a set of parameters related to the galvanic distortion, which were presented in Rung-Arunwan et al. (2016).

*e*and

*s*, which are also called the geometric distortion, than the det impedance, the amplitude of which is always biased downward by these two parameters (Rung-Arunwan et al. 2016). After re-examination with the Groom–Bailey model of galvanic distortion, the Berdichevsky average is written as

*i*th observed (perhaps distorted) MT impedance at the position \(\mathbf {r}_i\); \(e_i\) and \(s_i\) are the shear and splitting parameters at the

*i*th station, respectively;

*N*is the total number of observations; \(\omega\) is the angular frequency; and \(Z^{\text {R}}_{\mathrm {det}}(\omega )\) is the regional det impedance. Note that the twist parameter has no effect on the det and ssq impedances (also discussed in Gómez-Treviño et al. 2013; Rung-Arunwan et al. 2016). If geometric distortion is contained in the data, the coefficient in Eq. (6) becomes effective and is always smaller than unity. Hence, the use of the Berdichevsky average always gives a downward-biased regional 1D impedance, which yields an inverted model of the structure that is more conductive than the true structure.

Additionally, a set of parameters related to the galvanic distortion is defined as follows.

*local distortion indicator*(LDI) indicates the strength of the shear and splitting parameters at a single station individually and is defined as the squared ratio of the ssq impedance to the det impedance:

Defined in this way, the LDI is intrinsically independent of the site gain. As the twist parameter has no effect on the det and ssq impedances, the presence of the twist effect cannot be ascertained from the LDI. Employing the fact that the shear and splitting distortion affects the det and ssq impedances differently (Gómez-Treviño et al. 2013; Rung-Arunwan et al. 2016), the LDI represents the effects of the shear and splitting parameters as a combination, which is unlike the decomposition approaches (e.g., Groom and Bailey 1989; McNeice and Jones 2001; Gómez-Treviño et al. 2014), where the distortion parameters, twist and shear in particular, are estimated.

*regional distortion indicator*(RDI) also indicates the strength of the shear and splitting parameters but on a regional scale, i.e., it quantitatively indicates how strongly distorted the dataset is on average. It is defined as the geometric mean of the LDIs:

*apparent gain*is defined as the ratio of a rotational invariant at a given position to its regional average. As we are interested in two rotational invariants, the corresponding apparent det and ssq gains are derived as

*i*th observation site. Obviously, if the data are strongly distorted, the apparent det gain underestimates the site gain because of the shear and splitting parameters. Thus, the apparent ssq gain is expected to be the more accurate approximation of the site gain when the data are strongly distorted. In the following sections, the characteristic and behaviors of these parameters are synthetically examined.

## Estimation of a model of the regional mean 1D profile

Rung-Arunwan et al. (2016) proposed a modification to the Berdichevsky average—the use of the average ssq impedance instead of the average det impedance—to avoid biasing from galvanic distortion. This section examines whether the proposed method can reliably estimate a model of the regional mean 1D profile from synthetically distorted data.

*g*and the twist

*t*, shear

*e*, and splitting

*s*of the parameters of the Groom–Bailey model were generated following a normal distribution (Fig. 3). The distorted impedances were then calculated by applying these random parameter values to the synthetic impedances. More explicitly, we assumed that each set of distortion parameters has a mean of zero and is bounded by \((-1,+1)\). If any values are outside the bound, random numbers were generated again so that the set of random distortion parameters conforms with the bound. The random site gain was generated on a logarithmic scale without a bound. To quantitatively control the strength of the galvanic distortion, the standard deviation (SD) of the normal distribution of each parameter was varied. Five SD values of 0.1, 0.2, 0.3, 0.4, and 0.5 were used. Finally, five MT datasets with 25 stations each and different galvanic distortion strengths were considered.

### 1D example

### 3D example

*i*th station is given by

*s*is the typical site spacing, which is 32 km in this case; and \(r_x\) and \(r_y\) are uniform random numbers bounded by \((-0.5,+0.5)\).

## Examination of the consistency between the theoretical and estimated models of the regional mean 1D profile

According to the fact that the host layer earth or background is absolutely unknown in reality, the estimated models of the regional mean 1D conductivity profiles from 3D models should not be compared with the synthetic layered-earth model (the model in Fig. 2a, for example). Instead, it should be compared with the theoretical regional mean 1D conductivity profiles, the linear and logarithmic averages of the lateral conductivity distribution (Eqs. 3 and 4). Obviously, the regional mean 1D profiles, either theoretical or estimated, depend on the array size and location when the subsurface structure is laterally heterogeneous. This section aims to examine the consistency between the defined and estimated models of the regional mean 1D profile and the effect of the consistency on the location of the array and its size relative to the anomaly size through synthetic modeling.

In this situation, where the array is much larger than the typical anomaly, the regional mean 1D conductivity profiles from different array locations, both theoretical and estimated, are shown to be almost identical to the theoretical model. This is also a consequence of the application of the averaging approach, in which the effects of the positive and negative anomalous conductivities are averaged out. In other words, the theoretical and estimated models of the regional mean 1D profile are nearly independent of the array location when the array size is much larger than the typical anomaly size.

In general, the observation array should be designed to cover the structure of interest if its size is known a priori. However, if the anomaly size is found *a posteriori* to be comparable to or even larger than the size of observation array, 3D inversion of any approach will fail to accurately image the heterogeneity. To obtain more reliable results, one suggestion in such a case is to add more MT observations to make the array size sufficiently greater than the anomaly size.

## Local and regional distortion indicators

On the basis of the fact that the galvanic distortion has different effects on the det and ssq impedances, the LDI and RDI given by Eqs. (8) and (9), respectively, were constructed to quantify the strength of the geometric distortion that can be described by the shear and splitting parameters. This section examines the numerical results of LDIs and RDIs derived from the synthetic 1D and 3D examples presented in Sections 3.1 and 3.2, respectively.

*i*th station,

*M*is the number of periods. The percentage error in the mean LDI is calculated with

One possible practical usage of the LDI is the omission of some stations with heavily distorted impedances from the interpretation or inversion if the number of such sites is small. If a limited number of sites showing heavy distortion are removed, the RDI after removal is supposed to be small. Conversely, if the RDI still exhibits a high value, a proper treatment for the galvanic distortion, such as inversion including the galvanic distortion (e.g., DeGroot-Hedlin 1995; Ogawa and Uchida 1996; Sasaki and Meju 2006; Avdeeva et al. 2015) or an MT data analysis (e.g., Weaver et al. 2000; Caldwell et al. 2004), will be essential. The combination of LDIs and RDI helps to provide insight, at least to some extent, as to which approach should be applied to a set of MT impedances obtained from observation.

## Apparent gains

From the theoretical derivation, the apparent ssq gain is expected to correctly estimate the site gain in 1D cases and to yield a good approximation of it in 3D cases, whereas the apparent det gain underestimates the synthetic site gain if the data are strongly affected by geometric distortion. In this section, we demonstrate the use of the apparent gains obtained from the synthetic 1D and 3D examples described in Sections 3.1 and 3.2, respectively.

*M*, the mean apparent gains can be written as

*i*th station. In spite of the large site-to-site variation in the synthetic site gain of nearly one order of magnitude (two orders of magnitude in terms of the static shift in the apparent resistivity), its estimation error by the mean apparent ssq site gains is as small as only a few percent.

Unlike the 1D case, the existing 3D anomalies may cause further uncertainty, as the apparent ssq gain has been demonstrated to be affected by the induction effect from the underlying 3D structure. For example, the mean apparent ssq gains from stations over the conductive structure (e.g., stations syn07 and syn19) tend to be slightly smaller than the synthetic site gains (Fig. 23b). In spite of this, the percentage differences (Eq. 18) still remain about 10% which is within the statistical uncertainty (Fig. 23a). The regional distortion indicator in this case (Fig. 19b) shows a feature consistent with the distorted 1D case (Fig. 19a) at periods shorter than 15 s. If we estimate the mean ssq gain from this period band instead of Eq. (16), the percentage gain difference becomes as small as 5%.

In previous works, the site gain is considered or regarded to be an indeterminable distortion parameter if other independent geophysical data, e.g., transient electromagnetic (TEM) data (Beamish and Travassos 1992; Groom et al. 1993; Bibby et al. 2005; Árnason 2015), are not available. However, the TEM data may not be available at all MT stations. In addition, the static shift could be corrected with the TEM data with some limitations (see Watts et al. 2013; Tournerie et al. 2007; Wilt and Williams 1989), e.g., when the heterogeneity is smaller than the transmitter loop. Utada and Munekane (2000) attempted to solve this problem by introducing Faraday’s law as a constraint, but the solution was not practical. The numerical examples presented here show that the concept of the apparent gain can be used to approximate the site gain in the assumed situation.

This paper considers the galvanic distortion caused only by small-scale heterogeneities (smaller than the typical site spacing and confined within a near-surface layer shallower than the inductive scale length of interest). Thus, the effect of galvanic distortion is considered as spatial aliasing in the MT data. The apparent gain can then be regarded as a shift in the magnitude of the impedance relative to the average value. For the case where the data are systematically shifted by some near-surface structure larger than or comparable to the array size (see Section 4), e.g., a valley environment such as of the Rhine Graben model (see Chp. 6 in Chave and Jones 2012), the apparent ssq gain may be distributed around some biased central values or may not be normally distributed on a logarithmic scale. In such cases, the concept of the apparent gain should be used with caution.

## Dependence on the distortion model

The proposed method (Rung-Arunwan et al. 2016) is theoretically formulated on the basis of the Groom–Bailey model of galvanic distortion. In this paper, it is shown numerically that the use of the average ssq impedance is a reliable method for estimating the regional mean 1D conductivity profile, and the combination of the two rotational invariant impedances helps to detect the geometric distortion and to approximate the site gain. Although the Groom–Bailey model is well known and adopted by a number of studies, it is not the only model. The distortion operator \(\mathbf {C}\) can be parameterized using other models (e.g., Bahr 1988; Chave and Smith 1994; Smith 1995; Tietze et al. 2015). Therefore, the galvanic distortion model dependence of the proposed methods may be questionable.

Under the PIM model, the site gain is likely to have a skewed distribution, which can compensate for the downward bias of det impedances caused by geometric distortion (shear and splitting), e.g., the case of weak distortion in the PIM model. Therefore, it is possible that there may be a case where the average det impedance is more appropriate than the average ssq impedance for a particular condition. However, searching for a special case where the skewed site gain and the bias due to the geometric distortion are balanced is out of the scope of this paper.

From results shown above, it is clear that the proposed method does not strongly depend on the choice of galvanic distortion model. The use of the average ssq impedance is a promising method for estimating the regional mean 1D conductivity profile and a good approximation for the site gain. The results confirmed that a combination of the rotational invariants (det and ssq) is useful for detecting the galvanic distortion. More details of the PIM model test can be found in Section 4 of the Additional file 1.

## Conclusions

This paper presents numerical examples of the properties and galvanic-distortion-related parameters (a model of the regional mean 1D conductivity profile, the local and regional distortion indicators, and the apparent gains) that can be obtained from a set of distorted MT impedances. By correcting the traditional Berdichevsky average, this study has shown that a model of the regional mean 1D profile can be correctly estimated by using the average ssq impedance. Regardless of the galvanic distortion strength, the average ssq impedance gives a reliable model of the regional mean 1D conductivity profile. The local and regional distortion indicators were defined to detect the effects of the shear and splitting parameters in the Groom–Bailey model of galvanic distortion, and the apparent gains were used to approximate the magnitude of the site gain in some cases presented in this paper. The use of these parameters may help to quantify the intensity of the galvanic distortion contained in MT data and determine the need for the proper treatment of the galvanic distortion. For example, if the distortion in a given dataset is proven to be solely caused by the site gain, only gain correction is required. In addition to gain correction, if the data at only a few stations include strong geometric distortion, as revealed by the local and regional distortion indicators, the data could be omitted or weighted less during 3D inversion. Note that a model of galvanic distortion is applicable for a particular frequency band where the distortion is expressed by a real-valued \(2\times 2\) tensor. This can be tested by checking whether the apparent ssq gain and LDI are real-valued and almost frequency independent at each site. This test also shows that the LDI can be used to justify the use of the impedance phase tensor as well. Because the apparent gain is a good approximation of the site gain, it can be used as an initial guess for the static shift in a 3D inversion. All of the results of the present study would resolve several difficulties encountered during the inversion of a set of MT impedances that are contaminated by galvanic distortion.

## Declarations

### Authors' contributions

TR conducted the numerical experiments and wrote this manuscript. WS critically revised this manuscript. HU provided basic ideas and critically revised this manuscript. All authors read and approved the final manuscript.

### Acknowledgements

We wish to thank Yasuo Ogawa (the Editor), Ian Ferguson, one anonymous reviewer for constructive comments, and Alexey Kuvshinov for discussion. This work was partially supported by the Thailand Center of Excellence in Physics (ThEP) for WS and TR and was carried out as a part of an internship program for TR partially supported by the Earthquake Research Institute, the University of Tokyo. Most of the figures were produced using the Generic Mapping Tools (GMT) software (Wessel and Smith 1998).

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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