Physics-Driven Deep Learning Inversion with Application to Magnetotelluric

SUMMARYBecause the magnetotelluric (MT) method uses natural sources, the electric and magnetic fields recorded in the field acquisition are not directly used but usually converted into other MT response functions for interpretation such as inversion. Considering that inversion results are dependent on types of input data, it can be helpful to analyse different characteristics of MT response functions for inversion. In this study, we examine sensitivity patterns of MT response functions used commonly in MT inversion, which are the impedance tensor, apparent resistivity, phase, tipper, effective impedance and phase tensor; and investigate how their sensitivity patterns affect inversion results. We first describe overall tendencies of 3-D sensitivity patterns of the MT response functions, and then classify the MT response functions into six groups based on 2-D sensitivity patterns computed at the surface, which are briefly called ‘surface-sensitivity patterns’ in this study. The ’diagonal components of the impedance’ and ‘off-diagonal components of the phase tensor’, which have four petals-shaped surface-sensitivity patterns along the diagonal directions, belong to Group 1, and contribute to imaging 3-D subsurface structures from receivers installed evenly at the surface. Group 2 contains the ‘xy-components of the impedance, apparent resistivity and phase’ and ‘yy-component of the phase tensor’ whose surface-sensitivity patterns are linear in the y-axis. The ‘yx-components of the impedance, apparent resistivity and phase’ and ‘xx-component of the phase tensor’ that have strong linear surface-sensitivity patterns along the x-axis are classified into Group 3. The MT response functions of Groups 2 and 3 are useful for inversion of structures close to 2-D, whose strike extends along the y- and x-axes, respectively. Groups 4 and 5 include the ‘x- and y-components of tipper’ that possess linearly aligned two petals-shaped surface-sensitivity patterns in the x- and y-axes, respectively. The tipper can be helpful in imaging both 2-D and 3-D structures. The ‘effective impedance’ belongs to Group 6, whose surface-sensitivity patterns appear as a small circle. The surface-sensitivity patterns allow the effective impedance to have an advantage in interpretation of 1-D structures. By using several MT response functions for specific cases of 1-D, 2-D and 3-D interpretation of MT data, we investigate whether characteristics of the sensitivity patterns are reflected in modelling (simulating field data) and inversion results, and then suggest optimal MT response functions for those cases. In doing so, we show how to utilize the characteristics of the sensitivity patterns in inversion, and recommend the input MT response functions for inversion according to MT exploration situations. Our study provides basic information on similarities and differences of major MT response functions for inversion and insights on which MT response functions are suitable to increase the feasibility of MT inversion for different field situations based on the sensitivity patterns.

Magnetotelluric (MT) inversion algorithms are an essential tool in exploration geophysics because they provide us with resistivity models of the subsurface. Inverse problems are ill-posed and non-unique. Resistivity models produced by geologically unconstrained MT inversions are typically smooth, which means that the geological interpretations obtained based on the resistivity models will be ambiguous and uncertain. To solve the above problems, geophysicists have made many efforts over the years to introduce any prior information (such as geological structure information and physical property data) into models, which could help to better constrain the inverted model with the inversion algorithms. In the last few years, there has been renewed interest in machine learning techniques. Various machine learning methods have been applied to inversions. Using the fuzzy c-means (FCM) clustering method, geophysicists proposed new inversion algorithms that are capable of building petrophysical information into the inversion. Although the FCM has been used and advanced in previous work, geophysicists concluded that a priori information of the correct number and value of cluster centers is very important for a successful FCM inversion. In fact, it is difficult to obtain the appropriate clustering information in some geophysical survey areas. In this study, we present an effective way to build the petrophysical information for the MT inversion based on FCM clustering algorithm. When the actual petrophysical information is insufficient, considering the characteristics of MT data, we perform a one-dimensional blocky inversion that can clearly identify the interface with distinct electrical property contrast; then, we obtain the number and value of cluster centers from 1D blocky inversion for the further MT inversion with FCM clustering. The algorithm uses guided FCM clustering to improve the model within the iterative minimization during MT inversion. In our MT inversion method, we integrate the geophysical inversion and geological differentiation into a unified scheme, which interact and enhance each other. The resistivity models obtained from the inversion respect the geophysical and petrophysical data and are easy to geologically interpret. We tested the algorithm using two synthetic examples and a field data example.

Magnetotelluric (MT) inversion is widely used to image subsurface electrical resistivity structures, but three-dimensional (3-D) MT inversion is inherently non-unique, and the resulting models can depend strongly on the choice of the initial model. Despite this well-known sensitivity, systematic evaluation of initial-model-induced variability remains uncommon in practical 3-D MT studies due to the high computational cost of running multiple inversions. In this study, we propose a practical and computationally efficient framework to quantify inversion uncertainty arising specifically from differences in the initial model. The approach employs a low-dimensional parameterization based on representative points and uses Kriging interpolation to generate an ensemble of smooth, geologically plausible starting models. Each realization is inverted independently using identical inversion settings, allowing initial-model effects to be isolated under fixed regularization and data-error assumptions. The method is applied to a 3-D MT dataset from the Yuzawa geothermal field in northeastern Japan. A total of 100 inversions were performed, from which 55 well-converged realizations (final RMS ≤ 2.15) were selected for analysis. Ensemble statistics reveal that shallow conductive structures are reproduced consistently across realizations, whereas variability increases with depth and exhibits strong spatial dependence. Bootstrap resampling confirms that the depth-dependent variability patterns are statistically stable under the present ensemble size. Although the proposed framework does not account for all possible sources of inversion uncertainty, it provides an operationally realistic lower-bound estimate of model variability associated with initial-model choice. By identifying which parts of the resistivity structure are robust and which are weakly constrained, the method supports uncertainty-aware interpretation of 3-D MT inversion results and demonstrates its practical applicability through a geothermal case study.

We present a novel 2‐D magnetotelluric (MT) inversion scheme, in which the local weights of the regularizing smoothness constraints are based on the envelope attribute of a reflection seismic image. The weights resemble those of a previously published seismic modification of the minimum gradient support method. We measure the directional gradients of the seismic envelope to modify the horizontal and vertical smoothness constraints separately. Successful application of the inversion to MT field data of the Collisional Orogeny in the Scandinavian Caledonides (COSC) project using the envelope attribute of the COSC reflection seismic profile helped to reduce the uncertainty of the interpretation of the main décollement by demonstrating that the associated alum shales may be much thinner than suggested by a previous inversion model. Thus, the new model supports the proposed location of a future borehole COSC‐2 which is hoped to penetrate the main décollement and the underlying Precambrian basement.

The gradient-based optimization methods are preferable for the large-scale three-dimensional (3D) magnetotelluric (MT) inverse problem. Compared with the popular nonlinear conjugate gradient (NLCG) method, however, the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method is less adopted. This paper aims to implement a L-BFGS-based inversion algorithm for the 3D MT problem. And we develop our code on top of the ModEM package, which is highly extensible and popular among the MT community. To accelerate the convergence speed, the preconditioning technique by the affine linear transformation of the original model parameters is used. Two modifications of the conventional L-BFGS algorithm are also made to get a comparable convergence rate with the NLCG method. The impacts of the preconditioner parameters, the regularization parameters, the starting model, etc., on the inversion are evaluated by synthetic examples for both L-BFGS and NLCG methods. And the real MT Kayabe dataset is also inverted by the inversion algorithms. The synthetic tests show that through our L-BFGS inversion algorithm the similar resistivity models can be obtained with that from the NLCG method. For the real data inversion, the L-BFGS method performs more efficiently and reasonable results could be obtained by less iterations of the inversion process than the NLCG method. Thus, we suggest the common usage of the L-BFGS method for the 3D MT inverse problem.

Inversion is a fundamental step in magnetotelluric (MT) data routine analysis to retrieve a subsurface geoelectrical model that can be used to inform geological interpretations. To reduce the effect of non-uniqueness and local minimum trapping problems and improve calculation speeds, a data-driven mathematical method with a deep neural network was developed to estimate the subsurface resistivity. In this study, a deep learning (DL) inversion technique using a revised multi-head convolutional neural network (CNN) architecture was investigated for MT data analysis. We created synthetic datasets consisting of 100,000 random samples of resistivity layers to train the network’s parameters. The trained model was validated with independent noised datasets, and the predicted results displayed reasonable accuracy and reliability, which demonstrates the potential application of DL inversion for real-world MT data. The trained model was used to analyze MT data collected in the southwestern Athabasca Basin, Canada. The calculated results from the DL method displayed a detailed subsurface resistivity distribution compared to traditional iterative inversion. Since this approach can predict a resistivity model without multiple forward modeling operations after the CNN model is created, this framework is suitable to speed up the computation of multidimensional MT inversion for subsurface resistivity.

Magnetotelluric studies in and adjacent to the Northumberland Basin, Northern England

Deep learning (DL) inversion methods have showcased promising applications in solving geophysical inverse problems. In this work, we develop a DL model (SwinTUNet) to enable accurate reconstruction of the nonlinear mapping between the network inputs and outputs and achieve two-dimensional (2-D) magnetotelluric (MT) inversion. SwinTUNet is built by employing the strong Swin Transformer as the backbone network and adopting the UNet architecture. Moreover, to create a training dataset that aligns with practical MT inverse problems and ensuring effective applications, we design and generate a set of random synthetic resistivity models with gradual-varying resistivity values. Inversion examples demonstrate that the proposed SwinTUNet inversion method holds a great promise in promoting the applicability of DL inversion methods in realistic MT prospecting scenarios.

An extensive magnetotelluric (MT) survey comprised of 85 sites has been acquired over the Darajat geothermal field in Indonesia to map the geothermal reservoir and the overlying clay cap. The rouged topography and the geometry of the margin of the clay cap makes the resistivity structure 3D at reservoir depth. Although 3D MT inversion is now commonly used in geothermal interpretations 1D and 2D MT inversions are still effective tools for a variety of tasks such as quality assurance. Lower dimensional inversion can also play two critical roles in determining and assessing the resistivity model derived by 3D inversion: 1) by providing a good starting model to reduce the computational cost of the 3D inversion, and 2) by providing a computationally feasible path to stochastic inversion of the data that provides realistic parameter’s standard deviations for use in assessing reliability of the resistivity model. Using a spatially constrained 1D stochastic inversion of the MT data, we investigate the common claim that 1D inversion can provide a pseudo 3D model which closely matches the 3D inversion for the overburden and clay cap layers. The discrepancy between the pseudo 3D and true 3D inverse models increases with depth, however the presence of the core resistive feature of the field is still indicated at approximately the same depth as found in the true 3D model. Analysis of the 1D model parameter probability density functions shows that layer thicknesses are better determined than layer resistivities.

2D magnetotelluric inversion based on ResNet

A radio magnetotelluric (MT) field data set, acquired in scalar mode, over a buried waste site has been successfully analyzed using a 3D MT inversion scheme using nonlinear conjugate gradients. The results of this analysis demonstrate the utility of the scheme where more than 4800 data points collected on multiple measurement profiles have been inverted simultaneously. The resulting image clearly detects the buried waste; when receiver profiles cross pit boundaries, the image maps the lateral extent of the pit. However, the base of the pit is poorly resolved, and depends upon the starting model used to launch the inversion. Hence, critical information on whether contamination is leaching into a resistive gravel bed lining the base of the pit, as well as the deeper geological horizons consisting of brown coal, clay, and tertiary sands, is inconclusive. Nevertheless, by incorporating within the inversion process a priori information of the background media that is host to the waste, sharper images of the base of the pit are obtained, which are in good agreement with borehole data. The 3D analysis applied in this paper overcomes previous limitations in the radio magnetotelluric (RMT) method using 2D data analysis and inversion. With 3D analysis, it is unnecessary to make assumptions regarding geological strike, and near‐surface statics can be accommodated in both source polarizations. Our findings also indicate that 2D MT interpretation can overestimate the pit's depth extent. This may lead to the erroneous conclusion that the geological horizons beneath the pit have been contaminated.

How to get the rapid and stable inversion results and the clear geological interfaces is a focus problem in current Magnetotelluric (MT) inversion. Constable et al (1987,1990) provided the OCCAM as a smoothing inversion, and the focusing inversion which is based on the minimum gradient support functional provided by Portniaguine, O. and Zhdanov, M. S.(1999,2002,2004). We integrate the ideas of smoothing inversion and Focusing inversion to construct a new objective function for inversion. Meanwhile, We introduce a diagonal gradient support into the objective functional to enhance oblique geo‐electrical interface.

Seafloor and land magnetotelluric (MT) data were collected in the SWAGGIE (Southern Waters of Australia Geoelectric and Geomagnetic Induction Experiment) project in April-May 1998, from 30 seafloor and 23 land sites. The principal objective of the experiment was to delineate the strike and depth of a zone of high electrical conductivity, known as the Eyre Peninsula Anomaly (EPA) in South Australia. Three linear arrays of marine magnetotelluric instruments were deployed across the continental shelf and slope to locate the offshore extension of the EPA on the continental margin, and to image the continental-oceanic lithosphere-asthenosphere transition. A land array of magnetometers was deployed at the same time to better resolve the EPA in the southern Eyre Peninsula. Robust remote-reference processing of time-series magnetic and electric data gives good MT and geomagnetic depth sounding responses in the bandwidth of 101 to 105 s, corresponding to skin-depths of mid-crustal (10 km) to mantle transition (400 km) range. Initial processing of marine and land data clearly indicates that the EPA is continuous to the edge of the continental shelf, with a conductance greater than 15 000 S confined to a narrow, near-vertical zone. At sites distant from the EPA, one-dimensional MT inversions fit the data well and provide a background conductivity structure for two and three-dimensional forward and inverse modelling of the EPA.

In the magnetotelluric (MT) method, observed data consist of the sum of several types of signals. The technique of blind source separation (BSS) has been utilized to suppress the effect of artificial noises on MT response functions. When reconstructing the noise-reduced observed data by means of BSS, it is crucial to eliminate the specific components which correspond to noise. Yet improper subtraction of values from separated signals can lead to the loss of useful values of the natural signal or missing a large amount of noise-affected values, which may result in failure in deriving the true MT responses. This study proposes a novel algorithm that utilizes frequency-domain independent component analysis (FDICA). The observed data were classified into the XY and YX modes and the components of the reference magnetic field were appended to respective modes so that both the assumed source signal and the mixed signal have same number of components. The MT response functions, which are represented as the ratio of horizontal electric to magnetic field components, are substantially constant over the relatively short period of time. Moreover, the MT response functions vary smoothly in the frequency domain. An evaluation index with respect to these two characteristics of the natural signal was introduced when determining which values to subtract prior to the reconstruction of the observed data. We verify the performance of noise suppression by applying the proposed method to a set of MT time series affected by synthetic strong noise over the whole observation time.

The geoelectrical features of the Travale geothermal field (Italy), one of the most productive geothermal fields in the world, have been investigated by means of three-dimensional (3D) magnetotelluric (MT) data inversion. This study presents the first resistivity model of the Travale geothermal field derived from derivative-based 3D MT inversion. We analyzed MT data that have been acquired in Travale over the past decades in order to determine its geoelectrical dimensionality, directionality, and phase tensor properties. We selected data from 51 MT sites for 3D inversion. We carried out a number of 3D MT inversion tests by changing the type of data to be inverted, the inclusion of static-shift correction at some sites where new time-domain electromagnetic soundings (TDEM) were acquired, the grid rotation, as well as the starting model in order to assess the connection between the inversion model and the geology. The final 3D model herein presents deep elongated resistive bodies between the depths of 1.5 and 8 km. They are transverse to the Apennine structures and suggest a correlation with the strike-slip tectonics. Comparison with a seismic velocity model and well log data suggests a highly-fractured volume of rocks with vapor-dominated circulation. The outcome of this study provides new insights into the complex geothermal system of Travale.