Three-dimensional conjugate gradient inversion of magnetotelluric sounding data

Based on the analysis of the conjugate gradient algorithm, we implement a threedimensional （3D） conjugate gradient inversion algorithm with magnetotelluric impedance data. During the inversion process, the 3D conjugate gradient inversion algorithm doesn＇ t need to compute and store the Jacobian matrix but directly updates the model from the computation of the Jacobian matrix. Requiring only one forward and four pseudo-forward modeling applications per frequency to produce the model update at each iteration, this algorithm efficiently reduces the computation of the inversion. From a trial inversion with synthetic magnetotelluric data, the validity and stability of the 3D conjugate gradient inversion algorithm is verified.

Three-dimensional magnetotelluric regularized inversion based on smoothness-constrained model

How to get the rapid and stable inversion results and reconstruct the clear subsurface resistivity structures is a focus problem in current magnetotelluric inversion. A stable solution of an ill-posed inverse problem was obtained by the regularization methods in which some desired structures were imposed to stabilize the inverse problem. By the smoothness-constrained model and approximate sensitivity method, the stable subsurface resistivity structures were reconstructed. The synthetic examples show that the smoothness-constrained regularized inversion method is effective and can be reasonable to reconstruct three-dimensional subsurface resistivity structures.

Three-dimensional conjugate gradient inversion of magnetotelluric full information data

Based on the analysis of impedance tensor data, tipper data, and the conjugate gradient algorithm, we develop a three-dimensional （3D） conjugate gradient algorithm for inverting magnetotelluric full information data determined from five electric and magnetic field components and discuss the method to use the full information data for quantitative interpretation of 3D inversion results. Results from the 3D inversion of synthetic data indicate that the results from inverting full information data which combine the impedance tensor and tipper data are better than results from inverting only the impedance tensor data （or tipper data） in improving resolution and reliability. The synthetic examples also demonstrate the validity and stability of this 3D inversion algorithm.

Electrical Structure of Wulingshan and Middle Jiangnan Orogen by Three-Dimensional Magnetotelluric Data Inversion

Funded by The National Key Research and Development Program of China,China Deep Exploration(Sinoprobe)and The China Geological Suvery Project on 2009–2019,a large scale magnetotelluric sounding(MT)survey grid(Fig.1)has covered whole south China.

Parallel rapid relaxation inversion of 3D magnetotelluric data

We implement a parallel algorithm with the advantage of MPI （Message Passing Interface） to speed up the rapid relaxation inversion for 3D magnetotelluric data. We test the parallel rapid relaxation algorithm with synthetic and real data. The execution efficiency of the algorithm for several different situations is also compared. The results indicate that the parallel rapid relaxation algorithm for 3D magnetotelluric inversion is effective. This parallel algorithm implemented on a common PC promotes the practical application of 3D magnetotelluric inversion and can be suitable for the other geophysical 3D modeling and inversion.

3D magnetotelluric inversions with unstructured finite-element and limited-memory quasi-Newton methods

Traditional 3D Magnetotelluric(MT) forward modeling and inversions are mostly based on structured meshes that have limited accuracy when modeling undulating surfaces and arbitrary structures. By contrast, unstructured-grid-based methods can model complex underground structures with high accuracy and overcome the defects of traditional methods, such as the high computational cost for improving model accuracy and the difficulty of inverting with topography. In this paper, we used the limited-memory quasi-Newton(L-BFGS) method with an unstructured finite-element grid to perform 3D MT inversions. This method avoids explicitly calculating Hessian matrices, which greatly reduces the memory requirements. After the first iteration, the approximate inverse Hessian matrix well approximates the true one, and the Newton step(set to 1) can meet the sufficient descent condition. Only one calculation of the objective function and its gradient are needed for each iteration, which greatly improves its computational efficiency. This approach is well-suited for large-scale 3D MT inversions. We have tested our algorithm on data with and without topography, and the results matched the real models well. We can recommend performing inversions based on an unstructured finite-element method and the L-BFGS method for situations with topography and complex underground structures.

Nonlinear inversion for magnetotelluric sounding based on deep belief network

To improve magnetotelluric(MT)nonlinear inversion accuracy and stability,this work introduces the deep belief network(DBN)algorithm.Firstly,a network frame is set up for training in different 2D MT models.The network inputs are the apparent resistivities of known models,and the outputs are the model parameters.The optimal network structure is achieved by determining the numbers of hidden layers and network nodes.Secondly,the learning process of the DBN is implemented to obtain the optimal solution of network connection weights for known geoelectric models.Finally,the trained DBN is verified through inversion tests,in which the network inputs are the apparent resistivities of unknown models,and the outputs are the corresponding model parameters.The experiment results show that the DBN can make full use of the global searching capability of the restricted Boltzmann machine(RBM)unsupervised learning and the local optimization of the back propagation(BP)neural network supervised learning.Comparing to the traditional neural network inversion,the calculation accuracy and stability of the DBN for MT data inversion are improved significantly.And the tests on synthetic data reveal that this method can be applied to MT data inversion and achieve good results compared with the least-square regularization inversion.

3D Step-by-step inversion strategy for audio magnetotellurics data based on unstructured mesh

A three-dimensional(3D)step-by-step inversion strategy for audio magnetotellurics(AMT)is investigated in this study.The objective function is minimized by iteratively solving the Gauss-Newton normal equation,and the inversion region is discretized with unstructured tetrahedral elements.The inversion proceeds step-by-step from a coarse mesh to a fine mesh.In the inversion iteration process,a mesh is adaptively optimized according to the spatial gradient information about the model resistivity to fine delineate the boundaries of abnormal bodies.In the early stage of inversion execution,a coarse mesh is used for inversion,and the inversion stability is improved by reducing the number of inversion elements.In addition,mesh refinement is performed in the iterative inversion process.The inversion results obtained from the previous mesh are used as the reference and initial models for the next mesh iterative inversion.The step-by-step inversion strategy can ensure that the inversion is performed in the correct direction,improving the inversion stability and results gradually.Synthetic results show that the step-by-step inversion strategy with a Gauss-Newton method for 3D AMT inversion is stable and reliable,which lays a foundation for further practical 3D AMT data inversion.

Three-dimensional gravity inversion based on sparse recovery iteration using approximate zero norm

This research proposes a novel three-dimensional gravity inversion based on sparse recovery in compress sensing. Zero norm is selected as the objective function, which is then iteratively solved by the approximate zero norm solution. The inversion approach mainly employs forward modeling; a depth weight function is introduced into the objective function of the zero norms. Sparse inversion results are obtained by the corresponding optimal mathematical method. To achieve the practical geophysical and geological significance of the results, penalty function is applied to constrain the density values. Results obtained by proposed provide clear boundary depth and density contrast distribution information. The method's accuracy, validity, and reliability are verified by comparing its results with those of synthetic models. To further explain its reliability, a practical gravity data is obtained for a region in Texas, USA is applied. Inversion results for this region are compared with those of previous studies, including a research of logging data in the same area. The depth of salt dome obtained by the inversion method is 4.2 km, which is in good agreement with the 4.4 km value from the logging data. From this, the practicality of the inversion method is also validated.

Magnetotelluric extremum boundary inversion based on different stabilizers and its application in a high radioactive waste repository site selection

Geophysical inversion under different stabilizers has different descriptions of the target body boundary,especially in complex geological structures.In this paper,we present an extremum boundary inversion algorithm based on different stabilizers for electrical interface recognition.Firstly,we use the smoothest and minimum-support stabilizing functional to study the applicability of adaptive regularization inversion algorithm.Then,an electrical interface recognition method based on different stabilizers is developed by introducing extremum boundary inversion algorithm.The testing shows that the adaptive regularization inversion method does work for different stabilizers and has a low dependence on the initial models.The ratio of the smooth and focusing upper and lower boundaries obtained using the extremum boundary inversion algorithm can clearly demarcate electrical interfaces.We apply the inversion algorithm to the magnetotelluric(MT)data collected from a preselected area of a high-level-waste clay-rock repository site in the Tamusu area.We recognized regional structures with smooth inversion and the local details with focusing inversion and determined the thickness of the target layer combined with the geological and drilling information,which meets the requirement for the site of the high-level waste clay-rock repository.

Three-dimensional forward modeling for magnetotelluric sounding by finite element method

A finite element algorithm combined with divergence condition was presented for computing three-dimensional(3D) magnetotelluric forward modeling. The finite element equation of three-dimensional magnetotelluric forward modeling was derived from Maxwell's equations using general variation principle. The divergence condition was added forcedly to the electric field boundary value problem, which made the solution correct. The system of equation of the finite element algorithm was a large sparse, banded, symmetric, ill-conditioned, non-Hermitian complex matrix equation, which can be solved using the Bi-CGSTAB method. In order to prove correctness of the three-dimensional magnetotelluric forward algorithm, the computed results and analytic results of one-dimensional geo-electrical model were compared. In addition, the three-dimensional magnetotelluric forward algorithm is given a further evaluation by computing COMMEMI model. The forward modeling results show that the algorithm is very efficient, and it has a lot of advantages, such as the high precision, the canonical process of solving problem, meeting the internal boundary condition automatically and adapting to all kinds of distribution of multi-substances.

Three-dimensional frequency-domain full waveform inversion based on the nearly-analytic discrete method

The nearly analytic discrete(NAD)method is a kind of finite difference method with advantages of high accuracy and stability.Previous studies have investigated the NAD method for simulating wave propagation in the time-domain.This study applies the NAD method to solving three-dimensional(3D)acoustic wave equations in the frequency-domain.This forward modeling approach is then used as the“engine”for implementing 3D frequency-domain full waveform inversion(FWI).In the numerical modeling experiments,synthetic examples are first given to show the superiority of the NAD method in forward modeling compared with traditional finite difference methods.Synthetic 3D frequency-domain FWI experiments are then carried out to examine the effectiveness of the proposed methods.The inversion results show that the NAD method is more suitable than traditional methods,in terms of computational cost and stability,for 3D frequency-domain FWI,and represents an effective approach for inversion of subsurface model structures.

Three-dimensional gravity inversion based on 3D U-Net++

The gravity inversion is to restore genetic density distribution of the underground target to be explored for explaining the internal structure and distribution of the Earth.In this paper,we propose a new 3D gravity inversion method based on 3D U-Net++.Compared with two-dimensional gravity inversion,three-dimensional(3D)gravity inversion can more precisely describe the density distribution of underground space.However,conventional 3D gravity inversion method input is two-dimensional,the input and output of the network proposed in our method are three-dimensional.In the training stage,we design a large number of diversifi ed simulation model-data pairs by using the random walk method to improve the generalization ability of the network.In the test phase,we verify the network performance by using the model-data pairs generated by the simulation.To further illustrate the eff ectiveness of the algorithm,we apply the method to the inversion of the San Nicolas mining area,and the inversion results are basically consistent with the borehole measurement results.Moreover,the results of the 3D U-Net++inversion and the 3D U-Net inversion are compared.The density models of the 3D U-Net++inversion have higher resolution,more concentrated inversion results,and a clearer boundary of the density model.

Inversion of source mechanism of 1989 Loma Prieta earthquake by three-dimensional FEM Green's function

Based on three-dimensional joint finite element, this paper discusses the theory and methodology about inversionof geodetic data. The FEM and inversion formula is given in detail; also a related code is developed. By use of theGreen's function about 3-D FEM, we invert geodetic measurementS of coseismic deformation of the 1989 Ms=7. 1Loma Prieta earthquake to datermine itS source mechanism. The result indicates that the slip on the fault plane isvery heterogeneous. The maximum slip and shear stress are located about 10 kin to northwest of the eathquakesource, the stress drop is about more than 1 MPa.

3D joint inversion of controlled-source audio-frequency magnetotelluric and magnetotelluric data

Different geophysical exploration methods have significant differences in terms of exploration depth,especially in frequency domain electromagnetic(EM)exploration.According to the definition of skin depth,this difference will increase with the effective detection frequency of the method.As a result,when performing three-dimensional inversion on single type of EM data,it is not possible to effectively distinguish the subsurface geoelectric structure at the full scale.Therefore,it is necessary to perform joint inversion on different type of EM data.In this paper we combine the magnetotelluric method(MT)with the controlled-source audio-magnetotelluric method(CSAMT)to study the frequency-domain three-dimensional(3D)joint inversions,and we use the unstructured finite-element method to do the forward modeling for them,so that the numerical simulation accuracies of different electromagnetic methods can be satisfied.By combining the two sets of data,we can obtain the sensitivity of the electrical structure at different depths,and depict the full-scale subsurface geoelectric structures.In actual mineral exploration,the 3D joint inversion is more useful for identifying subsurface veins in the shallow part and blind mines in the deep part.It can delineate the morphological distribution of ore bodies more completely and provide reliable EM interpretations to guide the mining of minerals.

FAST APPROXIMATE INVERSION OF MAGNETOTELLURIC(FAIMT) DATA FOR A HORIZONTALLY LAYERED EARTH WITH NEAR-SURFACE INHOMOGENEITIES

This paper discusses use of approximations and the Integral Mean Value Theorem to show that 6 coefficients approximately describe the distortions of near surface inhomogeneities on the MT field of a horizontally layered earth model. When these 6 coefficients are considered together with those of the magnetic field of a horizontally layered earth model,the analytic and approximate wave impedance equations can be derived for the MT response of a horizontally layered earth model with near-surface 2-D and 3-D inhomogeneities. These approximate wave impedance equations are used with inverted MT data for 2-D and 3-D forward modelling. Although these 6 coefficients cannot be determined before inversion,initial estimates can be used. The 6 coefficients and the asistivity and thickness of each layer of a horizontally layered earth can be obtained by using published inversion methods. The 6 coefficients give important informaion (depths and resistivities) on the near-surface inhomogenelties.The authors inverted 2-D and 3-D theoretical models for Fast Approximate Inversion of Magnetotelluric (FAIMT) data for a horizontally layered earth with near-surface inhomogeneities compares favorably with traditional invrsion methods, especially for inverting regional or basin structures. This method simplifies computation and gives a reasonable 1 -D geological model with fewer nonuniquenas problems.

Magnetotelluric data inversion with seismic data constraint

In the paper we present a new method to invert the interior structure in the basement or ancient hidden hill by using magnetotelluric (MT) data with seismic data constraint. We first obtain the thickness and resistivity of each layer above the basement or buried hill by the inversion of seismic and log data and create a geoelectrical model for the layers above the basement or hidden hill. Then with the reference to the inversion of 1D MT data, a geoelectrical model for the layers below the basement or hidden hill is created. On the basis of the above initial model, we present an effective and practical forward method, i.e., a model-matched approach to conduct forward inversion arithmetic. Finally, by the method of conjugate gradient iteration, a forward and backward iterative calculation is made. Taking No. 618 profile of Shengli Oil Field as an example, we have found out that the tectonic information that is unreflective in the seismic data below the basement is better reflected in the inversion result.

2D magnetotelluric inversion based on ResNet

In this study,a deep learning algorithm was applied to two-dimensional magnetotelluric(MT)data inversion.Compared with the traditional linear iterative inversion methods,the MT inversion method based on convolutional neural networks(CNN)does not rely on the selection of the initial model parameters and does not fall into the local optima.Although the CNN inversion models can provide a clear electrical interface division,their inversion results may remain prone to abrupt electrical interfaces as opposed to the actual underground electrical situation.To solve this issue,a neural network with a residual network architecture(ResNet-50)was constructed in this study.With the apparent resistivity and phase pseudo-section data as the inputs and with the resistivity parameters of the geoelectric model as the training labels,the modified ResNet-50 model was trained end-to-end for producing samples according to the corresponding production strategy of the study area.Through experiments,the training of the ResNet-50 with the dice loss function effectively solved the issue of over-segmentation of the electrical interface by the cross-entropy function,avoided its abrupt inversion,and overcame the computational inefficiency of the traditional iterative methods.The proposed algorithm was validated against MT data measured from a geothermal field prospect in Huanggang,Hubei Province,which showed that the deep learning method has opened up broad prospects in the field of MT data inversion.

A Bayesian Inference Approach to Reduce Uncertainty in Magnetotelluric Inversion: A Synthetic Case Study

The deterministic geophysical inversion methods are dominant when inverting magnetotelluric data whereby its results largely depends on the assumed initial model and only a single representative solution is obtained. A common problem to this approach is that all inversion techniques suffer from non-uniqueness since all model solutions are subjected to errors, under-determination and uncertainty. A statistical approach in nature is a possible solution to this problem as it can provide extensive information about unknown parameters. In this paper, we developed a 1D Bayesian inversion code based Metropolis-Hastings algorithm whereby the uncertainty of the earth model parameters were quantified by examining the posterior model distribution. As a test, we applied the inversion algorithm to synthetic model data obtained from available literature based on a three layer model (K, H, A and Q). The frequency for the magnetotelluric impedance data was generated from 0.01 to 100 Hz. A 5% Gaussian noise was added at each frequency in order to simulate errors to the synthetic results. The developed algorithm has been successfully applied to all types of models and results obtained have demonstrated a good compatibility with the initial synthetic model data.

A regularized magnetotelluric inversion with a minimum support gradient constraint

Magnetotelluric(MT)inversion is an illposed problem and the standard way to address it is through regularization,by adding a stabilizing functional to the data objective functional in order to obtain a stable solution.The traditional stabilizing functionals,in which a low-order differential operator is used,yield a smooth solution that may not be appropriate when anomalies occur in block patterns.In some cases the focused imaging of a sharp electrical boundary is necessary.Even though various experiments have used stabilizing functionals that are suitable to obtain a clear and sharp boundary,such as the minimum support(MS)and the minimum gradient support(MGS)functionals,there are still some limitations in practice.In this paper,the minimum support gradient(MSG)is proposed as the stabilizing functional.Under the uniform regularization framework,a regularized inversion with a variety of stabilizing functionals is performed and the inversion results are compared.This study shows that MSG inversion can not only obtain a clearly focused inversion but also a quite stable and robust one.