Three-dimensional magnetotelluric inversion using L-BFGS

PreviousNext No AccessSEG Technical Program Expanded Abstracts 20013‐D magnetotelluric inversion for resource explorationAuthors: Randall L. MackieWilliam RodiM. Donald WattsRandall L. MackieGSY‐USA, Inc., William RodiMassachusetts Institute of Technology, and M. Donald WattsGeosystem srlhttps://doi.org/10.1190/1.1816392 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InReddit Permalink: https://doi.org/10.1190/1.1816392FiguresReferencesRelatedDetailsCited by3D Step-by-step inversion strategy for audio magnetotellurics data based on unstructured mesh17 February 2022 | Applied Geophysics, Vol. 18, No. 3Three‐dimensional electrical structure of the northwestern margin of the Karamay region, China, revealed by magnetotelluric data14 September 2020 | Geophysical Prospecting, Vol. 68, No. 9Three-dimensional electrical conductivity in the mantle beneath the Payún Matrú Volcanic Field in the Andean backarc of Argentina near 36.5°S: evidence for decapitation of a mantle plume by resurgent upper mantle shear during slab steepening25 June 2014 | Geophysical Journal International, Vol. 198, No. 2NLCG and L-BFGS optimization for 3D HEM inversionYunhe Liu and Changchun Yin19 August 2013Methods and algorithms for reconstructing three-dimensional distributions of electric conductivity and polarization in the medium by finite-element 3D modeling using the data of electromagnetic sounding8 May 2013 | Izvestiya, Physics of the Solid Earth, Vol. 49, No. 3Multiple-domain, simultaneous joint inversion of geophysical data with application to subsalt imagingMichele De Stefano, Federico Golfré Andreasi, Simone Re, Massimo Virgilio, and Fred F. Snyder5 May 2011 | GEOPHYSICS, Vol. 76, No. 3A compressed implicit Jacobian scheme for 3D electromagnetic data inversionMaokun Li, Aria Abubakar, Jianguo Liu, Guangdong Pan, and Tarek M. Habashy28 April 2011 | GEOPHYSICS, Vol. 76, No. 3Electrical resistivity structure at the northern margin of the Tibetan Plateau and tectonic implications7 December 2011 | Journal of Geophysical Research, Vol. 116, No. B12Application of the 3D magnetotelluric inversion code in a geologically complex area9 June 2010 | Geophysical Prospecting, Vol. 58, No. 6Three‐dimensional regularized Gauss‐Newton inversion algorithm using a compressed implicit Jacobian calculation for electromagnetic applicationsMaokun Li, Aria Abubakar, Jianguo Liu, Guangdong Pan, and Tarek M. Habashy21 October 2010Three-dimensional inversion of magnetotelluric data for mineral exploration: An example from the McArthur River uranium deposit, Saskatchewan, CanadaJournal of Applied Geophysics, Vol. 68, No. 43D magnetotelluric inversion using a limited-memory quasi-Newton optimizationDmitry Avdeev and Anna Avdeeva27 April 2009 | GEOPHYSICS, Vol. 74, No. 3Advancing process‐based watershed hydrological research using near‐surface geophysics: a vision for, and review of, electrical and magnetic geophysical methods11 March 2008 | Hydrological Processes, Vol. 22, No. 18A survey of current trends in near-surface electrical and electromagnetic methodsEsben Auken, Louise Pellerin, Niels B. Christensen, and Kurt Sørensen7 September 2006 | GEOPHYSICS, Vol. 71, No. 5Three-Dimensional Electromagnetic Modelling and Inversion from Theory to ApplicationSurveys in Geophysics, Vol. 26, No. 6Mining, Environmental, Petroleum, and Engineering Industry Applications of Electromagnetic Techniques in GeophysicsSurveys in Geophysics, Vol. 26, No. 5Multi-dimensional electromagnetic modeling and inversion with application to near-surface earth investigationsComputers and Electronics in Agriculture, Vol. 46, No. 1-3Accelerated integral equation inversion of 3‐D magnetotelluric data in models with inhomogeneous backgroundNikolay Golubev and Michael S. Zhdanov7 December 20053D inversion of a scalar radio magnetotelluric field data setGregory A. Newman, Stephan Recher, Bülent Tezkan, and Fritz M. Neubauer29 May 2003 | GEOPHYSICS, Vol. 68, No. 32-D and 3-D interpretation of magnetotelluric data in the Bajawa geothermal field, central Flores, IndonesiaBULLETIN OF THE GEOLOGICAL SURVEY OF JAPAN, Vol. 53, No. 2-3 SEG Technical Program Expanded Abstracts 2001ISSN (print):1052-3812 ISSN (online):1949-4645Copyright: 2001 Pages: 2135 publication data© 2001 Copyright © 2001 Society of Exploration GeophysicistsPublisher:Society of Exploration Geophysicists HistoryPublished Online: 03 Jan 2005 CITATION INFORMATION Randall L. Mackie, William Rodi, and M. Donald Watts, (2001), "3‐D magnetotelluric inversion for resource exploration," SEG Technical Program Expanded Abstracts : 1501-1504. https://doi.org/10.1190/1.1816392 Plain-Language Summary PDF DownloadLoading ...

In this chapter, we investigate recently proposed nonlinear conjugate gradient (NCG) methods for shape optimization problems. We briefly introduce the methods as well as the corresponding theoretical background and investigate their performance numerically. The obtained results confirm that the NCG methods are efficient and attractive solution algorithms for shape optimization problems.

Full-waveform inversion (FWI) is a nonlinear optimization problem, and a typical optimization algorithm such as the nonlinear conjugate gradient or limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) would iteratively update the model mainly along the gradient-descent direction of the misfit function or a slight modification of it. Based on the concept of meta-learning, rather than using a hand-designed optimization algorithm, we have trained the machine (represented by a neural network) to learn an optimization algorithm, entitled the “ML-descent,” and apply it in FWI. Using a recurrent neural network (RNN), we use the gradient of the misfit function as the input, and the hidden states in the RNN incorporate the history information of the gradient similar to an LBFGS algorithm. However, unlike the fixed form of the LBFGS algorithm, the machine-learning (ML) version evolves in response to the gradient. The loss function for training is formulated as a weighted summation of the L2 norm of the data residuals in the original inverse problem. As with any well-defined nonlinear inverse problem, the optimization can be locally approximated by a linear convex problem; thus, to accelerate the training, we train the neural network by minimizing randomly generated quadratic functions instead of performing time-consuming FWIs. To further improve the accuracy and robustness, we use a variational autoencoder that projects and represents the model in latent space. We use the Marmousi and the overthrust examples to demonstrate that the ML-descent method shows faster convergence and outperforms conventional optimization algorithms. The energy in the deeper part of the models can be recovered by the ML-descent even when the pseudoinverse of the Hessian is not incorporated in the FWI update.

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.

Purpose– Generally, iterative methods which have some instability solutions in complex structural and non-linear mechanical problems are used to compute reliability index. The purpose of this paper is to establish a non-linear conjugate gradient (NCG) optimization algorithm to overcome instability solution of the Hasofer-Lind and Rackwitz-Fiessler (HL-RF) method in first-order reliability analysis. The NCG algorithms such as the Conjugate-Descent (CD) and the Liu-Storey (LS) are used for determining the safety index. An algorithm is found based on the new line search in the reliability analysis.Design/methodology/approach– In the proposed line search for calculating the safety index, search direction is computed by using the conjugate gradient approach and the HL-RF method based on the new and pervious gradient vector of the reliability function. A simple step size is presented for the line search in the proposed algorithm, which is formulated by the Wolfe conditions based on the new and previous safety index results in the reliability analysis.Findings– From the current work, it is concluded that the proposed NCG algorithm has more efficient, robust and appropriate convergence in comparison with the HL-RF method. The proposed methods can eliminate numerical instabilities of the HL-RF iterative algorithm in highly non-linear performance function and complicated structural limit state function. The NGC optimization is applicable to reliability analysis and it is correctly converged on the reliability index. In the NCG method, the CD algorithm is slightly more efficient than the LS algorithm.Originality/value– This paper usefully shows how the HL-RF algorithm and the NCG scheme are formulated in first-order reliability analysis. The proposed algorithm is validated from six numerical and structural examples taken from the literature. The HL-RF method is not converged on several non-linear mathematic and complex structural examples, while the two proposed conjugate gradient methods are appropriately converged for all examples. The CD algorithm is converged about twice faster than the LS algorithm in most of the problems. Therefore, application of the NCG method is possible in reliability analysis.

We have formulated a 3-D inverse solution for the magnetotelluric (MT) problem using the non-linear conjugate gradient method. Finite difference methods are used to compute predicted data efficiently and objective functional gradients. Only six forward modelling applications per frequency are typically required to produce the model update at each iteration. This efficiency is achieved by incorporating a simple line search procedure that calls for a sufficient reduction in the objective functional, instead of an exact determination of its minimum along a given descent direction. Additional efficiencies in the scheme are sought by incorporating preconditioning to accelerate solution convergence. Even with these efficiencies, the solution’s realism and complexity are still limited by the speed and memory of serial processors. To overcome this barrier, the scheme has been implemented on a parallel computing platform where tens to thousands of processors operate on the problem simultaneously. The inversion scheme is tested by inverting data produced with a forward modelling code algorithmically different from that employed in the inversion algorithm. This check provides independent verification of the scheme since the two forward modelling algorithms are prone to different types of numerical error.

High Quality 4-dimensional Cone Beam CT by Deforming Prior Images

We study the development of nonlinear conjugate gradient methods, Fletcher Reeves (FR) and Polak Ribiere (PR). FR extends the linear conjugate gradient method to nonlinear functions by incorporating two changes, for the step length αk a line search is performed and replacing the residual, rk (rk=b-Axk) by the gradient of the nonlinear objective function. The PR method is equivalent to FR method for exact line searches and when the underlying quadratic function is strongly convex. The PR method is basically a variant of FR and primarily differs from it in the choice of the parameter βk. On applying the nonlinear Rosenbrock function to the MATLAB code for the FR and the PR algorithms we observe that the performance of PR method (k=29) is far better than the FR method (k=42). But, we observe that when the MATLAB codes are applied to general nonlinear functions, specifically functions whose minimum is a large negative number not close to zero and the iterates too are large values far off from zero the PR algorithm does not perform well. This problem with the PR method persists even if we run the PR algorithm for more iterations or with an initial guess closer to the actual minimum. To improve the PR algorithm we suggest finding a better weighing parameter βk, using better line search method and/or using specific line search for certain functions and identifying specific restart criteria based on the function to be optimized.

In this paper, we propose a three-parameter family of conjugate gradient methods for unconstrained optimization. The three-parameter family of methods not only includes the already existing six practical nonlinear conjugate gradient methods, but subsumes some other families of nonlinear conjugate gradient methods as its subfamilies. With Powell’s restart criterion, the three-parameter family of methods with the strong Wolfe line search is shown to ensure the descent property of each search direction. Some general convergence results are also established for the three-parameter family of methods. This paper can also be regarded as a brief review on nonlinear conjugate gradient methods.

Due to the strong capability of building complex nonlinear mapping without involving linearization theory and high prediction efficiency; the deep learning (DL) technique applied to solve geophysical inverse problems has been a subject of growing interest. Currently, most DL-based inversion approaches are fully data-driven (namely standard deep learning), the performance of which largely depends on the training sample sets. However, due to the heavy burden of time and computational resources, it can be challenging to supply such a massive and exhaustive training dataset for generic realistic exploration scenarios and to perform network training. In this work, based on the recent advances in physics-based networks, the physical laws of magnetotelluric (MT) wave propagation is incorporated into a purely data-driven DL approach (PlainDNN) and thus builds a physics-driven DL MT inversion scheme (PhyDNN). In this scheme, the forward operator modeling MT wave propagation is integrated into the network training loop, in the form of minimizing a hybrid loss objective function composed of the data-driven model misfit and physics-based data misfit, to guide the network training. Consequently, the proposed PhyDNN method will take the advantage of the fully data-driven DL and conventional physics-based deterministic methods, allowing it to deal with complex realistic exploration scenarios. Quantitative and qualitative analysis results demonstrate that the PhyDNN can honor the physical laws of the MT inverse problem, and with other conditions unchanged, the PhyDNN outperforms the PlainDNN and the classical deterministic Occam inversion method. When processing field data, the PhyDNN method yields considerably impressive inversion results compared to the Occam method, and the corresponding simulated MT responses agree well with the real measurements, which confirms the effectiveness and applicability of the PhyDNN method.

Nonlinear conjugate gradient methods are a class of techniques that are used for solving nonlinear optimisation problems frequently arising in many engineering applications such as machine learning, computer vision, least-square optimisations, to name a few. With so many surveys on the nonlinear conjugate gradient method (NLCG) available around, this paper addresses the current updates and sheds a new light on the evolution of hybrid conjugate gradient parameters with their global convergence properties.

• The thermotectonic architecture beneath the Vindhyan Basin. • The Malwa Plateau uplift is a high-conductivity feature formed by fissure eruptions. • Enhanced conductivity in the upper-mid crust during the late K-Pg periods. Magnetotelluric (MT) studies are employed to investigate the electrical structure beneath the Paleoproterozoic intracratonic Vindhyan rift basin, which is overlain by Deccan basalts. Deep thermotectonic activity, a geological process responsible for shaping a complex subsurface architecture, is more clearly revealed along a north-south MT profile that crosses geological faults and the Malwa Plateau. Using 25 broadband magnetotelluric (BBMT) stations and the non-linear conjugate gradient (NLCG) algorithm, a 2D electrical model was developed. The significant resistivity contrast in the model, ranging from ∼0.1 to 5000 Ωm, indicates altered resistivity in the Vindhyan basin sediments, which are exposed below 100 Ωm. Uncertainties in the basin thickness indicated along the profile, where it is approximately 2 km thick layer of sediments beneath stations vin01-vin08, are deposited on a resistivity feature, R2. Four conductors (C1, C2, C3, and C4) with resistivities between ∼0.1 and 1 Ωm are identified in the upper-mid crust, where the central conductors indicate the uplift of the Malwa Plateau, approximately 100 km wide. The east-west oriented faults could represent the earliest faults of the basin, with the Basoda-Barsingarh fault situated at the center of the plateau. Additionally, the Kannod-Damoh Fault and Ratlam-Shivpuri Fault surround it to the south and north, respectively. The plume activity associated with the supercontinents Columbia and Rodinia might have triggered the reactivation of these ancient faults. As tectonically weak zones, these faults provide pathways for heat, melts, and fluids to rise from the Reunion plume head, resulting in melting within the crust and upper mantle. The thermotectonic rejuvenation of the subsurface involves a crust-mantle boundary that becomes indistinct. Key factors contributing to increased crustal conductivity include carbonate-rich fluids trapped during the Paleoproterozoic, basaltic melts with fluid inclusions, and magmatic underplating during the late Cretaceous-Paleogene (K-Pg) periods. The melt fraction is estimated to range from 8% to 16% for each conductor with a bulk resistivity below 5 Ωm. As sulfides melt together with basaltic melts, they collectively boost the overall melt conductivity and decrease the partial melt fraction required to account for the high observed conductivity.

Because of the discovery of ultrahigh pressure metamorphic (UHPM) belt beneath the Sulu (Jiangsu Province-Shandong Province) orogen, this area has become a focused subject of current geoscience, as it has a close relationship with the evolution of the orogen and the neighboring North China craton. Probing the deep structure beneath this area would be of great significance for the geological interpretation of this issue. In this study, we make an analysis of magnetotelluric (MT) data along a profile across the Sulu orogen to provide evidence of deep structure below this region. The profile begins in west from the North China block, extending in S129°E, across the Tan-Lu fault, Sulu UHPM zone, and Sulu high pressure metamorphic (HPM) zone, and terminates in the Yangtze block in east. We use the nonlinear conjugate gradient method and TE-TM combined mode to perform inversion and interpretation of the MT data, and obtain an electrical structure image above depth of 150 km along the profile. It shows that the structure can be divided into seven sections in lateral direction, between which the electric boundaries coincide well with the major faults, such as the Tan-Lu, Haizhou-Siyang, and Jiashan-Xiangshui faults. In vertical direction the electrical structure can be subdivided into six layers of different resistivities. It is noted that there exist high-conductivity areas in crust below the North China block and Yangtze block, while such a feature is not present beneath the Sulu orogen, which is very different from the Dabie orogen. It is also observed that a fairly continuous zone of relatively low-resistivity exists at depths of 50–90 km of the electrical structure image, which is presumably a weak zone in the uppermost mantle. Just below this low-resistivity zone are the relatively high- resistivity layer of the North China block, relatively low-resistivity layer of the Sulu orogen, and relatively high-resistivity layer of the Yangtze block, all in the shallow upper mantle, respectively. From the whole 2D electrical structure image, there is no abnormally low-resistivity layer in the shallow upper mantle beneath the Sulu orogen and neighboring areas, indicating that no hot asthenoshperic material associated with lithospheric thinning exists at present.

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.

SummaryAcceleration schemes can dramatically improve existing optimization procedures. In most of the work on these schemes, such as nonlinear generalized minimal residual (N‐GMRES), acceleration is based on minimizing the ℓ2 norm of some target on subspaces of . There are many numerical examples that show how accelerating general‐purpose and domain‐specific optimizers with N‐GMRES results in large improvements. We propose a natural modification to N‐GMRES, which significantly improves the performance in a testing environment originally used to advocate N‐GMRES. Our proposed approach, which we refer to as O‐ACCEL (objective acceleration), is novel in that it minimizes an approximation to the objective function on subspaces of . We prove that O‐ACCEL reduces to the full orthogonalization method for linear systems when the objective is quadratic, which differentiates our proposed approach from existing acceleration methods. Comparisons with the limited‐memory Broyden–Fletcher–Goldfarb–Shanno and nonlinear conjugate gradient methods indicate the competitiveness of O‐ACCEL. As it can be combined with domain‐specific optimizers, it may also be beneficial in areas where limited‐memory Broyden–Fletcher–Goldfarb–Shanno and nonlinear conjugate gradient methods are not suitable.