Inverse Theory and Applications in Geophysics

Geophysical Inverse Theory and Regularization Problems

Inverse problems play a crucial role in geophysics because one of the main tasks in this field is to probe the Earth's interior both for economic reasons, such as oil prospecting, and for the pursuit of academic knowledge about our planet. A variety of different physical fields are used for this: elastic waves form the basis of seismic prospecting, electromagnetic or magnetic fields are used to make inferences about the electrical conductivity in the Earth and the magnetic properties of the core, the gravity field constrains the mass distribution within the Earth, and many other examples can be given where inverse problem theory is crucial for inferring the properties of the Earth from measurements at its surface. Examples of the theory and practice of geophysical inverse problems are given by Iyer and Hirahara (1993), Lines and Levin (1988) and Parker (1994). This special section is aimed at presenting the inverse problems community with the theoretical barriers that geophysicists encounter, and providing some unorthodox examples of geophysical inverse problems. Every image obtained from an inverse problem with an incomplete data set gives an incomplete impression of the object under consideration. Trampert shows how limitations in global tomography influence the interpretation and implications of global maps of the seismic velocity in the Earth. In addition he shows how difficult it is to quantify the limitation of the images that are obtained. As shown by Snieder, nonlinearity aggravates this problem. He makes the point that currently there are no satisfactory theoretical tools for the error and resolution analysis of truly nonlinear inverse problems. Mosegaard attacks this problem, not by seeking a single solution to inverse problems, but by constructing many models and making inferences on the likelihood of these models using Bayesian statistics. That such an approach is needed for many practical problems is shown by Sambridge who shows that in practical problems the misfit function (or probability density function) can be extremely complex. In addition he develops tools to characterize this complexity. That inverse problem theory can be applied to problems other than making images of the Earth's interior is shown by Peltier, who makes inferences about the viscosity in the Earth from the measurement of sea-level variations, and by Gallagher, who reconstructs the thermal history of oil reservoirs from measurements at the surface. A major challenge in geophysics is to use different datasets to make inferences about the Earth's interior. Barghazi and Sansó present theoretical tools that can be used for the joint inversion of seismological and gravity data. It will be clear from this special section that inverse problems is a rich and diverse field of research within geophysics. Despite the success of many of its applications (western industrial society would not exist without effective methods for seismic exploration), geophysicists run into theoretical barriers. Within the geophysics community, solutions are usually sought that rely heavily on numerical techniques. It could be that this is the wisest strategy. However, it is also likely that an increased interaction with theoreticians who are active in the field of inverse problems could be beneficial both for the geophysics community and for those active in inverse problem theory. The goal of this special section is to help to close the gap between theory and practice within the inverse problems community. References Iyer H M and Hirahara K (ed) 1993 Seismic Tomography; Theory and Practice (London: Chapman and Hall) Lines L R and Levin F L (ed) 1988 Inversion of Geophysical Data SEG reprint series, 9 Parker R L 1994 Geophysical Inverse Theory (Princeton, NJ: Princeton University Press)

Seismic inversion in geophysics is a method that uses certain prior information, such as known geological laws and well logging and drilling data, to infer the physical parameters of underground media, such as wave impedance, velocity, and density, from seismic observation data, and thereby obtain the spatial structure and physical properties of underground strata. Seismic inversion is a highly complex problem with multiple solutions, and with the advancement of collection equipment, the volume of geophysical observation data is increasing at an astonishing rate. This presents new challenges for the accuracy and speed of seismic data inversion methods. There is an urgent need to develop intelligent and efficient inversion technologies for seismic inversion.Deep learning networks have powerful nonlinear fitting capabilities and can be used to solve complex nonlinear problems, such as seismic inversion. However, the predictive ability of deep learning networks largely depends on the quantity of training data. In the early stages of oil and gas exploration and development, the amount of well logging label data available for training is very limited, which poses a challenge for the application of deep learning in seismic inversion. Semi-supervised learning seismic inversion methods consider both data mismatch issues and well logging data mismatch issues, and can better adapt to inversion problems in real-world scenarios. Unlike supervised learning approaches, semi-supervised learning does not require a large amount of labeled data, thus it can better handle situations of data scarcity or mismatch.This paper utilizes a semi-supervised learning workflow to perform inversion on post-stack seismic data and has conducted experimental validation on the Marmousi 2 model. The experimental results show that, compared to supervised learning networks, the semi-supervised learning network still exhibits good predictive performance with a limited amount of data, demonstrating better stability in the presence of noise and geological variations, and effectively learns the mapping relationship between seismic data and artificial intelligence. Furthermore, as the amount of training data increases, the performance of the network also improves, confirming the importance of data quantity for training deep learning networks. The application results of the network on actual data indicate that the network has broad application prospects and feasibility. However, since the network is based on a channel-by-channel inversion method, there is still a lack of representation in terms of lateral continuity, which requires further exploration and improvement in subsequent research.

PreviousNext No AccessSEG Technical Program Expanded Abstracts 1995Two‐dimensional regularized inversion of magnetotelluric dataAuthors: Michael S. ZhdanovPatricia P. de LugaoOleg N. PortniaguineMichael S. ZhdanovUniversity of Utah, Patricia P. de LugaoUniversity of Utah, and Oleg N. PortniaguineUniversity of Utahhttps://doi.org/10.1190/1.1887564 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InRedditEmail Permalink: https://doi.org/10.1190/1.1887564FiguresReferencesRelatedDetailsCited ByA Selection Algorithm of Survey Lines and the Interactive Mode of the Sounding Curve7 January 2015 SEG Technical Program Expanded Abstracts 1995ISSN (print):1052-3812 ISSN (online):1949-4645Copyright: 1995 Pages: 1566 publication data© 1995 Copyright © 1995 Society of Exploration GeophysicistsPublisher:Society of Exploration Geophysicists HistoryPublished: 10 Feb 2005 CITATION INFORMATION Michael S. Zhdanov, Patricia P. de Lugao, and Oleg N. Portniaguine, (1995), "Two‐dimensional regularized inversion of magnetotelluric data," SEG Technical Program Expanded Abstracts : 803-806. https://doi.org/10.1190/1.1887564 Plain-Language Summary PDF DownloadLoading ...

One of the most important stages in addressing geophysical problems is the inversion procedure of the observed data, which involves incorporating several interdependent model parameters that describe the geophysical properties of the Earth’s subsurface. In this article, we propose a new modification to the inversion technique by applying global optimization method based on systematic search. Global optimization methods are particularly useful because geophysical inverse problems are often nonlinear making traditional local optimization methods less effective. A systematic search using the permutation method for inverting geophysical data refers to a comprehensive and methodical approach that explores different parameter combinations or configurations. The guided permutation method ensures that all possible configurations of model parameters are tested to find the best match between the geophysical model and the observed data. While this method is highly reliable in finding global solutions, its computational cost can be prohibitive for large-scale problems. This issue can be mitigated by conducting an analysis of geophysical parameter sensitivity, which helps determine which model parameters can be reliably estimated, and which are less influential. Furthermore, this analysis enhances the understanding of model uncertainties, guides the inversion process, reduces computational costs, and improves the reliability of the final geophysical model by employing this new inversion technique with the specific method used that is systematic search, permutation, sensitivity analysis. The computer program codes based on systematic search and guided permutation method demonstrates the robustness and accuracy of geophysical inversions. The inversion results from this program, applied to multimode Love wave dispersion curves, show a significant improvement in the quality of the match between the observed and modelled dispersion curves. This improvement is quantitatively confirmed by a decrease in root mean square (RMS) error from 13.435 m/s to 2.887×10-5 m/s, representing a significant reduction in misfit. Based on the obtained results, the program implementing this new inversion technique, incorporating systematic search, guided permutation, and sensitivity analysis, indicates the robustness and accuracy of geophysical inversions. Finally, we ensure that this inversion technique can be applied to solve other geophysical data problems.

In the last 25 years, mathematical and numerical modeling of cardiovascular problems has experienced a spectacular progress. Moving progressively from the stage of idealized models supporting traditional in vitro and in vivo approaches – basically with proofs of concept – Cardiovascular Mathematics has been transforming into a unique tool for patient-specific analysis. This has been made possible by the strong integration with imaging devices. Nowadays, the work pipeline starting from a medical image in the Digital Imaging and Communications in Medicine (DICOM) format to retrieve the patient-specific geometry of an artery or the left ventricle to be used as computational domain in a numerical simulation is well established. This has initiated a paradigm shift in the interactions between mathematicians, bioengineers, and medical doctors. The latter are progressively getting used to in silico analysis as an important component of their research [3]. The introduction of numerical procedures as a part of an established clinical routine and more in general of a consolidated support to the decision-making process of physicians still requires some steps both in terms of infrastructures (to bring computational tools to the operating room or the bedside) and methods. In particular, the quality of the numerical results needs to be assessed and certified. The reliability of simulations calls for an accurate quantification and possibly reduction of uncertainty. In this scenario and in view of terrific advancements of measuring techniques, an important research line – quite established in other fields – is data assimilation [4,6,9], that is, the integration of numerical simulations and measurements. We may say that numerical models provide a background knowledge (based on physical principles and constitutive laws, not patient-specific), whereas measures give a foreground (individual) information; accuracy of in silico procedures relies on the correct integration of these two levels of knowledge of the problem [9,10]. As a matter of fact, numerical models depend on parameters and boundary/initial conditions. Their knowledge is in practice usually incomplete, inaccurate, and noisy. In some cases, the sensitivity itself of the final results to these parameters is unclear and needs to be quantified. Such an uncertainty may be significantly reduced by the availability of patient-specific measures; on the other hand, the quality of measures can be strongly enhanced by comparison with mathematical models (after all, noise is in general not educated, it does not know PDEs . . . ) [4,5,7,8,9,11]. We think therefore that, in the future, a crucial step for the clinical application of Cardiovascular Mathematics will be the passage from numerical simulation to data assimilation. To be concrete, borrowing a motto by Michel Fortin supporting mesh adaptivity in finite element simulations, stating that a ‘grid should not be considered a data, but an unknown of the problem’, we think that many parameters currently tuned on the basis of nonpatient specific criteria (even in patient-specific geometries) should be considered part of the unknowns to be found by assimilation procedures.

Much of our knowledge of the Earth's interior is perforce based on the interpretation of measurements made at the surface, rather than sampling of the material in the interior. In the past few years there have been great advances in the mathematical aspects of this problem, and the topic has come to be called geo­ physical inverse theory. To apply these ideas, there must be a valid mathematical model of the physics of the system under study, so that one would be able to calculate the values of observations made on an exactly known structure: the calculation of the behavior of a specified system is the solution of the or direct problem. Frequently it is the forward problem that presents a difficult challenge to the theoretical geophysicist. Illustrations include the mechanism of earthquake rupture or the generation of the Earth's magnetic field; in problems like these, inverse theory is normally quite inappropriate. When the forward problem has been completely solved, there are of course unknown parameters in the mathematical model representing physical properties of the Earth such as Lame parameters, density or electrical conductivity. The goal of inverse theory is to determine the parameters from the observations or, in the face of the inevitable limitations of actual measurement, to find out as much as possible about them. The quality that distinguishes inverse theory from the parameter estimation problem of statistics (Bard 1974, Rao 1973) is that the unknowns are junctions, not merely a handful of real numbers. This' means that the solution contains in principle an infinite number of variables, and therefore with real data the problem is as under­ determined as it can be. Naturally, there are geophysical problems containing a relatively small number of free parameters: for example, in describing the relative instantaneous motion of N lithospheric plates, we find that the assumption of internal rigidity reduces the number of unknowns to 3N 3 for the N 1 relative angular velocity vectors (McKenzie & Parker 1974). Sometimes, however, unknown structures are conceived in terms of small numbers of homogeneous layers for reasons of computational simplicity rather than on any convincing geophysical or

Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

Geophysicists have been working on solutions to the inverse problem since the dawn of our profession. An interpreter infers subsurface properties on the basis of observed data sets, such as seismograms or potential field recordings. A rough model of the process that produces the recorded data resides within the interpreter's brain; the interpreter then uses this rough mental model to reconstruct subsurface properties from the observed data. In modern parlance, the inference of subsurface properties from observed data is identified with the solution of a so-called “inverse problem.” In contrast, the “forward problem” consists of the determination of the data that would be recorded for a given subsurface configuration and under the assumption that given laws of physics hold. Until the early 1960s, geophysical inversion was carried out almost exclusively within the geophysicist's brain. Since then, we have learned to make the geophysical inversion process much more quantitative and versatile by recourse to a growing body of theory, along with the computer power to reduce this theory to practice. We should point out the obvious, however, namely that no theory and no computer algorithm can presumably replace the ultimate arbiter who decides whether the results of an inversion make sense or nonsense: the geophysical interpreter. Perhaps our descendants writing a future third Millennium review article can report that a machine has been solving the inverse problem without a human arbiter. For the time being, however, what might be called “unsupervised geophysical inversion” remains but a dream.

Three types of inverse problem are proposed and defined in solid mechanics, and their solutions and properties are investigated. The solutions of the inverse problems are transformed into those of an optimization problem, and the solutions are obtained by a non-linear programming method. Based on this work, the parameter optimization method for inversion of a structural stress field is presented, and a method for evaluating solutions of the non-linear inverse problem is discussed. The feasibility of these methods is demonstrated by numerical tests. Finally, by using the proposed method and geodetic data, the non-uniform rupture pattern of the Tangshan earthquake fault is analysed. It is shown that the parameter optimization method for inversion of the tectonophysical model will improve the present methods used in the studies of structural stress fields.

Inversion of geophysical data relies on knowledge about how to solve the forward problem, that is, computing data from a given set of model parameters. In many applications of inverse problems, the solution to the forward problem is assumed to be known perfectly, without any error. In reality, solving the forward model (forward-modeling process) will almost always be prone to errors, which we referred to as modeling errors. For a specific forward problem, computation of crosshole tomographic first-arrival traveltimes, we evaluated how the modeling error, given several different approximate forward models, can be more than an order of magnitude larger than the measurement uncertainty. We also found that the modeling error is strongly linked to the spatial variability of the assumed velocity field, i.e., the a priori velocity model. We discovered some general tools by which the modeling error can be quantified and cast into a consistent formulation as an additive Gaussian observation error. We tested a method for generating a sample of the modeling error due to using a simple and approximate forward model, as opposed to a more complex and correct forward model. Then, a probabilistic model of the modeling error was inferred in the form of a correlated Gaussian probability distribution. The key to the method was the ability to generate many realizations from a statistical description of the source of the modeling error, which in this case is the a priori model. The methodology was tested for two synthetic ground-penetrating radar crosshole tomographic inverse problems. Ignoring the modeling error can lead to severe artifacts, which erroneously appear to be well resolved in the solution of the inverse problem. Accounting for the modeling error leads to a solution of the inverse problem consistent with the actual model. Further, using an approximate forward modeling may lead to a dramatic decrease in the computational demands for solving inverse problems.

Seismic oceanography can provide a two- or three-dimensional view of the water column thermocline structure at a vertical and horizontal resolution from the multi-channel seismic dataset. Several seismic imaging methods and techniques for seismic oceanography have been presented in previous research. In this study, we suggest a new formulation of the frequency-domain reverse-time migration method for seismic oceanography based on the analytic Green's function. For imaging thermocline structures in the water column from the seismic data, our proposed seismic reverse-time migration method uses the analytic Green's function for numerically calculating the forward- and backward-modeled wavefield rather than the wave propagation modeling in the conventional algorithm. The frequency-domain reverse-time migration with analytic Green's function does not require significant computational memory, resources, or a multifrontal direct solver to calculate the migration seismic images as like conventional reverse-time migration. The analytic Green's function in our reverse-time method makes it possible to provide a high-resolution seismic water column image with a meter-scale grid size, consisting of full-band frequency components for a modest cost and in a low-memory environment for computation. Our method was applied to multi-channel seismic data acquired in the Arctic Ocean and successfully constructed water column seismic images containing the oceanographic reflections caused by thermocline structures of the water mass. From the numerical test, we note that the oceanographic reflections of the migrated seismic images reflected the distribution of Arctic waters in a shallow depth and showed good correspondence with the anomalies of measured temperatures and calculated reflection coefficients from each XCDT profile. Our proposed method has been verified for field data application and accuracy of imaging performance.

Inverse problems in geophysics require the introduction of complex a priori information and are solved using computationally expensive Monte Carlo techniques (where large portions of the model space are explored). The geostatistical method allows for fast integration of complex a priori information in the form of covariance functions and training images. We combine geostatistical methods and inverse problem theory to generate realizations of the posterior probability density function of any Gaussian linear inverse problem, honoring a priori information in the form of a covariance function describing the spatial connectivity of the model space parameters. This is achieved using sequential Gaussian simulation, a well-known, noniterative geostatisticalmethod for generating samples of a Gaussian random field with a given covariance function. This work is a contribution to both linear inverse problem theory and geostatistics. Our main result is an efficient method to generate realizations, actual solutions rather than the conventional least-squares-based approach, to any Gaussian linear inverse problem using a noniterative method. The sequential approach to solving linear and weakly nonlinear problems is computationally efficient compared with traditional least-squares-based inversion. The sequential approach also allows one to solve the inverse problem in only a small part of the model space while conditioned to all available data. From a geostatistical point of view, the method can be used to condition realizations of Gaussian random fields to the possibly noisy linear average observations of the model space.

The paper presents methods overview and software for forward and inverse gravity problems solution for contact surfaces. Forward calculations are improved by introducing the finite elements method. This allows one to calculate the gravity field of shallow boundaries. Inversion is performed by the method of local corrections. Both methods are optimized with CUDA, which significantly increased efficiency. The software CUGRACID is distributed under open license. Usage information for forward and inverse problems solution programs is provided.

As we seek to make geophysical models more consistent with geological models, more and more a priori information is incorporated into geophysical inverse problems. How we make the most use of a priori information is still a key issue in geophysical inversion, particularly for effectively representing such information in inversion problems. To address this problem, we propose a symmetric polynomial constrained inversion algorithm that can represent multiple known physical properties for the lithology. The electrical resistivity synthetic model tests show that the inversion results constrained with symmetric polynomials can make the model resistivity be in line with a priori information and obtain more realistic sharp boundaries. Finally, we give a practical verification of our inversion algorithm with field data.