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Title: Optimization and geophysical inverse problems

Abstract

A fundamental part of geophysics is to make inferences about the interior of the earth on the basis of data collected at or near the surface of the earth. In almost all cases these measured data are only indirectly related to the properties of the earth that are of interest, so an inverse problem must be solved in order to obtain estimates of the physical properties within the earth. In February of 1999 the U.S. Department of Energy sponsored a workshop that was intended to examine the methods currently being used to solve geophysical inverse problems and to consider what new approaches should be explored in the future. The interdisciplinary area between inverse problems in geophysics and optimization methods in mathematics was specifically targeted as one where an interchange of ideas was likely to be fruitful. Thus about half of the participants were actively involved in solving geophysical inverse problems and about half were actively involved in research on general optimization methods. This report presents some of the topics that were explored at the workshop and the conclusions that were reached. In general, the objective of a geophysical inverse problem is to find an earth model, described by a setmore » of physical parameters, that is consistent with the observational data. It is usually assumed that the forward problem, that of calculating simulated data for an earth model, is well enough understood so that reasonably accurate synthetic data can be generated for an arbitrary model. The inverse problem is then posed as an optimization problem, where the function to be optimized is variously called the objective function, misfit function, or fitness function. The objective function is typically some measure of the difference between observational data and synthetic data calculated for a trial model. However, because of incomplete and inaccurate data, the objective function often incorporates some additional form of regularization, such as a measure of smoothness or distance from a prior model. Various other constraints may also be imposed upon the process. Inverse problems are not restricted to geophysics, but can be found in a wide variety of disciplines where inferences must be made on the basis of indirect measurements. For instance, most imaging problems, whether in the field of medicine or non-destructive evaluation, require the solution of an inverse problem. In this report, however, the examples used for illustration are taken exclusively from the field of geophysics. The generalization of these examples to other disciplines should be straightforward, as all are based on standard second-order partial differential equations of physics. In fact, sometimes the non-geophysical inverse problems are significantly easier to treat (as in medical imaging) because the limitations on data collection, and in particular on multiple views, are not so severe as they generally are in geophysics. This report begins with an introduction to geophysical inverse problems by briefly describing four canonical problems that are typical of those commonly encountered in geophysics. Next the connection with optimization methods is made by presenting a general formulation of geophysical inverse problems. This leads into the main subject of this report, a discussion of methods for solving such problems with an emphasis upon newer approaches that have not yet become prominent in geophysics. A separate section is devoted to a subject that is not encountered in all optimization problems but is particularly important in geophysics, the need for a careful appraisal of the results in terms of their resolution and uncertainty. The impact on geophysical inverse problems of continuously improving computational resources is then discussed. The main results are then brought together in a final summary and conclusions section.« less

Authors:
; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
Earth Sciences Division
OSTI Identifier:
939130
Report Number(s):
LBNL-46959
TRN: US200823%%117
DOE Contract Number:
DE-AC02-05CH11231
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
58; EVALUATION; GEOPHYSICS; MEDICINE; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; PHYSICAL PROPERTIES; PHYSICS; RESOLUTION; ROUGHNESS

Citation Formats

Barhen, J., Berryman, J.G., Borcea, L., Dennis, J., de Groot-Hedlin, C., Gilbert, F., Gill, P., Heinkenschloss, M., Johnson, L., McEvilly, T., More, J., Newman, G., Oldenburg, D., Parker, P., Porto, B., Sen, M., Torczon, V., Vasco, D., and Woodward, N.B.. Optimization and geophysical inverse problems. United States: N. p., 2000. Web. doi:10.2172/939130.
Barhen, J., Berryman, J.G., Borcea, L., Dennis, J., de Groot-Hedlin, C., Gilbert, F., Gill, P., Heinkenschloss, M., Johnson, L., McEvilly, T., More, J., Newman, G., Oldenburg, D., Parker, P., Porto, B., Sen, M., Torczon, V., Vasco, D., & Woodward, N.B.. Optimization and geophysical inverse problems. United States. doi:10.2172/939130.
Barhen, J., Berryman, J.G., Borcea, L., Dennis, J., de Groot-Hedlin, C., Gilbert, F., Gill, P., Heinkenschloss, M., Johnson, L., McEvilly, T., More, J., Newman, G., Oldenburg, D., Parker, P., Porto, B., Sen, M., Torczon, V., Vasco, D., and Woodward, N.B.. Sun . "Optimization and geophysical inverse problems". United States. doi:10.2172/939130. https://www.osti.gov/servlets/purl/939130.
@article{osti_939130,
title = {Optimization and geophysical inverse problems},
author = {Barhen, J. and Berryman, J.G. and Borcea, L. and Dennis, J. and de Groot-Hedlin, C. and Gilbert, F. and Gill, P. and Heinkenschloss, M. and Johnson, L. and McEvilly, T. and More, J. and Newman, G. and Oldenburg, D. and Parker, P. and Porto, B. and Sen, M. and Torczon, V. and Vasco, D. and Woodward, N.B.},
abstractNote = {A fundamental part of geophysics is to make inferences about the interior of the earth on the basis of data collected at or near the surface of the earth. In almost all cases these measured data are only indirectly related to the properties of the earth that are of interest, so an inverse problem must be solved in order to obtain estimates of the physical properties within the earth. In February of 1999 the U.S. Department of Energy sponsored a workshop that was intended to examine the methods currently being used to solve geophysical inverse problems and to consider what new approaches should be explored in the future. The interdisciplinary area between inverse problems in geophysics and optimization methods in mathematics was specifically targeted as one where an interchange of ideas was likely to be fruitful. Thus about half of the participants were actively involved in solving geophysical inverse problems and about half were actively involved in research on general optimization methods. This report presents some of the topics that were explored at the workshop and the conclusions that were reached. In general, the objective of a geophysical inverse problem is to find an earth model, described by a set of physical parameters, that is consistent with the observational data. It is usually assumed that the forward problem, that of calculating simulated data for an earth model, is well enough understood so that reasonably accurate synthetic data can be generated for an arbitrary model. The inverse problem is then posed as an optimization problem, where the function to be optimized is variously called the objective function, misfit function, or fitness function. The objective function is typically some measure of the difference between observational data and synthetic data calculated for a trial model. However, because of incomplete and inaccurate data, the objective function often incorporates some additional form of regularization, such as a measure of smoothness or distance from a prior model. Various other constraints may also be imposed upon the process. Inverse problems are not restricted to geophysics, but can be found in a wide variety of disciplines where inferences must be made on the basis of indirect measurements. For instance, most imaging problems, whether in the field of medicine or non-destructive evaluation, require the solution of an inverse problem. In this report, however, the examples used for illustration are taken exclusively from the field of geophysics. The generalization of these examples to other disciplines should be straightforward, as all are based on standard second-order partial differential equations of physics. In fact, sometimes the non-geophysical inverse problems are significantly easier to treat (as in medical imaging) because the limitations on data collection, and in particular on multiple views, are not so severe as they generally are in geophysics. This report begins with an introduction to geophysical inverse problems by briefly describing four canonical problems that are typical of those commonly encountered in geophysics. Next the connection with optimization methods is made by presenting a general formulation of geophysical inverse problems. This leads into the main subject of this report, a discussion of methods for solving such problems with an emphasis upon newer approaches that have not yet become prominent in geophysics. A separate section is devoted to a subject that is not encountered in all optimization problems but is particularly important in geophysics, the need for a careful appraisal of the results in terms of their resolution and uncertainty. The impact on geophysical inverse problems of continuously improving computational resources is then discussed. The main results are then brought together in a final summary and conclusions section.},
doi = {10.2172/939130},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Sun Oct 01 00:00:00 EDT 2000},
month = {Sun Oct 01 00:00:00 EDT 2000}
}

Technical Report:

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  • The focus of research was: Developing adaptive mesh for the solution of Maxwell's equations; Developing a parallel framework for time dependent inverse Maxwell's equations; Developing multilevel methods for optimization problems with inequality constraints; A new inversion code for inverse Maxwell's equations in the 0th frequency (DC resistivity); A new inversion code for inverse Maxwell's equations in low frequency regime. Although the research concentrated on electromagnetic forward and in- verse problems the results of the research was applied to the problem of image registration.
  • In recent years there has been a proliferation of modeling techniques for forward predictions of crack propagation in brittle materials, including: phase-field/gradient damage models, peridynamics, cohesive-zone models, and G/XFEM enrichment techniques. However, progress on the corresponding inverse problems has been relatively lacking. Taking advantage of key features of existing modeling approaches, we propose a parabolic regularization of Barenblatt cohesive models which borrows extensively from previous phase-field and gradient damage formulations. An efficient explicit time integration strategy for this type of nonlocal fracture model is then proposed and justified. In addition, we present a C++ computational framework for computing in- putmore » parameter sensitivities efficiently for explicit dynamic problems using the adjoint method. This capability allows for solving inverse problems involving crack propagation to answer interesting engineering questions such as: 1) what is the optimal design topology and material placement for a heterogeneous structure to maximize fracture resistance, 2) what loads must have been applied to a structure for it to have failed in an observed way, 3) what are the existing cracks in a structure given various experimental observations, etc. In this work, we focus on the first of these engineering questions and demonstrate a capability to automatically and efficiently compute optimal designs intended to minimize crack propagation in structures.« less
  • 'It is the objective of this proposed study to develop and field test a new, integrated Hybrid Hydrologic-Geophysical Inverse Technique (HHGIT) for characterization of the vadose zone at contaminated sites. This fundamentally new approach to site characterization and monitoring will provide detailed knowledge about hydrological properties, geological heterogeneity and the extent and movement of contamination. HHGIT combines electrical resistivity tomography (ERT) to geophysically sense a 3D volume, statistical information about fabric of geological formations, and sparse data on moisture and contaminant distributions. Combining these three types of information into a single inversion process will provide much better estimates of spatiallymore » varied hydraulic properties and three-dimensional contaminant distributions than could be obtained from interpreting the data types individually. Furthermore, HHGIT will be a geostatistically based estimation technique; the estimates represent conditional mean hydraulic property fields and contaminant distributions. Thus, this method will also quantify the uncertainty of the estimates as well as the estimates themselves. The knowledge of this uncertainty is necessary to determine the likelihood of success of remediation efforts and the risk posed by hazardous materials. Controlled field experiments will be conducted to provide critical data sets for evaluation of these methodologies, for better understanding of mechanisms controlling contaminant movement in the vadose zone, and for evaluation of the HHGIT method as a long term monitoring strategy.'« less
  • 'The objective of this study is to develop and field test a new, integrated Hybrid Hydrologic-Geophysical Inverse Technique (HHGIT) for characterization of the vadose zone at contaminated sites. This new approach to site characterization and monitoring can provide detailed maps of hydrogeological heterogeneity and the extent of contamination by combining information from electric resistivity tomography (ERT) surveys, statistical information about heterogeneity and hydrologic processes, and sparse hydrologic data. Because the electrical conductivity of the vadose zone (from the ERT measurements) can be correlated to the fluid saturation and/or contaminant concentration, the hydrologic and geophysical measurements are related. As of themore » 21st month of a 36-month project, a three-dimensional stochastic hydrologic inverse model for heterogeneous vadose zones has been developed. This model employs pressure and moisture content measurements under both transient and steady flow conditions to estimate unsaturated hydraulic parameters. In this model, an innovative approach to sequentially condition the estimate using temporal measurements has been incorporated. This allows us to use vast amounts of pressure and moisture content information measured at different times while keeping the computational effort manageable. Using this model the authors have found that the relative importance of the pressure and moisture content measurements in defining the different vadose zone parameters depends on whether the soil is wet or dry. They have also learned that pressure and moisture content measurements collected during steady state flow provide the best characterization of heterogeneity compared to other types of hydrologic data. These findings provide important guidance to the design of sampling scheme of the field experiment described below.'« less