Simulation of fine-scale electrical conductivity fields using resolution-limited tomograms and area-to-point kriging
TL;DR: In this article, a two-step methodology based on area-to-point kriging is proposed to generate fine-scale multi-Gaussian realizations from smooth tomographic images.
Abstract:
Deterministic geophysical inversion approaches yield tomographic images with strong imprints of the regularization terms required to solve otherwise ill-posed inverse problems. While such tomograms enable an adequate assessment of the larger-scale features of the probed subsurface, the finer-scale details tend to be unresolved. Yet, representing these fine-scale structural details is generally desirable and for some applications even mandatory. To address this problem, we have developed a two-step methodology based on area-to-point kriging to generate fine-scale multi-Gaussian realizations from smooth tomographic images. Specifically, we use a co-kriging system in which the smooth, low-resolution tomogram is related to the fine-scale heterogeneity through a linear mapping operation. This mapping is based on the model resolution and the posterior covariance matrices computed using a linearization around the final tomographic model. This, in turn, allows us for analytical computations of covariance and cross-covariance models. The methodology is tested on a heterogeneous synthetic 2-D distribution of electrical conductivity that is probed with a surface-based electrical resistivity tomography (ERT) survey. The results demonstrate the ability of this technique to reproduce a known geostatistical model characterizing the fine-scale structure, while simultaneously preserving the large-scale structures identified by the smoothness-constrained tomographic inversion. Small discrepancies between the geophysical forward responses of the realizations and the reference synthetic data are attributed to the underlying linearization. Overall, the method provides an effective and fast alternative to more comprehensive, but computationally more expensive approaches, such as, for example, Markov chain Monte Carlo techniques. Moreover, the proposed method can be used to generate fine-scale multivariate Gaussian realizations from virtually any smoothness-constrained inversion results given the corresponding resolution and posterior covariance matrices.
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TL;DR: This work proposes the use of a new EKI-based framework for ERT which estimates a resistivity model and its uncertainty at a modest computational cost and highlights its readiness and applicability to similar problems in geophysics.
Abstract: Electrical resistivity tomography (ERT) is widely used to image the Earth's subsurface and has proven to be an extremely useful tool in application to hydrological problems. Conventional smoothness-constrained inversion of ERT data is efficient and robust, and consequently very popular. However, it does not resolve well sharp interfaces of a resistivity field and tends to reduce and smooth resistivity variations. These issues can be problematic in a range of hydrological or near-surface studies, e.g. mapping regolith-bedrock interfaces. While fully Bayesian approaches, such as those employing Markov chain Monte Carlo sampling, can address the above issues, their very high computation cost makes them impractical for many applications. Ensemble Kalman Inversion (EKI) offers a computationally efficient alternative by approximating the Bayesian posterior distribution in a derivative-free manner, which means only a relatively small number of 'black-box' model runs are required. Although common limitations for ensemble Kalman fillter-type methods apply to EKI, it is both efficient and generally captures uncertainty patterns correctly. We propose the use of a new EKI-based framework for ERT which estimates a resistivity model and its uncertainty at a modest computational cost. Our EKI framework uses a level-set parameterization of the unknown resistivity to allow efficient estimation of discontinuous resistivity fields. Instead of estimating level-set parameters directly, we introduce a second step to characterize the spatial variability of the resistivity field and infer length scale hyper-parameters directly. We demonstrate these features by applying the method to a series of synthetic and field examples. We also benchmark our results by comparing them to those obtained from standard smoothness-constrained inversion. Resultant resistivity images from EKI successfully capture arbitrarily shaped interfaces between resistivity zones and the inverted resistivities are close to the true values in synthetic cases. We highlight its readiness and applicability to similar problems in geophysics.
18 citations
Cites background from "Simulation of fine-scale electrical..."
...Similarly, to speed up the generation of MCMC proposals and hence convergence, an area-to-point kriging approach has been proposed recently to generate fine-scale multi-Gaussian realizations from smooth tomographic images obtained from smoothness-constrain inversions (Nussbaumer et al. 2019)....
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01 Jan 2019
TL;DR: This thesis examines various aspects of uncertainty in ERT and develops new methods to better use geophysical data quantitatively and proposes that the various steps in the general workflow of an ERT study can be viewed as a pipeline for information and uncertainty propagation and suggested some areas have been understudied.
Abstract: Our knowledge and understanding to the heterogeneous structure and processes occurring in the Earth’s subsurface is limited and uncertain. The above is true even for the upper 100m of the subsurface, yet many processes occur within it (e.g. migration of solutes, landslides, crop water uptake, etc.) are important to human activities. Geophysical methods such as electrical resistivity tomography (ERT) greatly improve our ability to observe the subsurface due to their higher sampling frequency (especially with autonomous time-lapse systems), larger spatial coverage and less invasive operation, in addition to being more cost-effective than traditional point-based sampling. However, the process of using geophysical data for inference is prone to uncertainty. There is a need to better understand the uncertainties embedded in geophysical data and how they translate themselves when they are subsequently used, for example, for hydrological or site management interpretations and decisions. This understanding is critical to maximize the extraction of information in geophysical data. To this end, in this thesis, I examine various aspects of uncertainty in ERT and develop new methods to better use geophysical data quantitatively. The core of the thesis is based on two literature reviews and three papers. In the first review, I provide a comprehensive overview of the use of geophysical data for nuclear site characterization, especially in the context of site clean-up and leak detection. In the second review, I survey the various sources of uncertainties in ERT studies and the existing work to better quantify or reduce them. I propose that the various steps in the general workflow of an ERT study can be viewed as a pipeline for information and uncertainty propagation and suggested some areas have been understudied. One of these areas is measurement errors. In paper 1, I compare various methods to estimate and model ERT measurement errors using two long-term ERT monitoring datasets. I also develop a new error model that considers the fact that each electrode is used to make multiple measurements. In paper 2, I discuss the development and implementation of a new method for geoelectrical leak detection. While existing methods rely on obtaining resistivity images through inversion of ERT data first, the approach described here estimates leak parameters directly from raw ERT data. This is achieved by constructing hydrological models from prior site information and couple it with an ERT forward model, and then update the leak (and other hydrological) parameters through data assimilation. The approach shows promising results and is applied to data from a controlled injection experiment in Yorkshire, UK. The approach complements ERT imaging and provides a new way to utilize ERT data to inform site characterisation. In addition to leak detection, ERT is also commonly used for monitoring soil moisture in the vadose zone, and increasingly so in a quantitative manner. Though both the petrophysical relationships (i.e., choices of appropriate model and parameterization) and the derived moisture content are known to be subject to uncertainty, they are commonly treated as exact and error‐free. In paper 3, I examine the impact of uncertain petrophysical relationships on the moisture content estimates derived from electrical geophysics. Data from a collection of core samples show that the variability in such relationships can be large, and they in turn can lead to high uncertainty in moisture content estimates, and they appear to be the dominating source of uncertainty in many cases. In the closing chapters, I discuss and synthesize the findings in the thesis within the larger context of enhancing the information content of geophysical data, and provide an outlook on further research in this topic.
14 citations
TL;DR: In this paper, a coupled hydrogeophysical model is used to assess the discrepancies in recovered salinity caused by the geophysical inversion and the application of the chosen petrophysical model when heterogeneity is ignored.
12 citations
Cites methods from "Simulation of fine-scale electrical..."
...3.5 ERI inversion 345 For the ERI inversion the simulated forward electrical responses (quadrupoles of 346 Wenner-alpha arrays of 72 electrodes) from the four modelled scenarios were contaminated 347 with 3% uncorrelated Gaussian noise (e.g., Nussbaumer et al., 2019)....
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...…detail the effect of the different types of heterogeneities and the relation 686 between property variation scales and resolution of the geophysical method and provide 687 alternative methodologies to improve the estimation of properties from electrical tomograms 688 (e.g. Nussbaumer et al., 2019)....
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TL;DR: In this paper, the authors use approximate Bayesian computations and the Kullback-Leibler divergence measure to quantify to what extent horizontal and vertical equivalent electrical conductivity time-series observed during tracer tests constrain the 2-D geostatistical parameters of multivariate Gaussian log-hydraulic conductivity fields.
8 citations
TL;DR: In this article, a stochastic inversion procedure for common-offset ground-penetrating radar (GPR) reflection measurements was developed for a common-offered radar system.
Abstract: We have developed a stochastic inversion procedure for common-offset ground-penetrating radar (GPR) reflection measurements. Stochastic realizations of subsurface properties that offer an a...
4 citations
References
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TL;DR: The usefulness of the electrical resistivity log in determining reservoir characteristics is governed largely by: (1) the accuracy with which the true resistivity of the formation can be determined; (2) the scope of detailed data concerning the relation of resistivity measurements to formation characteristics; (3) the available information concerning the conductivity of connate or formation waters; and (4) the extent of geologic knowledge regarding probable changes in facies within given horizons, both vertically and laterally, particularly in relation to the resultant effect on the electrical properties of the reservoir as mentioned in this paper.
Abstract: THE usefulness of the electrical resistivity log in determining reservoir characteristics is governed largely by: (I) the accuracy with which the true resistivity of the formation can be determined; (2) the scope of detailed data concerning the relation of resistivity measurements to formation characteristics; (3) the available information concerning the conductivity of connate or formation waters; (4) the extent of geologic knowledge regarding probable changes in facies within given horizons, both vertically and laterally, particularly in relation to the resultant effect on the electrical properties of the reservoir. Simple examples are given in the following pages to illustrate the use of resistivity logs in the solution of some problems dealing with oil and gas reservoirs. From the available information, it is apparent that much care must be exercised in applying to more complicated cases the methods suggested. It should be remembered that the equations given are not precise and represent only approximate relationships. It is believed, however, that under favorable conditions their application falls within useful limits of accuracy.
6,411 citations
Book•
01 Jan 1989
TL;DR: In this article, the authors describe a number of different types of inverse problems, such as the least squares problem, the purely underdetermined problem, and the Mixed*b1Determined problem.
Abstract: Preface. Introduction. DESCRIBING INVERSE PROBLEMS Formulating Inverse Problems. The Linear Inverse Problem. Examples of Formulating Inverse Problems. Solutions to Inverse Problems. SOME COMMENTS ON PROBABILITY THEORY Noise and Random Variables. Correlated Data. Functions of Random Variables. Gaussian Distributions. Testing the Assumption of Gaussian Statistics Confidence Intervals. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 1:THE LENGTH METHOD The Lengths of Estimates. Measures of Length. Least Squares for a Straight Line. The Least Squares Solution of the Linear Inverse Problem. Some Examples. The Existence of the Least Squares Solution. The Purely Underdetermined Problem. Mixed*b1Determined Problems. Weighted Measures of Length as a Type of A Priori Information. Other Types of A Priori Information. The Variance of the Model Parameter Estimates. Variance and Prediction Error of the Least Squares Solution. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 2: GENERALIZED INVERSES Solutions versus Operators. The Data Resolution Matrix. The Model Resolution Matrix. The Unit Covariance Matrix. Resolution and Covariance of Some Generalized Inverses. Measures of Goodness of Resolution and Covariance. Generalized Inverses with Good Resolution and Covariance. Sidelobes and the Backus-Gilbert Spread Function. The Backus-Gilbert Generalized Inverse for the Underdetermined Problem. Including the Covariance Size. The Trade-off of Resolution and Variance. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 3: MAXIMUM LIKELIHOOD METHODS The Mean of a Group of Measurements. Maximum Likelihood Solution of the Linear Inverse Problem. A Priori Distributions. Maximum Likelihood for an Exact Theory. Inexact Theories. The Simple Gaussian Case with a Linear Theory. The General Linear, Gaussian Case. Equivalence of the Three Viewpoints. The F Test of Error Improvement Significance. Derivation of the Formulas of Section 5.7. NONUNIQUENESS AND LOCALIZED AVERAGES Null Vectors and Nonuniqueness. Null Vectors of a Simple Inverse Problem. Localized Averages of Model Parameters. Relationship to the Resolution Matrix. Averages versus Estimates. Nonunique Averaging Vectors and A Priori Information. APPLICATIONS OF VECTOR SPACES Model and Data Spaces. Householder Transformations. Designing Householder Transformations. Transformations That Do Not Preserve Length. The Solution of the Mixed-Determined Problem. Singular-Value Decomposition and the Natural Generalized Inverse. Derivation of the Singular-Value Decomposition. Simplifying Linear Equality and Inequality Constraints. Inequality Constraints. LINEAR INVERSE PROBLEMS AND NON-GAUSSIAN DISTRIBUTIONS L1 Norms and Exponential Distributions.
3,592 citations
TL;DR: In this article, the authors proposed a smoothest model which fits the data to within an expected tolerance for the inversion of both magnetotelluric and Schlumberger sounding field data.
Abstract: The inversion of electromagnetic sounding data does not yield a unique solution, but inevitably a single model to interpret the observations is sought. We recommend that this model be as simple, or smooth, as possible, in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models.To obtain smooth models, the nonlinear forward problem is linearized about a starting model in the usual way, but it is then solved explicitly for the desired model rather than for a model correction. By parameterizing the model in terms of its first or second derivative with depth, the minimum norm solution yields the smoothest possible model.Rather than fitting the experimental data as well as possible (which maximizes the roughness of the model), the smoothest model which fits the data to within an expected tolerance is sought. A practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models, resulting in a stable and rapidly convergent algorithm. The inversion of both magnetotelluric and Schlumberger sounding field data, and a joint magnetotelluric-resistivity inversion, demonstrate the method and show it to have practical application.
2,438 citations
TL;DR: In this article, a general definition of the nonlinear least squares inverse problem is given, where the form of the theoretical relationship between data and unknowns may be general (in particular, nonlinear integrodierentia l equations).
Abstract: We attempt to give a general definition of the nonlinear least squares inverse problem. First, we examine the discrete problem (finite number of data and unknowns), setting the problem in its fully nonlinear form. Second, we examine the general case where some data and/or unknowns may be functions of a continuous variable and where the form of the theoretical relationship between data and unknowns may be general (in particular, nonlinear integrodierentia l equations). As particular cases of our nonlinear algorithm we find linear solutions well known in geophysics, like Jackson’s (1979) solution for discrete problems or Backus and Gilbert’s (1970) a solution for continuous problems.
1,800 citations
TL;DR: In this paper, the authors propose an extension of the existing 1-D algorithm, Occam's inversion, to smooth 2-D models using an extension to the existing Occam inversion.
Abstract: Magnetotelluric (MT) data are inverted for smooth 2-D models using an extension of the existing 1-D algorithm, Occam’s inversion. Since an MT data set consists of a finite number of imprecise data, an infinity of solutions to the inverse problem exists. Fitting field or synthetic electromagnetic data as closely as possible results in theoretical models with a maximum amount of roughness, or structure. However, by relaxing the misfit criterion only a small amount, models which are maximally smooth may be generated. Smooth models are less likely to result in overinterpretation of the data and reflect the true resolving power of the MT method. The models are composed of a large number of rectangular prisms, each having a constant conductivity. Apriori information, in the form of boundary locations only or both boundary locations and conductivity, may be included, providing a powerful tool for improving the resolving power of the data. Joint inversion of TE and TM synthetic data generated from known models al...
1,411 citations