Geophysical Inverse Theory and Regularization Problems

The unique solvability of linear inverse coefficient problems for the evolutionary equation in a Banach space with the Caputo Fabrizio derivative is studied. An operator at the unknown function in the equation is assumed to be bounded, the equation is endowed with the Cauchy condition. For the inverse problem with a constant unknown coefficient and with an integral overdefinition condition in the sense of Riemann Stieltjes, which includes the condition of final overdefinition as a special case, a well-posedness criterion is obtained. Sufficient conditions for unique solvability and an estimate of the well-posedness for the solution are obtained for a linear inverse problem with a time-dependent unknown coefficient. The abstract results obtained are used in the study of inverse problems with an unknown coefficient depending only on spatial variables or only on time, for equations with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.

Mathematical formulations of applied inverse problems often involve operator equations in normed functional spaces. In many cases, these spaces can, in addition, be endowed with a partial order relation, which turns them into Banach lattices. The fact that two tools, such as a partial order relation and a monotone (with respect to this partial order) norm, are available to the researchers, gives them a clear advantage of having more freedom in the problem formulations. For instance, errors in the approximate data are sometimes easier to describe in terms of pointwise bounds. Inverse problems in partially ordered normed spaces (Banach lattices) have been studied before in the case when a compact set containing the unknown exact solution is available a priori. It turned out that under this assumption it is possible, even in the ill-posed case, to compute ‘pointwise’ bounds for the unknown exact solution (or rather, bounds by means of the appropriate partial order), thus providing an error estimate by means of the partial order. Also, a useful property of this approach was that one was able to quantify the uncertainty in the operator by means of linear inequalities that were included in the corresponding optimization problems as (linear) constraints and made the computations easier. However, the compactness assumption might sometimes be too demanding in practice. This paper aims at revealing the possibilities and advantages of using partial order in solving inverse problems in the case when no compact set of prior restrictions is available, concentrating on linear inverse (possibly ill-posed) problems.

The Newsletter is a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting forum by sending to our Bristol office material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome. Book review Introduction to Inverse Problems in Imaging M Bertero and P Boccacci 1998 Bristol: Institute of Physics 362 pp ISBN 0-7503-0435-9 (pbk) £25.00, $49.00 This book shows that several problems in imaging are in fact linear inverse problems. In general inverse problems are ill-posed in the sense that small perturbations in the (measured) data have a significant influence on the output. Consequently, numerical methods developed in a general setting of inverse problems which cure the ill-posedness (such methods are called regularization methods) can be applied to solve problems in imaging. This is an excellent textbook on the principle of linear inverse problems, methods of their numerical solution and practical applications in imaging. It introduces basic ideas and methods for the solution of inverse and ill-posed problems while avoiding the mathematics of functional treatment of operator equations. This goal is achieved by a bottom-up presentation: the authors focus on the two problems of image deconvolution and tomography (bottom) and develop regularization theory specifically for these two problems. Most other books which are concerned with the numerical solution of inverse problems present a top-down approach. That is, a general regularization theory (top) is developed which is then applied to particular problems. The distinguished presentation makes the book very useful to students in applied mathematics, practitioners in engineering science and image processing. This book has the best prerequisites to establish a close link between the mathematical fields of imaging and inverse problems. Mainly the book focuses on two topics: image deconvolution and linear imaging systems. 1. In image deconvolution the goal is to restore a space invariant blurred image. This part of the book contains mathematical tools in image deconvolution, examples of space invariant imaging systems, a comparison of image deconvolution techniques and low-pass filtering techniques, the analysis of the ill-posedness of the deconvolution problem, regularization methods for deconvolution such as constrained least squares regularization, Tikhonov regularization, iterative regularization and statistical methods. Since the authors focus on particular problems they are able to analyse various methods for deconvolution on a very concrete level. As an example I would like to mention that the authors analyse Fourier methods for deconvolution. Thus they provide an optimal starting point to the theory of inverse problems for people working in the area of signal and image processing. 2. The second part of the book considers linear imaging systems that are not of a convolution type such as tomography problems, diffraction problems and inverse scattering problems. Efficient numerical methods for the solution of this class of problems such as singular value decomposition, Tikhonov regularization methods and Fourier-based methods are presented. O Scherzer Universität Linz

The Bayesian inference is widely used in many scientific and engineering problems, especially in the linear inverse problems in infinite-dimensional setting where the unknowns are functions. In such problems, choosing an appropriate prior distribution is an important task. Especially when the function to infer has much detail information, such as many sharp jumps, corners, and the discontinuous and nonsmooth oscillation, the so-called total variation-Gaussian (TG) prior is proposed in function space to address it. However, the TG prior is easy to lead the blocky (staircase) effect in numerical results. In this work, we present a fractional order-TG (FTG) hybrid prior to deal with such problems, where the fractional order total variation (FTV) term is used to capture the detail information of the unknowns and simultaneously uses the Gaussian measure to ensure that it results in a well-defined posterior measure. For the numerical implementations of linear inverse problems in function spaces, we also propose an efficient independence sampler based on a transport map, which uses a proposal distribution derived from a diagonal map, and the acceptance probability associated to the proposal is independent of discretization dimensionality. And in order to take full advantage of the transport map, the hierarchical Bayesian framework is applied to flexibly determine the regularization parameter. Finally we provide some numerical examples to demonstrate the performance of the FTG prior and the efficiency and robustness of the proposed independence sampler method.

In this study, the relationship between nonlinear model properties and inverse problem solutions is analyzed using a numerical technique based on the inverse problem theory formulated by Mosegaard and Tarantola. According to this theory, the inverse problem and solution are defined via convolution and conjunction of probability density functions (PDFs) that represent stochastic information obtained from the model, observations, and prior knowledge in a joint multidimensional space. This theory provides an explicit analysis of the nonlinear model function, together with information about uncertainties in the model, observations, and prior knowledge through construction of the joint probability density, from which marginal solution functions can then be evaluated. The numerical analysis technique derived from the theory computes the component PDFs in discretized form via a combination of function mapping on a discrete grid in the model and observation phase space and Monte Carlo sampling from known parametric distributions. The efficacy of the numerical analysis technique is demonstrated through its application to two well-known simplified models of atmospheric physics: damped oscillations and Lorenz’s three-component model of dry cellular convection. The major findings of this study include the following: (i) Use of a nonmonotonic forward model in the inverse problem gives rise to the potential for a multimodal posterior PDF, the realization of which depends on the information content of the observations and on observation and model uncertainties. (ii) The cumulative effect of observations over time, space, or both could render the final posterior PDF unimodal, even with the nonmonotonic forward model. (iii) A greater number of independent observations are needed to constrain the solution in the case of a nonmonotonic nonlinear model than for a monotonic nonlinear or linear forward model for a given number of degrees of freedom in control parameter space. (iv) A nonlinear monotonic forward model gives rise to a skewed unimodal posterior PDF, implying a well-posed maximum likelihood inverse problem. (v) The presence of model error greatly increases the possibility of capturing multiple modes in the posterior PDF with the nonmonotonic nonlinear model. (vi) In the case of a nonlinear forward model, use of a Gaussian approximation for the prior update has a similar effect to an increase in model error, which indicates there is the potential to produce a biased mean central estimate even when observations and model are unbiased.

The paper is devoted to the study of the solvability of inverse coefficient problems for degenerate parabolic equations of the second order. We study both linear inverse problems – the problems of determining an unknown right-hand side (external influence), and nonlinear problems of determining an unknown coefficient of the equation itself. The peculiarity of the studied work is that its unknown coefficients are functions of a time variable only. The work aims to prove the existence and uniqueness of regular solutions to the studied problems (having all the generalized in the sense of S.L. Sobolev derivatives entering the equation).

Millions of digital images are captured by imaging devices on a daily basis. The way imaging devices operate follows an integral process from which the information of the original scene needs to be estimated. The estimation is done by inverting the integral process of the imaging device with the use of optimization techniques. This linear inverse problem, the inversion of the integral acquisition process, is at the heart of several image processing applications such as denoising, deblurring, inpainting, and super-resolution. We describe in detail the use of linear inverse problems in these applications. We review and compare several state-of-the-art optimization algorithms that invert this integral process. Linear inverse problems are usually very difficult to solve. Therefore, additional prior assumptions need to be introduced to successfully estimate the output signal. Several priors have been suggested in the research literature, with the Total Variation (TV) being one of the most prominent. In this thesis, we review another prior, the l0 pseudo-norm over the gradient domain. This prior allows full control over how many non-zero gradients are retained to approximate prominent structures of the image. We show the superiority of the l0 gradient prior over the TV prior in recovering genuinely piece-wise constant signals. The l0 gradient prior has shown to produce state-of-the-art results in edge-preserving image smoothing. Moreover, this general prior can be applied to several other applications, such as edge extraction, clip-art JPEG artifact removal, non-photorealistic image rendering, detail magnification, and tone mapping. We review and evaluate several state-of-the-art algorithms that solve the optimization problem based on the l0 gradient prior. Subsequently we apply the l0 gradient prior to two applications where we show superior results as compared to the current state-of-the-art. The first application is that of single-image reflection removal. Existing solutions to this problem have shown limited success because of the highly ill-posed nature of the problem. We show that the standard l0 gradient prior with a modified data-fidelity term based on the Laplacian operator is able to sufficiently remove unwanted reflections from images in many realistic scenarios. We conduct extensive experiments and show that our method outperforms the state-of-the-art. In the second application of haze removal from visible-NIR image pairs we propose a novel optimization framework, where the prior term penalizes the number of non-zero gradients of the difference between the output and the NIR image. Due to the longer wavelengths of NIR, an image taken in the NIR spectrum suffers significantly less from haze artifacts. Using this prior term, we are able to transfer details from the haze-free NIR image to the final result. We show that our formulation provides state-of-the-art results compared to haze removal methods that use a single image and also to those that are based on visible-NIR image pairs.

We study the convergence rates of a non-stationary asymptotical regularization method for linear statistical inverse problems from indirect noisy measurements. The convergence rates are usually derived under some range-type source conditions, which are difficult to be verified rigorously in general, especially for the models governed by some partial differential equations. We shall propose a new strategy associated with conditional stability estimates to obtain the error bounds in terms of mean-squared error. To the best of our knowledge, it is the first time to apply conditional stability estimates for convergence rates in statistical inverse problems. As an application, an inverse source problem in a wave equation is considered and its conditional stability estimate is verified rigorously, and therefore the convergence rates are derived immediately. Several numerical experiments for the inverse source problem are also provided to demonstrate the accuracy and efficiency of the non-stationary asymptotical regularization method.

This study compares the efficiency of 3-D transient electromagnetic forward modeling schemes on the multi-resolution grid for various modeling scenarios. We developed time-domain finite-difference modeling based on the explicit scheme earlier. In this work, we additionally implement 3-D transient electromagnetic forward modeling using the backward Euler implicit scheme. The iterative solver is used for solving the system of equations and requires a proper initial guess that has significant effect on the convergence. The standard approach usually employs the solution of a previous time step as an initial guess, which might be too conservative. Instead, we test various initial guesses based on the linear extrapolation or linear combination of the solutions from several previous steps. We build up the implicit scheme forward modeling on the multi-resolution grid, which allows for the adjustment of the horizontal resolution with depth, hence improving the performance of the forward operator. Synthetic examples show the implicit scheme forward modeling using the linearly combined initial guess estimate on the multi-resolution grid additionally reduces the run time compared to the standard initial guess approach. The result of comparison between the implicit scheme developed here with the previously developed explicit scheme shows that the explicit scheme modeling is more efficient for more conductive background models often found in environmental studies. However, the implicit scheme modeling is more suitable for the simulation with highly resistive background models, usually occurring in mineral exploration scenarios. Thus, the inverse problem can be solved using more efficient forward solution depending on the modeling setup and background resistivity.

A fast, approximate 3D TEM inversion scheme has been developed, based on EM modeling at the resistive limit. The TEM data are converted to magnetic moments via time integration. In effect, the moment transformation converts the nonlinear 3D TEM inversion problem into a linear 3D magnetic inversion problem. The resistive limit response is realized as a linear combination of a discretized 3D target response and a continuous host response. A starting model is constructed from conductivity depth images of the TEM profiles. In addition, the inversion is constrained by geological information and by standard potential field inversion devices such as depth weights. The underlying model is both geological and petrophysical. The inverse problem is solved using a fast steepest‐descent algorithm. The approximate inversion scheme has been successfully tested on synthetic fixed‐loop TEM examples and on real fixed‐loop TEM field data. Typically, for ground‐TEM, the approximate 3D inversion completes in minutes on a Pentium 4 2.8‐Ghz (from 2006).

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 surface nuclear magnetic resonance (SNMR) method is a geophysical method designed for non-invasive groundwater investigations. Inversion of experimental data provides the spatial distribution of the water content in the subsurface. However, SNMR inversion is ill-posed and admits many solutions because of the imaging equation properties that are compounded by experimental error. SNMR data sets are conveniently presented as complex numbers, thus possessing phase and amplitude components. Subsurface electroconductive formations and fluctuations of the Earth's magnetic field cause non-negligible phase shifts. Consequently, the forward modelling of the SNMR signal generated by 3-D water saturated formations is achieved in the complex domain. Nevertheless, in many cases, phase measurements are less reliable than amplitude measurements and water content rendering cannot be carried out using the complex SNMR signal. This problem is resolved by performing inversion using complex forward modelling whose resulting signal amplitude is used for comparison with the data. Along with water content boundaries ranging from 0 to 1, this property turns the linear initial value inversion problem into a non-linear one. In such a situation, the comprehensive analysis of inversion uncertainties is achieved by performing a solution space exploration based on a Monte Carlo approach. An adapted Metropolis-Hastings algorithm has been used on SNMR 3-D data sets to perform such an exploration. Computing time depends on the problem dimensions. With a standard laptop computer about 10 hr were necessary for the inversion of our field data set. The resulting model collection is used to calculate the probability density functions of the water content. From there, it is possible to estimate the uncertainty of the water content imagery. Using both synthetic and experimental data, we show that our routine provides robust estimates of the spatial distribution of the water content for the SNMR 3-D initial amplitude inversion.

Stochastic inverse modeling deals with the estimation of functions from sparse data, which is a problem with a nonunique solution, with the objective to evaluate best estimates, measures of uncertainty, and sets of solutions that are consistent with the data. As finer resolutions become desirable, the computational requirements increase dramatically when using conventional solvers. A method is developed in this paper to solve large-scale stochastic linear inverse problems, based on the hierarchical matrix (or ℋ 2 matrix) approach. The proposed approach can also exploit the sparsity of the underlying measurement operator, which relates observations to unknowns. Conventional direct algorithms for solving large-scale linear inverse problems, using stochastic linear inversion techniques, typically scale as 𝒪(n 2 m+nm 2), where n is the number of measurements and m is the number of unknowns. We typically have n ≪ m. In contrast, the algorithm presented here scales as 𝒪(n 2 m), i.e., it scales linearly with the larger problem dimension m. The algorithm also allows quantification of uncertainty in the solution at a computational cost that also grows only linearly in the number of unknowns. The speedup gained is significant since the number of unknowns m is often large. The effectiveness of the algorithm is demonstrated by solving a realistic crosswell tomography problem by formulating it as a stochastic linear inverse problem. In the case of the crosswell tomography problem, the sparsity of the measurement operator allows us to further reduce the cost of our proposed algorithm from 𝒪(n 2 m) to $\mathcal {O}(n^{2} \sqrt {m} + nm)$ . The computational speedup gained by using the new algorithm makes it easier, among other things, to optimize the location of sources and receivers, by minimizing the mean square error of the estimation. Without this fast algorithm, this optimization would be computationally impractical using conventional methods.

The geomagnetic components from eight low-latitude ground stations and the ATS 1 satellite are studied for substorms occurring on December 24 and 25, 1967, and on August 15, 1968. A simple, four-parameter, field-aligned line current model is employed, and the linear inverse problem is constructed to determine the set of model parameters that best fit the data. A straightforward review of linear inversion theory is included in the theory section. The four model parameters are the magnitude of the current, the L shell on which the current flows, and the azimuthal positions of the eastern and western portions of the circuit. Both growth phases and expansion phases are treated. The best fits to the ground and satellite data are shown to be independent of the L parameter for values of L ≳ 3.5 when only ground data are treated and for values of L ≳ 8.5 when satellite data are included. The four-parameter model, although it produces good qualitative fits to expansion phase data, is unable to fit the data within 2 standard deviations of the error in the observations. The model is able to fit the growth phase of one substorm within 1 standard deviation. In one substorm with an apparent growth phase a better fit of the data during the expansion phase was obtained by combining the line current model of the growth phase with an additional four-parameter line current model.

We apply the natural pixel (NP) approach to the single‐photon emission computed tomography (SPECT) problem. The NP approach allows us to split the tomographic problem into two subproblems. The first is a linear inverse problem. The data are the measured projections and the linear operator is described by a Gram matrix, and provides a set of coefficients. The second consists of computing the solution of the tomographic problem as a linear combination of the elements of the NP basis with respect to the coefficients obtained by solving the first problem, and it provides a solution for any given grid of points. The spatially varying geometric response of the system is taken into account by properly choosing the elements of the basis. The rotational invariance shown by the elements of the considered basis induces a block circulant structure in the Gram matrix. This structure can be used to reduce the computational efforts needed for solving the inverse problem. In particular, we diagonalize (blockwise) the Gram matrix by applying the discrete Fourier transform and we solve the inverse problem in the frequency domain associated with the rotation angles. We develop numerical validation with synthetic data in order to test the performance of the NP approach and to assess the reliability of the results. A reconstruction of a two‐dimensional image requires 45–94 s, which is an acceptable time for clinical purposes. Finally, we apply the method to acquired clinical data that consist of a three‐dimensional brain scan. © 2002 John Wiley & Sons, Inc. Int J Imaging Syst Technol 12, 1–8, 2002