A Fast Approach to Magnetic Equivalent Source Processing Using an Adaptive Quadtree Mesh Discretization

The use of two to four magnetometers in airborne magnetic surveys is becoming more frequent, allowing for the observation of the total‐field as well as its horizontal gradients (Redford, et al., 2006). There are advantages to these data types because of the increased and complementary information content in them. Yet little has been done to take full advantage of these data beyond simple enhancement of the total‐field anomaly. As a first step, we have developed an equivalent source processing algorithm by making use of a well‐known but little used relationship between the derivative of the total‐field with the derivative of its source. Utilizing gradient and total‐field data allows for a constructed susceptibility model for total‐field data, constrained by the observed gradients. In this paper, we discuss using gradients in order to construct a more representative dataset through equivalent source processing. The methodology of the joint equivalent source processing technique is presented and a synthetic example is shown.

Fostered by advances in computing, simulation and data science have emerged as the third and fourth paradigms enabling scientific discovery. Many mathematical challenges arise when integrating these paradigms to produce new knowledge. One of the most important challenges is the inverse or calibration problem: find the inputs of a simulation such that its outputs agree with a given set of measurements. As both the cost of the complex simulations and the volume of measurement data sets increase, intuition-driven trial-and-error methods for calibration quickly become untenable; modern scientists need more computationally efficient and mathematically rigorous methods. When measurements and simulated predictions contain uncertainty, the Bayesian perspective on statistical inference provides a popular mathematical framework for developing computational methods to solve inverse problems. Markov chain Monte Carlo (MCMC) methods are a mainstay of this framework; they characterize a probability distribution on the space of model inputs that is conditioned on the given measurements. However, despite recent advances, MCMC methods remain impractical for expensive simulations with more than a handful of inputs. At its heart, MCMC searches the space of inputs for corresponding outputs that match measurements, and so it is afflicted with the curse of dimensionality. Additionally, MCMC methods become less efficient as the volume of the measurement data increases, particularly if the measurements are redundant. To make MCMC—and Bayesian methods, in general—amenable to today’s computer models with millions of inputs and terabytes of measurement data, one must devise methods to reduce the dimension of both the simulation inputs and the measurements. For many models used in practice, the outputs vary primarily on a low-dimensional manifold of the nominally high-dimensional input space. Recently developed active subspace methods discover this low-dimensional manifold and exploit it to enable analyses typically suited for low-dimensional settings (e.g., statistical inversion of complex systems). We hypothesize that active subspace methods can provide the necessary dimension reduction, enabling existing Bayesian methods to find solutions for a large class of otherwise intractable statistical inverse problems. We will test the hypothesis with a series of mathematical analyses and computational experiments that will proceed along three research veins: (i) integrating active subspace methods into methods for statistical inversion, (ii) addressing current challenges in active subspace methods, and (iii) incorporating methods for nonlinear dimensionality reduction into active subspace methods. We will apply the methods we develop to real inverse problems in chemical kinetics and turbulent flame modeling. Our team combines expertise in methods for inverse problems, dimension reduction, and reduced order modeling. With a history of active collaboration and thoughtful engagement, we bring enthusiasm and fresh perspectives to a well studied subject. We are excited by the potential opportunity to work together to extend and apply active subspace methods to data-intensive inverse problems.

ABSTRACTLeast squares Fourier reconstruction is basically a solution to a discrete linear inverse problem that attempts to recover the Fourier spectrum of the seismic wavefield from irregularly sampled data along the spatial coordinates. The estimated Fourier coefficients are then used to reconstruct the data in a regular grid via a standard inverse Fourier transform (inverse discrete Fourier transform or inverse fast Fourier transform).Unfortunately, this kind of inverse problem is usually under‐determined and ill‐conditioned. For this reason, the least squares Fourier reconstruction with minimum norm adopts a damped least squares inversion to retrieve a unique and stable solution.In this work, we show how the damping can introduce artefacts on the reconstructed 3D data. To quantitatively describe this issue, we introduce the concept of “extended” model resolution matrix, and we formulate the reconstruction problem as an appraisal problem. Through the simultaneous analysis of the extended model resolution matrix and of the noise term, we discuss the limits of the Fourier reconstruction with minimum norm reconstruction and assess the validity of the reconstructed data and the possible bias introduced by the inversion process. Also, we can guide the parameterization of the forward problem to minimize the occurrence of unwanted artefacts. A simple synthetic example and real data from a 3D marine common shot gather are used to discuss our approach and to show the results of Fourier reconstruction with minimum norm reconstruction.

The solution of a Rayleigh-wave inverse problem may potentially deviate from the realistic shear-wave velocity structure due to nonuniqueness. To overcome such deviation, it is necessary to understand the source of nonuniqueness and situations that may give rise to the nonuniqueness. In this study, the existence and formation of the nonunique solutions in an inverse problem are demonstrated by modeling the solution space of a synthetic surface-wave inverse problem and investigating the major causes that might engender nonuniqueness, namely (1) the inversion convergence threshold, (2) ambient noise, (3) corner frequency of the recordings, and (4) the water level. Regarding the severity of nonuniqueness in the phase-velocity inverse problems, a technique is proposed to improve the inversion that exploits the match between the synthetic and observed time series used as a posteriori information for constraining the realistic velocity structure. Through a synthetic example, the effectiveness of such method is tested and demonstrated effective.

Simulation of hyperelastic materials in real-time using deep learning.

Parallel inverse-problem solver for time-domain optical tomography with perfect parallel scaling

For many inverse parameter problems for partial differential equations in which the domain contains only well-separated objects, an asymptotic solution to the forward problem involving ‘polarization tensors’ exists. These are functions of the size and material contrast of inclusions, thereby describing the saturation component of the non-linearity. In this paper, we show how such an asymptotic series can be applied to non-linear least-squares reconstruction problems, by deriving an approximate diagonal Hessian matrix for the data misfit term. Often, the Hessian matrix can play a vital role in dealing with the non-linearity, generating good update directions which accelerate the solution towards a global minimum, but the computational cost can make direct calculation infeasible. Since the polarization tensor approximation assumes sufficient separation between inclusions, our approximate Hessian does not account for non-linearity in the form of lack of superposition in the inverse problem. It does, however, account for the non-linear saturation of the change in the data with increasing material contrast. We, therefore, propose to use it as an initial Hessian for quasi-Newton schemes. We present numerical experimentation into the accuracy and reconstruction performance of the approximate Hessian for the case of electrical impedance tomography, providing a proof of principle of the reconstruction scheme.

Regularization is usually necessary to guarantee a solution to a given inverse problem. When constructing a model that gives an adequate fit to the data, some suitable method of regularization which provides numerical stability can be used. When investigating the resolution and variance of the computed model parameters, the character of regularization should be specified by the a priori information available. This avoids arbitrary variation of the damping to suit the interpreter. For geophysical inverse problems we determine the appropriate level of regularization (in the form of parameter covariances) from power spectral analysis of well‐log measurements. For resistivity data, well logs indicate that the spatial variation with depth can be described adequately by a scaling noise model, that is, one in which the power spectral density is proportional to some power (α) of the frequency. We show that α, the scaling exponent, controls the smoothness of the final model. For α < 0, the model becomes smoother as α becomes more negative. As a specific example, this approach is applied to the magnetotelluric inverse problem. A synthetic example illustrates the smoothing effect of α on inversion. Comparison between the scaling noise approach and a previous Backus‐Gilbert type inversion on some field data shows that using the appropriate value of α (−1.8 for this example) results in a model which is structurally simple and contains only those features well resolved by the data.

Amplitude variation with offset (AVO) inversion is a typical ill-posed inverse problem. To obtain a stable and unique solution, regularization techniques relying on mathematical models from prior information are commonly used in conventional AVO inversion methods (hence the name model-driven methods). Due to the difference between prior information and the actual geology, these methods often have difficulty achieving satisfactory accuracy and resolution. We have developed a novel data-driven inversion method for the AVO inversion problem. This method can effectively extract useful knowledge from well-log data, including sparse dictionaries of elastic parameters and sparse representation of subsurface model parameters. Lateral continuity of subsurface geology allows for the approximation of model parameters for a work area using the learned dictionaries. Instead of particular mathematical models, a sparse representation is used to constrain the inverse problem. Because no assumption is made about the model parameters, we consider this a data-driven method. The general process of the algorithm is as follows: (1) using well-log data as the training samples to learn the sparse dictionary of each elastic parameter, (2) imposing a sparse representation constraint on the objective function, making the elastic parameters be sparsely represented over the learned dictionary, and (3) solving the objective function by applying a coordinate-descent algorithm. Tests on several synthetic examples and field data demonstrate that our algorithm is effective in improving the resolution and accuracy of solutions and is adaptable to various geologies.

The gradient-based optimization methods are preferable for the large-scale three-dimensional (3D) magnetotelluric (MT) inverse problem. Compared with the popular nonlinear conjugate gradient (NLCG) method, however, the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method is less adopted. This paper aims to implement a L-BFGS-based inversion algorithm for the 3D MT problem. And we develop our code on top of the ModEM package, which is highly extensible and popular among the MT community. To accelerate the convergence speed, the preconditioning technique by the affine linear transformation of the original model parameters is used. Two modifications of the conventional L-BFGS algorithm are also made to get a comparable convergence rate with the NLCG method. The impacts of the preconditioner parameters, the regularization parameters, the starting model, etc., on the inversion are evaluated by synthetic examples for both L-BFGS and NLCG methods. And the real MT Kayabe dataset is also inverted by the inversion algorithms. The synthetic tests show that through our L-BFGS inversion algorithm the similar resistivity models can be obtained with that from the NLCG method. For the real data inversion, the L-BFGS method performs more efficiently and reasonable results could be obtained by less iterations of the inversion process than the NLCG method. Thus, we suggest the common usage of the L-BFGS method for the 3D MT inverse problem.

A method is proposed for estimating the particle size distribution (PSD) of a latex with particle diameters in the sub-micrometer range, from combined elastic light scattering (ELS) and dynamic light scattering (DLS) measurements. The method is implemented through a general regression neural network (GRNN), that estimates the PSD from the ELS measurement carried out at several angles together with the average diameters of the PSD predicted by the DLS measurement at the same angles. The GRNN was trained with several measurements simulated on the basis of typical asymmetric PSDs. The ability of the trained GRNN was tested on the basis of two synthetic examples. The estimated PSDs are more accurate than those obtained through standard numerical techniques for `ill-conditioned' inverse problems.

In this study we develop a practical method for the location of microseismic events in anisotropic media. The velocity anisotropy is taken into account in the additional step after the conventional location procedure. In this step, the isotropic velocity model and corresponding estimated locations serve as an initial model, and the anisotropy parameters as well as resulting location misfits are inverted simultaneously due to the coupling of velocity model and hypocenter parameters in calculation. After obtaining the anisotropic velocity model, one can relocate all the events around their isotropic velocity based locations. The method is ray theory based and only consider transversely isotropic layered model. The nonlinear inversion problem is solved by using Gauss-Newton method. The results of synthetic example closely fit with the true values and event depths are better determined than radial distances. Real data test shows that the estimated Thomsen parameters are consistent with previous study, and the isotropic velocity based locations are significantly corrected with large decrease of average misfit from 52m to 6m.

In diffuse optical tomography (DOT), the discretization error in the numerical solutions of the forward and inverse problems results in error in the reconstructed optical images. In this first part of our work, we analyse the error in the reconstructed optical absorption images, resulting from the discretization of the forward and inverse problems. Our analysis identifies several factors which influence the extent to which the discretization impacts on the accuracy of the reconstructed images. For example, the mutual dependence of the forward and inverse problems, the number of sources and detectors, their configuration and their orientation with respect to optical absorptive heterogeneities, and the formulation of the inverse problem. As a result, our error analysis shows that the discretization of one problem cannot be considered independent of the other problem. While our analysis focuses specifically on the discretization error in DOT, the approach can be extended to quantify other error sources in DOT and other inverse parameter estimation problems.

Current deep learning-based solutions for image analysis tasks are commonly incapable of handling problems to which multiple different plausible solutions exist. In response, posterior-based methods such as conditional Diffusion Models and Invertible Neural Networks have emerged; however, their translation is hampered by a lack of research on adequate validation. In other words, the way progress is measured often does not reflect the needs of the driving practical application. Closing this gap in the literature, we present the first systematic framework for the application-driven validation of posterior-based methods in inverse problems. As a methodological novelty, it adopts key principles from the field of object detection validation, which has a long history of addressing the question of how to locate and match multiple object instances in an image. Treating modes as instances enables us to perform mode-centric validation, using well-interpretable metrics from the application perspective. We demonstrate the value of our framework through instantiations for a synthetic toy example and two medical vision use cases: pose estimation in surgery and imaging-based quantification of functional tissue parameters for diagnostics. Our framework offers key advantages over common approaches to posterior validation in all three examples and could thus revolutionize performance assessment in inverse problems.

A key step in sparsifying signals is the choice of a sparsity-promoting dictionary. There are two basic approaches to design such a dictionary: the analytic approach and the learning-based approach. Although the analytic approach enjoys the advantage of high efficiency, it lacks adaptivity to various data patterns. On the other hand, the learning-based approach can adaptively sparsify different data sets but has a heavier computational complexity and involves no prior-constraint pattern information for particular data. We have developed a double-sparsity dictionary (DSD) for seismic data to combine the benefits of both approaches. We have evaluated two models to learn the DSD: the synthesis model and the analysis model. The synthesis model learns DSD in the data domain, and the analysis model learns DSD in the model domain. We tested the analysis model and proposed to use the seislet transform and data-driven tight frame (DDTF) as the base transform and adaptive dictionary, respectively, in the DSD framework. The DDTF obtains an extra structure regularization by learning dictionaries, whereas the seislet transform obtains a compensation for the transformation error caused by slope dependency. The DSD aims to provide a sparser representation than the individual transform and dictionary and, therefore, can help to achieve better performance in denoising applications. Although for the purpose of compression, the DSD is less sparse than the seislet transform, it outperforms the seislet and DDTF in distinguishing the signal and noise. Two simulated synthetic examples and three field data examples confirm a better denoising performance of the proposed approach.