Linear Inversion Research Articles

Efficient computation of extended surface sources

Source extension is a reformulation of inverse problems in wave propagation, that at least in some cases leads to computationally tractable iterative solution methods. The core subproblem in all source extension methods is the solution of a linear inverse problem for a source (right hand side in a system of wave equations) through minimization of data error in the least squares sense with soft imposition of physical constraints on the source via an additive quadratic penalty. For an acoustic formulation for sources supported on a surface, with a soft contraint enforcing concentration at a point, a variant of the time reversal method from photoacoustic tomography provides an approximate solution. This approximate inverse can be used to precondition Krylov space iteration for rapid convergence to the solution of the core subproblem in this setting. Numerical examples illustrate the effectiveness of this preconditioner.

High Resolution Model of the Vinton Salt-Dome Cap Rock by Joint Inversion of the Full Tensor Gravity Gradient Data with the Simulated Annealing Global Optimization Method

We present a 3D high-resolution modeling methodology based on the interpretation of gravity gradient data and its joint inversion with the simulated annealing (SA) global optimization method. The geometry of the model, used as computational domain in the solution of the forward and inverse problems, is defined with an irregular ensemble of cubic prisms that recreates the interpreted shape of the target, derived from the results of applying different interpretation methods to the gravity gradient data. In our inversion approach, the linear inverse problem resulting from the domain discretization is not solved. Instead, the cost function is explored with the SA algorithm, its low misfit region is identified, and models belonging to it are selected for obtaining the mean model, which represents the most likely model among them, as well as for estimating its uncertainty. The SA inversion algorithm we applied was numerically optimized to reduce the computational burden required by the forward problem, and it was driven by optimal tuning parameters, determined by a parametric analysis. Tests on synthetic data show the efficiency of our methodology to obtain a model that approximates the synthetic target and the usefulness of the estimated uncertainty to complement the interpretation. Finally, by applying our methodology to gravity gradient data acquired over the Vinton dome located in Louisiana, USA, we obtained results that are in agreement with geological information and previous studies.

Bayesian Inversion of Finite‐Fault Earthquake Slip Model Using Geodetic Data, Solving for Non‐Planar Fault Geometry, Variable Slip, and Data Weighting

AbstractA precise finite‐fault model including the fault geometry and slip distribution is essential to understand the physics of an earthquake. However, the conventional linear inversion of geodetic data for a finite‐fault model cannot fully resolve the fault geometry. In this study, we developed a Bayesian inversion framework that can comprehensively solve a non‐planar fault geometry, the corresponding fault slip distribution with spatially variable directions, and objective weighting for multiple data types. In the proposed framework, the probability distributions of all the model parameters are sampled using the Monte Carlo method. The developed methodology removes the requirement for manual intervention for the fault geometry and data weighting and can provide an ensemble of plausible model parameters. The performance of the developed method is tested and demonstrated through inversions for synthetic oblique‐slip faulting models. The results show that the constant rake assumption can significantly bias the estimates of fault geometry and data weighting, whereas additional consideration of the variability of slip orientations can allow plausible estimates of a non‐planar fault geometry with objective data weighting. We applied the method to the 2013 Mw 6.5 Lushan earthquake in Sichuan province, China. The result reveals dominant thrust slips with left‐lateral components and a curved fault geometry, with the confidence interval of the dip angles being between 20°–25° and 56°–58°. The proposed method provides useful insights into the scope of imaging a non‐planar fault geometry, and could help to interpret more complex earthquake sources in the future.

Nondimensional river linear inversion code for ksn_prior

Nondimensional river linear inversion code for ksn_prior

Nondimensional river linear inversion code for ksn_prior

Nondimensional river linear inversion code for ksn_prior

The influence of physical and algorithmic factors on simulated far-field waveforms and source–time functions of underground explosions using unsupervised machine learning

SUMMARY Characterizing explosion sources and differentiating between earthquake and underground explosions using distributed seismic networks becomes non-trivial when explosions are detonated in cavities or heterogeneous ground material. Moreover, there is little understanding of how changes in subsurface physical properties affect the far-field waveforms we record and use to infer information about the source. Simulations of underground explosions and the resultant ground motions can be a powerful tool to systematically explore how different subsurface properties affect far-field waveform features, but there are added variables that arise from how we choose to model the explosions that can confound interpretation. To assess how both subsurface properties and algorithmic choices affect the seismic wavefield and the estimated source functions, we ran a series of 2-D axisymmetric non-linear numerical explosion experiments and wave propagation simulations that explore a wide array of parameters. We then inverted the synthetic far-field waveform data using a linear inversion scheme to estimate source–time functions (STFs) for each simulation case. We applied principal component analysis (PCA), an unsupervised machine learning method, to both the far-field waveforms and STFs to identify the most important factors that control variance in the waveform data and differences between cases. For the far-field waveforms, the largest variance occurs in the shallower radial receiver channels in the 0–50 Hz frequency band. For the STFs, both peak amplitude and rise times across different frequencies contribute to the variance. We find that the ground equation of state (i.e. lithology and rheology) and the explosion emplacement conditions (i.e. tamped versus cavity) have the greatest effect on the variance of the far-field waveforms and STFs, with the ground yield strength and fracture pressure being secondary factors. Differences in the PCA results between the far-field waveforms and STFs could possibly be due to near-field non-linearities of the source that are not accounted for in the estimation of STFs and could be associated with yield strength, fracture pressure, cavity radius and cavity shape parameters. Other algorithmic parameters are found to be less important and cause less variance in both the far-field waveforms and STFs, meaning algorithmic choices in how we model explosions are less important, which is encouraging for the further use of explosion simulations to study how physical Earth properties affect seismic waveform features and estimated STFs.

A rapid multispectral endoscopic imaging system for in vivo assessment of the morphological and physiological characteristics of mouse intestines.

Accurate assessment of blood content in biological tissues is critical for the diagnosis and monitoring of various diseases, including cardiovascular disease, tumors, trauma, and the success rate of organ transplants. In this study, a multispectral endoscopic imaging system was built for capturing tissue reflection optical images in 18 bands across the wavelength range from 400 nm to 760 nm, non-invasively. The system was characterized by six tri-channel narrowband filters installed in front of the light source to achieve spectral separation and was equipped with a specially designed color CCD for achieving a speed of 24 multispectral imaging cubes per second. A method based on linear matrix inversion was proposed to calibrate the CCD spectral response overlaps, while a spectral analysis algorithm was developed for evaluating blood content and detecting tissue composition. The developed system was implemented in an in vivo mouse model for illustrating the blood volume, blood oxygen saturation index, and scattering particle size of the intestinal wall mucosa. The observations not only helped us to understand the blood supply situation in the intestinal mucosa, but also further testified the feasibility of our presented system. Meanwhile, the developed system could provide critical non-invasive optical information for intracavitary cancer diagnosis, surgery guidance, and treatment assessment.

An Exact Zoeppritz Based Prestack Inversion Using Whale Optimization Particle Filter Algorithm Under Bayesian Framework

Conventional AVO (Amplitude Versus Offset) inversion methods are mainly based on various Zoeppritz approximations. The assumptions of small contrast and linear relationship lead the most inversion methods being difficult to have high inversion accuracy. In this paper, the exact Zoeppritz equation is used to establish the prestack inversion method under the Bayesian framework. It integrates multi-source information to generate posterior distributions of P-, S-wave velocity and density. In the Bayesian theory, the prior model works as the regularization term which has a strong effect on the inversion results. The strategy to obtain a relatively accurate prior model can improve the inversion accuracy. Therefore, an exact Zoeppritz equation based nonlinear AVO inversion algorithm combing whale optimization particle filtering (WOPF) is proposed. The WOPF method can generate a relatively stable and accurate initial model for the Bayesian prestack inversion. We validate the new method through two synthetic models and field data. Comparisons are made with the conventional linear and nonlinear AVO inversion methods. The results show that the proposed method can provide much more accurate inverted elastic parameters in different geological conditions.

A Deep Generative Model-Integrated Framework for 3-D Time-Difference Electrical Impedance Tomography

The time-difference image reconstruction problem of electrical impedance tomography (EIT) refers to reconstructing the conductivity change in a human body part between two time points using the boundary impedance measurements. Conventionally, the problem can be formulated as a linear inverse problem. However, due to the physical property of the forward process, the inverse problem is seriously ill-posed. As a result, traditional regularized least-squares (LSs)-based methods usually produce low-resolution images that are difficult to interpret. This work proposes a framework that uses a deep generative model (DGM) to constrain the unknown conductivity. Specifically, this framework allows the inclusion of a constraint that describes a mathematical relationship between the generative model and the unknown conductivity. The resultant constrained minimization problem is solved using an extended alternating direction method of multipliers (ADMM). The effectiveness of the framework is demonstrated by the example of 3-D time-difference chest EIT imaging. Numerical experiment shows a significant improvement of the image quality compared with total variation (TV)-regularized LSs method (peak signal-to-noise ratio (PSNR) is improved by 4.3% for 10% noise and 4.6% for 30% noise; structural similarity (SSIM) is improved by 4.8% for 10% noise and 6.0% for 30% noise). Human experiments show improved correlation between the reconstructed images and images from reference techniques.

Three-Dimensional Ray-Based Tomographic Approach for Contactless GPR Imaging

This paper proposes a three-dimensional imaging approach for contactless ground penetrating radar surveys. The imaging problem is formulated in the linear inverse scattering context and solved by using the Singular Values Decomposition tool. A ray-based model accounting for the electromagnetic signal propagation into an inhomogeneous medium is developed to accurately evaluate the kernel of the integral equation to be inverted. Under the proposed model, an analysis of the spatial resolution performance is carried out as a function of the geometrical and electromagnetic parameters of the scenario. To this end, theoretical concepts based on diffraction tomography and the Singular Value Decomposition of the scattering operator are exploited. Reconstruction results based on full-wave simulated data assess the feasibility of the imaging approach.

Решение обратной задачи, описывающей медленную тепловую конвекцию во вращающемся слое

The linear inverse initial-boundary value problem arising when modeling the rotational motion of a viscous heat-conducting liquid in a flat layer is solved. It is shown that the problem has two different integral identities. Based on these identities, a priori estimates of the solution in a uniform metric are obtained and its uniqueness is proved. The conditions for the input data are also determined, under which this solution goes to the stationary mode with increasing time according to the exponential law. In the final part, the existence of a unique classical solution of the inverse problem is proved. To do this, differentiating the problem by a spatial variable, we come to a direct non-classical problem with two integral conditions instead of the usual boundary conditions. The new problem is solved by the method of separation of variables, which makes it possible to find a solution in the form of rapidly converging series on a special basis.

Sampling-Free Bayesian Inference for Local Refinement in Linear Inversion Problems With a Latent Target Property

Sampling-Free Bayesian Inference for Local Refinement in Linear Inversion Problems With a Latent Target Property

Frequency-Dependent Nonlinear AVO Inversion for Q-Factors in Viscoelastic Media

Viscoelastic theory-based frequency-dependent amplitude variation with offset (AVO) inversions are a useful tool for identifying fluids based on their dispersion properties. When used to identify reservoir oil and gas qualities, the quality factors ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factors) derived from the inversion have produced good results. But currently, the majority of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factor inversions are linear inversions based on linear approximations, whereas nonlinear inversions based on exact equations with higher precision and fewer assumptions are only occasionally carried out. Meanwhile, in broadband seismic inversion, the low-frequency model estimated by complex frequency inversion can fully utilize seismic data’s low-frequency information and provide improved robustness and rationality for inversion. Therefore, we derive a frequency-dependent reflection coefficient equation that varies with angle by replacing the nearly constant <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> model suggested by Aki and Richards into the exact Zoeppritz equation and simplifying. Additionally, using the Bayesian framework as a foundation, we created a novel two-stage broadband frequency-dependent nonlinear inversion approach for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factors that can estimate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factors, produce accurate fluid identification results, and serve as a solid foundation for oil and gas exploration and reservoir identification. Utilizing examples from both synthetic and real-world data, we verify the applicability of oil and gas indicators.

A Numerical Method using Collocation Approach for the Solution of Volterra-Fredholm Integro-Differential Equations

This paper consider collocation approach for the numerical solution of Volterra-Fredholm Integro-differential equation using collocation method. We transformed the problem into a system of linear algebraic equations and matrix inversion is adopted to solve the algebraic equations. We substituted the solution algebraic equations into the approximate equation to obtain the numerical result. Some numerical problems are solved to demonstrate the efficiency and consistency of the method.

Vertical transversely isotropic elastic least-squares reverse time migration based on elastic wavefield vector decomposition

Anisotropic elastic reverse time migration (RTM) is a promising technique for imaging complex oil and gas reservoirs. However, the migrated images often suffer from low spatial resolution, migration artifacts, wave-mode crosstalk, and unbalanced amplitude response. Conventional vertical transversely isotropic elastic least-squares reverse time migration (VTI-elastic LSRTM) defines stiffness parameter perturbations as elastic images, which have different physical meanings from VTI-elastic RTM images. We have developed a VTI-elastic LSRTM method based on elastic wavefield vector decomposition that is a natural extension of VTI-elastic RTM. More specifically, our method applies least-squares inversion to VTI-elastic RTM and defines the compressional- and shear-wave reflectivity as elastic images (PP, PS, SP, and SS images). When computing the elastic images, we decompose the elastic wavefields into compressional and shear wavefields and cross-correlate the corresponding wave modes. We derive the reverse time demigration operator by taking the adjoint of the RTM operator. Using the migration and demigration operators, we formulate the VTI-elastic LSRTM as a linear inverse problem with the least-squares criterion. The conjugate gradient method is used to solve the optimization problem. Three numerical examples are presented to test the feasibility of our method. The VTI-elastic LSRTM images have higher resolution, fewer migration artifacts and wave-mode crosstalk, and improved amplitude response when compared with VTI-elastic RTM images.

Bayesian Linear Seismic Inversion Integrating Uncertainty of Noise Level Estimation and Wavelet Extraction

Seismic impedance inversion is an important method to identify the spatial characteristics of underground rock physical properties. Seismic inversion results and uncertainty evaluation are the important scientific basis for risk decision-making in oil and gas development. Under the assumption that the impedance and the error of the observed seismic data meet the Gaussian distribution or log–Gaussian distribution, the Bayesian linear seismic inversion can analytically obtain the posterior probability distribution of impedance. However, errors from observation, calculation, model and other factors can lead to an inaccurate and incomplete uncertainty evaluation. In this paper, the noise variance is used to represent the noise level of seismic data and the uncertainties from seismic wavelet extraction and noise level estimation are considered in inversion. Assuming that the probability distribution of the noise variance meets the inverse gamma distribution and the seismic wavelet meets the Gaussian distribution, we could obtain the conditional distribution for one variable given another analytically using well-log data and seismic data. In order to integrate the uncertainty from noise level estimation and wavelet extraction into the seismic impedance inversion, the Gibbs sampler algorithm was applied to draw a set of realizations of noise variance and wavelet. For each realization, the corresponding posterior probability model of impedance was achieved by Bayesian linear inversion and the final posterior probability of the impedance model was obtained by integrating all the single posterior probabilities for each pair of wavelet and noise variance. Synthetic and real data experiments showed that the uncertainties of seismic wavelet extraction and noise level estimation have an important influence on inversion results and their uncertainties. The proposed method could effectively integrate the uncertainty of wavelet and noise estimation to obtain a more accurate and comprehensive uncertainty evaluation. Under the assumption that the model meets the linear relationship and the parameters meet some specified distribution, the proposed method has high calculation efficiency. However, it also loses some accuracy when the assumptions are not completely consistent with the actual situation.

Sparse inference and active learning of stochastic differential equations from data

Automatic machine learning of empirical models from experimental data has recently become possible as a result of increased availability of computational power and dedicated algorithms. Despite the successes of non-parametric inference and neural-network-based inference for empirical modelling, a physical interpretation of the results often remains challenging. Here, we focus on direct inference of governing differential equations from data, which can be formulated as a linear inverse problem. A Bayesian framework with a Laplacian prior distribution is employed for finding sparse solutions efficiently. The superior accuracy and robustness of the method is demonstrated for various cases, including ordinary, partial, and stochastic differential equations. Furthermore, we develop an active learning procedure for the automated discovery of stochastic differential equations. In this procedure, learning of the unknown dynamical equations is coupled to the application of perturbations to the measured system in a feedback loop. We show that active learning can significantly improve the inference of global models for systems with multiple energetic minima.

Study on Seismic Dynamic Response of Deep-Buried Tunnel Based on Tectonic Stress

The southeast region of Tibet in China is prone to earthquakes at high seismic levels. Since the construction of tunnels in this region faces a seismic hazard, it is necessary to explore the seismic responses of related construction areas there. In this study, a high-precision finite model of a mountain in a construction area was established based on a deep-buried tunnel in Southeast Tibet. Core specimens were obtained through on-site drilling, with the initial tectonic stress in the tunnel area measured by using the hydraulic fracturing method. Dynamic mechanical parameters of the rocks were acquired through in-situ testing. Moreover, indoor tests were conducted to obtain the physical and mechanical properties of the rocks. The tectonic stress, identified through multiple linear regression inversions, was used as the initial condition for dynamic calculation. The dynamic responses of the tunnel-engineering area under seismic action were calculated, followed by an analysis of the dynamic response characteristics. In addition to dramatic fluctuations in mountain topography, complex dynamic response characteristics of the mountain structure in earthquake environments were also revealed. Although the displacement time-history curve of each observation point displayed a consistent fluctuating pattern, the dynamic displacement extremes in different areas of the mountain varied significantly when the time arrived for their appearance. A maximum dynamic displacement of 0.53 m was observed at the observation point with the highest elevation (5,265.8 m). The acceleration amplification effect inside the mountain body was obvious, with the acceleration increasing as the elevation rose. The time for the occurrence of PGA (acceleration extreme) of each observation point in different directions differed substantially. The extreme value of dynamic stress along the tunnel axis conformed to the distribution pattern of the initial tectonic stress. Compared to the initial tectonic stress, the dynamic stress increased significantly at all observation points along the tunnel axis, with the dynamic stress increasing at a larger rate in the areas at a greater buried depth. Horizontal dynamic stress Sx at the maximum buried depth and Sy increased to 13.86 MPa and 10.32 MPa, respectively, and vertical dynamic stress Sz increased to 4.30 MPa. Variations in horizontal stress became more apparent as the elevation rose, in contrast to weakened variations in vertical stress as the elevation dropped. This study aims to explore the dynamic responses of mountains and the variation law of tectonic stress in deep-buried tunnels in earthquake environments in Tibet through qualitative analysis. The results revealed drastic changes in acceleration, dynamic displacement, and ground stress in the tunnel-engineering area. It is hoped that the contents of the study will contribute to the prevention and control of seismic geological hazards in the tunnel-engineering region and render technical support for the design and construction of similar projects in this region.

Linear multi-scale modeling of diffusion MRI data: A framework for characterization of oriented structures across length scales.

Diffusion-weighted magnetic resonance imaging (DW-MRI) has evolved to provide increasingly sophisticated investigations of the human brain's structural connectome in vivo. Restriction spectrum imaging (RSI) is a method that reconstructs the orientation distribution of diffusion within tissues over a range of length scales. In its original formulation, RSI represented the signal as consisting of a spectrum of Gaussian diffusion response functions. Recent technological advances have enabled the use of ultra-high b-values on human MRI scanners, providing higher sensitivity to intracellular water diffusion in the living human brain. To capture the complex diffusion time dependence of the signal within restricted water compartments, we expand upon the RSI approach to represent restricted water compartments with non-Gaussian response functions, in an extended analysis framework called linear multi-scale modeling (LMM). The LMM approach is designed to resolve length scale and orientation-specific information with greater specificity to tissue microstructure in the restricted and hindered compartments, while retaining the advantages of the RSI approach in its implementation as a linear inverse problem. Using multi-shell, multi-diffusion time DW-MRI data acquired with a state-of-the-art 3T MRI scanner equipped with 300 mT/m gradients, we demonstrate the ability of the LMM approach to distinguish different anatomical structures in the human brain and the potential to advance mapping of the human connectome through joint estimation of the fiber orientation distributions and compartment size characteristics.

Increasing quantitation in spatial single-cell metabolomics by using fluorescence as ground truth.

Imaging mass spectrometry (MS) is becoming increasingly applied for single-cell analyses. Multiple methods for imaging MS-based single-cell metabolomics were proposed, including our recent method SpaceM. An important step in imaging MS-based single-cell metabolomics is the assignment of MS intensities from individual pixels to single cells. In this process, referred to as pixel-cell deconvolution, the MS intensities of regions sampled by the imaging MS laser are assigned to the segmented single cells. The complexity of the contributions from multiple cells and the background, as well as lack of full understanding of how input from molecularly-heterogeneous areas translates into mass spectrometry intensities make the cell-pixel deconvolution a challenging problem. Here, we propose a novel approach to evaluate pixel-cell deconvolution methods by using a molecule detectable both by mass spectrometry and fluorescent microscopy, namely fluorescein diacetate (FDA). FDA is a cell-permeable small molecule that becomes fluorescent after internalisation in the cell and subsequent cleavage of the acetate groups. Intracellular fluorescein can be easily imaged using fluorescence microscopy. Additionally, it is detectable by matrix-assisted laser desorption/ionisation (MALDI) imaging MS. The key idea of our approach is to use the fluorescent levels of fluorescein as the ground truth to evaluate the impact of using various pixel-cell deconvolution methods onto single-cell fluorescein intensities obtained by the SpaceM method. Following this approach, we evaluated multiple pixel-cell deconvolution methods, the 'weighted average' method originally proposed in the SpaceM method as well as the novel 'linear inverse modelling' method. Despite the potential of the latter method in resolving contributions from individual cells, this method was outperformed by the weighted average approach. Using the ground truth approach, we demonstrate the extent of the ion suppression effect which considerably worsens the pixel-cell deconvolution quality. For compensating the ion suppression individually for each analyte, we propose a novel data-driven approach. We show that compensating the ion suppression effect in a single-cell metabolomics dataset of co-cultured HeLa and NIH3T3 cells considerably improved the separation between both cell types. Finally, using the same ground truth, we evaluate the impact of drop-outs in the measurements and discuss the optimal filtering parameters of SpaceM processing steps before pixel-cell deconvolution.