Seismic tomography is a methodology to image the interior of solid or fluid media, and is often used to map properties in the subsurface of the Earth. In order to better interpret the resulting images it is important to assess imaging uncertainties. Since tomography is significantly nonlinear, Monte Carlo sampling methods are often used for this purpose, but they are generally computationally intractable for large datasets and high-dimensional parameter spaces. To extend uncertainty analysis to larger systems we use variational inference methods to conduct seismic tomography. In contrast to Monte Carlo sampling, variational methods solve the Bayesian inference problem as an optimization problem, yet still provide probabilistic results. In this study, we applied two variational methods, automatic differential variational inference (ADVI) and Stein variational gradient descent (SVGD), to 2D seismic tomography problems using both synthetic and real data and we compare the results to those from two different Monte Carlo sampling methods. The results show that variational inference methods can produce accurate approximations to the results of Monte Carlo sampling methods at significantly lower computational cost, provided that gradients of parameters with respect to data can be calculated efficiently. We expect that the methods can be applied fruitfully to many other types of geophysical inverse problems.
Seismic traveltime tomography using transmission data is widely used to image the Earth's interior from global to local scales. In seismic imaging, it is used to obtain velocity models for subsequent depth-migration or full-waveform inversion. In addition, cross-hole tomography has been successfully applied for a variety of applications, including mineral exploration, reservoir monitoring, and CO2 injection and sequestration. Conventional tomography techniques suffer from a number of limitations, including the use of a smoothing regularizer that is agnostic to the physics of wave propagation. …
We introduce a `double-difference' method for the inversion for seismic wavespeed structure based on adjoint tomography. Differences between seismic observations and model predictions at individual stations may arise from factors other than structural heterogeneity, such as errors in the assumed source-time function, inaccurate timings, and systematic uncertainties. To alleviate the corresponding nonuniqueness in the inverse problem, we construct differential measurements between stations, thereby reducing the influence of the source signature and systematic errors. We minimize the discrepancy…
In this article, we develop a 1-D traveltime tomography method to calculate the seismic P-wave velocity of a medium. We use the results of 1-D tomography to obtain linear inhomogeneity parameters in a specific layer. To get the trustworthiness of the method, we perform several synthetic experiments. We show that the inverted model parameters are reasonably accurate and stable. To examine the results of linear inhomogeneity parameters using a different method, we also develop an inversion method based on a two-parameter velocity model. Finally, we apply both the methods to Vertical Seismic Prof…
This paper introduces novel deep recurrent neural network architectures for Velocity Model Building (VMB), which is beyond what Araya-Polo et al 2018 pioneered with the Machine Learning-based seismic tomography built with convolutional non-recurrent neural network. Our investigation includes the utilization of basic recurrent neural network (RNN) cells, as well as Long Short Term Memory (LSTM) and Gated Recurrent Unit (GRU) cells. Performance evaluation reveals that salt bodies are consistently predicted more accurately by GRU and LSTM-based architectures, as compared to non-recurrent architec…
Seismotectonics and Slip Behavior of a Submarine Plate Boundary Fault from Seismicity Repeaters and Tomography using a high-resolution earthquake catalog from machine learning
The effects of several nonlinear regularization techniques are discussed in the framework of 3D seismic tomography. Traditional, linear, $\ell_2$ penalties are compared to so-called sparsity promoting $\ell_1$ and $\ell_0$ penalties, and a total variation penalty. Which of these algorithms is judged optimal depends on the specific requirements of the scientific experiment. If the correct reproduction of model amplitudes is important, classical damping towards a smooth model using an $\ell_2$ norm works almost as well as minimizing the total variation but is much more efficient. If gradients (e…
The purpose of this work is to generalize the frozen Gaussian approximation (FGA) theory to solve the 3-D elastic wave equation and use it as the forward modeling tool for seismic tomography with high-frequency data. FGA has been previously developed and verified as an efficient solver for high-frequency acoustic wave propagation (P-wave). The main contribution of this paper consists of three aspects: 1. We derive the FGA formulation for the 3-D elastic wave equation. Rather than standard ray-based methods (e.g. geometric optics and Gaussian beam method), the derivation requires to do asymptot…
Deep Earth imaging through neutrino and seismic tomography
Institute of Physics（印度物理研究所）；Homi Bhabha National Institute（霍米·巴巴国立学院）；Indian Institute of Science Education and Research, Pune（印度科学教育研究学院浦那分校）；Tata Institute of Fundamental Research（塔塔基础研究院）；CSIR Fourth Paradigm Institute（印度科学工业研究理事会第四范式研究所）；Department of Physics, Aligarh Muslim University（阿利加尔穆斯林大学物理系）
We propose a local earthquake tomography method that applies a structured regularization technique to determine sharp changes in Earth's seismic velocity structure using arrival time data of direct waves. Our approach focuses on the ability to better image two common features that are observed in Earth's seismic velocity structure: sharp changes in velocities that correspond to material boundaries, such as the Conrad and Moho discontinuities; and gradual changes in velocity that are associated with pressure and temperature distributions in the crust and mantle. We employ different penalty term…
Variational Bayesian experimental design for geophysical applications: seismic source location, amplitude versus offset inversion, and estimating CO2 saturations in a subsurface reservoir
本文将变分贝叶斯实验设计引入地球物理，通过震源定位、AVO 反演和 CO2 储层饱和度估计验证其可同步计算并优化实验价值、降低计算成本。
Seismic Traveltime Tomography with Label-free Learning
College of Big Data and Software Engineering, Zhejiang Wanli University（浙江万里学院大数据与软件工程学院）；Key Laboratory of Geoscience Big Data and Deep Resource of Zhejiang Province, School of Earth Sciences, Zhejiang University（浙江大学地球科学学院浙江省地球科学大数据与深部资源重点实验室）
We present the development of extended diffraction tomography, a new approach to the solution of the linear seismic waveform inversion problem. This method has several appealing features, such as the use of arbitrary depth-dependent reference models and the decomposition of the full 2D or 3D inverse problem into a large number of independent 1D problems. This decomposition makes the method naturally highly parallelizable. Careful implementation yields significant robustness with respect to noise. Several synthetic examples are shown which characterize the benefits of our method and demonstrate…
Solving inverse problems involving measurement noise and modeling errors requires regularization in order to avoid data overfit. Geophysical inverse problems, in which the Earth's highly heterogeneous structure is unknown, present a challenge in encoding prior knowledge through analytical expressions. Our main contribution is a generative-model-based regularization approach, robust to out-of-distribution data, which exploits the prior knowledge embedded in existing data and model pairs. Utilizing an amortized variational inference objective, a conditional normalizing flow (NF) is pretrained on…
King Abdulllah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）
Uncertainty quantification is crucial to inverse problems, as it could provide decision-makers with valuable information about the inversion results. For example, seismic inversion is a notoriously ill-posed inverse problem due to the band-limited and noisy nature of seismic data. It is therefore of paramount importance to quantify the uncertainties associated to the inversion process to ease the subsequent interpretation and decision making processes. Within this framework of reference, sampling from a target posterior provides a fundamental approach to quantifying the uncertainty in seismic …
Seismic full-waveform inversion (FWI) provides high resolution images of the subsurface by exploiting information in the recorded seismic waveforms. This is achieved by solving a highly nonnlinear and nonunique inverse problem. Bayesian inference is therefore used to quantify uncertainties in the solution. Variational inference is a method that provides probabilistic, Bayesian solutions efficiently using optimization. The method has been applied to 2D FWI problems to produce full Bayesian posterior distributions. However, due to higher dimensionality and more expensive computational cost, the …
Georgia Institute of Technology（佐治亚理工学院）；Utrecht University（乌得勒支大学）
Bayesian inference for high-dimensional inverse problems is computationally costly and requires selecting a suitable prior distribution. Amortized variational inference addresses these challenges via a neural network that approximates the posterior distribution not only for one instance of data, but a distribution of data pertaining to a specific inverse problem. During inference, the neural network -- in our case a conditional normalizing flow -- provides posterior samples at virtually no cost. However, the accuracy of amortized variational inference relies on the availability of high-fidelit…
Quantum computing is in its early stage of implementation. Its capacity has been growing in the last years but its application in several fields of sciences is still restricted to oversimplified problems. In this stage, it is important to identify the situations where quantum computing presents the most promising results to be prepared when the technology is ready to be deployed. The geophysics field has several areas which are limited by the current computation capability, among them the so-called seismic inversion is one of the most important ones, which are strong candidates to benefit from…
Uncertainty quantification provides quantitative measures on the reliability of candidate solutions of ill-posed inverse problems. Due to their sequential nature, Monte Carlo sampling methods require large numbers of sampling steps for accurate Bayesian inference and are often computationally infeasible for large-scale inverse problems, such as seismic imaging. Our main contribution is a data-driven variational inference approach where we train a normalizing flow (NF), a type of invertible neural net, capable of cheaply sampling the posterior distribution given previously unseen seismic data f…
A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal
IFP Energies nouvelles（IFP能源新动力）；Univ. Paris-Est, LIGM UMR CNRS 8049（巴黎东大学，LIGM UMR CNRS 8049）；Aix-Marseille Univ., I2M UMR CNRS 7373（艾克斯-马赛大学，I2M UMR CNRS 7373）
Incorporating prior knowledge on model unknowns of interest is essential when dealing with ill-posed inverse problems due to the nonuniqueness of the solution and data noise. Unfortunately, it is not trivial to fully describe our priors in a convenient and analytical way. Parameterizing the unknowns with a convolutional neural network (CNN), and assuming an uninformative Gaussian prior on its weights, leads to a variational prior on the output space that favors "natural" images and excludes noisy artifacts, as long as overfitting is prevented. This is the so-called deep-prior approach. In seis…
We propose the use of $\ell_1$ regularization in a wavelet basis for the solution of linearized seismic tomography problems $Am=d$, allowing for the possibility of sharp discontinuities superimposed on a smoothly varying background. An iterative method is used to find a sparse solution $m$ that contains no more fine-scale structure than is necessary to fit the data $d$ to within its assigned errors.
In a variety of scientific applications we wish to characterize a physical system using measurements or observations. This often requires us to solve an inverse problem, which usually has non-unique solutions so uncertainty must be quantified in order to define the family of all possible solutions. Bayesian inference provides a powerful theoretical framework which defines the set of solutions to inverse problems, and variational inference is a method to solve Bayesian inference problems using optimization while still producing fully probabilistic solutions. This chapter provides an introductio…
California Institute of Technology（加州理工学院）；Purdue University（普渡大学）
We introduce a scheme for probabilistic hypocenter inversion with Stein variational inference. Our approach uses a differentiable forward model in the form of a physics informed neural network, which we train to solve the Eikonal equation. This allows for rapid approximation of the posterior by iteratively optimizing a collection of particles against a kernelized Stein discrepancy. We show that the method is well-equipped to handle highly multimodal posterior distributions, which are common in hypocentral inverse problems. A suite of experiments is performed to examine the influence of the var…
Utrecht University（乌得勒支大学）；Georgia Institute of Technology（佐治亚理工学院）；Microsoft（微软）
Obtaining samples from the posterior distribution of inverse problems with expensive forward operators is challenging especially when the unknowns involve the strongly heterogeneous Earth. To meet these challenges, we propose a preconditioning scheme involving a conditional normalizing flow (NF) capable of sampling from a low-fidelity posterior distribution directly. This conditional NF is used to speed up the training of the high-fidelity objective involving minimization of the Kullback-Leibler divergence between the predicted and the desired high-fidelity posterior density for indirect measu…
