Uncertainty Quantification in Seismic Inversion by Nonlinear Sampling

在科学和工程的许多分支中，从遥感数据中构建目标图像是进行有意义的推断的基本任务。然而，数据往往远不是理想的，因为它们通常被噪声污染，并且可能由于记录站相对于要成像的目标的几何形状或密度等问题而不够充分。此外，目标的解决办法通常是以临时方式选择的，这给结果答案带来了不确定性。因此，对不确定性的量化衡量对于建立对数据分析结果的信心至关重要。现有的表征不确定性的方法往往基于简单化的假设，主要是因为计算能力的限制。这项建议的目标是开发一种使用非线性采样来估计不确定性的技术，该技术将应用于地震数据的成像。地震层析成像是从地震走时、振幅和波形数据估计地球？S地下图像的主要工具。数据往往是不充分和噪声的，正演模拟通常是基于近似物理的。地下模型参数的特殊参数化增加了地下特征估计的复杂性。过去已经很好地认识到解决方案估计中的非唯一性，地球物理学界一直在提倡不确定性量化的必要性。描述逆问题的贝叶斯方法已被发现适合于这一目的。它使我们能够用称为后验概率密度(PPD)的概率密度函数来描述我们的答案。然而，通常无法获得对PPD的简单功能描述。因此，从通常高度多模的PPD中估计样本是一项具有挑战性的任务。通常的做法是推导出最大后验概率(MAP)模型，并在MAP点处用黑森表示不确定性。该方法假定PPD为高斯分布。由于正问题的非线性性质和数据中的噪声特征，这一假设经常被违反。另一方面，基于Metropolis-Hastings的马尔可夫链蒙特卡罗方法计算非常昂贵，通常需要超过一百万次的正向模型评估。在这里，我们建议开发计算高效的不确定量化的MCMC方法，并将其应用于地震层析成像。一种可逆跳跃蒙特卡罗方法(RJMCMC)，它允许数据本身找到所需的适当数目的模型参数，解决了常用方法的一些缺点。然而，这种方法在计算上是昂贵的。研究人员提出并实现了一种新的地震反演方法--可逆跳哈密顿蒙特卡罗方法(RJHMC)。这种方法可以证明比传统的RJMCMC快两倍，因为它使用梯度信息在MCMC步长中进行大跳跃。将其应用于海洋二维多道地震数据集。

In many branches of science and engineering construction of an image of targets from remotely sensed data is an essential task for drawing meaningful inferences. The data, however, are often far from being ideal in that they are generally contaminated with noise and may be inadequate because of issues such as the geometry or density of the recording stations with respect to the targets to be imaged. In addition, resolution of the target is often chosen in an ad-hoc manner, introducing uncertainty in the resulting answer. Quantitative measures of uncertainty are, therefore, crucial to establishing confidence in the results of data analysis. Existing methods of characterizing uncertainty are often based on simplistic assumptions primarily because of limitations of computing powers. The objective of this proposal is to develop a technique for estimation of uncertainty using nonlinear sampling that will be applied to imaging of seismic data.Seismic tomography is the primary tool for estimating Earth?s subsurface images from seismic travel time, amplitude and waveform data. The data are often inadequate and noisy, and the forward modeling is generally based on approximate physics. Ad hoc parameterization of subsurface model parameters adds further complication in estimation of subsurface characteristics. The non-uniqueness in the solution estimates has been well recognized in the past and the need for uncertainty quantification has been promoted by the geophysics community. The Bayesian approach to describing our inverse problems has been found appropriate for this purpose. It enables us to describe our answer in terms of a probability density function, called the posterior probability density (PPD). A simple functional description of the PPD is generally not available, however. Thus, estimating samples from the PPD which is generally highly multi-modal is a challenging task. The common practice is to derive the maximum a posterioi (MAP) model and represent the uncertainty using the Hessian at the MAP point. This method assumes that the PPD is Gaussian ? an assumption often violated due to the nonlinear nature of the forward problem and the noise characteristics in the data. On the other hand, Metropolis-Hastings based Markov chain Monte Carlo methods are computationally very expensive, often requiring over a million forward model evaluations. Here we propose to develop computationally efficient MCMC methods for uncertainty quantification with application to seismic tomography. A Reversible jump Monte Carlo method (RJMCMC) in which the data themselves are allowed to find suitable number of model parameters required, addresses some of the shortcomings of the commonly used methods. The method, however, is computationally expensive. The researchers propose to develop and implement a new method called Reversible jump Hamiltonian Monte Carlo (RJHMC) method to seismic inversion. This method can be demonstrated to be two times faster than the conventional RJMCMC since it uses gradient information to take large jumps in MCMC steps. It will be applied to a 2D marine multi-channel seismic dataset.