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得出有意义推论的基本任务。然而，这些数据往往远不是理想的，因为它们通常受到噪音的污染，并且可能由于诸如记录站相对于待成像目标的几何形状或密度等问题而不充分。此外，目标的分辨率通常以一种特殊的方式选择，从而在结果答案中引入不确定性。因此，不确定性的定量测量对于建立对数据分析结果的信心至关重要。由于计算能力的限制，现有的表征不确定性的方法往往基于简单的假设。本提案的目的是开发一种利用非线性采样估计不确定性的技术，该技术将应用于地震数据成像。地震层析成像是估计地球地震的主要工具。S地下图像从地震传播时间，振幅和波形数据。数据往往不充分且有噪声，正演模拟一般基于近似物理。地下模型参数的特别参数化增加了地下特征估计的复杂性。在过去，解估计的非唯一性已经得到了很好的认识，并且地球物理界已经提出了对不确定性量化的需要。贝叶斯方法描述我们的反问题已经被发现适合于这个目的。它使我们能够用概率密度函数描述我们的答案，称为后验概率密度（PPD）。然而，一般无法对PPD进行简单的功能描述。因此，从通常高度多模态的PPD中估计样本是一项具有挑战性的任务。通常的做法是推导最大后验（MAP）模型，并在MAP点使用黑森函数表示不确定性。这种方法假设PPD是高斯分布。由于正演问题的非线性性质和数据中的噪声特性，这个假设经常被违反。另一方面，基于Metropolis-Hastings的马尔可夫链蒙特卡罗方法在计算上非常昂贵，通常需要超过一百万次的正演模型评估。在此，我们建议开发计算效率高的MCMC方法，用于地震层析成像的不确定性量化。可逆跳跃蒙特卡罗方法（RJMCMC）允许数据本身找到所需的适当数量的模型参数，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。研究人员提出了一种新的地震反演方法，称为可逆跳变哈密顿蒙特卡罗（RJHMC）方法。由于该方法使用梯度信息在MCMC步骤中进行大的跳跃，因此可以证明该方法比传统的RJMCMC快两倍。该方法将应用于二维海洋多通道地震数据集。

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.