Uncertainty Quantification in Seismic Inversion by Nonlinear Sampling

在科学和工程的许多分支中，从遥感数据构建目标图像是进行有意义的推断的基本任务。然而，这些数据通常远非理想，因为它们通常被噪声污染，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之类的问题而不充分。此外，目标的分辨率通常是以特定的方式选择的，从而在得到的答案中引入了不确定性。因此，对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。现有的方法表征不确定性往往是基于简单的假设，主要是因为计算能力的限制。这项建议的目的是开发一种技术，用于估计的不确定性，使用非线性采样，将适用于成像的地震data.Seismic层析成像的主要工具，用于估计地球？从地震走时、振幅和波形数据中提取地下图像。数据往往是不充分的和有噪声的，并且正演模拟通常基于近似物理。地下模型参数的特别参数化在地下特性的估计中增加了进一步的复杂性。在解决方案估计的非唯一性已被公认在过去和不确定性量化的需要已被推动的物理学界。贝叶斯方法来描述我们的逆问题已被发现适合于此目的。它使我们能够用概率密度函数来描述我们的答案，称为后验概率密度（PPD）。然而，PPD的简单功能描述通常不可用。因此，从通常高度多模态的PPD估计样本是一项具有挑战性的任务。通常的做法是导出最大后验（MAP）模型，并在MAP点使用Hessian表示不确定性。这种方法假设PPD是高斯分布的？由于前向问题的非线性性质和数据中的噪声特性，经常违反该假设。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个前向模型评估。在这里，我们建议开发计算效率的MCMC方法的不确定性量化与应用地震层析成像。可逆跳蒙特卡罗方法（RJMCMC），其中数据本身被允许找到所需的模型参数的适当数量，解决了一些常用方法的缺点。然而，该方法在计算上昂贵。研究人员提出并实现了一种新的方法，称为可逆跳跃哈密顿蒙特卡罗（RJHMC）方法的地震反演。这种方法可以被证明是两倍的速度比传统的RJMCMC，因为它使用梯度信息采取大的跳跃MCMC步骤。将其应用于一个二维海洋多道地震数据集。

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.