地球物理反演问题中的贝叶斯方法研究

The research on Bayesian inference for geophysical inversion

基于统计理论的贝叶斯反演方法在先验信息和观测数据的约束下,以后验概率分布的形式表征模型参数在不同区间的可能性大小.相对于确定性反演理论,贝叶斯反演通过提取模型参数边缘概率分布、最大后验解、平均解、相关系数等定量评价反演结果的不确定性以及模型参数之间的相互关系,通过模型参数后验概率分布反映观测数据和先验信息对模型参数的约束能力.本文基于贝叶斯方法在地球物理反演中的应用,总结了贝叶斯反演的基本流程,详细介绍了不同背景条件下的先验信息概率分布选择、似然函数建立、后验概率公式求解.在优化参数方面,介绍了模型参数的固定维和变维反演概念,以及超参数的优化方法;在反演方法方面,着重介绍了固定维和变维反演马尔科夫链蒙特卡罗采样方法;在模型参数评价方面,介绍了不同情况下贝叶斯统计参数的求取.然后讨论了贝叶斯反演方法采样效率提升的具体措施.最后对贝叶斯方法在地球物理反演中的应用作出总结.

Based on statistical theory, the Bayesian inversion method adopts the posterior probability distribution to evaluate the model parameters under the constraints of prior information and observation data. Compared to deterministic inversion theory, Bayesian inference is beneficial to quantitative evaluation of inversion uncertainty by the model parameter marginal probability distribution, maximum a posterior estimation(MAP), mean model estimation and correlation coefficient, which accurately reflects the constraint ability of observation data and prior information on model parameters. We systematically summarized the application of the Bayesian inference in geophysical inversion and proposed the basic flowchart to realize Bayesian model evaluation. Firstly, the Bayesian theory is simply introduced. The prior information probability distribution, the likelihood function formula, and the construction of the posterior probability equation are explained in detail. Secondly, the implementation process of Bayesian inversion is discussed in detail. As for model parameter updates, the concepts of fixed and trans-dimensional inversion with the hyperparameter optimization are discussed. In terms of inversion methods, the Markov Chain Monte Carlo(MCMC) sampling methods of fixed and trans-dimensional inversion are highlighted. In consideration of model parameter evaluation, the calculation of Bayesian statistical parameters under different conditions is introduced. Then the specific measures to improve the sampling efficiency of Bayesian inversion are discussed. Finally, the application of Bayesian inference in geophysical inversion is summarized.