基于贝叶斯MCMC方法的资料同化技术研究

Bayesian MCMC Method for the Solution of Data Assimilation

摘要: 在贝叶斯理论框架下，提出基于马尔科夫链蒙特卡罗(MCMC)算法估计非线性模型初始状态和模式误差概率密度分布的一种新方法。首先利用贝叶斯方法，导出了非线性动力系统中未知初始状态和模式误差分布规律的后验概率密度函数(PDF)，将每个参数的后验边缘PDF的数学期望当作未知参数估计值。其次采用自适应Metropolis算法以后验PDF分布为极限不变分布来构造Markov链，即对未知参数进行重要性抽样，并利用收敛后的样本序列计算数学期望，从而得到初始状态和模式误差的估计值。然后利用初始状态和模式误差样本序列定量计算了未知参数的一维后验分布和相互之间的二维后验分布，后者定量描述了初始状态和模式误差之间的相关关系。最后通过数值试验结果说明该方法能有效地估计非线性动力系统的初始条件，具有较好的同化效果。

Abstract: In the framework of Bayesian theorem, a new method is proposed to find the solution of data as-similation problem based on Markov Chain Monte Carlo (MCMC) algorithm, which can quantify the posterior probability density function (PPDF) for initial states and model errors of nonlinear dy-namical system. Firstly, the PPDF for unknown initial states and model errors which are derived with the Bayesian method and parameters to be estimated can be thought as the mathematic ex-pectation of corresponding marginal PPDF. Secondly, taking the posterior probability as the inva-riant distribution, the Adaptive Metropolis algorithm is used to construct the Markov Chains of unknown initial states and model errors, respectively. That is, importance sampling of the posterior distribution is carried out. And the converged samples are used to calculate the mathematic expectation. So far, initial states and model errors of nonlinear dynamical system are estimated by Bayesian MCMC method successfully. Then, one and two-dimensional posterior distributions are constructed from the converged samples of initial states and model errors. And two-dimensional posterior distributions depict the interactions and correlations quantitatively between two dif-ferent and arbitrary parameters. Finally, the results of numerical experiments show that the new data assimilation method can estimate initial conditions of nonlinear dynamical system very con-veniently and accurately.