A Bayesian data assimilation scheme is formulated for advection-dominated advective and diffusive evolutionary problems, based upon the Dynamic Likelihood (DLF) approach to filtering. The DLF was developed specifically for hyperbolic problems -waves-, and in this paper, it is extended via a split step formulation, to handle advection-diffusion problems. In the dynamic likelihood approach, observations and their statistics are used to propagate probabilities along characteristics, evolving the likelihood in time. The estimate posterior thus inherits phase information. For advection-diffusion the advective part of the time evolution is handled on the basis of observations alone, while the diffusive part is informed through the model as well as observations. We expect, and indeed show here, that in advection-dominated problems, the DLF approach produces better estimates than other assimilation approaches, particularly when the observations are sparse and have low uncertainty. The added computational expense of the method is cubic in the total number of observations over time, which is on the same order of magnitude as a standard Kalman filter and can be mitigated by bounding the number of forward propagated observations, discarding the least informative data.

翻译：本文基于动态似然滤波方法，为平流主导的平流-扩散演化问题构建了一种贝叶斯数据同化方案。动态似然滤波最初专为双曲问题（波动问题）而开发，本文通过分裂步格式将其扩展至平流-扩散问题的处理。在动态似然方法中，观测数据及其统计量被用于沿特征线传播概率，实现似然函数的时间演化。因此，估计的后验分布继承了相位信息。对于平流-扩散问题，时间演化中的平流部分仅基于观测数据进行处理，而扩散部分则同时结合模型信息与观测数据。我们预期并在此证明：在平流主导的问题中，动态似然滤波方法能比其他同化方法产生更优的估计结果，尤其在观测数据稀疏且不确定性较低的情况下。该方法增加的计算成本与时间维度上观测数据总量的立方成正比，其量级与标准卡尔曼滤波相当，且可通过限制前向传播的观测数据量（剔除信息量最低的数据）来降低计算负担。