Filtering a statistically exactly solvable test model for turbulent tracers from partial observations

A statistically exactly solvable model for passive tracers is introduced as a test modelfor the authors’ Nonlinear Extended Kalman Filter (NEKF) as well as other filteringalgorithms. The model involves a Gaussian velocity field and a passive tracer governedby the advection-diffusion equation with an imposed mean gradient. The model hasdirect relevance to engineering problems such as the spread of pollutants in the air orcontaminants in the water as well as climate change problems concerning the transportof greenhouse gases such as carbon dioxide with strongly intermittent probability distri-butions consistent with the actual observations of the atmosphere. One of the attractiveproperties of the model is the existence of the exact statistical solution. In particular,this unique feature of the model provides an opportunity to design and test fast andefficient algorithms for real-time data assimilation based on rigorous mathematical the-ory for a turbulence model problem with many active spatiotemporal scales. Here, weextensively study the performance of the NEKF which uses the exact first and secondorder nonlinear statistics without any approximations due to linearization. The role ofpartial and sparse observations, the frequency of observations and the observation noisestrength in recovering the true signal, its spectrum, and fat tail probability distributionare the central issues discussed here. The results of our study provide useful guidelinesfor filtering realistic turbulent systems with passive tracers through partial observations.