Application of the ensemble Kalman smoothing in the inverse modeling for transport and diffusion models

The study of the spread of greenhouse gases in space and time, as well as the assessment of fluxes from the Earth's surface of these gases using a data assimilation system is an urgent task of monitoring the state of the environment. One of the approaches to estimating greenhouse gas fluxes is an approach based on the assumption that fluxes are constant in a given subdomain and over a given time interval (on the order of a week). This is due to both the need for an effective implementation of the algorithm and the properties of the observational data used in such problems.
Modern problems of estimating greenhouse gas fluxes from the Earth's surface have a large dimension, therefore, a variant is usually considered in which the estimated variable is fluxes, and the transport and diffusion model is included in the observation operator. At the same time, there is a problem of using large assimilation windows, within which the flow values are estimated at several time intervals.
The paper considers an algorithm for estimating fluxes based on observations from a given time interval. The algorithm is a variant of the ensemble smoothing algorithm, widely used in such problems. It is shown that when using the assimilation window, in which the flow values are estimated for several time intervals, the algorithm can become unstable, while the observability condition is violated.