A study of the nonlinear filter and its application in the data assimilation with physical “onoff” processes

Using an idealized model of partial differential equation with discontinuous “on-off” switches in the forcing term, the feasibility and effectivity of the nonlinear filter in data assimilation with discontinuous physical “on-off” processes is investigated. First, the general nonlinear filter algorithm was discussed in the Bayesian theory framework, then two kinds of assimilation approaches based respectively on the particle filter (PF) and the ensemble Kalman filter (EnKF) were analyzed and compared. Since the EnKF utilizes only first two statistic moments in the analysis step, it is able to deal with the noises with probability density function well matching the Gaussian distribution, and will produce larger error when the true probability density functions badly approximated by the Gaussian distribution. Although the PF approximates the error distribution utilizing particles, it uses the whole error statistic moments, so it is able to deal with the problem caused by the nonGaussian observation error in the data assimilation. Last, numerical experiments are conducted for the four cases of linear and nonlinear observation operators, Gaussian and nonGaussian observation errors. The assimilation results show that no matter the observation operator is linear or nonlinear, both the PF and EnKF can overcome the difficulty brought by the discontinuous “on-off” switches and obtain satisfying assimilation results in the case of Gaussian observation error. But for the nonGaussian observation error, the PF is more effective with satisfactory results than the EnKF and the filtering being unstable during the process of assimilation using the EnKF.