A novel Gaussian sum filter method for accurate solution to the nonlinear filtering problem

The paper presents two methods of updating the weights of a Gaussian mixture to account for the density propagation within a data assimilation setting. The evolution of the first two moments of the Gaussian components is given by the linearized model of the system. When observations are available, both the moments and the weights are updated to obtain a better approximation to the a posteriori probability density function. This can be done through a classical Gaussian Sum Filter. When the measurement model offers little or no information in updating the states of the system, better estimates may be obtained by updating the weights of the mixands to account for the propagation effect on the probability density function. The update of the forecast weights proves to be important in pure forecast settings, when the frequency of the measurements is low, when the uncertainty of the measurements is large or the measurement model is ambiguous making the system unobservable. Updating the weights not only provides us with better estimates but also with a more accurate probability density function. The numerical results show that updating the weights in the propagation step not only gives better estimates between the observations but also gives superior performance for systems when the measurements are ambiguous.