Data assimilation (DA) for the non-differentiable parameterized moist physical processes is a complicated and difficult problem, which may result in the discontinuity of the cost function (CF) and the emergence of multiple extreme values. To solve the problem, this paper proposes an inner/outer loop ensemble-variational algorithm (I/OLEnVar) to DA. It uses several continuous sequences of local linear quadratic functions with single extreme values to approximate the actual nonlinear CF so as to have extreme point sequences of these functions converge to the global minimum of the nonlinear CF. This algorithm requires no adjoint model and no modification of the original nonlinear numerical model, so it is convenient and easy to design in assimilating the observational data during the non-differentiable process. Numerical experimental results of DA for the non-differentiable problem in moist physical processes indicate that the I/OLEnVar algorithm is feasible and effective. It can increase the assimilation accuracy and thus obtain satisfactory results. This algorithm lays the foundation for the application of I/OLEnVar method to the precipitation observational data assimilation in the numerical weather prediction (NWP) model.
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