Initial error and model error are key factors restricting the accuracy of numerical weather prediction (NWP). The purpose of the present study is to estimate the errors of spatiotemporal evolution model by using recent observations. By considering the continuous evolution of atmosphere, the observed data (ignoring the measurement error) can be viewed as a series of solutions of accurate model governing the actual atmosphere, and the model errors can be objectively assumed to be an unknown functional term (a missing forcing term) of the numerical model, thus the NWP can be considered as an inverse problem to uncover the unknown model error term by using the long periods of observed data. In this study, we first construct an inverse problem model with its optimization problem, which is constrained by the numerical model, to estimate the errors of spatiotemporal evolution model, then we present a derivative-free optimization (DFO) method to find the minimum solution of the optimization problem by running the numerical model with an external forcing term. The DFO method does not need to compute the gradient of the objective functional and the tangent linear model or adjoint model of the original numerical model. The numerical study of Burgers equation indicates that the presented methods can effectively uncover the model errors from the past data and evidently improve the numerical prediction. The precedures described in this paper open up possibilities for utilizing the past observation data to extract useful information about model errors and enhance the prediction efficiency in the operational models.