The use of forecast gradients in 3DVar data assimilation

In this paper, we propose an optimization approach for data assimilation by the use of forecast gradients. The proposed objective function consists of two data-fitting terms. The first term is based on the difference between the gradients of the forecast and the analysis, and the second term is based on the difference between the observations and the analysis in observation space. The motivation for using forecast gradients is that the forecast values provide an estimation of the system state, but they may not be accurate enough. We therefore propose to construct analysis gradients driven by the forecast gradients, instead of the forecast values. The associated data-fitting term can be interpreted by using the second-order finite difference matrix as the inverse of the background error covariance matrix in the 3DVar setting. In the proposed approach, it is not necessary to estimate the background covariance matrix and to deal with its inverse in the 3DVar algorithm. The existence and uniqueness of the analysis solution of the proposed objective function are established in this paper. The solution can be calculated by using the conjugate gradient method iteratively. Experimental results based on Community Multiscale Air Quality (CMAQ) and Weather Research and Forecasting (WRF) simulations are presented. We show in our air quality data assimilation experiment that the performance of the proposed method is better than that of the 3DVar method and the En3DVar method. The average improvements over the CMAQ simulation results for single-species NO₂, O₃, SO₂, NO, and CO are 18.9%, 34.0%, 22.2%, 4.3%, and 91.9%, respectively; and for the multiple-species PM2.5 and PM10, the improvements are 61.2% and 70.1%, respectively. In addition, the performance of the proposed method in temperature data assimilation is improved by 45.1% compared with the 3DVar method.