A SVD-based ensemble projection algorithm for calculating the conditional nonlinear optimal perturbation

Conditional nonlinear optimal perturbation(CNOP) is an extension of the linear singular vector technique in the nonlinear regime.It represents the initial perturbation that is subjected to a given physical constraint,and results in the largest nonlinear evolution at the prediction time.CNOP-type errors play an important role in the predictability of weather and climate.Generally,when calculating CNOP in a complicated numerical model,we need the gradient of the objective function with respect to the initial perturbations to provide the descent direction for searching the phase space.The adjoint technique is widely used to calculate the gradient of the objective function.However,it is difficult and cumbersome to construct the adjoint model of a complicated numerical model,which imposes a limitation on the application of CNOP.Based on previous research,this study proposes a new ensemble projection algorithm based on singular vector decomposition(SVD).The new algorithm avoids the localization procedure of previous ensemble projection algorithms,and overcomes the uncertainty caused by choosing the localization radius empirically.The new algorithm is applied to calculate the CNOP in an intermediate forecasting model.The results show that the CNOP obtained by the new ensemble-based algorithm can effectively approximate that calculated by the adjoint algorithm,and retains the general spatial characteristics of the latter.Hence,the new SVD-based ensemble projection algorithm proposed in this study is an effective method of approximating the CNOP.