An ensemble-based approach is proposed to obtain conditional nonlinear optimal perturbation (CNOP), which is a natural extension of linear singular vector to a nonlinear regime. The new approach avoids the use of adjoint technique during maximization and is thus more attractive. Comparisons among CNOPs of a simple theoretical model generated by the ensemble-based, adjoint-based, and simplex-search methods, respectively, not only show potential equivalence of the first two approaches in application according to their very similar spatial structures and time evolutions of the CNOPs, but also reveal the limited performance of the third measure, an existing adjoint-free algorithm, due to its inconsistent spatial distribution and weak net growth ratio of norm square of CNOP comparing with the results of the first two methods. Because of its attractive features, the new approach is likely to make it easier to apply CNOP in predictability or sensitivity studies using operational prediction models.

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A B S T R A C TThe authors propose to implement conditional non-linear optimal perturbation related to model parameters (CNOP-P) through an ensemble-based approach. The approach was first used in our earlier study and is improved to be suitable for calculating CNOP-P. Idealised experiments using the Lorenz-63 model are conducted to evaluate the performance of the improved ensemble-based approach. The results show that the maximum prediction error after optimisation has been multiplied manifold compared with the initial-guess prediction error, and is extremely close to, or greater than, the maximum value of the exhaustive attack method (a million random samples). The calculation of CNOP-P by the ensemble-based approach is capable of maintaining a high accuracy over a long prediction time under different constraints and initial conditions. Further, the CNOP-P obtained by the approach is applied to sensitivity analysis of the Lorenz-63 model. The sensitivity analysis indicates that when the prediction time is set to 0.2 time units, the Lorenz-63 model becomes extremely insensitive to one parameter, which leaves the other two parameters to affect the uncertainty of the model. Finally, a serial of parameter estimation experiments are performed to verify sensitivity analysis. It is found that when the three parameters are estimated simultaneously, the insensitive parameter is estimated much worse, but the Lorenz-63 model can still generate a very good simulation thanks to the relatively accurate values of the other two parameters. When only two sensitive parameters are estimated simultaneously and the insensitive parameter is left to be non-optimised, the outcome is better than the case when the three parameters are estimated simultaneously. With the increase of prediction time and observation, however, the model sensitivity to the insensitive parameter increases accordingly and the insensitive parameter can also be estimated successfully.

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