Optimizing parameterization schemes with ensemble-based parameter estimation

State augmentation in a data assimilation cycle can be used as an objective method to estimate uncertain empirical constants in parameterization schemes. In this approach, empirical parameters are appended to the model state vector. They cannot be observed, but, like any other unobserved state variable, they can be updated based on their correlations with the model equivalents of observable quantities. State-parameter correlations are likely flow-dependent, therefore they are best estimated with an ensemble of simulations.

Despite its potential usefulness in parameterization design, ensemble-based parameter estimation has been used so far as a way of accounting for model errors in the assimilation process, and as a method to increase ensemble spread. In this study, we discuss if and how it can aid parameterization optimization. As a case study, we consider a simple first-order parameterization of turbulence in the atmospheric boundary layer. We run several idealized assimilation experiments, partly in a perfect-model scenario (the forecast ensemble and the nature run providing the assimilated observations are instances of the same model), partly in a more realistic imperfect-model scenario (the models providing the forecast ensemble and the nature run the have different formulations).

We demonstrate that, in our case, sensible parameter estimation results are obtained only under restrictive conditions. First, initial conditions must be very accurate, so that the spread of the forecast ensemble is determined primarily by the uncertain parameter. Second, the error variance of the assimilated observations must be low enough for the state perturbations induced by the estimated parameter to be accurately sampled. We show that, when these conditions are met, optimized parameters can compensate for sources of model error, and argue that this property can be used to extend the flexibility of parameterization schemes. For instance, this could be achieved by using parameter estimation experiments to populate lookup tables for adaptive parameters.