Ensemble Transformation and Adaptive Observations

Suppose that the geographical and temporal resolution of the observational network could be changed on a daily basis. Of all the possible deployments of observational resources, which particular deployment would minimize expected forecast error? The ensemble transform technique answers such questions by using nonlinear ensemble forecasts to rapidly construct ensemble-based approximations to the prediction error covariance matrices associated with a wide range of different possible deployments of observational resources. From these matrices, estimates of the expected forecast error associated with each distinct deployment of observational resources are obtained. The deployment that minimizes the chosen measure of forecast error is deemed optimal. The technique may also be used to find the perturbation that evolves into the leading eigenvector or singular vector of an ensemble-based prediction error covariance matrix. This time-evolving perturbation “explains” more of the ensemble-based prediction error variance than any other perturbation. It may be interpreted as the fastest growing perturbation on the subspace of ensemble perturbations. The ensemble-based approximations to the prediction error covariance matrices are constructed from transformation matrices derived from estimates of the analysis error covariance matrices associated with each possible deployment of observational resources. The authors prove that the ensemble transform technique would precisely recover the prediction error covariance matrices associated with each possible deployment of observational resources provided that (i) estimates of the analysis error covariance matrix were precise, (ii) the ensemble perturbations span the vector space of all possible perturbations, and (iii) the evolution of errors were linear and perfectly modeled. In the absence of such precise information, the ensemble transform technique links available information on analysis error covariances associated with different observational networks with error growth estimates contained in the ensemble forecast to estimate the optimal configuration of an adaptive observational network. Tests of the technique will be presented in subsequent publications. Here, the objective is to describe the theoretical basis of the technique and illustrate it with an example from the Fronts and Atlantic Storm Tracks Experiment (FASTEX).