ECO-RAP, Part 2: Inexpensive huge ensembles

A general theorem describing the relationship between the square roots of two matrices and the square root of their elementwise (Hadamard) product is introduced. We use the theorem to define huge ensembles whose sample covariance matrices equal the four-dimensional high rank forecast error covariance CALECO matrices whose properties have been discussed in part 1. The theorem enables the production of analytic formulae for the maximum possible rank of CALECO matrices. It is shown that each member of the huge ensemble whose covariance gives a CALECO matrix can be generated by a specific set of element-wise products between columns of the square root of the Ensemble COrrelations Raised to A Power (ECO-RAP) matrix and the raw ensemble perturbations. The ensemble enlargement enabled by this method is shown to greatly improve the accuracy of predictions of forecast error variance reduction made with the Ensemble Transform Kalman Filter (ETKF) adaptive sampling technique.