A tool for new Hybrid models: a swift algorithm to calculate the left inverse of a square root of a horizontally localized climatological ensemble covariance matrix

RESUMO

Computing matrix inverses (or, e.g., left inverses) using known algorithms is in general computationally prohibitive in high dimension, such as in an operational weather forecasting context. In this article, a swift, scalable algorithm to calculate the left inverse (through singular values decompositions techniques) of a square root of a horizontally localized ensemble covariance matrix is introduced. In this algorithm, calculations involving model variables located in different grid vertical columns can be performed separately. This is the aspect that makes this algorithm scalable by allowing for parallel computing. The algorithm is developed in the case when the number of ensemble members equals the number of model variables in each grid vertical column (this number is of order $10^3$ for current high-resolution models). The algorithm is tested in a synthetic experiment on a three-dimensional grid, and it is found to have very good accuracy. Additionally, a version of the algorithm outputting the left-multiplication of a generic vector by that above-mentioned left inverse is also introduced. For operational applications, if executed on a single processor, that version of the algorithm is already $9$ orders of magnitude faster than a traditional algorithm, and, with multiple, parallel processors, it scales until the number of processors reaches the order of $10^6$. In addition, an application of the algorithm is pointed out: performing the transform from model variables to an alternative set of variables, defined herein as standardized variables, being climatologically uncorrelated and each having unit climatological error variance. Furthermore, it is pointed out that the standardized variables allow constructing a new Hybrid covariance model, which is positive definite and climatologically unbiased, and it features a geographically dependent climatological error covariance matrix.