Optimal decomposition of covariance matrices for multivariate stochastic models in hydrology

A new method is proposed for decomposing covariance matrices that appear in the parameter estimation phase of all multivariate stochastic models in hy drology. This method applies not only to positive definite covariance matri ces (as do the typical methods of the literature) but to indefinite matrice s, too, that often appear in stochastic hydrology. It is also appropriate f or preserving the skewness coefficients of the model variables as it accoun ts for the resulting coefficients of skewness of the auxiliary (noise) vari ables used by the stochastic model, given that the latter coefficients are controlled by the decomposed matrix. The method is formulated in an optimiz ation framework with the objective function being composed of three compone nts aiming at (1) complete preservation of the variances of variables, (2) optimal approximation of the covariances of variables, in the case that com plete preservation is not feasible due to inconsistent (i.e., not positive definite) structure of the covariance matrix, and (3) preservation of the s kewness coefficients of the model variables by keeping the skewness of the auxiliary variables as low as possible. Analytical expressions of the deriv atives of this objective function are derived, which allow the development of an effective nonlinear optimization algorithm using the steepest descent or the conjugate gradient methods. The method is illustrated and explored through a real world application, which indicates a very satisfactory perfo rmance of the method.