Improvement of the PEST parameter estimation algorithm through Extended Kalman Filtering

Abstract

During the last decades, a number of methods have been developed for the estimation of hydrologic model parameters. One frequently used and relatively simple algorithm is the parameter estimation (PEST) method. A close examination of this algorithm shows that it is very similar to the Extended Kalman Filter (EKF). The differences between the methods are caused by the derivation of the algorithms: the EKF is derived through a minimization of the square difference between the true and the estimated model state, while PEST has been derived through a minimization of an objective function related to, but not equal to, the root mean square error between the model results and the observations. The objective of this paper is to analyze the performance of these two algorithms. A synthetic-data experiment has been developed for this purpose. It has been found that under high observation errors and/or temporally sparse observations the EKF can lead to a stable parameter estimation, while it is possible that under the same circumstances PEST does not yield a solution. Also, the choice of the initial guess for the parameter values can be an important issue in the application of PEST, while this is not so important for the EKF. The application of the Marquardt algorithm can lead to stable parameter estimates in case the PEST algorithm fails (meaning that nonphysical parameter values were obtained which lead to a premature abortion of the model simulations), but numerically the EKF is still superior. In order to solve this problem, a simple alternative to the Marquardt algorithm has been developed, which leads to a quicker convergence. Application of both methods to a conceptual rainfall–runoff model with 10 parameters shows the robustness of the EKF for parameter calibration. The overall conclusion from this work is that generally PEST and the EKF will lead to similar results, but that under high observation errors, infrequent observations, and/or strongly erroneous initial parameter values, the PEST method can fail because of excessive parameter updates caused by the conceptual error in the objective function minimized by the calibration algorithm, while the EKF can still yield stable parameter estimates.