A robust Gauss-Newton algorithm and its application to the calibration of conceptual rainfall-runoff hydrological model

Abstract

Parameter estimation remains an ongoing challenge in hydrological modeling for a number of reasons. One severe challenge is that of numerically rough objective function surfaces characterized by discontinuities and pitting. This prevents the use of efficient gradient-based methods which converge prematurely. This has motivated development of probabilistic gradient-free search methods such as SCE-UA (shuffled complex evolution method developed at The University of Arizona), which although robust require substantially greater function evaluations. This paper introduces a robust Gauss-Newton method which can robustly search over rough objective function surfaces. It introduces a curve fitting strategy, an inexact line search and box constraints. The filtering technique is used to smooth the Jacobian matrix; the inexact line search method provides a self-adaptive method to update the scaling used in the filter; the box constraint minimizes the influence of a poor quadratic approximation where the Hessian matrix is near singular. The robust Gauss-Newton algorithm converges very rapidly in the vicinity of the global optimum. A case study involving conceptual hydrological models demonstrates the performance of the robust Gauss-Newton algorithm by comparing function evaluations and convergence success rate against the SCE-UA method in an exhaustive set of trials. The efficiency of the algorithm is measured by the number of function evaluations. The results show an improvement in efficiency more than 75% compared to the SCE-UA algorithm with comparable accuracy.