More Efficient Derivative-based Watershed Model Calibration

Recent publications have demonstrated the efficacy of model independent, derivative-based calibration of watershed models (Skahill and Doherty 2006; Doherty and Skahill 2006; Guti¨¦rrez-Magness and McCuen 2005). In the works referenced above, the Jacobian at each optimization iteration was approximated using software that requires between m and 2m forward model calls (see below). The objective of this work is to describe and demonstrate a potentially more efficient approach to approximating the Jacobian. Let the matrix X represent the action of a linear model. Let the vector p represent its m parameters, the vector h represent the n observations comprising the calibration dataset, and the n-dimensional vector ¦Å represent the observation noise. These quantities are related by: Xp = h + ¦Å. Minimization of a weighted least squares objective function is achieved for p calculated as p = ((XtQX)-1XtQh (1), where Q is proportional to the inverse of the covariance matrix of measurement noise. For a nonlinear model, implementation of equation 1 becomes an iterative process starting from a user- supplied set of initial parameter estimates. Furthermore, the Jacobian matrix replaces X. The nonlinear parameter estimation process then becomes one of successive linearization, requiring construction of the Jacobian matrix at each optimization iteration for computation of the upgrade vector. Although the Levenburg- Marquardt method of computer-based parameter estimation can generally complete a single inversion run with a high level of efficiency, even if the column space of the Jacobian must be populated based on model runs with incrementally varied parameter values, the process of constructing the Jacobian is usually the most computationally demanding aspect of the inversion process. Where model run times are high, model run efficiency of the calibration process becomes of paramount concern. One way to reduce the cost of each optimization iteration is to iteratively update the Jacobian matrix by using a secant approximation to the derivative along each search direction. Over the course of many iterations, the accuracy of the approximated Jacobian may degrade, so the Jacobian is occasionally recalculated, as necessary, using finite differences. As we demonstrate by calibrating two different surface hydrology models, this reduced the total number of model calls by approximately 45 percent relative to conventional updating of the Jacobian, and in each case with no loss in objective function improvement. Skahill, B., and Doherty, J. 2006. Efficient accommodation of local minima in watershed model calibration. Journal of Hydrology (in press). Doherty, J., and Skahill, B.E. 2006. "An Advanced Regularization Methodology for Use in Watershed Model Calibration." Journal of Hydrology, (327), 564¨C 577. Guti¨¦rrez-Magness, A.L., and McCuen, R.H. 2005. "Effect of Flow Proportions on HSPF Model Calibration Accuracy." Journal of Hydrologic Engineering, Vol. 10, No. 5.