The Hyperbolic Smoothing Approach for Automatic Calibration of Rainfall-Runoff Models

Authors: Adilson Elias Xavier, Otto Corrêa Rotunno Filho, Paulo Canedo de Magalhães

Abstract:

This paper addresses the issue of automatic parameter estimation in conceptual rainfall-runoff (CRR) models. Due to threshold structures commonly occurring in CRR models, the associated mathematical optimization problems have the significant characteristic of being strongly non-differentiable. In order to face this enormous task, the resolution method proposed adopts a smoothing strategy using a special C∞ differentiable class function. The final estimation solution is obtained by solving a sequence of differentiable subproblems which gradually approach the original conceptual problem. The use of this technique, called Hyperbolic Smoothing Method (HSM), makes possible the application of the most powerful minimization algorithms, and also allows for the main difficulties presented by the original CRR problem to be overcome. A set of computational experiments is presented for the purpose of illustrating both the reliability and the efficiency of the proposed approach.

Keywords: Rainfall-runoff models, optimization procedure, automatic parameter calibration, hyperbolic smoothing method.

References:

[1] Dawdy, D. R., and T. O'Donnell, Mathematical models of catchment behaviour, Journal of the Hydraulics Division, 91, 123-137, 1965.
[2] Holland, J. H., Adaptation in natural and artificial systems, University of Michigan Press, Ann Arbor, 1975.
[3] Ibbitt, R. P., Systematic parameter fitting for conceptual models of catchment hydrology, Ph. D. thesis, Univ. of London, 1970.
[4] Ibbitt, R. P., and T. O'Donnell, Fitting methods for conceptual catchment, Journal Hydraul. Eng., 97(HY9), 1331-1342, 1971.
[5] Johnston, P. R., and D. Pilgrim, Parameter optimization for watershed models, Water Resour. Res., 12(3), 477-486, 1976.
[6] Pickup, G., Testing the efficiencies of algorithms and strategies for automatic calibration of rainfall-runoff models, Hydrol. Sci. Bull., 22(2), 257-274, 1977.
[7] Brazil, L. E., and M. D. Hudlow, Calibration procedures used with the National Weather Service River Forecast System, in Water and Related Land Resource Systems, edited by Y. Haimes and J. Kindler, 447-466, Pergamon, New York, 1981.
[8] Sorooshian, S., and F. Arfi, Response surface parameter sensitivity analysis methods for post calibration studies, Water Resour. Res., 18, 1531-1538, 1982.
[9] Sorooshian, S., V. K. Gupta, and J. L. Fulton, Evaluation of maximum likelihood parameter estimation techniques for conceptual rainfall-runoff models: influence of calibration data variability and length on model credibility, Water Resour. Res., 19, 251-259, 1983.
[10] Gupta, V. K., and S. Sorooshian, Uniqueness and observability of conceptual rainfall-runoff model parameters: the percolation process examined, Water Resour. Res., 19(1), 269-276, 1983.
[11] Gupta, V. K., and S. Sorooshian, The automatic calibration of conceptual catchment models using derivative-based optimization algorithms, Water Resour. Res., 21(4), 473-485, 1985.
[12] Hendrickson, J. D., S. Sorooshian, and L. E. Brazil, Comparison of Newton-type and direct search algorithms for calibration of conceptual rainfall-runoff models, Water Resour. Res., 24(5), 691-700, 1988.
[13] Duan, Q., S. Sorooshian, and V. K. Gupta, Effective and efficient global optimization for conceptual rainfall-runoff models, Water Resour. Res., 28, 1015-1031, 1992.
[14] Sorooshian, S., Q. Duan, and V. K. Gupta, Calibration of rainfall-runoff models: application of global optimization to the Sacramento Soil Moisture Accounting Model, Water Resour. Res., 29, 1185-1194, 1993.
[15] Rotunno Filho, O.C., Soil moisture mapping using remote sensing and geostatistics applied to rainfall-runoff models, Ph. D. thesis, Department of Civil Engineering, University of Waterloo, Canada, 1995.
[16] Thibault, M., V. Andréassian, C. Perrin and C. Michel, Intercomparison of local and global optimization techniques for conceptual rainfall-runoff models: an extensive test of 4 models, over 313 watersheds, VII th IAHS Scientific Assembly, April 03th to 09th, Foz do Iguaçu, Brazil, 2005.
[17] Bagirov, A. M., A. Al Nuaimat and N. Sultanova, Hyperbolic smoothing function method for minimax problems, Optimization, 62, pp. 759-782, 2013.
[18] Xavier, A. E. and A. A. F. Oliveira, Optimal covering of plane domains by circles via hyperbolic smoothing, Journal of Global Optimization, 31, pp. 493-504, 2005.
[19] Venceslau, H. M., D. C. Lubke and A. E. Xavier, Optimal covering of solid bodies by spheres via hyperbolic smoothing technique, Optimization Methods & Software, 30, Issue 2, pp. 391-403, 2015.
[20] Xavier, A. E., The hyperbolic smoothing clustering method, Pattern Recognition, 43, pp. 731-737, 2010.
[21] Bagirov, A. M., B. Ordin, G. Ozturk and A. E. Xavier, An incremental clustering algorithm based on hyperbolic smoothing, Comp. Optim. Appl., 61 (1), pp. 219-241, 2015.
[22] Xavier, V. L. and A. E. Xavier, Accelerated hyperbolic smoothing method for solving the multisource Fermat-Weber and k-median problems, Knowledge-Based Systems, 191, 2020 (https://doi.org/10.1016/j.knosys.2019.105226).
[23] Lopes, J. E. G., B. P. F. Braga, and J. G. L. Conejo, SMAP, a simplified hydrologic model, in V. Singh (Ed.), Applied modeling in catchment hydrology, Water Resources Publication, Littleton, CO, 1982.
[24] Huyer, W., and Neumaier, A. , A new exact penalty function, SIAM J., 13 , 1141-1158, 2003.
[25] Kitanidas, P. K., and R. L. Bras, Real-time forecasting with a conceptual hydrologic model, 1, Analysis of uncertainty, Water Resour. Res., 16(6), 1025-1033, 1980.
[26] U.S. Soil Conservation Service, Urban hydrology for small watersheds, Technical Release no. 95, U. S. Department of Agriculture, 1975.
[27] U.S. Soil Conservation Service, Urban hydrology for small watersheds, Technical Release no. 95, U. S. Department of Agriculture, 1975.
[28] Gill, P., W. Murray, and M. M. Wright, Practical optimization, Princeton University Press, N. Y., 1981.
[29] ANEEL, “Sistema de Informações de Geração da ANEEL (Siga)”, 2021. Available at https://www.aneel,gov.br/siga.