Description

Non-linear regression techniques stem from the consideration of the hydrological model as a regression function and need for improved computation techniques due to the bias introduced by Standard Least Squares (SLS) parameter estimation (Kavetski et al., 2003) caused both by input uncertainty and the non-linearity in the model. In general the techniques works by specifying a likelihood function relating the model output and observed data series. The likelihood function selected is often based on time-series analysis (Box and Tiao, 1973). Two schools of computational techniques exist. The first school implements Bayesian inference schemes to derive posterior parameter distributions from which samples are generated (Kuczera and Parent, 1998; Tanner, 1993; Thiemann et al., 2001; Vrugt et al., 2003). In some cases the sampling algorithm is deigned to highlight aspects of the data set, for example the information content of the data (Vrugt et al., 2002). The second school aims to find the Maximum likelihood Estimator (MLE), often through the use of a Gauss-Marquard-Levenberg methodology (Doherty and Johnston, 2003). The parameter uncertainty around this MLE can be approximated by linearization techniques or regularisation (Doherty and Johnston, 2003). It has also been shown (Kuczera, 1983) that with a carefully chosen likelihood function that the posterior distribution can be calculated in close form for little computational cost.

Software

• PEST http://www.sspa.com/pest/download.html

• UCODE_2005 and Six Other Computer Codes for Universal Sensitivity Analysis, Calibration, and Uncertainty Evaluation: http://water.usgs.gov/software/ucode.html

Advantages

• Many tools available for analysis

• Computational complexity can be tailored by the selection of the likelihood function (error model)

• Potential for the representation of the uncertainties in a posterior distribution.

Disadvantages

• Without strong assumptions as to the likelihood function computation may be expensive, especially for large numbers of unknowns;

• Different sources of error are not explicitly represented;

• Assumptions about the likelihood function are hard to justify in terms of beliefs about the sources of error.

Case studies

• Fitting Flood Frequency Distributions (case study)

References and Further reading

Box, G.E.P. and G.C. Tiao, Bayesian inference in Statistical analysis. 1973, Reading, Massachusetts: Addison-Wesley.

Clarke, R.T., 1994. Statistical modelling in hydrology. Wiley & Sons, Chichester, xii,412p. pp.

Doherty, J., 2002. PEST Model-Independent Parameter Estimation, http://www.sspa.com/pest/download.html.

Doherty, J. and Johnston, J.M., 2003. Methodologies for calibration and predictive analysis of a watershed model. Journal Of The American Water Resources Association, 39(2): 251-265.

Kavetski, D.N., Franks, S.W. and Kuczera, G., 2003. Confronting input uncertainty in environmental modelling. In: Q. Duan, H. Gupta, S. Sorooshian, A.N. Rousseau and R. Turcotte (Editors), Calibration of Watershed Models. Water Science and Applications Series. American Geophysical Union, pp. 49-68.

Kuczera, G., 1983. Improved Parameter Inference In Catchment Models.1. Evaluating Parameter Uncertainty. Water Resources Research, 19(5): 1151-1162.

Kuczera, G. and Parent, E., 1998. Monte Carlo assessment of parameter uncertainty in conceptual catchment models: the Metropolis algorithm. Journal of Hydrology, 211(1-4): 69-85.

Poeter, E.P. and Hill, M.C., 1999. UCODE, A computer code for universal inverse modeling. Computers in Geosciences, 25(4): 457-462.

Tanner, M.A., 1993. Tools for Statistical Inference: Methods for the exploration of posterior distributions and likelihoods. Springer, New York, 156 pp.

Thiemann, M., Trosset, M., Gupta, H. and Sorooshian, S., 2001. Bayesian recursive parameter estimation for hydrologic models. Water Resources Research, 37(10): 2521-2535.

Vrugt, J.A., Bouten, W., Gupta, H.V. and Sorooshian, S., 2002. Toward improved identifiability of hydrologic model parameters: The information content of experimental data. Water Resources Research, 38(12): art. no.-1312.

Vrugt, J.A., Gupta, H.V., Bouten, W. and Sorooshian, S., 2003. A Shuffled Complex Evolution Metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters. Water Resources Research, 39(8): art. no.-1201.