Views
Uncertainty in Rating Curve Estimation and Effect on Flood Risk (Fuzzy Methods)
Introduction
The traditional approach to fitting stage-discharge rating curve has been statistical regression. This often involves transformations of the data (e.g. using log-log regressions, or power law regressions) or, in more complex cases, fitting piecewise regressions to different stage ranges. There is are alternatives including the use of artificial neural networks and fuzzy set methods, that are flexible in representing complex relations. Fuzzy methods are particularly interesting since they can be used to produce uncertainty ranges for the discharge estimates (e.g. Lohani et al., 2006, Shrestha et al., 2007).
Case study
The study by Shrestha et al. (2007) is an application of fuzzy regression methods to rating curve estimation in an application to sites on the River Neckar in Germany. The paper gives a good introduction to the principles of fuzzy regression, which is also covered in Bardossy et al. (1990). The resulting uncertainties in the estimated discharge are then used to determine the spatial patterns of flood by the use of α level cuts within a hydraulic model of flood inundation.
Fuzzy regression methods
Fuzzy regression is based upon replacing the crisp parameters of a traditional regression by fuzzy numbers. The fuzzy numbers can be represented in quite flexible ways by using left-right membership functions around the measurement of discharge at any stage (Figure 1). Both linear and nonlinear relationships, including multiple segments can be included. The regression is then set up as an optimisation problem to minimise an objective function subject to fuzzy constraints. In the paper by Shrestha et al. (2007) the objective function is the minimisation of a vagueness criterion on the range of the fuzzy parameters, and the constraints are provided by the range of support for each discharge measurement treated as a fuzzy number. Stage is assumed to be known crisply. When more than one segment of the rating curve is included then this is a multiobjective optimisation problem with continuity constraints at each change of segment. Results of the process are shown in Figure 2.
Fig. 1 L-R fuzzy number with α level cut.
Fig. 2 Uncertainty bound curves from fuzzy linear regression with scattered data.
Fuzzy Alpha-cut uncertainty propagation
Fuzzy set theory also provides a mechanism for the propagation of uncertainties through another model using the concept of the α level cut (Figure 1). The α level cut can be used to convert a fuzzy number into a crisp range, that can then be used in propagating an uncertain quantity through another function or model. For quasi-linear problems this requires only 2 model simulations per α level cut. For highly nonlinear problems involving multiple fuzzy parameters it cannot be assured that the limits on the output will be at the combinations of the values provided by the range of each α level cut. A more sophisticated search algorithm may then be necessary. The result of the propagation in this Case Study is illustrated in Figure 3, where a fuzzy rating curve (Figure 2) has been used to estimate the uncertain patterns of inundation resulting from uncertain discharge based on the 1-D HD model of the CARIMA modelling system of Cunge et al. (1980) which uses a modified implicit Preissman solution algorithm.
(a) 
(b) 
(c) 
Fig. 3 Differences in inundation areas for different membership levels (a) 0.5(L) (b) 1.0 (c) 0.5(R).
Comment
Fuzzy methods provide an interesting alternative to statistical regression and uncertainty propagation methods. They can also be used in model evaluation where the uncertainty in observed variables is not easily represented by a statistical distribution (e.g.
References
-
Bárdossy, A., Bogardi, I. & Duckstein, L. (1990) Fuzzy regression in hydrology. Water Resour. Res. 26(7), 1497-1508.
-
Cunge, J.A., Holly, F.M., & Verway, A. (1980) Practical Aspects of Computational River Hydraulics. Pitman, London, UK.
-
Shrestha, R R, Bardossy, A and Nestmann, F, 2007, Analysis and propagation of uncertainties due to stage-discharge relationship: a fuzzy set approach, Hydrol. Sci. J., 52(4), 595-610, doi: 10.1623/hysj.52.4.595.
tzwbu vyusqoljf --Sun, 02 Sep 2007 01:48:09 +0100
tazkyfesg kwjr sczvel jeic tszar pxoml mdxqj
reliability --Sun, 20 Apr 2008 10:19:34 +0100