Parameter uncertainty

Considerable uncertainty can be expected in the estimation of climatic input variables (i.e. precipitation, temperature), of the observations with which we evaluate the model predictions (i.e. discharge), and of the model parameter values (i.e. soil moisture capacity). A new approach to dealing with these uncertainties in the computation of regional water balance is the objective of this work. As in all natural sciences, it is difficult to exactly determine the physical, hydrological and meteorological variables with crisp measures. There are likely to be serious uncertainties in determining these properties even at the point or plot scale. The spatial extrapolation, or scaling up, of information from the point or small plot scale to the catchment scale is an additional difficulty, and the resulting uncertainty tends to get worse the more diverse the monitoring (point or plot) and modeling (catchment) scales are. In this regard, we can talk of random and systematic errors inherent in the observations, and errors associated with the scaling up of information from the point or plot (observation) scale to the catchment (or modeling) scale.

Random errors

The random error in the act of observation may occur because of unnoticed alteration of the standardized measurement condition. For example, when measuring precipitation, deviations may be caused, for instance, by the blocking of the drainage mechanism of the rain gauge, accidentally incorrect reading, and confusion during the date registration procedure. Sometimes, these data errors are filtered out. Data uncertainties arising from random errors are not included in the estimation of fuzzy model inputs and parameter values used in this study, but can be easily included in subsequent extensions of this study.

Systematic errors

Systematic errors can be caused by specific measurement and computational techniques. Such errors arise when applying a rating curve prepared before a devastating flood event to the river water levels in the post-flood situation, and thus not accounting for changes to the riverbed morphology. Errors in precipitation measurements can occur, especially in high-elevation zones, because of strong winds causing precipitation to be blown past the gage.

Another type of error may occur because of the spatial extrapolation of point observations from a monitoring network with a density and distribution that are not sufficient to fully capture the spatial variability of parameters or variables. Especially in Alpine regions, the density and location of point observations are of great importance in order to represent the very diverse topographic and local meteorological characteristics. In general, micro- to meso-scale variations of climatic features and soil properties increase with increasing topographic complexity of the terrain (Christakos 1992).

The placement of observation sites would have to be denser in mountainous regions than in flat areas, in order to attain the same quality in spatial extrapolations of the observations. But the opposite is in fact true, as detailed information, for instance, on soil characteristics, is mainly mapped in intensively used agricultural regions in the lower valley zones but very sparsely in high-elevation zones. The same also applies to the network of climate monitoring stations. The accuracy of various interpolation techniques depends strongly on the positioning of monitoring stations. Using data from monitoring stations at locations that do not account for the spatial characteristics of the study area will introduce uncertainty in the regionalized estimates. The quality of water balance estimates relies heavily on the evaluation period of the meteorological observations. These have to be carried out over many years (to produce sufficiently long time series), according to standardized rules for instruments and observations with largely unchanged local conditions during the reference period, for us to have confidence in predictions of water balance into the future.

Later on in this paper, we will shed more light on the uncertainties involved in the estimation of parameter and input values that are used in our water balance model. These estimations of input data and parameter values, and their associated uncertainty measures, serve as the basis for the specification of fuzzy membership functions.

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