Rainfall network design through comparative kriging methods

The methods of simple and disjunctive kriging are applied and compared in the estimation of optimum locations of recording raingauges as part of a network for the determination of storm characteristics to be used in forecasting and design. Some advantages are shown but problems arise when there are large differences in storm structures and movements. Another source of uncertainty is in the modelling of the semi-variogram. Application is made to the management of an area of the Severn-Trent water basin, UK, with 13 autographic raingauges. La planification des réseaux de pluviomètres par les méthodes comparatives de krigeage Les méthodes de krigeage simple et disjonctif sont appliquées et comparées pour la détermination des meilleurs sites pour l'installation des pluviomètres enregistreurs faisant partie à un réseau pour la détermination des caractéristiques des averses. Elles seront utilisées pour les prévisions météorologiques et les projets. Les avantages sont évidents, mais quelques problèmes surviennent quand il y a des variations considérables au niveau de la structure et du déplacement de l'averse. D'autres sources d'incertitude dérivent du modèle semivariogramme. On a appliqué ces méthodes dans l'exploitation pour cette recherche d'une zone d'un bassin situé à Severn-Trent, UK, avec 13 pluviomètres enregistreurs. MTRODUCnON An optimum raingauge network will vary with location and the purpose for which the data are collected. For example, networks can be designed to monitor rainfall for use in resource assessment, design, operations and flood warning schemes. A basic problem arises because a rainfall process cannot be sufficiently explained in terms of a mathematical model in which the rainfall at one point is a time-invariant function of the rainfall at another point. To estimate a storm rainfall process, for instance, one has to design a network so as to minimize uncertainties in the inferences drawn from the *Now at: Institute di Idraulica, Politecnico di Milano, 32 Piazza Leonardo da Vinci, 1-20133 Milano MI, Italy. Open for discussion until 1 December 1991 223 A. H. Kassim & N. T. Kottegoda 224 measurements of high intensity rainfall, at optimum cost. The main source of error lies in the location of the raingauges, and a well designed network will reduce the error associated with the estimation of storm rainfall. The location of raingauges is important, for example, in analysing storm characteristics. Several methods of siting raingauges have been suggested but only a few of these are applicable to an ungauged area. It is very common to find that the sites of raingauges are not representative of the sub-areas within a catchment thus resulting in uncertainties in engineering designs applied to a particular area. Even in the case of a homogeneous zone, the allocation of a specified number of gauges per unit area may not be optimal since the network design is dependent on the nature of the storms experienced. The methods of allocation depend partly on the type of raingauges which are used. Also, one needs to take account of the mean annual rainfall distribution which is closely related to the topographical features of the surrounding area. A particular region has its own characteristics and these vary from one area to another. Network design using indirect methods Rainfall-runoff modelling is basic to hydrology and is important if the spatial variation of runoff for a given input is required. Bras (1979) reviewed four articles dealing with the sampling of the interrelated random fields. Areal rainfall can be estimated reasonably well with a lumped model if the number of raingauges meets the requirements of the model. However, the approach is dependent on the availability of accurate rainfall models. The models in use do not generally account for the uncertainty in rainfall estimates at a particular raingauge site. An additional disadvantage in these models is that the assumptions and simplifications in their formulation usually result in overestimates or underestimates when applied to other regions, i.e. the original model performs reasonably well only in the area under study. Standard error method of network design A common practice in network design is to evaluate the error associated with a particular sampling density. Quite often an error criterion is applied. Some of the methods which are being used are simple random sampling, correlation functions and regression techniques (Caffey, 1965), structural functions (Hutchinson, 1972), and the spatial application of time series analysis (Rodriguez-Iturbe & Mejia, 1974). These methods, involving mathematical statistics, assume that the value recorded by each gauge is independent of the values recorded by other gauges in the area, i.e. the sample of events is treated as random. Methodology for kriging and objectives An optimum interpolation between gauge values can be implemented by 225 Rainfall network design through comparative kriging kriging. This is essentially a method of estimation. The assumption here is that there is a local structure in the sampling domain and the accuracy of the estimated values depends on how this is accounted for. Also, a weighted sum of the observations is assumed to provide the best estimate. Thus kriging is a method of determining the optimal weights. It is relevant to note here some of the principles of the estimation process. Given a set of rainfall observations and the exact locations of the gauging sites one needs to estimate a set of values at an ungauged site. It is assumed that the properties are stationary over the field studied and there are no significant trends, for example. Nevertheless, there are differences in the structure between one site and another. Also, each site will have a local area of influence. Observations made within this area will affect the estimate at the site. It means that one must identify the range of influence and determine the best method of estimation. This is implemented by the system of weights. The method of kriging is well suited because the data points available are rarely well-located with respect to the points to be estimated, and the observed values are weighted to provide the best estimates. In addition to this procedure, the weights can be obtained by the methods of optimal interpolation or by simple Thiessen coefficients. Previously, Hughes & Lettenmaier (1981) suggested the potential use of kriging in network design. Kriging was originated by Matheron (1971) and first applied in mining (see Journel & Huijbregts, 1978, and Delhomme, 1979); basic concepts are explained by Huijbregts (1973); also Virdee & Kottegoda (1984) applied the technique to optimal well selection. In this study, the methods of kriging of the simple and disjunctive types are applied and compared since they not only cover the application of most of the methods mentioned above but they can be associated with the physical characteristics of storm rainfall processes. The main properties of the simple linear kriging estimator are that it is unbiassed, optimal and linear. PROBLEM FORMULATION FOR THE KRIGING SYSTEM Consider an area, C, as shown in Fig. 1, over which a number of raingauges are sited with some additional observation points. The unknown mean precipitation for the area is defined as: P= (1/C)jcq(x)dx ( 1 ) where: x = location of a point of observation; and q(x) = a function describing depth of storm event over region C. Equation (1) can be presented in discretized form to give the estimated mean depth as follows: è=Bçi (2) where B = transpose of n x 1 vector of weights, X̂, applied to n raingauges, and g = n x l vector of observations. A. H. Kassim & N. T. Kottegoda 226