WATER RESOURCES RESEARCH, VOL. 20, NO. 4, PAGES 463-470, APRIL 1984 Optimal Estimationof the AverageAreal Rainfall and Optimal Selection of Rain Gauge Locations G. BASTIN, B. LORENT, C. DUQU!•,AND M. GEVERS Laboratoired'Automatique et d'Analyse desSystdmes, Louvain University We propose a simple procedure for the real-time estimation of the average rainfallover a catchment area. The rainfall is modeledas a two-dimensional random field. The average areal rainfall is computed by a linear unbiased minimum variance estimation method (kriging) whichrequires knowledge of the variogram of the random field. We propose a time-varying estimator for the variogram which takes into account the influences of both the seasonal variations and the rainfall intensity. Our average areal rainfallestimator hasbeen implemented in practice. We illustrate its application to real data in two river basins in Belgium. Finally,it is shown how the method canbe used for the optimal selection of the rain gauge locations in a basin. INTRODUCTION We propose a simpleprocedure for the real-time estimation of the average rainfall over a catchmentarea from rainfall measurements made at a few measurement stations in that area. The estimationof suchareal rainfall is an important step in many hydrological applications, such as evaluation of hy- draulic balances, management of surfacewater resources, or real-time forecastingof river flows. For this last application the rainfall over the river basinis, of course, the main input to any rainfall-river flow fo.recasting model [Lorent and Gevers, 1976]. Following previous contributions [Creutin and Obled,1982; Rodri•iuez-Iturbe and Mefia, 1974; Chua and Bras, 1982], the rainfall over a basin is modeled as a two-dimensional random field.This approachallows us to take into account, in a rigor- ousand systematic way, the seasonal and spatialvariability of the rainfall process. The estimatorfor the averageareal rainfall is then a linear minimum variance unbiased estimator (also called BLUE) [Papoulis,1965], which is obtained by a straightforward ex- tensionof the well-known kriging approach [Delftnet and Del- hornme, 1975; Journel and Huijbregts, 1978; Delhornrne, 1978]. The optimal estimator requires knowledge of the variogramof the rainfall random field as a function of spaceand time. In order to obtain realistic rainfall estimates, a theoretical vario- gram model must be chosen, and its parameters must be esti- mated.This is the most difficult step. The main contribution of this paper is in the design of a procedure for the real-time estimationof a variogram model. The spatial variability of rainfall data has beenanalyzedunder different sets of assumptions; the seasonal trends of the vario- gram and the influence of the rainfall intensity have been examined. This has led to the adoption of a simple variogram model, in which the time nonstationarity of the rainfall func- tion is entirely concentratedin a time-varying scaling factor which can be adapted in real time. The advantageis that the weighting coefficients of the optimal rainfall estimator can now be computed once and for all, while the estimation vari- ance is computed in real time using a very simple adaptive procedure. The objective of our research was to design an adaptive Copyright1984by the American Geophysical Union. Paper number 3W1935. 0043-1397/84/003 W- 1935505.00 estimatorfor the average areal rainfall which is simpleenough to be used in real time and which does not have to rely on delicate meteorological interpretations. We believe that the proposed procedure achieves these objectives. Our estimator has been practicallyimplemented; we presentan application to real data in two river basins. Finally, as an interesting by-product, we showhow the opti- mal estimation method developedin this paper can also be usedto optimally select the location of rainfall gauges in the catchment area. 2. DEFINITIONS AND NOTATION The point rainfall depth is denotedp(k, z), with z = (x, y) • R2, a Cartesian space coordinate, and k • N+, an integer index.We consider the discrete sequence (indexed by k): {p(k, z)lk = 1, 2,..., K} of K nonzero point rainfall depths during K (not necessarily successive) time intervals, eachone of duration Ts. In line with previous works [Creutin and Obled, 1982; Rodriguez-lturbe and Mefia, 1974; Chua and Bras, 1982; Delfiner and Delhomme, 1975], for a fixed k, p(k, z) is viewed as a realization of a two-dimensional random field (RF) on R2 denoted P(k, z). The mean and the variogram of this field are written as m(k, z)= E[P(k, z)] (1) 7(k, z,,z•) = «E[{P(k, zi) - P(k, z•)} 2] (2) with (z•, zj)a pair of current points in R 2.It willbeassumed, in this paper, that for any k the field P(k, z) is isotropicand fulfills the "intrinsic assumption":(1) the mean is space- stationary (independent of z): m(k, z)= m(k) (3) and (2) the variogram is isotropicand space-stationary (it de- pends onlyon theEuclidean distance d o between z•andz j): 7(k, zi, zj) --- 7(k, do) (4) Consider a catchmentarea (it is most often a river basin) rlcR 2 with rainfall measurement stations numbered 1 to N. For each value of the index k, the measurements are thus specific numerical valuesof the function p(k, z): z,), ..., (5) 463