VOL. 12, NO. 2 WATER RESOURCESRESEARCH APRIL 1976 Evaluation of Mean Square Error Involved in Approximating the Areal Average of a Rainfall Event by a DiscreteSUmmation RAFAEL L. BRAS Department of Civil Engineering, University of Puerto Rico,Mayaguez, Puerto Rico 00708 IGNACIO RODRfGUEZ-ITURBE Universidad Simon Bolivar, Caracas, Venezuela Two-dimensional areal processes are commonly evaluated in hydrology through a discretization in space over theregion in which theprocess isbeing studied. Such a discretization involves anerrorin, going from the continuous process to the discrete one. This error is studied theoretically, and graphs are presented for its evaluation as function of the sizeof the area, the functional form of the correlation equation in space, and the level of discretization orsize of the sample. Correlation structures ofthe BesSel type and of the single and doubleexponential kind are considered, and their different implications are discussed. INTRODUCTION The areal average depth of a rainfall event is defined as = /(x) ax where f(x) is a function defining the rainfall eventover fixed area A, A is the area of interest, and P is the areal average rainfall depth. In evaluating the above expression the usual procedure is to discretize in space. With a uniform discretization, (1) is ap- proximatedby (2) where f(xi) is the value of f(x) at point xi, N is the degree of discretization, and /5 is the approximated arealaverage rain- fall. It is of interest to the hydrologist to have a measure of the error introducedin using(2) [see Bras and Roddguez-lturbe, 1975]. Onesuch criterion isthe mean square error,defined as E(P- p)2 = E(PP)+ E(PP)- 2E(PP) (3) The evaluation of the above expression depends on the sampling technique used.If it is assumed that sampling is being made in a fixed uniform grid with a known starting point, as was previously implied, (3) becomes N A cov (x,xi)dx (4) where cov(x,, x•,)= E[f(x,)f(xj)] Copyright ¸ 1976 by the American Geophysical Union. 181 For simplicity it is assumed that we are dealing with a zero mean process. EVALUATION The two integral expressions in (4) require a functional definition of the covariance of the process. Furthermore, the integrals in (4) have no closed form solu- tion for most valid covariance functions. Traditional numer- icalintegrations lackthe necessary accuracy, andmost impor- tant, they are very expensive computationally, especially for the case of the double integral in space appearingin (4). Fortunately, integrals of the form at hand canbeconverted to one-dimensional expectation operations which lend them- selves tO relatively easy and cheap numerical integration. Roddguez-lturbe and Mejœa [1974] pointed out the following equivalence: coy .= coy (r)G(r) dr = E[cov (r)/A] (5) where G(r) is the probability density function of the distance r between two randomly chosen points inthe region ofinterest and is dependent on the shape and sizeof the regionand d is the longest distance in the regionof interest (e.g., the diagonal for the rectangular area). In the aboveexpression it is inherentlyassumed that the user is dealing with functional covariances which dependonly on the distance between points; i.e., they are homogeneous. No isotropicassumption is made. In this work all areasare approximatedby rectangles (rec- tangulargrid pattern). For thisgeometric figure,Ghosh [ 1951 ] and Matern [1960] define G(r) as G(r) - (A1)•/•. GIrl(A) •/2, (l,/l•.) •/•] (6) where G(a, b) = 2a[G•(a, b) q- G2(ab, b) q- G•.(a/b, I/b)]