A methodology for the design of precipitation networks is formulated. The network problem is discussed in its general conception, and then focus is made on networks to provide background information for the design of more specific gaging systems. The rainfall process is described in terms of its correlation structure in time and space. A general framework is developed to estimate the variance of the sample long‐term mean areal precipitation and mean areal rainfall of a storm event. The variance is expressed as a function of correlation in time, correlation in space, length of operation of the network, and geometry of the gaging array. The trade of time versus space is quantitatively developed, and realistic examples are worked out showing the influence of the network design scheme on the variance of the estimated values.
The disaggregation model proposed by Schaake et al. (1972) is revised to include linkages with the past at the different levels of aggregation. This modification produces a more realistic hydrologic model.
Jos•} M. MEJ[A Instituto Venezolano de Investigaciones Ciendficas, Caracas, VenezuelaA general methodology is developed for the transformation of point rainfall to areal rainfall. The reduction factor is shown to depend solely on the expected correlation coefficient between the point rainfall at two randomly chosen points in the area in consideration. The methodology can be used to characterize the input to rainfall-runoff models, and it includes the case in which multiple inputs are used in the model in the form of a subdivision in modules of the whole catchment. An example is worked out discussing the different methodologies for the estimation of total mean volume of rainfall over an area.The estimation of return periods for floods has traditionally been done by using a probability distribution that is fitted to the yearly maximums of the streamflow data. These data are in general of shorter duration than those of precipitation. It has been argued that the additional information in precipitation can be used through the application of rainfall-runoff models that as a by-product can take into account nonstationarities due to physical changes in the catchment. Such models usually operate by a disaggregation of the catchment area in modules over which the input is assumed to be lumped. Since rainfall information is taken at a point, it is necessary to have a tool that can transform this information into areal information in order for the results obtained from the models to be meaningful.The differences between the areal process and the point process and the use of the latter in the calibration of the model, added to the lack of conformance of the model to the physical world, may have as an effect that the results obtained from rainfall-runoff models are not better than the ones obtained by using the classical scheme. Besides that, there is not a general procedure for checking the accuracy of the results obtained from rainfall-runoff models as exists for the classical sampling theory in statistics. This paper tries to find the relationship existing between the point data and the areal data for isotropic processes having a separable correlation function in order to have a more realistic input for rainfall-runoff type of models. RELATED TECHNIQUESU.S. Weather Bureau methodology. Point rainfall depths are usually transformed to average areal precipitation by using a correction factor developed by the U.S. Weather Bureau. This factor was developed to transform point rainfall frequency curves into areal frequency curves. The factor is presented in Figure 1 and is expressed by equation (1) as reported by Leclerc and Schaake [1972]:
A procedure for the synthesis of processes exhibiting temporal and spatial variability is presented. The method involves the addition of harmonics of random frequencies that are sampled from the spectral density function or the radial spectral density function. The process obtained is asymptotically Gaussian and ergodic. An estimate of the error in time or space averages due to the nonergodicity of the process as a function of the number of harmonics is also included. Spatial patterns of variation of geophysical phenomena canbe studied by means of stochastic processes representing these variations in a continuous way over the space considered or at discrete points on it. The stochastic process associated with the second of these representations is called a multivariate process by some authors, whereas others prefer the name multidimensional process. It represents a family of random vectors X(t) = {X•, (t), .--, Xn(t)} v depending on the parameter t, where each one of the components is the value assumed by the process at specified points on the area at time t. In contrast, the continuous representation of such processes requires the theory of random fields (some authors prefer the name multidimensional processes, but it is confusing) in which the stochastic process X(u, t) represents a random variable depending on the parameter u = {X•, ..-, X,•} r, representing the coordinates of a point in an n-dimensional space. The applications of' the theory of multivariate processes to the description of' geophysical phenomena have increased significantly over the last few years, whereas random fields have been almost completely ignored by the applied hydrostatistician. However, the study of random fields is justified, since the effect of coarse discrete representations of the phenomenon under study may not capture adequately the physical reality, whereas the use of a larger number of points, especially over large areas, involves operations with matrices who• size may render the problem impractical, particularly for precipitation due to the highly variable nature of the phenomenon.
This paper deals with the theory of the broken line process and its applications in the simulation of stochastic sequences.
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