0 10 20 30

Lag (in Ax units)

Figure 9 Comparison of correlation function for monthly rainfall (from Kang and Ramirez46). See ftp site for color image.

scale environment. Being able to parameterize the scaling characteristics of precipitation as a function of such variables is a prerequisite for implementing of down-scaling methodologies based on random cascades. Perica and Foufoula-Georgiou77 introduced a spatial downscaling scheme that is able to statistically reproduce the spatial heterogeneity of observed precipitation fields at subgrid scales while being conditioned on large-scale averages and physical properties. They computed multi-scale standardized fluctuations using an orthogonal Haar wavelet decomposition and found that, at least for the range of scales of their analysis, these fluctuations exhibited normality and simple scaling. They also found that the scale-independent parameter H characterizing the simple scaling behavior of the standardized fluctuations was strongly dependent on the convective instability of the prestorm environment, namely on the convective available potential energy. The utility of the model in reproducing the small-scale statistical variability of precipitation as well as the fraction of area covered by rain at all subgrid scales was demonstrated,77 and the relationship between H and the convective available potential energy of the prestorm

TABLE 2 Comparison of Basic Statistics for Observed (July 1997) and Simulated Precipitation Fields


Nonparametric Superposition






Standard deviation




Skewness coefficient








From Kang and Ramirez.46

From Kang and Ramirez.46

environment established.77 On the other hand, the relationship between the /S-log-normal random cascade model parameters and the mean of the large-scale precipitation intensity was also observed and established.46'75

Most downscaling methodologies proposed in the literature only deal with the spatial variability of the precipitation field. The temporal evolution of the fields is usually described independently of the spatial downscaling, so that these schemes do not properly account for the temporal correlation structure, i.e., persistence, of the precipitation fields at subgrid scales. Recently, the linkage between the spatial and temporal scaling of precipitation fields has been explicitly addressed.12'75'113'114 Over and Gupta75 propose a model for space-time description of rainfall distributions based on multiplicative random cascades with independent weights in space, which are time varying according to an imposed structure. Carsteanu and Foufoula-Georgiou12 argue that space and time variations of rainfall are necessarily connected. They postulate and experimentally verify a Taylor-like hypothesis stating that the power law variation for the moments is the same in time and space. Venugopal et al.114 found that for spatial scales of 2 to 30 km and for temporal scales of 10 min to several hours, the evolution of precipitation remained statistically invariant under a transformation of the type t ~ L~, where z is a so-called dynamic scaling exponent. That is, they found that the space-time organization of rainfall fields is scale-invariant and that its characteristics can be obtained by a simple renormaliza-tion of the space and time coordinates as implied by the t ~ Lz transformation. They used the above results to develop a space—time precipitation downscaling scheme that is capable of preserving not only the spatial correlation of precipitation but also the temporal correlation at subgrid scales.

Finally, Seed et al." have modeled the space-time behavior of radar precipitation using a multiplicative bounded (multifractal) cascade, each level of which was linked to the same level at the next time step via a different ARMA( 1,1) model. Also Pegram and Clothier,76 developed the so-called string-of-beads model in which power-law filtering of Gaussian random fields in space and time is used to capture the correlation structure of the rainfall process. Two autoregressive models, one at the image scale, the other at the pixel scale, drive the string-of-beads model. The spatial power-law filtering then ensures that the generated fields scale correctly in space and time.


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