The rainfall areal reduction factor: A multifractal analysis

The areal reduction factor (ARF) η is a key parameter in the design for hydrologic extremes. For a basin of area A, η(A, D, T) is the ratio between the area-average rainfall intensity over a duration D with return period T and the point rainfall intensity for the same D and T. Besides depending on A, D and possibly T, the ARF is affected by the shape of the basin and by a number of seasonal, climatic and topographic characteristics. Commonly used formulas and charts for η have been derived by smoothing or curve-fitting empirical area reduction factors extracted from raingage network records. We derive results on η(A, D, T) based on modeling intense space-time rainfall as a conserved multifractal field. A key parameter is the ratio r = vhyd/vatm between the hydrologic velocity vhyd= (√A)/D and the atmospheric velocity vatm= (√Arain)/Drain, where √Arain and Drain are the characteristic linear dimension and duration of rainfall “structures”. While η depends on A, D and T in a complicated way, some simple asymptotic results hold: For v small (say v < 0.2), η is close to 1 and for v large (say v > 5) η ~ A^(-0.5) for given A, D and T ->oo, whereas η ~ A^(-0.5γ1) for given D, T and A -> 0. The constant γ1 in the exponent of the latter expression is the order of singularity for which the co-dimension function c(γ) of the multifractal rainfall process equals 1; hence γ1 < 1. These analytical scaling results are in generally good agreement with empirical findings. Discrepancies for small A and D are due to the finite range of scaling behavior of physical rainfall and bias in the empirically estimated extreme area rainfall due to sparseness of the raingage network. These effects are evaluated through numerical simulation and corrections to empirical ARF formulas are suggested.