Calibration and Testing of a Daily Rainfall Erosivity Model

A general procedure is presented for calibrating a model for rainfall erosivity based on daily rainfall. The approach is based on probability distributions of wet-day precipitation amount and monthly erosivities which are inferred from published data summaries. The calibrated model was tested by comparisons with erosivities computed from hourly precipitation data. Model results were generally consistent with values based on hourly data and explained over 85% and 70%, respectively, of the variations in annual and event erosivities. Model results for extreme values (annual erosivities exceeded in 5% of the years and l-in-20-year event erosivities) often substantially exceeded values computed for hourly data. To facilitate general use of the daily model, calibration coefficients were calculated for 33 sites in the eastem and central U.S. INTRODUCTION A general equation for estimating the erosivity term in the Universal Soil Loss Equation from daily rainfall data was proposed by Richardson et al. (1983). The equation provides a simple model for calculating event soil losses from daily, rather than hourly, precipitation data. Since daily weather records are more commonly available than hourly records, the equation is potentially a valuable tool for erosion, sediment yield, and nonpoint source pollution studies. It was subsequently tested by Haith and Merrill (1987) for 23 locations in the eastem and central U.S. Long-term synthetic daily rainfall records were generated at each location, and these were used in the Richardson et al. model to compute erosivities. The testing involved comparisons of these model results with rainfall erosivities reported by Wischmeier and Smith (1978). Although the results of the comparisons were generally favorable, they were not conclusive. The use of synthetic weather introduced an additional level of uncertainty because discrepancies between the two sets of erosivities may have been associated with defects in the weather generating procedure. Difficulties were also apparent in the estimation of parameters for the Richardson et al. model. The two required coefficients were available only for the 11 original Richardson et al. sites. Coefficients for other locations were arbitrarily assigned the values of the closest Article was submitted for publication in April 1990; reviewed and approved for publication by the Soil and Water Div. of ASAE in August 1990. The authors are J. S. Selker, Graduate Research Assistant, D. A. Haith, Professor, and J. E. Reynolds, Undergraduate Research Assistant, Dept. of Agricultural Engineering, Cornell University, Ithaca, NY. of the 11 original sites. This article describes a more complete testing of the erosivity model. The objectives of the study were to develop a general procedure for calibrating the model to any U.S. location and to test the calibrated model for selected locations in the eastern and western U.S., respectively. Although the testing procedures were similar to those used in Haith and Merrill (1987), model erosivities were computed using historic daily weather records rather than synthetic records. CALIBRATION METHODS The erosivity model given by Richardson et al. (1983) is the regression equation: Loĝ o EÎ = Logĵ a + 1.81 Log^^^ + ^ (1) or equivalently, (2) EI, = a lO^R/*' with lower and upper bounds EI^j„ and EI^jj^, respectively: ^̂ min = ^t ^ [0-00364 LogjRt) 0.000062] (3a) E U = Rt'[0-291+0.1746 Logj^Rj] ifR,<38 EI = 0.566 R max t ifRj>38 (3b) (3c) where EÎ = daily rainfall erosivity on day t (MJ-mm/ha-hr), a = seasonal erosivity coefficient, € = normally distributed random variable with mean zero and standard deviation 0.34, and Rj = rainfall amount on day t (mm). The coefficient a is given by two values, a^ and â ,, where a^ is used for the warm months of April through September, and â , is used for the cool months of October through March. The random term € , which corresponds to the € ' variable in Richardson et al. (1983), is a residual or error term for the regression equation. The lower and upper bounds on EÎ given by equations 3a-3c limit erosivity to physically realistic values. The lower bound EIĵ ĵĵ corresponds to a minimum rainfall intensity case in which R̂ is distributed over 24 hours. Conversely, EÎ ^^ ̂ ^ produced when R̂ occurs in a single 1612 © 1990 American Society of Agricultural Engineers 0001-2351 / 90 / 3305-1612 TRANSACTIGNS OF THE ASAE half-hour period (Richardson et al., 1983). The erosivity model can be calibrated for U.S. locations by appropriate selection of the coefficients a^ and â .. The calibration procedure is based on information published in Agriculture Handbook 537 (Wischmeier and Smith, 1978). The handbook provides mean annual erosivities for the U.S. over the 22-year period, 1937-1958. The expected monthly fractions of mean annual erosivity for various regions are also provided. These fractions can be multiplied by mean annual erosivity to obtain ERĵ ,̂ the mean erosivity in month m (MJ-mm/ha-hr). The Wischmeier and Smith English units are converted to SI units by: 100 ft-ton-in/ac-hr = 17.0195 MJ-mm/ha-hr. The expected monthly erosivity can also be approximately estimated using equation 2. Assuming that in any given month the daily mean precipitation is constant throughout the month, the expected erosivity is: ER ' = d E (a. 10^ P. 1.81) (4) where d^ = number of days in month m, Pm = precipitation on any day in month m (mm), a^ = montfily value of a, and E = expectation operator. Equation 4 is an approximation since the upper and lower bounds of EI are neglected and rainfall has been replaced by precipitation (which may include both rain and snow). The random variables e and P̂ ^ are independent and equation 4 may be rewritten as: ER„' = d „ a „ E ( l 0 ' ) E ( p J ' ' ) (5) The exponent G is normally distributed and hence 10^ is lognormally distributed. The expectation of a base ten lognormal random variable is given by Hald (1952) as: E ( I O ^ ) = 10^ 10̂ '̂̂ '̂ ^̂ ô̂ L 1.359 (6) where |Li and a^ are the mean and variance of the normally distributed random variable (0 and 0.1156, respectively). The erosivity parameter â ^ is obtained by setting ER̂ ^ equal to ER^̂ 'and solving: a = m ER 1.359 d E(P '•*') m V m / (7) The seasonal coefficients a^ and â , are given by weighting the monthly values by ERj„: Sept I VER„ _ m = Apr Mar fl — m = Oct **c Sept *Mar I ER„ X ER„ m = Apr m = Oct / g \ The expected value in equation 7 can be determined from the unconditional probability distribution of daily precipitation: F^*(p) = Prob{P^0} (10) If w„ is the probability of a wet day in month m, then: F„*(p) = ( l w J + w„F„(p) = WTM[F„(P)-1] + 1 (11) Letting fm*(P) be the density function corresponding to F^*(P). then: E ( p : « % f ' p ' f ; ( p ) d p (12) WET-DAY PRECIPITATION PROBABILITY DISTRIBUTIONS The calibration procedure requires two probability measures. The wet-day probability Wj„ for any month is given by the average number of wet days divided by the number of days in the month. The conditional, or wet-day probability distribution of precipitation, ^^(p) can be estimated from analysis of daily precipitation records. However, if a single-parameter distribution is assumed, the distribution can be obtained from |a^ the mean wet-day precipitation in month m (mm). These monthly means are computed from precipitation summaries such as Climates of the States (National Oceanic and Atmospheric Administration, 1985) and Statistical Abstract of The United States (Bureau of the Census, 1982) by dividing mean monthly precipitation by mean number of wet days for any month. Here we evaluate the integral in equation 12 and the resulting aĵ for three of these single-parameter distributions. EXPONENTIAL DISTRIBUTION The exponential distribution is probably the most widely used, single-parameter distribution of daily precipitation amount (Todorovic and Woolhiser, 1974; Richardson, 1981; Pickering et al., 1988). It is given by: F (P) = l e •viK (13) The associated unconditional distribution is given by equation 11 as: F j ( p ) = l w „ e * ' ' ' (14) and the density function is: f,„*(P) = (w„/^Je-'''^"> (15) Substituting into equation 12, we obtain: E(pJ") = wJV'/Oe-'^dp (16) Jo VOL. 33(5): SEPTEMBER-OCTOBER 199