ON THE EMPIRICAL-FORMULA OF WILLETT ET-AL RELATING LIGHTNING RETURN-STROKE PEAK CURRENT AND PEAK ELECTRIC-FIELD

The empirical formula proposed by Willett et al. (1989) for the estimation of lightning return-stroke peak current, I, from measured peak electric field, E, at a range D, is analyzed and discussed. The formula of Willett et al. (1989), obtained from the regression of E on I, is not the least squares fit and hence is not the best expression for predicting the peak current from the measured peak electric field. Based on the same data, the least squares fit and, hence, the best expression is obtained from the regression of I on E, given by I = 1.5 - 0.037DE, where I is in kA and taken as negative, E is positive and in V/m, and D is in km. The Willett et al. (1989) formula results in an error with respect to the best peak current estimating expression that varies from -15% to +2.6% over the range of peak field values of 1.9 to 11 V/m (normalized to 100 km) used to derive the two relations. When the data of Willett et al. (1989) are separated into a high return-stroke speed group (1.5 x 10(8) to 1.9 x 10(8) m/s) and a low return-stroke speed group (1.2 x 10(8) to 1.4 x 10(8) m/s), the I-E regression lines differ for the two groups, with the difference in the regression line slopes being statistically significant at the 0.01 significance level. If the difference between intercepts of these two regression lines, found to be statistically insignificant, is neglected, the observed difference in slopes suggests that the group with the higher measured return-stroke speed is associated with a lower peak electric field for the same peak current. Finally, the practical applications of the Willett et al. (1989) formula presently found in the literature are reviewed, and the several cases of improper use, mostly related to misinterpretation of Willett et al.'s (1989) sign convention, are corrected.