The inverse-cubic polynomial as a probability function in climatology particularized to monthly and annual precipitation

Abstract

The objective of this study was to determine in detail the reliability of the inverse-cubic polynomial as an approximation function for probability distributions of monthly and annual precipitation. In addition, 20 other climatological distributions were fitted to this model to obtain an indication of its general usefulness as an approximation function for climatology is general. The study was restricted to the non-extreme portions of the probability distributions. Non-extreme was defined as being from the 10th to the 90th percentiles. However, some of the variables were extremes, such as maximum temperature. The inverse-cubic polynomial is defined by: [see PDF for equation]. P(Y) is the probability of a value equal to or less than Y. Each distribution was fitted by the least square method. Decile residuals were defined as estimated minus observed values. These residuals are no doubt greater than the errors of estimating the parent populations, since they include small sampling errors. For precipitations, the method of interpolation of missing decile values tended to underestimate the approximation efficiency of the model. For the other climatological distributions this interpolation was linear. January, April, July, October, and annual precipitation data for 11 stations east of the Rocky Mountains were tested in detail. The record lengths were from 77 to 164 years. The largest magnitude of the monthly residuals was 0.8 inch. Only 4 of the 396 were greater than 0.4 inch. The average absolute value was 0.1 inch and the algebraic average was 0.0 inch. The absolute values for 94 percent of residuals were 0.2 inch or less. The largest magnitude of annual precipitation residuals was 1.5 inches, and 92 percent were an inch or less. The average absolute value was 0.6 inch, and the algebraic average was +0.1 inch..