Standard Asymptotic Response and Expected Runoff from Curve Number Theory

The Standard Asymptotic response is a common finding with rainfall-runoff analysis with the Curve Number method. Widely-observed in data analyses, it is the secondary relationship between the data-defined CN and the causative rainfall P, for which the empirical fitting equation CN(P) = CN∞+ (100-CN∞)exp(-kP) has been found to fit well. This is an unexpected variation in the handbook methodology. It is investigated here on a “theoretical” basis by using the handbook tabled values of CN for the three AMC classes, I, II, and III; the storm-to-storm S (=1000/CN-10); and the observed 12-50-88 percent conditional probabilities for the ARC classes. With this, and treating S as a variable across the AMC classes, the standardized values of S, or S/SII =S*, are found to well describe a lognormal distribution with σn=0.713. Using standardized (dimensionless) values of P (P*=P/SII) and the found lognormal distribution, the expected values of Q* (or E(Q/SII)) are calculated for an array of P*. Converting back to dimensioned rainfall and E(Q), resulting CNs generate the standard asymptotic phenomenon. The effects are most prominent for smaller storms, in the range of P/S< ~0.5. These findings revolve around 1) the seldom-appreciated difference between the median CN (CN50) and median Q (Q50) that is defined with the traditional CN method, theory, and practice, and the expected values (E(CN), E(Q)) that arise from most rainfall-runoff data analysis techniques; and 2) the common practice of using only runoff-producing events in event rainfall-runoff analysis. A serendipitous result is that the E(Q*) and P* array offers a direct alternative to the median-based CN method/equation as currently practiced, with an expected value of Q>0 for any P>0. This simple shift to expected values from medians allows 1) explanation of the Standard response, and 2) an alternate and more enlightened application of the Curve Number method, especially with smaller runoff-producing rainstorms.