Modeling Direct Runoff Hydrographs with the Surge Function

Abstract

A surge function is a mathematical function of the form f(x)=axpe-bx. We simplify the surge function by holding p constant at 1 and investigate the simplified form as a potential model to represent the full peak of a stream discharge hydrograph. The previously studied Weibull and gamma distributions are included for comparison. We develop an analysis algorithm which produces the best-fit parameters for every peak for each model function, and we process the data with a MATLAB script that uses spectral analysis to filter year-long, 15-minute, stream-discharge data sets. The filtering is necessary to locate the concave-upward inflection points used to separate the data set into its constituent, individual peaks. The Levenberg-Marquardt algorithm is used to iteratively estimate the unknown parameters for each version of the modeled peak by minimizing the sum of squares of residuals. The results allow goodness-of-fit comparisons between the three model functions, as well as a comparison of peaks at the same gage through the year of record. Application of these methods to five rivers from three distinct hydrologic regions shows that the simple surge function is a special case of the gamma distribution, which is known to be useful as a modeling function for a full-peak hydrograph. The study also confirms that the Weibull distribution produces good fits to 15-minute hydrograph data