Optimality of antecedent precipitation index and its application

The decay constant k in the antecedent precipitation index (API) formula is generally determined arbitrarily and empirically, which causes great confusion in API calculation and comparison of different research results. In response to these problems, two hypotheses related to API (that is, k has the properties of optimality and multivaluedness) are proposed in this study. The daily runoff and daily precipitation of a typical watershed in the Yellow River Basin from January1956-December 2013 is taken as an objective investigated. The results show that the API has a stable antecedent day t(0). When antecedent day t >= t(0), the API remains stable; and when t <= t(0), there are the optimal value of k (k(opt)) and the optimal t (t(opt)) in the k(opt) scenario. The k(opt) ranges between 0.906 and 0.957, which is different from the traditional empirical single value suggested between 0.80 and 0.98. The existence of k(opt) verifies the optimality of k or API (Hypothesis 1). The k(opt) varies with t, which verifies that the k or API has the property of multivaluedness (Hypothesis 2). The two hypotheses of API exist objectively and are related to the runoff generation and soil moisture content of the watershed. The correlation coefficient and its T statistic, sliding average, approximate entropy (ApEn), and four error indicators are used to determine the optimal pair among four pairs of low flow and API. Based on the two hypotheses and the optimal pair, the daily low flow from January 2011 to December 2013 is simulated. The simulation results from the power function are better than those from the logarithmic function, and the k(opt) scenario is better than the other k scenarios. The simulation accuracy is high and the results are reliable. The contributions of this study lie in (I) the findings of two properties of API (i.e., the optimality and multivaluedness of k); (II) the proposal of an optimal standard for the selection of k, which overcomes the shortcomings of traditional arbitrary selection of k; and (III) the successful application of the two properties and the complexity (ApEn) to low flow simulation, which has theoretical value and practical significance.