The study of the method for the Φ values across the entire regional range of α in the frequency calculation of the P-Ⅲ distribution - Qifeng, Song; Xi, Chen; Zhicai, Zhang (Journal of hydrology (Amsterdam), 2025)
•Proposed a method for the Φ values across the entire regional range of α in the frequency calculation of the P-Ⅲ distribution.•Resolved the issue of slow convergence at the critical point x = α + 1.•Addressed the data overflow problem when α is greater than 100.•Achieved a balance between computational speed and accuracy in the calculation of Φ values. The frequency computation of the P-Ⅲ distribution plays a pivotal role in various applications such as rainfall estimation, flood forecasting, and ecological flow analysis across various countries. The determination of Φ values stands as the crux of this computation. Traditionally, Φ values are derived through interpolation from a table, which introduces computational inaccuracies due to the complexity involved. To address this, an improved high-precision numerical integration algorithm has been developed. This enhanced method segments the integration interval into three distinct parts. It employs a series expansion, substitution, and continued fraction techniques to perform numerical integration across these segments, thereby obtaining Φ values for α values in the range just above zero to 100. It resolves the issue of slow convergence at the critical point x = α + 1. For α values between 100 and 1600, the distribution function is meticulously constructed, and Φ values are calculated using both the improved high-precision numerical integration algorithm and the constructed distribution function. It tackles the data overflow issue when α is greater than 100.When α exceeds 1600, an approximate function method is utilized to determine Φ values. By employing this comprehensive approach, a full-range calculation method for Φ values of the P-Ⅲ distribution is established. The accuracy of this method is validated by comparing its results with established truth values, revealing that the method’s calculations match the truth values to three decimal places. The method not only boasts high accuracy but also exhibits swift computational speed, making it a viable alternative to traditional Φ value tables and improving the accuracy of the calculation of the P-III distribution frequency curve in the field of hydrology.