From Discrete Streamflow Components to Continuous Delay Spectra: A Mixed Weibull CDF Parameterization of Characteristic Delay Curves

Streamflow is traditionally regarded as the superposition of binary or a limited number of discrete components. Under this framework, the Baseflow Index (BFI) is a widely used hydrological signature for characterizing subsurface water and catchment behavior. However, streamflow generation is inherently a continuous, spectrum-like delayed response across multiple timescales, rendering such discrete representations a methodological simplification. Building on the Characteristic Delay Curves (CDCs), this study finds that a mixed Weibull cumulative distribution function (CDF) effectively fits CDCs. Drawing on interpretations of the Weibull distribution from Reliability and Survival Analysis, we propose three parameters within the CDCs: the ratio of baseline, the delayed discharge mode (corresponding to the Weibull shape parameter), and the characteristic delayed time (corresponding to the Weibull scale parameter), to characterize streamflow generation mechanisms and catchment behavior. This study examines the relationships and causal structure among these parameters using long-term streamflow records from 60 catchments in Taiwan and 671 catchments in the United Kingdom. The results indicate that characteristic delayed time acts as a common driver of both the ratio of baseline and the delayed discharge mode, while the latter two exhibit weak mutual dependence. Time- and frequency-domain analyses further demonstrate that the proposed parameters better discriminate streamflow dynamic regimes than the BFI. Overall, this study provides a continuous compositional framework for interpreting streamflow hydrographs and establishes a Weibull-based foundation for advancing theories of streamflow generation and aquifer drainage.