Insight into karst hydrological processes in the frequency domain: critical frequency, phase difference, causality, and machine learning model

The complexity of karst terrains, combined with aquifer heterogeneity and extensive anthropogenic activities, results in the diversity of groundwater response, which hinders our understanding of the karst hydrological processes of precipitation-driven spring discharge. This study first transforms precipitation and karst spring discharge signals into the time–frequency domain using wavelet analysis. This helps identify key frequencies and phase differences (or time-lags) between the two signals. Then, the Liang-Kleeman information flow method is used to quantify the causal relationship between them. Finally, a causal machine learning model is developed based on the information flow at the identified critical frequencies. The results of applying this analysis to the Niangziguan Springs catchment, China, show that the 1-year critical frequency is more potent than that of 6.5-year and 15-year frequencies. Simultaneously, the information flow detects four typical causal links corresponding to the four migration velocities of the karst aquifer: fast flow, intermediate fast flow, intermediate slow flow, and slow flow. The information flow of precipitation driving spring discharge mainly delivers through fast flow and intermediate slow flow (i.e., conduits and fractures) in the karst aquifer, which occupy about 61%-76% in the karst-developed areas, 50% in moderately developed areas, and 38.3% in the low-developed areas. Meanwhile, at the 1-year critical frequency, groundwater overexploitation reduces the information flow by approximately 63% compared with the natural state. Moreover, the results of the casual machine learning model show that it is superior to the traditional deep learning model in the multi-step forecasting of spring discharge. Overall, these approaches in the frequency domain can depict temporal variations of rainfall-runoff and spatial heterogeneity of karst aquifers, allowing for improved understanding of the nonlinear and complex karst hydrological processes.