A Process-Based Probabilistic View on Hydrological Drought Formation

Hydrological drought is commonly described using standardized indices derived from observed streamflow. While such indices are useful for monitoring purposes, they often rely on empirical distribution fitting and fixed thresholds, which makes their physical interpretation and transferability across climates and time periods difficult. In particular, it remains unclear how changes in climate forcing or catchment properties are reflected in drought characteristics defined by these indices.

In this study, hydrological drought is examined from a process-based probabilistic perspective, starting from stochastic rainfall–runoff dynamics. Daily rainfall is represented as a marked Poisson process, with storm arrivals occurring at a constant frequency and rainfall depths following an exponential distribution. Infiltration produces random increments of soil moisture, while evapotranspiration leads to continuous moisture losses from the root zone. These losses vary linearly with soil moisture between the wilting point and an upper threshold associated with soil water holding capacity. When this threshold is exceeded, runoff pulses are generated, with their occurrence and magnitudes described by stochastic processes. The resulting runoff feeds a lumped catchment storage, which is drained through the river network according to a nonlinear storage–discharge relationship that reflects the combined contribution of different flow components.

Based on this framework, the stationary probability distribution of streamflow is analytically derived, allowing hydrological drought to be interpreted as a left-tail behavior of the flow distribution rather than as an empirical anomaly. By mapping this theoretical distribution into a standardized probability space, drought conditions can be evaluated in a way that remains comparable with conventional approaches, while keeping an explicit link to physically meaningful parameters. The emphasis of this work is therefore not on defining a new drought index, but on improving the physical understanding of why and how hydrological drought characteristics change under different climatic and catchment conditions.