Analytical derivation of Horton index using a conceptual soil water balance model by cumulant expansion theory

In this study a new stochastic model for investigating the variability of the Horton index which is ratio of vaporization and wetting water is proposed. The proposed model is based on a conceptual soil water balance model. Using cumulant expansion theory, the steady-state soil water probabilistic density function is derived through meteorological and watershed characteristics. The steady-state soil water probability density function is then used to further derive the Horton index. As a result from observation data, it is shown that the inter-annual variability of the Horton index is lower than that of precipitation and they showed the strong negative correlation. As a model application, the sensitivity of Horton index to the precipitation occurrence rate and the mean of wet day precipitation is examined. Although the precipitation amount is not varied, the Horton index can be varied due to the fluctuation of the precipitation occurrence rate and the mean of wet day precipitation. In addition, it is presented that there is a non-linear relationship which has a critical point switching proportional or inverse relationship between the Horton index and two main characteristics of precipitation process. Figure 1. Sensitivity of Horton index to precipitation parameters. The Pm is the daily average precipitation rate [L/T] when precipitation falls in a day, and the lambda is the wet day probability. These statistics are estimated in the growing season.