Travel time distributions of subsurface flow along complex hillslopes with exponential width functions

Recently, Troch et al. [10] introduced the hillslope-storage Boussinesq (hsB) equation to describe subsurface flow and saturation along complex hillslopes. They demonstrated that numerical solutions of the hsB equation account explicitly for plan shape of the hillslope, by introducing the hillslope width function, and for profile curvature through bedrock angle and the hillslope soil depth function (see also Paniconi et al. [7] and Hilberts et al. [4]). This paper presents an analytical solution of the linearized hsB equation applicable to hillslopes with constant bedrock slope and exponential width functions. Then, the first four moments of the impulse response function (travel time distribution) of such hillslopes are derived using the Laplace transform of the partial differential equation describing the groundwater flow. These moments are explicitly related to the hydraulic (porosity, conductivity) and geometric (bedrock slope, width function, and soil depth) parameters of a hillslope. The zeroth moment can obviously be interpreted as the total outflow volume, and the (normalized) first moment as the mean response time of the hillslope. The second, third, and fourth moments have interpretations in terms of the relative width, asymmetry, and peakedness, respectively, of the hydrograph. The paper examines the effect of the geometric properties of hillslopes on the travel time distribution in a dimensionless context.