Use of an Efficient Proxy Solution for the Hillslope‐Storage Boussinesq Problem in Upscaling of Subsurface Stormflow

Abstract

Boussinesq representation of saturated subsurface flow that accommodates the impact of converging and diverging flows resulting from a hillslope with nonconstant width. Here, a surrogate model, the hsB Proxy, is developed which closely emulates the results of high-quality numerical solutions of the hsB to reproduce the drainage response of a wide range of wedge-shaped hillslopes in response to a recharge time series at a fraction of the computational cost of a numerical solution and with minimal loss of fidelity. The efficacy of the Proxy in emulating the hsB response of a single hillslope and the aggregate response of a network of hillslopes is demonstrated. The Proxy applies to wedge-shaped hillslopes with homogenous hydraulic conductivity under uniform recharge, with the expectation that these hillslopes may inform future investigations into the upscaling behavior that results in the known powerlaw relationship between subsurface storage and outflow in a basin. The utility of the hsB as a tool for evaluating upscaling problems is demonstrated in two applications. First, the Proxy response is compared to the known basin-scale recession behavior in the Panola Mountain Research Watershed, under both transient and steady-state conditions. Second, the Proxy is applied to derive the single best effective hillslope in a basin. The Proxy is expected to be a useful tool for rapid simulation of subsurface flow from hillslopes for use in hydrological models and land surface schemes.

A statistical approach for identifying factors governing streamflow recession behaviour

“…These include (among others): one‐dimensional saturated representation of a hillslope, developed by Troch et al (2003), three‐dimensional Richards Equation based numerical model developed by Paniconi et al (2003) or linearized hillslope‐storage Boussinesq model (hsB) which was solved analytically (e.g., Dralle et al, 2014) and numerically (e.g., Hazenberg et al, 2015). More recently, Ranjram and Craig (2021) developed a proxy solution for the hsB model that can upscale and solve a network of heterogenous hillslopes, using a hybrid numerical‐probabilistic approach, in order to estimate catchment‐scale recession parameters. Given the large data requirement for bottom‐up process‐based approaches, here we employ a “top‐down” large sample hydrology empirical path that relies solely on globally available data and prioritizes generalization and simplification in model design (McDonnell et al, 2007).…”

Upscaling Hillslope‐Scale Subsurface Flow to Inform Catchment‐Scale Recession Behavior

“…The recharge‐independence of the signature coefficients and appropriateness of superposition of the hsB response (Ranjram & Craig, 2021) enables the application of the upscaling relationships to any recharge time series in a basin with mean slope greater than two degrees. The final control on the recession response, the subsurface conductivity, is thus left to be evaluated.…”

“…Each hillslope toe is assumed to drain to a surface water channel network, where the travel time through the channel network is assumed to be negligible. The hillslope elements are derived directly from topography, as detailed in Ranjram and Craig (2021). Extracting hillslopes from a basin results in distributions of four hillslope properties: the length ($$L$$) along the primary axis; the constant bed slope of the hillslope ($$\theta $$); the width at the downslope end of the hillslope ($${W}_{b}$$); and a nondimensional parameter representing the upslope width of the hillslope as a fraction of the downslope width ($$X$$), which can also be considered as a measure of the degree of hillslope convergence (upslope width greater than downslope width, $$X$$ > 1) or divergence (upslope width smaller than downslope width, $$X$$ < 1).…”

“…𝐴𝐴 𝐴𝐴 is the flow through the hillslope [L³/T], 𝐴𝐴 𝐴𝐴 is the uniform recharge rate [L/T], and 𝐴𝐴 𝐴𝐴 is porosity [-]. This work uses the computationally efficient hsB proxy (Ranjram & Craig, 2021) to rapidly produce accurate emulated solutions to the hsB for wedge-shaped hillslopes (i.e., those with linear width functions) without having to solve the governing equations (Equations 2 and 3) numerically. Here, the use of the hsB proxy as an emulator is necessary in that we require thousands of simulations of transient hillslope drainage.…”

“…These relationships demonstrate a simple time and discharge scaling proportional to the new conductivity, where the magnitude of the discharge scaling is also dependent on the mean slope of the basin. It should be noted that the discharge scaling is effectively linear for small slopes (as seen in Ranjram & Craig, 2021), but the data indicated that this does not hold for larger slopes, and the power law exponent can get as low as 0.9 for the steepest slope in the training data set. The efficacy of this scaling function is demonstrated by comparing the simulated signature response to a single recharge event (10 mm/d) at various homogenous conductivity values with the signature predicted by the scaling rules across the set of 47 basins with mean slope greater than 2° (Figure 7, NSE > 0.965).…”