A new two‐step stochastic modeling approach: Application to water transport in a spatially variable unsaturated soil

Abstract

Abstract. A new two-step stochastic modeling approach based on stochastic parameter inputs to a deterministic model system is presented.Step I combines a Stratified sampling scheme with a deterministic model to establish a deterministic response surface (DRS).Step II combines a Monte Carlo sampling scheme with the DRS to establish the stochastic model response. The new two-step approach is demonstrated on a one-dimensional unsaturated water flow problem at field scale with a dynamic surface flux and two spatially variable and interdependent parameters: The Campbell [1974] soil water retention parameter (b) and the saturated hydraulic conductivity (Ks). The new two-step stochastic modeling approach provides a highly time efficient way to analyze consequences of uncertainties in stochastic parameter input at field scale. The new two-step approach is competitive in analyzing problems with time consuming deterministic model runs where the stochastic problem can be adequately described by up to two spatially variable parameters. IntroductionThe concept of soil as an inhomogeneous media was established already at the turn of the century [Vieira et al., 1981]. Despite this, only a limited number of studies focusing specifically on the spatially variable nature of soil were published until approximately 2• decades ago. Since then, however, research within this field has increased dramatically, and the results find applications within, for example, water resource management, pollution control, and agriculture. In recent years, there seems to be an increased bias toward development and validation of predictive modeling tools accounting for the effect of spatial variability.An accurate description of both water and solute transport in the unsaturated zone relies upon good descriptions of the soil-water retention properties and unsaturated hydraulic conductivity of the soil. In recognition of this many studies have dealt with the characterization of natural variability in parameters describing soil hydraulic properties [e.g., Nielsen et al., [1994a] point out that the equations arising from application of the stochastic theory are very complicated to solve (even numerically) because of intercoupling, nonlinearity, and high dimensionality. Furthermore, the estimation of the necessary statistics from measured field data, namely, correlation lengths and covariance functions between parameters, often requires larger data sets than the ones typically available at this point [Jensen and Mantoglou, 1992;Mantoglou, 1992]. Together with the fact that the stochastic theory is very mathematically complex, these points of critique have limited the practical use of the theory to a very small number of studies.Comparatively, there have been a much higher number of applied studies based on deterministic modeling with statistical sampling. Most often, these studies are based on a numerical or analytical solution to the governing partial differential equation of unsaturated water flow subject to appropriate initial and boundary conditions....

Effects of Soil Bulk Density on Gas Transport Parameters and Pore‐Network Properties across a Sandy Field Site

“…[19]) and the Campbell b . This result is in accordance with the work of Loll and Moldrup (1998) who mentioned the strong dependency of Campbell b on texture.…”

Predicting saturated hydraulic conductivity from air permeability: Application in stochastic water infiltration modeling

“…A simplified version of the Two-Step method [Loll and Moldrup, 1998] is adopted for evaluation of the stochastic scenarios, as described below.…”

A strategy for improved computational efficiency of the method of anchored distributions

“…First, bundling is the only approach that combines simulations from different samples in the training data matrix. Second, compared to LHS or DRS, where a single representative sample's likelihood value is assigned to the entire LH [ McKay et al ., ] or likelihood values are evaluated from the DRS linearly interpolated over the likelihood values of a stratified subset of samples [ Loll and Moldrup , ], bundling utilizes all the samples in determining the approximate likelihood value. Finally, with bundling, none of the individual likelihood values of the samples in a bundle are ever directly inferred, but rather it is the likelihood of the bundle which is inferred , which is different than traditional MAD, LHS, and DRS, where at least one sample's individual likelihood value is always directly inferred.…”

“…There is a body of literature with a focus on reducing the computational cost of model inversion, most of which can be roughly categorized as either model reduction or intelligent sampling. Briefly, model reduction is the replacement of a more expensive numerical FM with a cheaper analog; this category includes techniques like the response surface method (RSM) [ Downing et al ., ], the high‐dimensional model representation (HDMR) [ Rabitz et al ., ], the stochastic response surface method (SRSM) [ Isukapalli et al ., ], and the deterministic response surface [ Loll and Moldrup , ], or the temporal reduction of transient models to steady state via temporal moments (TMs) [ Harvey and Gorelick , ; Cirpka and Kitanidis , ; Leube et al ., ]. The approach presented later is not a model reduction technique but could be utilized in a complimentary manner, which is detailed after the method is presented.…”