## 1. Introduction

[2] Soil thickness, or depth to bedrock, is generated through interactions between weathering and erosion. It is a key controlling factor for understanding hillslope hydrologic processes. Early studies have shown that different spatial distribution patterns of soil thickness can markedly influence the rates of rainfall-runoff [*Hoover and Hursh*, 1943]. Soil thickness and the pore space within soils determine the amount of rainfall that can be stored in hillslope soil profiles [*Tromp-van Meerveld and McDonnell*, 2006]. Hence, soil thickness is used as an important variable in many physically based hydrologic models such as Institute of Hydrology Distributed Model (IHDM) [*Beven et al*., 1987], distributed hydrology soil vegetation model (DHSVM) [*Wigmosta et al*., 1994], and hillslope storage dynamics models (HSDMs) [*Fan and Bras*, 1998; *Troch et al*., 2002]. It can also be used to derive needed parameters (e.g., soil water storage capacity) in many hydrologic conceptual models such as the HBV model [*Bergström and Forsman*, 1973] and the Xianjiang model [*Zhao et al*., 1980]. Hydrological studies and practices are now faced with increasing demands for soil thickness map of high quality [*Dahlke et al*., 2009; *Tesfa et al*., 2009].

[3] So far, hydrologists have depended largely on soil survey databases for maps such as the U.S. Department of Agriculture's national soils databases (including SSURGO and STATSGO) and the Soil Information System of China (SISChina) [*Smith et al*., 2004; *Shi et al*., 2007]. However, standard soil surveys were not designed to provide high-resolution soil thickness maps, which have limited reliable application for distributed hydrologic modeling [*Moore et al*., 1993; *Tesfa et al*., 2009]. Therefore, quantitative prediction or measurement methods for high-resolution soil thickness maps are crucial for reliable and precise hydrologic modeling.

[4] Various models for predicting soil thickness have been developed over time, which can be categorized into two main types: the stochastically based models and the physically based models. The former includes regressions (e.g., multiple or stepwise regression), geostatistical techniques, neural networks, and others. For instance, *Moore et al*. [1993] applied linear regression to relate soil and topographic factors for predicting soil attributes including soil thickness. *Ziadat* [2010] also adopted multiple linear regression models for soil thickness prediction and found that DEM-derived factors (slope, curvature, and topographic index) were useful for spatial estimation of soil thickness. *Tesfa et al*. [2009] used generalized additive models and random forests statistical techniques to predict soil thickness from some newly built topographic attributes (e.g., Pythagoras distance to ridge/stream, vertical rise/drop to ridge/stream, and others). *Catani et al*. [2010] present an empirical geomorphology-based model that links soil thickness to gradient, curvature, and relative position within a hillslope.

[5] Recently developed geographical information systems (GIS) terrain analysis techniques, and widely available digital elevation models (DEMs) have facilitated the use of these statistically based models. Because of their simple structure, fewer input requirement, other types of stochastic models, e.g., geostatistical co-kriging model [e.g., *Penížek and Bor ůvka*, 2006], and neural networks [*Zhu*, 1999], have also been used to establish quantitative relationships between landform factors (e.g., topography) and soil attributes (e.g., soil thickness). However, all of the statistically based models have to follow the basic assumption that the relationship between sample soils and topographic indices can be extended to infer soil attributes of another place. They require a large amount of field data for calibration of parameters and because of their empirical nature it can only be applied to areas where it has been developed [*Dietrich et al*., 1995]. These have greatly limited the model's capability of extrapolating and predicting soil thickness for other watersheds without data or with scarcity of sample points.

[6] In contrast, physically based models such as landscape evolution models concentrate on the understanding of the evolution of soil thickness by applying equations of soil production, sediment transport, and deposition over geological timescales. For example, by assuming that the rate of soil transport is proportional to slope (i.e., simple soil creep), *Dietrich et al*. [1995] proposed a numerical model running on grid-based DEM to predict colluvial soil thickness. *Roering* [2008] further expanded the method of *Dietrich et al*. [1995] by introducing three published linear and nonlinear transport formula into soil thickness evolution models. But these soil thickness evolution models are usually taken as one part of landscape evolution dynamic models that depend on sophisticated solution techniques, e.g., numerical methods for simulating landscape evolution in response to climatic and tectonic forces over a geological timescale [*Saco et al*., 2006; *Pelletier et al*., 2011].

[7] To facilitate hydrologic practices, simple and yet versatile expressions of soil thickness as a function of landscape are needed. Many existing studies established their theories on the balanced steady-state assumption between soil production and erosion. For instance, under this assumption, *Bertoldi et al*. [2006] gave a simple formula for soil thickness prediction by using the simple soil creep model for describing soil transport. *Pelletier and Rasmussen* [2009] further developed *Bertoldi et al*. [2006] balanced steady method by using three nonlinear soil transport equations for describing soil particle movement, respectively. However, the balanced steady state is only valid for some locations under certain conditions but does not hold generally, e.g., the Shale Hills Catchment [*Ma et al*., 2010]. So far, there is no widely used, geomorphically based analytical method of unsteady state equations for predicting soil thickness for hydrologic applications [*Pelletier and Rasmussen*, 2009]. In this study, we developed a simple geomorphic-based model based on an unsteady state solution of dynamic equations of landscape evolution. This model works on grid DEMs as *Dietrich et al*. [1995] and *Pelletier and Rasmussen* [2009], but employs simplified analytic formulations to predict soil thickness temporal evolution and spatial distribution. We demonstrated this model by applying it to predict soil thickness evolution over 13 kyr in the 7.9 ha Shale Hills catchment.