A simple geomorphic-based analytical model for predicting the spatial distribution of soil thickness in headwater hillslopes and catchments

Authors

  • Jintao Liu,

    Corresponding author
    1. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, China
    2. Department of Ecosystem Science and Management, Pennsylvania State University, University Park, Pennsylvania, USA
    3. Department of College of Hydrology and Water Resources, Hohai University, Nanjing, China
    • Corresponding author: J. Liu, State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, 1 Xikang Rd., Nanjing 210098, China. (jtliu@hhu.edu.cn)

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  • Xi Chen,

    1. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, China
    2. Department of College of Hydrology and Water Resources, Hohai University, Nanjing, China
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  • Henry Lin,

    1. Department of Ecosystem Science and Management, Pennsylvania State University, University Park, Pennsylvania, USA
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  • Hu Liu,

    1. Department of Ecosystem Science and Management, Pennsylvania State University, University Park, Pennsylvania, USA
    2. Linze Inland River Basin Research Station, Chinese Ecosystem Research Network and Key Laboratory of Ecohydrology & River Basin Science, Chinese Academy of Sciences, Lanzhou, China
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  • Huiqing Song

    1. State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing, China
    2. Department of College of Hydrology and Water Resources, Hohai University, Nanjing, China
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Abstract

[1] Soil thickness acts as an important control for headwater hydrologic processes. Yet, its spatial distribution remains one of the least understood in catchment hydrology. Analytic methods are desirable to provide a simple way for predicting soil thickness distribution over a hillslope or a catchment. In this paper, a simple geomorphic-based analytical model is derived from the dynamic equations of soil thickness evolution in areas with no tectonic uplift or lowering since the recent geological past. The model employs terrain attributes (slope gradient, curvature, and upstream contributing area) as inputs on grid-based DEMs for predicting soil thickness evolution over time. The analytic model is validated first on nine abstract hillslopes through comparing 10 kyr simulation results between our proposed model and the numerical solution. The model is then applied to predict soil thickness evolution over 13 kyr in the 7.9 ha Shale Hills catchment (one of the Critical Zone Observatories in the U.S. located in central Pennsylvania). Field observed and model predicted values of soil thickness are in good agreement (with a root mean squared error of 0.39 m, R2 = 0.74, and absolute errors <0.10 m in 70% of 106 sample points). Moreover, our model verifies that terrain shape and position are the first-order control on soil thickness evolution in the headwater catchment. Therefore, the derived geomorphic-based analytical model can be helpful in understanding soil thickness change over geological time and is useful as a simple tool for deriving spatially distributed soil thickness needed for hydrologic modeling.

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.

2. Methodology

2.1. Basic Equations

[8] The dynamic equations for describing soil evolution as a result of the mass balance of soil production and transport can be written as

display math(1)

where h is soil thickness, e is bedrock elevation, q is soil transport flux, and η is the ratio of rock bulk density to soil bulk density, i.e., η = ρr/ρs. The first term on the right-hand side of equation (1), ∂e/∂t, is the rate of soil production from bedrock by weathering. As reported by Heimsath et al. [2000], soil production rate generally declines exponentially with increasing soil thickness as

display math(2)

where P0 is the soil production rate for bare bedrock (i.e., zero soil thickness) and h0 is a characteristic decay depth.

[9] As reported by McKean et al. [1993], Small et al. [1999], Dietrich et al. [1995], etc., in many humid or semiarid catchments, the dominant transport processes are slope dependent, i.e., creep model dominates the main processes of soil transport. Moreover, these linear transport laws tend to lead to a simple analytical solution for mass balance [Braun et al., 2001]. So, in this study, two extensively used linear sediment transport laws [Roering, 2008], were employed, i.e., a simple soil creep (qd) and a fluvial term (rainfall-runoff driven transport, qt) [Willgoose et al., 1991a]:

display math(3)

[10] Soil creep qd is described by a linear creep function:

display math(4)

where kd is diffusion coefficient and z is soil surface elevation.

[11] According to Willgoose et al. [1991b] and Saco et al. [2006], the general equation for rainfall-runoff driven transport qt can be written as

display math(5)

where kt is sediment transport coefficient, A is the contributing area, and m, n are exponent constants for a given landscape. The values of m and n are 1.4 and 2.1, respectively, based on the definition of Fagherazzi et al. [2002] definition.

[12] Substituting equations (2), (4), and (5) into equation (1), we can obtain a soil transport dynamic model for describing the evolution of soil thickness:

display math(6)

where math formula, and

display math(7)

where f is divergence of downhill soil flux for soil creep and rainfall-runoff driven transport. In order to compute f required by equations (6) and (7), a drainage network based on digital elevation model (DEM) was used. Figure 1 shows how soil flux is routed through such a raster network. According to equation (4), math formula in equation (7) can be expressed as math formula, in which the value of math formula equals to the topographic curvature, C. For raster i in Figure 1b, Ci can be computed according to Heimsath et al. [1999] as

display math(8)

where zi is the elevation for the study grid i, z1z8 are elevations of surrounding grids as shown in Figure 1b, and Δx is grid size. math formula in equation (7) can be expressed as a function of slope and upstream contributing area:

display math(9)

where math formula and math formula are inflow and outflow soil flux at grid i, respectively, math formula and math formula are ratio coefficients determined according to D∞ method [Tarboton, 1997]. For example, the coefficients for outflow, math formula and math formula can be defined as

display math(10)

where α1, α2 are the angle between cardinal, diagonal, and the triangular facet direction math formula, in Figure 1c, respectively.

Figure 1.

Soil flux routing in a 3 × 3 window: (a) the 3D structure of the 3 × 3 window; (b) the ranked grid cell for computation of curvature; and (c) computational mesh for soil flux derivative, f, where q represents soil flux, and gray arrow represents the triangular facet direction math formula of grid i, as calculated by Tarboton [1997] algorithm, and α1 and α2 are the angle between cardinal, diagonal, and math formula, respectively.

[13] Equations (6) and (7) can be solved by numerical methods. Here, an explicit difference method of MacCormack scheme [MacCormack, 1971] is employed to solve the governing differential equations (6) and (7) and to evaluate the results of the following analytical solution.

2.2. Analytical Solution

[14] Two assumptions were made for further simplifying the dynamic equations. First, we limit our study to humid or semihumid climates, where subsurface storm flow instead of Hortonian overland flow is the major runoff component and the underlying bedrock is mechanically strong [Dietrich et al., 1995]. Second, we assume no tectonic uplift or base level lowering over relatively short geologic timescales (e.g., 10–15 kyr), i.e., landscape terrain stability, for the catchment under study. Hence, the evolution of topographic characteristics (e.g., curvature) is driven by rock weathering and regolith transportation, of which the relative speed is supposed to be slower than that of soil thickness evolution, and equation (6) can be regarded as a first-order nonlinear and nonhomogenous ordinary differential equation. The general solution for equation (6) can be derived through the following procedures.

[15] First, the equation is rearranged as

display math(11)

where

display math(12)

[16] Then, it is transformed into a linear ordinary differential equation as

display math(13)

where math formula and math formula.

[17] Multiplying eat in both sides of equation (13) and rearranging yields

display math(14)

[18] Integration of equation (14), the general solution for this linear ordinary differential equation can be obtained as

display math(15)

where C1 is an integration constant.

[19] Substitution of equation (12) into equation (15), then the general expression for soil thickness can be derived as

display math(16)

[20] In equation (16), coefficient C1 can be determined by setting the initial soil thickness. So, if the initial soil thickness is assumed as

display math(17)

then the constant coefficient, C1, is derived as

display math(18)

[21] Substituting equation (18) into equation (16), we get

display math(19)

[22] Equation (19) is a common form of soil thickness evolution. As noted by Dietrich et al. [1995], the soil thickness evolution model is not sensitive to initial soil thickness. So, if the initial soil thickness is assumed as

display math(20)

then the constant coefficient, C1, is derived as

display math(21)

[23] Hence, equation (16) can be transformed to a more simplified expression as below:

display math(22)

where h0 and c are soil production parameters (which can be initially defined as the values through their physical meaning), and f is an integrative soil erosion rate, which can be determined according to equation (7).

[24] As soil thickness generally increases in swale areas within a catchment, the production rate may reach zero according to equation (2), so equation (6) can be rewritten as

display math(23)

[25] By integration of equation (23) with time limit from ts to t, and soil thickness from hs to h, we obtain

display math(24)

where ts is the time when soil production rate P ≈ 0 and hs is soil thickness at time ts.

[26] In the analytical model, parameters in soil production function, i.e., equation (2), can be measured through radioactive isotopes (e.g., uranium-series isotopes adopted by Ma et al. [2010]). Diffusion coefficients (e.g., kd) in equations (4) and (5), though with some empirical traits in the soil transport formulae, theoretically could be determined through field experiments if the rate of migration of stream channel can be assessed [Culling, 1963]. As reviewed by Chiang and Hsu [2006], for example, parameter kd ranges around 10−5 to 10−2 m2/yr according to published research results.

3. Model Assessments on Abstract Hillslopes

3.1. Nine Abstract Hillslopes

[27] We assessed the analytical solution for nine basic abstract hillslopes shown in Figure 2. All the hillslopes were constructed with a smooth bedrock surface, which was defined as a specific form of the bivariate quadratic function, i.e., equation (25) of Evans [1980] as below:

display math(25)

where z is elevation, x is horizontal distance in length direction of the hillslope surface, y is horizontal distance in width direction, E is the minimum elevation of the hillslope surface, H is the maximum elevation difference, L is the total length, γ is a profile curvature parameter, and ω is a plan curvature parameter.

Figure 2.

3D view of nine basic hillslopes: #1, Convergent-concave hillslope; #2, Parallel-concave hillslope; #3, Divergent-concave hillslope; #4, Convergent-straight hillslope; #5, Parallel-straight hillslope; #6, Divergent-straight hillslope; #7, Convergent-convex hillslope; #8, Parallel-convex hillslope; and #9, Divergent-convex hillslope.

[28] Nine basic hillslopes were determined by changing the values of parameters γ and ω as listed in Table 1 and Figure 2. The same height and slope were set for all the hillslopes with L = 100 m, E = 100 m, H = 30 m, and the average slope is 30%. The shapes for these nine basic hillslopes with concave, planar, and convex longitudinal profiles and convergent, parallel, and divergent platforms represent a wide range of natural hillslopes, and thus are amenable for experimental scenarios. In order to generate grid-DEMs for analytical solution evaluation, continuous hillslope surfaces were discretized into nrows × ncols DEM (nrows and ncols are the numbers of rows and columns, respectively) with horizontal resolution hr (hr = 1 m in this study). The DEMs for all these nine hillslopes were the same size.

Table 1. Geometrical Parameters for the Nine Abstract Hillslopes Considered in This Study
NumberProfilePlanγω × 10−2 (m−1)
1ConcaveConvergent1.52
2ConcaveParallel1.50
3ConcaveDivergent1.5−2
4StraightConvergent1.02
5StraightParallel1.00
6StraightDivergent1.0−2
7ConvexConvergent0.52
8ConvexParallel0.50
9ConvexDivergent0.5−2

3.2. Assessments

[29] The results were evaluated using the root mean square error (RMSE) as follows:

display math(26)

and the mean square error (MSE) as

display math(27)

where n is the number of data series, hN is soil thickness predicted by the numerical solution, hA is soil thickness predicted by the analytical solution, and math formula is the averaged soil thickness predicted by the analytical solution.

[30] We compared 10 kyr simulation results of the derived analytical model with those of the numerical model. In both the models, the same set of parameters of soil production and transport were used for the nine abstract hillslopes (Figure 3), i.e., P0 = 10−5 m/yr, h0 = 0.50 m, η = 2, kd = 10−4 m2/yr, kt = 10−7 m2/yr. On all the hillslopes, soil thickness was initially assumed to be zero. As shown in Figure 4, RMSEs of all the simulated soil thickness series for the nine hillslopes generally increase with time. For the final results of RMSEs, i.e., t = 10 kyr (see second column in Table 2), the maximum errors among all the time steps for the nine hillslopes are all below 1.50 × 10−6 m, which means that the analytical solution matches well with the numerical simulation results. From Figure 4, one can conveniently figure out that all the hillslopes can be divided into three groups according to the simulated results of RMSEs. The first group, with the largest average values of RMSEs, includes hillslopes #3, #6, and #9 that belong to divergent type of hillslopes; the second group includes all the convergent hillslopes (namely #1, #4, and #7), and the last group with the lowest RMSEs representing parallel hillslopes (i.e., #2, #5, and #8). So, our analytical model leads to a larger error on divergent terrain than on convergent or parallel areas because soil flux within each grid cell is more likely assigned to two different downstream grid cells on the divergent hillslopes according to the D∞ algorithm, which increases the amount of computation and hence it is more likely to increase rounding errors.

Figure 3.

Results of soil thickness simulation for the nine hillslopes after 10 kyr simulation by the analytical model, in the figure, different scales of color bars are used to differentiate those small discrepancies of predicted results among different types of hillslopes. The same set of parameters of soil production and transport used in simulations are: P0 = 10−5 m/yr, h0 = 0.50 m, η = 2, kd = 10−4 m2/yr, and kt = 10−7 m2/yr.

Figure 4.

10 kyr time series plots of RMSE (root mean square error) between simulation results of analytical and numerical solutions for the nine hillslopes.

Table 2. Final Results (t = 10 kyr) for Soil Depth Simulation on the Nine Abstract Hillslopes
Hillslope NumberRMSEa (×10−6 m)MSEb (×10−3 m)Predicted Soil Depth (m)c
Max.Ave.Min.
  1. a

    RMSE is the root mean square error.

  2. b

    MSE is the mean square error.

  3. c

    Max., Ave., and Min. represent maximum, average, and minimum value in the predicted soil depth.

11.058.440.390.210.18
20.953.030.190.170.17
30.993.720.160.140.13
40.967.590.200.200.11
50.820.130.170.170.17
61.112.770.130.130.12
71.0337.70.200.190
80.6834.80.170.150
91.0624.80.130.120

[31] We also found some interesting facts about soil thickness distribution under different terrain reliefs and hillslope shapes. We noticed that the spatial distribution of soil thickness on the nine hillslopes is quite different due to influences of hillslope relief and shape. As shown in Figure 3, soil particles tend to aggrade/aggregate in concave and convergent hillslopes (e.g., #1), but tend to erode in convex and divergent hillslopes (e.g., #9). Table 2 lists part of statistical results for the nine hillslopes. It can be seen that the maximum, average, and minimum soil thickness on concave or convergent hillslopes are always larger than those on convex or divergent hillslopes. For example, the average soil thickness for convergent hillslopes #1, #4, and #7 is 0.21 m, 0.20 m, and 0.19 m, respectively, which are thicker than 0.14 m, 0.13 m, and 0.12 m for the divergent hillsopes #3, #6, and #9. Similar trends are observed between the concave and convex hillslopes, where the simulated soil thickness is 0.21 m, 0.17 m, and 0.14 m for the concave hillslopes #1, #2, and #3, respectively, and is 0.19 m, 0.15 m, and 0.12 m for the convex hillslopes #7, #8, and #9, respectively. Maximum and minimum soil thicknesses in these hillslopes also have the same distribution pattern as averaged soil thickness. In particular, minimum soil thickness on the three convex hillslopes (#7, #8, and #9) is as small as zero. This phenomenon is caused by the gradually steepening slope on downslope edges of the convex hillslopes where local soil erosions have been gradually increased. On the contrary, soil thickness is a little thicker on downslope edges of the concave hillslopes due to relatively steady mild terrain.

[32] Within each hillslope, soil thickness distribution is still influenced by local relief. In Table 2, for instance, MSEs for #7, #8, and #9 (convex hillslopes) are 37.7 × 10−3 m, 34.8 × 10−3 m, and 24.8 × 10−3 m, respectively, which are far greater than that of the concave and planar hillslopes. This means that the ranges of soil thickness variations in the convex hillslopes are the largest among all the hillslopes, on which some areas are without soil cover. The MSE value of #4 is larger as compared to that of other two straight hillslopes (#5 and #6) because it has a distinct erosion groove (just like a river channel) with intensive concentration of upstream inflow in #4 as well as other two convergent hillslopes (Figure 3). However, on the convergent hillslopes, soil thickness distribution in the erosion groove is not always the same. For #4 and #7, soil thickness within the grooves is below the average thickness of surrounding areas, and the difference will be more obvious on the lower areas. This is because these two hillslopes possess gradually increasing slope from upstream to downstream. For #1's groove, soil erosion mainly occurs in the head area (in deep blue color) where contributing area has an abrupt increase in that part of the area. Thereafter, soil particles tend to deposit on the downstream groove as its slope remarkably decreases along downstream direction (see #1 in Figure 3).

4. Application in a Real Catchment

4.1. Study Catchment Description and Materials

[33] The 7.9 ha Shale Hills Catchment in central Pennsylvania (40°39′52.39″N 77°54′24.23″W) was chosen for testing the model proposed in this study (Figures 5a and 5b). This forested headwater catchment is one of the six U.S. Critical Zone Observatories (CZOs). It is a tectonically quiescent catchment [Jin et al., 2010], which fully satisfies our basic assumptions. This catchment is a first-order basin characterized by relatively steep slopes (average slope 26%, see in Figure 5c), with deciduous trees on hillsides and ridges and coniferous trees in the valley floor. Elevation ranges from 256 m a.s.l. at the outlet to 310 m a.s.l. at the highest point in the catchment. The mean annual temperature is 10°C and the mean annual precipitation is 1070 mm.

Figure 5.

Sampling points, elevation, and maps of terrain attributes in the Shale Hills catchment located in central Pennsylvania, USA, showing (a) the catchment location; (b) soil thickness sampling points, and topographic maps; (c) slope; (d) curvature; (e) upstream contributing area (unit in ha) in LOG; and (f) topographic wetness index (TWI).

[34] An accurate survey of catchment terrain was performed using LiDAR, and a 1 m digital elevation model is used here (Figure 5b). There are seven topographic swales embedded on otherwise relatively uniform hillsides (Figures 5c–5f).

[35] Detailed soil survey in the catchment have identified five soil series, i.e., the Weikert, Berks, Rushtown, Blairton, and Ernest soil series (with the first three being Inceptisols and the last two being Alfisols) [Lin et al., 2006]. The Weikert and Berks soils are mainly distributed on sideslopes and are shallow (less than 1 m), while other series are located on swales or valley floor and are deeper than 1 m. Soil sampling for all the five series showed that soils texture in the catchment is generally silt loam, and soil dry bulk density has an catchment-averaged value of 1.39 g/cm3 but has a general increasing trend with soil depth [Lin, 2006; Lin et al., 2006]. In this study, 106 soil thickness data points, collected by the Penn State's Hydropedology Laboratory using handheld auger, were adopted for model validation (Figure 5b).

[36] Geophysical and geochemical observations have shown that the last periglacial conditions at the Shale Hills Catchment begin to change to modern conditions at about 15 kyr ago [Gardner et al., 1991; Jin et al., 2010]. Afterward, at about 14 kyr ago, the present warm interval occurred and ice sheet began to melt. Hence, the last phase of landscape instability ended until about 13 kyr ago [Watts, 1979].

4.2. Model Calibration

[37] Considering the basic assumption of landscape terrain stability, we chose a run time of 13 kyr to simulate the soil thickness evolution. The initial soil thickness for the model simulation was set to zero for calibration phase. The model's geophysical and geochemical parameters, e.g., soil production and soil transport related parameters are available for the study catchment. In our model simulations, soil production rate was estimated using the results from Ma et al. [2010]:

display math(28)

where P0 = 100.8 m Myr−1, h0 = 1/2.79 m. In this catchment, for h larger than 2 m, soil production rate is negligible and is assumed as zero. In addition, the bedrock-soil density ratio (η) was set to be 1.87 according to the results reported by Lin [2006] and Jin et al. [2010]. Grid-based topographical parameters, such as slope, curvature, and upstream contributing area, were computed based on the 1 m (i.e., x = 1 m) LiDAR DEM (Figures 5c–5e).

[38] The only two parameters needing calibrations were soil diffusion coefficient, kd, and sediment transport coefficient, kt. These two parameters have some physical meaning; however, both are empirical in nature and hence are difficult to measure in the field. In order to simplify the calibrations, we define another parameter, the ratio of kd and kt as

display math(29)

where κ indicates the relative effect of the two kinds of soil transport. We can obtain an error field through performing model calibration by inputting different value combinations of kd and κ. So the only parameter in our soil thickness model that needs to be calibrated is kd, which ranges around 10−5 to 10−2 m2/yr according to existing research results reviewed in detail by Chiang and Hsu [2006].

[39] Through many times of numerical experiments, we found that our model gave consistent results regardless of the time step such as 100 years or less, hence t = 100 years was used in our simulations. We also found that the value of κ, regardless of kd, should be less than 10−2.1 for our study catchment. Otherwise, it would lead to floating-point overflow and the simulated soil thickness loses their normal meaning. As shown in the error field (Figure 6), RMSEs of the soil thickness between the results of analytical model and the measured values have a wide range from as high as about 1.1 m to relatively lower values that is less than 0.4 m. The distribution of RMSEs is greatly influenced by the parameter kd. As we can see from Figure 6, there exists an apparently narrow and long canal, i.e., the blue area, which is approximately parallel to the horizontal axis and vertical to the longitudinal axis crossing at Log(kd) ≈ −0.25. Parameter kd seems to exert stronger effects on prediction results than kt. However, with an increasing value of κ (equivalent to increasing kt), it shows a more remarkable impact of kt on the prediction results (see right part of Figure 6). Our results of the parameter calibration revealed that soil creep driven by terrain relief and surface shape largely dominates the whole process of soil formation in this catchment, while the impact of sediment erosion on soil thickness evolution increases as kt increases.

Figure 6.

RMSE field of soil depth between the results of the analytical model and the measured soil depth by handheld auger. In the figure, x and y axis represent parameter κ and kd, respectively, and their values are taken the logarithm for the sake of displaying.

[40] The optimal result with RMSE (=0.394 m) falls at the point (κ = 10−2.8, kd = 10−2.6) in Figure 6, i.e., kd = 2.51 × 10−3 m2/yr and kt = 3.98 × 10−6 m2/yr. According to this set of parameters, soil thickness map for the Shale Hills catchment was predicted as shown in Figure 7. Figure 8 gives a direct comparison between the measured and predicted soil thickness for this catchment. A clear linear relationship is noticeable (y = 0.80x + 0.19), with R2 = 0.74. The slope of this linear fit is 0.80, meaning that the predicted soil thickness generally underestimated the measured values. The maximum positive error (predicted minus measured) was 1.36 m, and the maximum negative error was −2.25 m. Among all of the 106 actual measured points, 53% was overpredicted, while the mean absolute error and the cumulative error for all the points were 0.18 m and −2.20 m, respectively. The absolute errors for nearly 70% of the measured points were less than 0.10 m. Overall, the model predicted results are generally satisfactory. Errors affected by terrain factors will be discussed in detail in the following section.

Figure 7.

Spatial distribution of soil thickness for the Shale Hills catchment after 13 kyr simulation under the optimal scenario o (refer to the text and Figure 6 for more details about this scenario).

Figure 8.

Scatterplot of measured versus predicted soil depth (SD) for the 106 sample points in the Shale Hills catchment.

5. Discussion

5.1. The Trend of Soil Evolution and the Effect of Initial Condition

[41] The soil evolution dynamics was firstly evaluated according to the optimal parameters. We found that the simulated average soil depth has been increasing since the periglacial period. But the growth rate significantly reduced around 5 kyr as shown in Figure 9a. During the period of 13 kyr, the average soil thickness is about 0.80 m, while the average soil thickness increases by 0.98 m for 20 kyr's evolution. This result indicates that the balance period of soil evolution seems far from being reached until nowadays and in the near future (e.g., t = 20 kyr). It means that the assumption of steady state of soil evolution is invalid in the Shale Hills catchment. As shown in Figure 9b, the best simulated results occur around t = 13 kyr, i.e., the root mean squared error (RMSE) between simulated and measured values is relative small.

Figure 9.

20 kyr simulated results under three different initial conditions (e.g., hi = 0 cm, 5 cm, 10 cm): (a) average soil thickness (Ave. ST) as a function of time, (b) root mean squared error (RMSE) between simulated and measured values as a function of time.

[42] The impact of initial conditions on soil evolution was investigated for three scenarios, e.g., hi = 0 cm, 5 cm, and 10 cm. We found that the simulated bias among the cases reduced dramatically from 10 cm at the beginning time to 1.4 cm (about 1.8% of average soil thickness) at t = 13 kyr, then further reduced to 1.0 cm (about 1.0% of average soil thickness) at t = 20 kyr (Figure 9a). Obviously, the influence of initial soil thickness is greatly reduced as the simulated period is extended. From Figure 9b, we can further conclude that different initial sets of soil thickness will have even less influence on the RMSEs. For instance, at t = 13 kyr, the RMSEs for the selected three scenarios are 0.394 m, 0.394 m, and 0.393 m, respectively. Thus, the effect of initial condition is relatively small for the simulation length used, and hence can be neglected for the Shale Hill catchment.

5.2. The Effect of Soil Transports on Soil Thickness Evolution

[43] The predictive capacity of the derived model has been partly justified above. We found in Figure 6 that soil creep and sediment erosion act quite differently in the process of soil thickness evolution. If one of the optimized parameters, i.e., κ = 10−2.8 or kd = 10−2.6, was fixed separately, by changing another parameter's value, then the RMSE results along the lines AB and CD in Figure 6 present a curve similar to an inverse-parabola (Figure 10). However, with the close range of these two parameters, the change of RMSE along the line AB shown in Figure 10a is relatively larger than that of the RMSE along the line CD shown in Figure 10b. This means that, in general, soil creep exerts a stronger impact on the soil thickness evolution in the study area as compared to sediment erosion.

Figure 10.

Plots of root mean squared error (RMSE) for the lines AB and CD in Figure 6: (a) RMSE plot for AB with κ = 10−2.8 and (b) RMSE plot for CD with kd = 10−2.6.

[44] Two pairs of scenarios (each with the same RMSE, including four points on the lines AB and CD in Figure 6, respectively) were adopted for further analysis (i.e., points a and b on AB, with RMSE = 0.82 m, and points c and d on CD, with RMSE = 0.43 m). Soil thickness fields simulated under these four scenarios were compared with the optimal results of the point o. These five scenarios (a, b, c, d, and o) as well as their coordinates are listed in Table 3. The bias field of soil thickness for the scenarios a, b, c, and d was obtained by using their simulated soil thickness, hsc, minus the predicted soil thickness, hop, under the scenario o (Figure 11). Within each pair of the four scenarios, soil thickness distribution and soil formation are greatly different (even in reverse direction). For example, under the scenario a, the effects of soil creep or sediment erosion dominate a large part of the entire area, and the soil is “peeled off” on hillsides; while deposits on swales and valley floor (Figure 11a) caused the largest inner-difference of soil thickness spatial distribution, and MSE for the scenario a was as high as 1.15 m (Table 3). Though the scenario a has the same RMSE value with that of the scenario b, the latter was assumed a smaller soil creep diffusion coefficient, which resulted in soil particles being deposited on hillsides. Compared to the scenario a, soil thickness tends to be distributed uniformly under the scenario b with a smallest value of MSE (0.08 m). It is obviously that soil formation and movement are more active under the scenario a, of which the values of the average soil depth (ASD) and the mean cumulative bias (MCB) are both larger than that under the scenario b (Table 3). In the scenarios c and d, we found similar distribution patterns as the scenarios a and b. Compared with a and b, soil thickness variations of c and d against the optimal results (Figure 7) were small, and MCBs are as small as about 1 cm, and MSEs and ASDs are also very close to the values of the scenario o (Table 3).

Table 3. Soil Depth Simulation Results Under the Five Scenarios (See the Text and Figure 6 for Details) Including the Optimized Results (Scenario o) for the Shale Hills Catchment
ScenariosaLog (κ)Log (kd)MSEb (m)ASDc (m)MCBd (m)
  1. a

    The scenarios a, b, c, d, and o are related five points in Figure 6 and their coordinates are listed in second and third column of this table.

  2. b

    MSE is the mean square error.

  3. c

    ASD is the average soil depth predicted under different scenarios.

  4. d

    MCB is the mean cumulative bias, MCB = ∑(hsc − hop)/n, where hsc is soil depth predicted under the scenarios ad, hop is soil depth predicted according to optimized parameters (i.e., the scenario o), n is number of grid cells within a catchment.

a−2.8−2.11.150.960.17
b−2.8−4.00.080.74−0.05
c−2.1−2.60.650.800.01
d−3.5−2.60.450.790
o−2.8−2.60.500.790
Figure 11.

Bias fields of soil thickness for the scenarios a, b, c, and d as compared to the scenario o (see Figure 6 and the text for more details). On the color bars, hsc − hop is the soil thickness bias between the scenarios a, b, c, or d, and o, where hsc is soil thickness predicted under the scenarios a–d, and hop is soil thickness predicted according to the optimized parameters (i.e., scenario o).

5.3. The Influence of Terrain Factors on Soil Thickness Evolution

[45] As indicated in section 3, spatial distribution of soil thickness depends heavily on terrain relief and shape. How does terrain affect soil thickness distribution in a natural catchment? In the following, we adopted four commonly used terrain factors, i.e., slope (Figure 5c), curvature (Figure 5d), upstream contributing area (Figure 5e), and topographical wetness index (TWI) (Figure 5f) for understanding better these effects. In Figure 12, terrain factors were compared with soil thickness at the 106 observed points as well as the 13 kyr simulated results at those points under the optimal scenario for the study catchment. As expected, all the factors affect the soil thickness to some extent. Generally, soil thickness in places of overall convergent, mild or flat slope, or with a larger contributing area and higher TWI tend to be thicker. From the correlation coefficients of each terrain factor, curvature is the most significant one among the four, while slope is the weakest in correlating with soil thickness. We found that the simulated and observed curves of terrain factors versus soil thickness have similar trends (despite the scattering of the points), which, in turn, further justified our derived model. For example, in Figures 12a and 12b, both the simulated and observed curves show a logarithmic trend, while the other two curves (Figures 12c and 12d) show a weak linear relationship. We also found that the effects on soil thickness evolution imposed by local slope or upstream contributing area on its own are relatively weak, as can be seen from Figures 12b and 12c. That is to say, areas with a similar slope (e.g., relatively flat areas on the ridge as well as in the swales or valley floor) would not necessarily have a similar soil thickness; rather, landscape position is important as it determines the upstream contribution areas. On the other hand, in the relatively larger contributing areas (e.g., toeslope areas on a hillslope), there may not be a large amount of soil deposition if local slope is steep. Hence, TWI exhibits a more significant impact on soil thickness, as shown in Figure 12d. Furthermore, curvature that reflects terrain relief and shape has the most significant impact on soil thickness evolution. Overall, soil thickness evolution is a complex process and cannot be interpreted simply by using empirical relationship between single terrain factor alone and soil thickness.

Figure 12.

Relationships between soil depth and various terrain attributes for observed (OBS) and simulated (SIM) results: (a) curvature versus soil depth, (b) contributing area versus soil depth, (c) slope versus soil depth, and (d) TWI versus soil depth.

5.4. The Influence of Terrain Factors on Prediction Errors

[46] Here, the predicted errors due to terrain factors were further analyzed for the Shale Hills catchment. We first reranked the 106 sample points in a declining order of the absolute errors between the predicted and observed soil thickness. Then, soil thickness along with their terrain attributes was plotted (Figure 13). The trend lines indicate that the predicted errors decrease consistently as the values of terrain factors decline. In the swale and valley areas with a large value of curvature, contributing area, and TWI, the predicted errors are also large. This is because in these areas terrain is relatively flat, a small change in soil thickness will more likely affect surface terrain relief and flow direction, hence the intensity of soil creep and sediment erosion. However, 20–30 cm increase or decrease of soil thickness is less likely to change flow directions and local relief for steep areas like ridges. These partly explain why larger errors occurred on the convergent terrains such as swale or valley floor. However, for a given slope gradient, we cannot tell the specific position or shape of the slope, all of which are important to determine soil thickness distribution. Therefore, though a weak trend (e.g., relative larger slope gradient tends to correspond to larger errors) is shown in Figure 13c, we cannot tell how it will impact the soil thickness. So, here, we can conclude that slope is the weakest factor to influence soil thickness distribution compared with curvature, upstream contributing area, and TWI.

Figure 13.

Ranked series plots of terrain factors in the order of declining predicted errors between measured and predicted soil depth for the 106 sample points: (a) curvature curve, (b) contributing curve, (c) slope curve, and (d) TWI.

6. Conclusions

[47] A simple geomorphic-based analytical model for predicting the spatial distribution of soil thickness in headwater hillslopes or catchments has been derived in this study. This model has been validated through comparing with numerical solutions on nine basic hillslope shapes and with 106 sampling points in the Shale Hills catchment. Our model has displayed the capacity to provide reasonable soil thickness spatial distribution across various hillslopes in the Shale Hill catchment and in other catchments similar to the one under this study, thus providing a practical tool to benefit hydrologists working in distributed hydrologic modeling.

[48] Moreover, we found that soil creep dominated soil thickness evolution in the Shale Hills in the past 13 kyr, as this is a catchment in humid climate with rare occurrence of overland flow. Terrain shape together with position, instead of slope gradient, has most strongly influenced soil thickness distribution. It is revealed that prediction errors at different places within the catchment were also affected by terrain attributes (e.g., larger errors always appear on the convergent and relatively flat terrains, such as swale or valley floor). This is because as the soil becomes thicker in these areas, the local relief and flow direction are more likely to change than that in steeper areas (not applicable to our second assumptions). It is therefore important to note that a catchment should be investigated regarding our model's basic assumptions before our model can be reliably applied for hydrologic modeling elsewhere.

Acknowledgments

[49] This work was supported by the National Natural Science Foundation of China (grants 51190091, 41271040) and by the U.S. National Science Foundation (grant 0725019). We are grateful to Lixin Jin and Lin Ma from the University of Texas at El Paso for their valuable suggestions about soil production in the Shale Hills catchment. We also wish to thank Zhongbo Yu for his kindly review that greatly improved the manuscript.

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