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  • View in gallery

    The DEM of (a) Maimai hillslope, (b) Tulpehocken Creek Watershed, and (c) Walnut Creek Watershed.

  • View in gallery

    The histograms of baseflow at (a) the Maimai hillslope, (b) the Tulpehocken Creek watershed, and (c) the Walnut Creek Watershed.

  • View in gallery

    Subsurface flow calculated based on Eq. (19): (a) with different values of ζ for n = 1 and (b) with different values of the power-law coefficient (n) for the Maimai hillslope. The corresponding flows based on the TOPMODEL formulations with a power-law transmissivity profile, Eq. (5), (dash–dotted line), an exponential transmissivity profile, Eq. (3), (dotted line), and the observations obtained based on the procedure in section 3a(1) are also included.

  • View in gallery

    (a) The subsurface flow for the Tulpehocken Creek Watershed calculated based on the new formulation (i.e., solid line) with n = 3.5 and ζ = 5. Also shown are the corresponding subsurface flows based on the power-law formulation with n = 3.5 (dashed line), and the exponential formulation (dot–dashed line). Both Dgwt and Qobsb (circles) are calculated from observations are included for comparison. (b) As in (a), but for the Walnut Creek Watershed with the subsurface flow calculated from the new formulation (i.e., solid line) with n = 3.5 and ζ = 30.

  • View in gallery

    Comparison of subsurface flows calculated from Eq. (19) with different values of the power-law coefficient (i.e., n) for (a) Tulpehocken Creek Watershed and (b) Walnut Creek Watershed. The corresponding subsurface flows based on Eqs. (5) and (3) are also shown. In the figure, the solid and dashed lines represent the subsurface flows from Eqs. (19) and (5), respectively, for different n values.

  • View in gallery

    Comparison of subsurface flows calculated from Eq. (19) for the Maimai hillslope with different values of ζ for (a) n = 1, (b) n = 3, and (c) as in (b), but for the case with pmin = 0.01 and pmax = 1.0.

  • View in gallery

    Comparison of subsurface flows calculated based on Eq. (19) with different values of ζ for (a) Tulpehocken Creek Watershed and (b) Walnut Creek Watershed.

  • View in gallery

    Comparison of subsurface flows calculated from Eq. (19) based on different synthetic DEMs for the three sites: (a)–(d), Maimai hillslope with n = 1 and ζ = 1.5, n = 1 and ζ = 5, n = 3 and ζ = 1.5 , and n = 3 and ζ = 5, respectively; (e) Tulpehocken Creek Watershed with n = 3.5 and ζ = 5; and (f) Walnut Creek Watershed with n = 3 and ζ = 30. The corresponding subsurface flows based on Eq. (5) are also included. Note that f = 1 represents a case with the original DEM.

  • View in gallery

    (a) Difference between subsurface flow values calculated from Eq. (19) (i.e., new) and Eq. (5) (i.e., power law) for the Tulpehocken Creek Watershed. (b) Difference between subsurface flow values calculated from different synthetic DEMs for the Tulpehocken Creek Watershed. Note that f = 1 represents the case with the original DEM.

  • View in gallery

    Same as Fig. 9, but for the Walnut Creek Watershed.

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A Generalized Subsurface Flow Parameterization Considering Subgrid Spatial Variability of Recharge and Topography

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  • 1 Department of Civil, Structural, and Environmental Engineering, University at Buffalo, State University of New York, Buffalo, New York, and Department of Global Ecology, Carnegie Institution for Science, Stanford, California
  • | 2 Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania
  • | 3 Atmospheric Science and Global Change Division, Pacific Northwest National Laboratory, Richland, Washington
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Abstract

Subsurface flow is an important hydrologic process and a key component of the water budget. Through its direct impacts on soil moisture, it can affect water and energy fluxes at the land surface and influence the regional climate and water cycle. In this study, a new subsurface flow formulation is developed that incorporates the spatial variability of both topography and recharge. It is shown through theoretical derivation and case studies that the power-law and exponential subsurface flow parameterizations and the parameterization proposed by Woods et al. are all special cases of the new formulation. The subsurface flows calculated using the new formulation compare well with values derived from observations at Tulpehocken Creek, Pennsylvania, and Walnut Creek, Iowa. Sensitivity studies show that when the spatial variability of topography or recharge, or both is increased, the subsurface flows increase at the two aforementioned sites and at the Maimai hillslope, New Zealand. This is likely due to enhancement of interactions between the groundwater table and the land surface that reduce the flow path. An important conclusion of this study is that the spatial variability of recharge alone, and/or in combination with the spatial variability of topography can substantially alter the behaviors of subsurface flows. This suggests that in macroscale hydrologic models or land surface models, subgrid variations of recharge and topography can make significant contributions to the grid mean subsurface flow and must be accounted for in regions with large surface heterogeneity. This is particularly true for regions with humid climate and a relatively shallow groundwater table where the combined impacts of spatial variability of recharge and topography are shown to be more important. For regions with an arid climate and a relatively deep groundwater table, simpler formulations, for example, the power law, for subsurface flow can work well, and the impacts of subgrid variations of recharge and topography may be ignored.

Corresponding author address: Dr. Xu Liang, Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, PA 15261. Email: xuliang@pitt.edu

Abstract

Subsurface flow is an important hydrologic process and a key component of the water budget. Through its direct impacts on soil moisture, it can affect water and energy fluxes at the land surface and influence the regional climate and water cycle. In this study, a new subsurface flow formulation is developed that incorporates the spatial variability of both topography and recharge. It is shown through theoretical derivation and case studies that the power-law and exponential subsurface flow parameterizations and the parameterization proposed by Woods et al. are all special cases of the new formulation. The subsurface flows calculated using the new formulation compare well with values derived from observations at Tulpehocken Creek, Pennsylvania, and Walnut Creek, Iowa. Sensitivity studies show that when the spatial variability of topography or recharge, or both is increased, the subsurface flows increase at the two aforementioned sites and at the Maimai hillslope, New Zealand. This is likely due to enhancement of interactions between the groundwater table and the land surface that reduce the flow path. An important conclusion of this study is that the spatial variability of recharge alone, and/or in combination with the spatial variability of topography can substantially alter the behaviors of subsurface flows. This suggests that in macroscale hydrologic models or land surface models, subgrid variations of recharge and topography can make significant contributions to the grid mean subsurface flow and must be accounted for in regions with large surface heterogeneity. This is particularly true for regions with humid climate and a relatively shallow groundwater table where the combined impacts of spatial variability of recharge and topography are shown to be more important. For regions with an arid climate and a relatively deep groundwater table, simpler formulations, for example, the power law, for subsurface flow can work well, and the impacts of subgrid variations of recharge and topography may be ignored.

Corresponding author address: Dr. Xu Liang, Department of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, PA 15261. Email: xuliang@pitt.edu

1. Introduction

Subsurface flow is an important hydrologic process and a key component of the water budget. Inaccurate estimation of subsurface flow can greatly alter the state of soil moisture, which plays an important role in the global energy and water balances in the land–atmosphere system (e.g., Koster and Suarez, 1996, 1999, 2001; Entekhabi et al. 1999; Dirmeyer 2000; Dirmeyer et al. 2000; Koster et al. 2000; Pal and Eltahir 2001). For example, Dirmeyer (2000) showed that the inclusion of reasonable soil moisture could significantly improve the simulation of water and energy fluxes globally and regionally, which would affect the growth of the planetary boundary layer, the generation of clouds, and the recirculation of precipitation.

In hydrological studies, subsurface flow from a watershed is often characterized by a recession curve. Tallaksen (1995) reviewed different ways of quantifying subsurface flow (Qb) recession rates and pointed out that the recession curves could be represented by a general relationship as follows:
i1525-7541-9-6-1151-e1
where S is the storage term and K and p are constants. Depending on whether p is greater or less than unity, the recession curve is concave upward or downward. Tallaksen (1995) suggested that p is large in typical catchments, and is rarely below unity. The form of Eq. (1) is consistent with theoretical and empirical studies of subsurface flows at hillslope and watershed scales (Zecharias and Brutsaert 1988; Troch et al. 1993; Brutsaert 1994; Duffy 1996; Brutsaert and Lopez 1998; Eng and Brutsaert 1999).

However, a nonlinear function has not been widely used in subsurface flow parameterizations in land surface models. For simplicity, the common practice in land surface schemes is to apply a gravity drainage (i.e., free drainage) condition (Yeh and Eltahir 2005a), where the drainage flux is equal to the unsaturated hydraulic conductivity at the bottom of the lowest soil layer, or a linear function of the unsaturated conductivity with an empirical coefficient accounting for factors such as topographic slope and amplitude, or the location of bedrock. Indeed, 13 out of the 16 land surface schemes that participated in the Program for the Intercomparison of Land-Surface Parameterization Schemes phase 2(c) [PILPS-2(c)] applied the gravity drainage condition for subsurface runoff generation (Lohmann et al. 1998). Although they produced reasonable results in the Red–Arkansas River basin in PILPS 2(c), Lohmann et al. (1998) noted that in humid groundwater-dominated basins, they would likely fail to reproduce runoff timing because of their lack of physical basis.

The nonlinear behavior of subsurface flow is further supported by Eltahir and Yeh (1999) and Liang et al. (2003), who found an asymmetric response of groundwater level to floods and droughts. Marani et al. (2001) suggested that the nonlinear behavior of subsurface flow might be key to the asymmetric response. Hence, Yeh and Eltahir (2005b) proposed a threshold-type nonlinear equation fitted to observed data to represent subsurface flow at the local scale. The equation can be integrated over the probability distribution function (PDF) of groundwater table depth to obtain a grid-scale subsurface flow parameterization. Other nonlinear functions have also been used to represent subsurface flow in land surface models. These include the ARNO formulation in the Variable Infiltration Capacity (VIC) model (Liang et al. 1994) and the topography-based hydrological model (TOPMODEL) formulation used in the Surface Energy and Water Balance (SEWAB) land surface model (Warrach et al. 2002). Huang and Liang (2006) proposed an alternative formulation suitable for ungauged basins. However, these parameterizations all have their own limitations. For example, the parameterization proposed by Yeh and Eltahir (2005b) and the ARNO formulation in the VIC model are empirically based and thus lack physical grounds. The TOPMODEL-based parameterizations, including the one proposed by Huang and Liang (2006), are subject to a few assumptions, as discussed in section 2a of this paper.

Therefore, a new subsurface flow parameterization is proposed in section 2 to overcome the limitations of previous formulations used in land surface models. The new parameterization makes use of the observed or parameterized dynamic groundwater table depth and takes into account the spatial variability of topography and recharge. The relationship of the new formulation with other parameterizations is also discussed. In addition, the TOPMODEL parameterization is briefly reviewed in section 2 as it provides the basis for the new method. In section 3, the new formulation is applied to three sites and compared to observations. Sensitivity analysis is also conducted to investigate the properties of the new formulation. Conclusion and a discussion are given in section 4.

2. Development of the new subsurface flow formulation

a. Brief review of the TOPMODEL parameterizations

Topographic wetness indices have been used in the past decades to estimate the hydrologic response of a catchment to runoff (Beven et al. 1984; O’Loughlin 1986; Burt and Butcher 1986; Moore et al. 1988a, b; Moore et al. 1993a, b; Barling et al. 1994; Ambroise et al. 1996). Among these indices, the topographic index in the topography-based hydrological model (TOPMODEL) was developed based on the kinematic approximation with a quasi-steady-state assumption. It has been shown to be a good approximation of the subsurface flow systems in a number of applications (e.g., Beven et al. 1984, 1995; Beven 1997) and because of its simplicity, has been adopted by some land surface models to represent subsurface flow.

In TOPMODEL, soil and topographic properties are used to describe the hydrological similarity of different points in a catchment. Kirkby (1975) introduced the topographic index, λ = ln(a)/tanβ, to describe the hydrological similarity over a watershed, where a is the contributing area to a point per unit contour length, and β is the local surface slope angle. The basic assumption is that all points in a catchment with the same topographic index will respond to rainfall events in a hydrologically similar way (Beven 2000). The original TOPMODEL also invoked other assumptions including: 1) approximating the saturated zone dynamics by successive steady-state representations; 2) assuming a homogeneous recharge rate to the water table over the catchment; 3) approximating the hydraulic gradient of the saturated zone by the local surface slope tanβ; and 4) assuming that the distribution of downslope transmissivity is an exponential function of storage deficit, which can be approximated by the depth to the water table. The function can be expressed as
i1525-7541-9-6-1151-e2
where T0 is the saturated lateral transmissivity, z is the local storage deficit or depth to the water table, and m is a scaling parameter describing the transmissivity profile with depth. Based on these assumptions, the subsurface baseflow per unit area (L/T) from TOPMODEL for a watershed with an area of A is given by
i1525-7541-9-6-1151-e3
where z is the mean storage deficit or mean depth to water table over the catchment; Λ1 is the spatial average of the logarithmic topographic index defined as Λ1 = A−1Aλ ds; and m is assumed to be homogeneous over the watershed. The choice of an exponential transmissivity profile was based on observations in upper soil layers (Beven 1984) and its attractive analytical expression. However, it could have a significant impact on the shape of the simulated subsurface flow recession curves (Ambroise et al. 1996). Duan and Miller (1997) and Iorgulescu and Musy (1997) used a generalized power function to represent the transmissivity profiles, where
i1525-7541-9-6-1151-e4
and M is the maximum storage deficit, and n is a nondimensional parameter between 1 and infinity. When n goes to infinity, Eq. (4) approaches the original exponential profile expressed in Eq. (2) with m = M/n.
Incorporating the transmissivity profile of Eq. (4) into the framework of TOPMODEL, the subsurface flow (L/T) from a watershed then becomes
i1525-7541-9-6-1151-e5
where Λ2 = A−1A(a/tanβ)1/n ds is the spatial average of the topographic index to the power 1/n over a watershed. It can be seen that Eq. (5) falls into the form of Eq. (1), with 1 − z/M being the storage term, and n and T0n2 being the two different constants.

b. A new subsurface flow formulation

In this section, a new subsurface flow parameterization is proposed that makes use of the dynamic groundwater table depth from either observations or model simulations. The new formulation also accounts for spatial variabilities of topography and recharge.

1) A subsurface flow parameterization incorporating spatial variability

Woods et al. (1997) proposed a topographic index for use in regions with rapid, spatially variable subsurface runoff. They suggested that the pattern of recharge to the saturated zone is controlled by both the prestorm watershed-average wetness and the pattern of saturated zone thickness, based on evidence from field measurements. Three major assumptions distinguish this index from others: 1) the soil lies above an impermeable layer, 2) the saturated hydraulic conductivity does not vary with depth, and 3) the pattern of recharge can be estimated using a simple nonlinear function of both local and watershed-average saturated zone thickness.

Based on this index, a subsurface flow formulation was proposed (Woods et al. 1997) that is capable of capturing the pattern of both subsurface runoff and saturated zone thickness. From field observations, Woods and Rowe (1996) and Woods et al. (1997) suggested that the spatially variable recharge rate in a watershed can be represented as
i1525-7541-9-6-1151-e6
where P is the average net rainfall rate at the ground surface, zi is the depth from the surface to the water table at point i, D is the soil depth, and ω is a function defined as
i1525-7541-9-6-1151-e7
where z is the average water table depth over the watershed, and ζ is an empirical constant.

For time scales shorter than the time it takes for rain to infiltrate to the water table, Eq. (6) does not apply because it neglects vertical transport processes. Equation (6) shows that the ratio of net rainfall to recharge and the time it takes to recharge the saturated zone at a location i depend on both the watershed-scale (ω) and local [1− (zi/D)] wetness conditions. It provides a model of the changing spatial patterns of recharge, without explicitly considering the processes of unsaturated flow and evapotranspiration.

In deriving Eq. (6), Woods et al. (1997) assumed that the transient behavior of the unsaturated soil moisture being replenished and depleted repeatedly is isolated within the unsaturated zone, and the influence of the unsaturated zone on recharge can be described using only the water table variables zi and z, without explicitly modeling the conditions in the unsaturated zone. These assumptions are consistent with observations (Woods and Rowe 1996) that when a watershed is wetter overall (i.e., smaller z), more subsurface runoff is generated everywhere, and that in wet areas where zi/D is smaller, relatively more subsurface runoff is generated than dry areas.

With Eq. (6), the spatially averaged recharge rate, ri, per unit area from all points upslope of a unit contour width at i can be expressed as
i1525-7541-9-6-1151-e8
where zi is the average zi over all points upslope of location i. By assuming a linear transmissivity profile and a uniform soil depth over the study area, Woods et al. (1997) derived the following expression for subsurface flow per unit contour length at location i:
i1525-7541-9-6-1151-e9
where ksx,i and βi are the saturated lateral hydraulic conductivity and the local surface slope angle at location i, respectively. The linear transmissivity profile is selected to facilitate theoretical derivation of the subsurface flow.

Although Eq. (9) has been shown to be appropriate for the Maimai hillslope site examined in Woods et al. (1997), the linear transmissivity profile is generally not a valid assumption for most watersheds. Based on recession curve analysis in previous studies (e.g., Tallaksen 1995; Zecharias and Brutsaert 1988; Brutsaert and Lopez 1998; Beven 1984), the transmissivity profile usually takes a nonlinear form such as the exponential profile in TOPMODEL, or more generally, a power-law profile as shown in Ambroise et al. (1996). Moreover, the relationship between regional subsurface flow and the storage term (e.g., aquifer depth) is nonlinear (e.g., Tallaksen 1995; Eltahir and Yeh 1999; Marani et al. 2001), it cannot be captured with a linear transmissivity profile assumption. In what follows, we propose a new topographic index that overcomes this limitation.

Instead of the linear transmissivity profile assumed in Eq. (9), we use a general power function, as shown in Eq. (4), to describe the transmissivity and the resulting subsurface flow at location i as follows:
i1525-7541-9-6-1151-e10
where n is a nondimensional parameter that ranges from 1 to infinity. Note that by using Eq. (10), we assume that the maximum storage deficit M in Eq. (4) is equal to the total soil depth D. Also, we assume that the modeling time step is long enough so that at each time step, a steady state could be reached. Thus, the subsurface flow at location i is given by
i1525-7541-9-6-1151-e11
where ai(t) is the contributing area to point i per unit contour length (L) at time t (Barling et al. 1994), and ri is defined by Eq. (8). Combining Eqs. (10) and (11), we obtain
i1525-7541-9-6-1151-e12
which can be further expressed, based on appendix A, as
i1525-7541-9-6-1151-e13
where Li = ai(t)/sinβi, c = c′(n − 1 + ω)/n, and g = c′(1 − ω)/n, where c′ = [P/(ksx,i D)]1/n.
From Appendix A, Eq. (13) can be simplified to a recursive form as follows:
i1525-7541-9-6-1151-e14
where
i1525-7541-9-6-1151-e15
and Hi = mean(Hj), and j is the collection of all the points upslope of i. If there are no points upslope of i, then Hi = 0, and Hi = L1/ni.
Integrating Eq. (14) over the watershed with an area A, we obtain
i1525-7541-9-6-1151-e16
i1525-7541-9-6-1151-e17
Let ai be the total drainage area upslope of i, and ai(t) be the dynamic contributing area upslope of i, the subsurface flow at point i is then given by
i1525-7541-9-6-1151-e18

Detailed derivation of the new topographic index and subsurface flow parameterization is shown in appendix A. The effective lateral saturated hydraulic conductivity is ksx and ksx = αks, where ks is the effective vertical saturated hydraulic conductivity, α is an anisotropic factor between ksx and ks. Also, ai(t) and ai should be different when a dynamic contributing area is considered (e.g., Beven 1997; Barling et al. 1994). This is a complicated issue that needs to be investigated separately. In this study, we assume that ai(t) = ai to simplify the calculation.

Applying Eq. (18) to the outlet, the subsurface flow from the watershed can be calculated as
i1525-7541-9-6-1151-e19
Equation (19) is a more general subsurface flow formulation that is applicable to a wider range of conditions compared to Eqs. (3) and (5) in this paper, and Eq. (11) in Woods et al. (1997). The spatial variability of subsurface flow is considered in Eq. (19) by taking into account the spatial variability of topography, recharge, and water table status. The subsurface flow rate of the entire watershed is given by Eq. (19), whereas at any given point in the watershed, it can be obtained from Eq. (18). When g goes to 0, the control of spatial variability of recharge and water table status on subsurface flow vanishes, H in Eq. (19) becomes Λ2 in Eq. (5), and Eq. (19) approaches Eq. (5). When n = 1, Eq. (19) becomes Eq. (11) in Woods et al. (1997) with a linear transmissivity profile, except that (i) we define the water table depth z as being measured downward from the surface, rather than upward from the base of the soil layer as in Woods et al. (1997); and (ii) while Woods et al. (1997) defined a similarity variable Hi so that zi = cHi, we have taken a dimensionless approach and defined [1 – (zi/D)] = cHi [i.e., Eq. (14)]. When g goes to 0 and n goes to infinity, Eq. (19) is reduced to Eq. (3), the original TOPMODEL formulation for subsurface flow, which does not consider the spatial variability of recharge and water table status, and assumes an exponential transmissivity profile. Duan and Miller (1997) and Iorgulescu and Musy (1997) provided detailed derivations to show that when n = ∞, the power-law transmissivity profile in Eq. (4) reduces to the exponential transmissivity profile in Eq. (2), and therefore Eq. (5) becomes Eq. (3).

2) Calculating the subsurface flow

In Eq. (19) the impacts of spatial variability on subsurface flow are represented by both the g and H terms. Variables z, P, H, and g are related to the “wetness” of a watershed. Derivations in appendix B show that the term gH in Eq. (19) can be uniquely determined by the mean depth of the groundwater table through
i1525-7541-9-6-1151-e20
where ω is a function of z/D as defined in Eq. (7), and the net rain rate P is also related to z through
i1525-7541-9-6-1151-e21

Equations (20) and (21) suggest that once z is known, none of the variables P, H, and g are independent variables. To calculate subsurface flow through Eq. (19), one can prepare a table that lists the relationships between g, H, and gH. Assigning different values of g, Hi can be calculated based on the DEM of the watershed through Eq. (15). Averaging Hi over the watershed yields the corresponding H for the entire watershed. From Eq. (15), it can be seen that Hi increases as g increases. Consequently, gH is a monotonic increasing function of g. For a given value of z, gH can be calculated through Eq. (20). Given gH, the corresponding value of g and H can be interpolated from the table, and the subsurface flow can then be calculated based on g and H through Eq. (19).

For any value of g, the Hi index can be calculated using existing topographic index methods (e.g., Quinn et al. 1991; Costa-Cabral and Burges 1994) with slight modifications. A procedure for calculating the Hi index with n = 1 is given in appendix D of Woods et al. (1997), which is also applicable for calculating the general Hi index proposed in this study, except that Eq. (15) should be used in the calculation. For completeness and consistency, the procedure for estimating Hi is provided in appendix C of this paper. The H index only needs to be computed once for each watershed or region based on DEM information. Therefore, the new subsurface flow formulation [i.e., Eq. (19)] can be easily implemented into hydrologic models or land surface models.

3. Model evaluation and sensitivity study

To understand the behavior of the new subsurface flow formulation, we applied it to three sites with different climate and hydrologic conditions. The simulated subsurface flow is evaluated using observations and compared against the TOPMODEL formulations with the transmissivity following an exponential and a power-law form, respectively, and the formulation of Woods et al. (1997). Sensitivity experiments were performed to understand the response of the new formulation to different parameters.

a. Site description

1) The Maimai hillslope

The Maimai hillslope is located in catchment M8 of the Maimai study area in Powerline Creek, which is in the headwaters of Mawheraiti River near Reefton in northwestern South Island, New Zealand (42o05′S, 171o48′E). This study area has provided valuable observations for a number of hydrological studies, including studies of the mechanisms for subsurface flow generation (e.g., Woods et al. 1997; Woods and Rowe 1996; McDonnell et al. 1991; McDonnell 1990; Sklash et al. 1986).

The climate of the area is superhumid with adequate rainfall in all seasons (Salinger et al. 2004; Garnier 1950). The average annual rainfall between 1975 and 1987 is 2450 mm, with less than 2 days of snow per year. Rain tends to be evenly distributed between April and December. Soils are shallow (with a typical depth of 0.6 m) and highly permeable (with saturated hydraulic conductivity on the order of 250 mm h−1) with dense vegetation.

The hillslope used in this study is located on the left bank of the main stream draining to catchment M8. It has an area of 0.2 ha. The local relief is about 40 m with short (60 m) and steep (30°) slope. Figure 1a shows the hillslope at the Digital Elevation Model (DEM) data resolution of 0.17 m. Thirty subsurface flow troughs were installed end-to-end across the base of the hillslope to measure subsurface flow. Rain and trough flow measurements were made continuously at the slope from January 1993 to July 1994 at a 10-min time step. In the period of December 1993 to February 1994, 16 tensiometers were installed at the slope in 6 “nests” (either two or three tensiometers at different depths at the same locations) to measure changes in soil moisture conditions at a 15-min time step. More detailed description of this site can be found in Woods and Rowe (1996) and Woods and Sivapalan (1995). As discussed in section 2a(1), our assumption is that the modeling time step is long enough so that at each time step, a steady state could be reached. Therefore, both the trough flow data and the tensiometer data were aggregated to a 6-h time step, as recommended in Woods et al. (1997), to facilitate the comparison between the observed and modeled subsurface flow.

For each 6-h interval, the total runoff from the slope was obtained by summing up the runoff from the 30 troughs across the bottom of the hillslope. The depth to groundwater table at each of the six locations of the tensiometer nests was inferred based on the measurements from the deepest tensiometer, which was installed at the base of the mineral soil layer (Woods and Sivapalan 1995). The mean depth to the groundwater table at the hillslope was then calculated by averaging over the groundwater depth of the six locations.

To allow comparison of the observed runoff with the calculated subsurface flow, the expectations of the groundwater table depth and the corresponding expected value of subsurface flow are used in this study. Total Q during the recession periods are estimated by calculating the time derivative:
i1525-7541-9-6-1151-eq1
where Δt is the time interval between streamflow measurements (6 hours in this case), and i is the index of the time series. All Qis with dQi/dt < 0 (i.e., recession) are added to a series called Q′. From the histogram of Q′ (Fig. 2a), we calculate the 95th percentile Q0.95. Flow values of Q′ higher than Q0.95 are removed to discard flow values that are contributed by surface flow during recession. The remaining Q′s form a new series called Qobsb and its corresponding Dgwt were grouped into pairs. The data pairs (Qobsb, Dgwt) were divided into nb bins based on the histogram of Dgwt. The mean values of Dgwt and Qobsb within each bin were then calculated and denoted as Dgwt(j) and Qobsb(j), where j = 1, . . . , nb and nb = 25 in this study. These calculations provide the expectations of observed subsurface flows Qobsb(j) and groundwater table Dgwt(j) used to test the new subsurface flow formulation.

Figure 2a shows the histogram of Qobsb at Maimai. It can be seen that the number of observations available at this site is small because the tensiometer and the trough flow measurements only overlap over a short period of time (i.e., 23 December 1993–8 February 1994). These data are included mainly for the completeness of our study. We also acknowledge the new method proposed by Rupp and Selker (2006) for conducting a more accurate recession analysis in which the time increment Δt varies. Here we keep Δt to be a constant because the streamflow data available at the site are at a 6-h time step, which is too coarse for an analysis with variable Δt.

2) Tulpehocken Creek watershed

The Tulpehocken Creek watershed, Pennsylvania has a drainage area of 172 km2 with an elevation ranging from 92 to 422 m. The resolution of DEM used for this site is 10 m from USGS (Fig. 1b). Daily observations of precipitation, streamflow, and groundwater table depth are available for a well inside the watershed from 1 October 1991 to 30 September 1998. Mean annual precipitation and streamflow are 1160 and 610 mm, respectively. For consistency with Liang et al. (2003), the same saturated hydraulic conductivity and soil depth, ks = 951.00 mm day−1 and D = 5 m, are used. The procedure described in section 3a(1) was also employed here to estimate the expectation of groundwater table depth and the corresponding expected value of subsurface flow. Because the data are at a daily time step, the method in Rupp and Selker (2006) was not used. Figure 2b shows the histogram of Qobsb of the Tulpehocken Creek watershed.

3) Walnut Creek watershed in Iowa

The Walnut Creek watershed is located in central Iowa with a total area of 52.2 km2. Seven groundwater monitoring wells were installed in this watershed from an upland recharge position to Walnut Creek at the valley bottom (Zhang and Schilling 2004). DEM at ⅓ arc-second (approximately 10 m) resolution is available from the U.S. Geological Survey (USGS) National Elevation Dataset (NED). This watershed ranges from 241 to 293 m in elevation. Figure 1c shows the DEM of the Walnut Creek watershed and locations of the groundwater monitoring wells.

Daily observations of precipitation, groundwater levels, and total runoff are available from January 1996 to March 2000. Zhang and Schilling (2004) estimated daily baseflow based on the total runoff for the Walnut Creek watershed using the hydrograph separation program (HYSEP; see Sloto and Crouse 1996). Their estimated baseflow time series and averaged groundwater level at the monitoring wells are used in this study. Again, we did not employ the method by Rupp and Selker (2006) for the recession analysis because of the daily time step of the runoff data.

The expectation of groundwater table depth and the corresponding expected value of subsurface flow are calculated following the procedures described in section 3a(1). However, since the baseflow Qobsb has been estimated by Zhang and Schilling (2004), we only need to estimate the expected groundwater table for the Walnut Creek watershed. The histogram of Qobsb for the Walnut Creek watershed is shown in Fig. 2c. Schilling et al. (2002) reported that most of the soils within the watershed are silty clay loams, silt loams, or clay loams formed in loess and till. In this study, ks = 172.8 mm day−1 and D = 3.5 m are used.

Because of the limited number of groundwater wells, the spatial distributions of groundwater table are not well represented at the Tulpehocken or Walnut Creek watersheds. The average groundwater table calculated from the available groundwater monitoring wells is used to approximate the average groundwater table of the study area. Although this may introduce errors, it represents the best estimate based on available data.

b. Case studies at the three sites

To evaluate the performance of the new subsurface flow formulation, we first applied it to the extensively studied Maimai site. Figure 3a shows the subsurface flow from Eq. (19) when n = 1 is used. The corresponding subsurface flow obtained from Eq. (5) with n = 1 (i.e., the TOPMODEL formulation with a power-law transmissivity), and Eq. (3) (i.e., the TOPMODEL formulation with an exponential transmissivity) are also included for comparison.

Comparing Eqs. (19) and (5), we can see that their main difference is determined by the term (1 + gH)/H, where g and H are functions of ω and the topography of the study site. From Eq. (7), we see that the shape of ω is controlled by the parameter ζ. Thus, the impacts of ζ on subsurface flow generation are introduced through (1 + gH)/H. Figure 3a shows the impacts of ζ on the subsurface flow when n = 1. When g = 0, that is, when the spatial variability of recharge and water table status is not considered, the topographic index H becomes Λ2 in Eq. (5), and Eq. (19) is reduced to Eq. (5), which assumes a uniform recharge rate.

The function ω describes how the overall wetness of the study area affects the spatial variability of recharge rate, as controlled by ζ, which is close to 0 if the recharge is relatively uniform, and vice versa. Figure 3a shows how the spatial variability of recharge impacts the generation of subsurface flow for a given topographic setting for the case of n = 1 in Eq. (19). When ζ increases, the degree of spatial variability of recharge increases. Therefore, the subsurface flow estimated by Eq. (19) deviates more significantly from the widely used Eq. (5) (i.e., the TOPMODEL formulation with a power-law transmissivity) for the same n.

It is worth mentioning that Fig. 3a here is essentially the same as Fig. 6d in Woods et al. (1997), except that we used linear, instead of logarithmic, scale and a wider range of ζ value to show the impacts of ζ on subsurface flow.

The subsurface flow based on the TOPMODEL formulation with an exponential transmissivity profile [i.e., Eq. (3)] is also included in Fig. 3a for comparison. In that formulation, the subsurface flow does not depend on n or ζ. The parameter m in Eq. (3) is set to D/8 in this study, where D is the soil depth, and an arbitrarily large value of 8 is used in the denominator, as we showed in section 2.1 that m = M/n when n goes to infinity. A larger n results in a larger curvature for the subsurface flow curve. In TOPMODEL applications, m is usually calibrated.

Figure 3b compares the subsurface flow from Eq. (19) for different values of n. The subsurface flows from Eqs. (5) and (3) are also shown. As n increases in Fig. 3b, the nonlinear response of the subsurface flow based on the new formulation [i.e., Eq. (19)] to the storage term [i.e., 1 − (z/D), which describes the overall wetness of the watershed or study area] becomes more evident. In other words, for the same change in water table depth Δz, the change in subsurface flow is greater when 1 − (z/D) is higher, and vice versa, at the Maimai site. This behavior holds also for the subsurface flow represented by Eq. (5) (i.e., the TOPMODEL formulation with a power-law transmissivity). However, for the same value of n, the subsurface flow obtained from Eq. (19) is always greater than that from Eq. (5) at this site. This suggests that for the same average groundwater table depth, the subsurface flow rate is larger when the spatial variability of recharge is considered at this site. Note that the observed subsurface flows obtained from the procedure in section 3a(1) are also included in Figs. 3a,b. However, because these observations only cover the low flow regime, and the number of points within each bin used to obtain the pairs of the observed subsurface flows Qobsb(j) and groundwater table Dgwt(j) is very small, it would not be appropriate to draw any statistically meaningful conclusions based on these observations. These observations are included for the completeness and consistency of this study, but they are not compared against the modeled flows, as we did for the other two sites.

The new formulation is also applied to the Tulpehocken Creek and Walnut Creek watersheds, which are larger than the Maimai site. A factor α = 10 is used to correct for the difference between the vertical and lateral saturated hydraulic conductivity for both watersheds. Figures 4a,b show the best fits of the subsurface flow based on Eqs. (19), (5), and (3), to observations at the Tulpehocken Creek and Walnut Creek watersheds, respectively. It is encouraging that the new formulation captured the trend better, as shown in the observation. That is, when the groundwater table becomes shallower, the subsurface flow rate generated from the new formulation is higher than those from the TOPMODEL formulations, as the observations suggested. This suggests that by capturing the spatial variability of recharge in the new formulation, interactions between the groundwater table and the land surface may be enhanced to result in higher subsurface flows that are closer to the observations. Hence, the new formulation should be able to capture subsurface flows in the higher flow regimes better than the TOPMODEL formulation with a power-law transmissivity [i.e., Eq. (5)], where a uniform recharge rate is assumed.

Similar to Fig. 3b, Figs. 5a,b compare the subsurface flows from Eqs. (19), (5), and (3) for the watersheds of Tulpehocken Creek and Walnut Creek, respectively. Similar behaviors as shown in Fig. 3b for the Maimai site are also found for the two watersheds. That is, for the same storage level [i.e., the same value of 1 − (z/D)], the subsurface flow calculated from Eq. (19) is higher than that based on Eq. (5) with the same n at both watersheds. An exception, however, is noted in the Walnut Creek watershed (Fig. 5c); when n = 2, Eq. (5) results in slightly higher subsurface flow than that from Eq. (19) when 1 − (z/D) is greater than 0.5.

These results suggest that spatial variability in recharge generally enhances subsurface flow rates for the same averaged groundwater table depths. Conceptually, spatial variability of the recharge may increase interactions between the groundwater table and the land surface. For example, when the groundwater table and land surface intersect, the flow paths of subsurface flow in the soil layer may be shortened, as they are replaced by the flow paths over the land surface. Therefore, through interactions between the groundwater table and the land surface, subsurface flow can reach the stream faster and result in greater subsurface flow rates for the same averaged groundwater table depths. However, exceptions are possible, depending on the topography of the watersheds, where the interactions between groundwater table and land surface may not increase the subsurface flow rates.

From Figs. 3b, 5a and 5b, the nonlinear dependence of subsurface flow on groundwater table depth or storage is evident. This nonlinear property has been discussed by previous studies (e.g., Tallaksen 1995; Eng and Brutsaert 1999; Marani et al. 2001), and may play an important role in determining subsurface runoff during droughts and floods.

In summary, the expression (with n = 1) proposed by Woods et al. (1997) results in a linear relationship between Qb and [1 − (z/D)], and it has been reported to work well in reproducing the temporal changes in spatial flow patterns in the Maimai hillslope site (Woods et al. 1997). The formulation in Woods et al. (1997) is a special case of the new subsurface flow parameterization. The TOPMODEL formulation, with an exponential distribution for transmissivity [i.e., Eq. (3)], considers nonlinear response but it is subject to the assumptions and limitations discussed in sections 1 and 2. It does not work well for the sites examined in this study.

It is worth mentioning that in Fig. 3a (and later in Figs. 6a and 8b) the subsurface flow decreases as the catchment gets wetter for cases with high values of ζ (i.e., ζ = 5) and n = 1. That is when a linear transmissivity profile is assumed, i.e., when Eq. (19) reduces to Eq. (11) in Woods et al. (1997), the subsurface flow may decrease when the groundwater table is high, which may not be realistic. This unrealistic behavior may be caused by the unrealistic combination of large value of ζ and n = 1 for the Maimai site. In other words, for the case of n = 1, there is a limitation on the value of ζ at this site. This is because ζ and the topography of the study site affect g and H, which are closely related to the subsurface flow of Qb.

c. Sensitivity analysis

1) Sensitivity to spatial variability of recharge and topography

This section investigates the sensitivity of the new subsurface flow parameterization to spatial variability of topography and recharge. Figure 6 shows the sensitivity of subsurface flow to spatial variability of recharge, as controlled by ζ. Note that a small ζ indicates a recharge close to a uniform case, and a large ζ implies a spatially variable recharge over the watershed. When the transmissivity shape parameter n increases, the spatial variability of the recharge has less impact on subsurface flow at the Maimai hillslope site (see Figs. 6a,b). Similar comparisons between subsurface flows calculated from the new formulation with different values of ζ for the Tulpehocken Creek and Walnut Creek watersheds are shown in Fig. 7. The Tulpehocken Creek and Maimai hillslope sites show similar sensitivity to ζ, but the sensitivity at the Walnut Creek site is opposite when n = 2. These results suggest that the sensitivity of subsurface flow to ζ depends also on other factors, as expected. Furthermore, Figs. 6 and 7 show that the spatial variability of recharge affects the subsurface flow Qb more significantly when 1 − (z/D) is between 0.5 and 1. When 1 − (z/D) approaches 1, Qb tends to converge to its corresponding maximum Qb for the same n with different ζ values. In other words, the spatial variability of recharge, as controlled by ζ, has minimal impact on Qb when the groundwater table approaches the surface [i.e., 1 − (z/D) approaches 1]. The same is true for Qb when it is close to zero. In addition, similar behaviors can be observed, as expected, in Fig. 5 between the new formulation and the TOPMODEL formulation with a power-law transmissivity [i.e., Eqs. (19) versus (5)] for the same n values, since Eq. (5) is a special case of Eq. (19) when spatial variability of recharge is ignored.

To investigate the impacts of spatial variability of topography on subsurface flow, new DEMs with different degrees of spatial variability are generated by multiplying the original DEM of each point by a factor f that ranges from 0.5 to 2. Values of f > 1 yield DEMs with larger spatial variability of topography, and vice versa.

Figures 8a–d show the sensitivity of subsurface flow to topography at the Maimai site, based on Eq. (19), with different sets of parameters for ζ and n, including the best set of parameters, ζ = 1.5 and n = 1, according to Woods et al. (1997). It can be seen that for the same n and ζ, the subsurface flow increases with increasing variability of topography (i.e., with larger f ). Figures 8e,f also show increasing subsurface flows at the Tulpehocken Creek and Walnut Creek watersheds, respectively, with increasing spatial variability of topography for Eq. (19). Furthermore, Fig. 8 shows that with increasing spatial variability of topography, subsurface flow also increases for the TOPMODEL formulation with a power-law transmissivity [i.e., Eq. (5)]. Marani et al. (2001) offers a possible explanation for why subsurface flow may increase with larger spatial variability of topography. They argued that the mean flow path for groundwater to reach the streams decreases with increasing spatial variability of topography. Figure 8 shows, in our case, that Eq. (19) is more sensitive to the variability of topography than Eq. (5). In other words, when n and ζ are fixed, there is a larger difference between the subsurface flow based on Eq. (19) and the TOPMODEL formulation with a power-law transmissivity that assumes no spatial variability in recharge [i.e., Eq. (5)]. This suggests that the spatial variability of recharge and topography may combine to result in greater interactions between the groundwater table and the land surface that further reduce the flow path. In this way, for the same mean groundwater table depth and topographic variability, greater subsurface flow is generated for Eq. (19) than for Eq. (5).

Note that in Fig. 8f, subsurface flow from Eq. (19) decreases slightly when 1 − (z/D) approaches 1 for f = 2. This is related to numerical calculation. When ω goes to 1, and g goes to 0, the term [(1 + gH)/H]n in Eq. (19) becomes very small, especially for watersheds with relatively large H and n, such as the Walnut Creek watershed. The smaller values of g and [(1 + gH)/H]n could introduce numerical errors to the topographic index obtained from the procedure described in appendix C, and thus to the baseflow calculated based on Eq. (19). However, this numerical error is normally small and hence, will not affect the overall effectiveness of the formulation proposed in this paper [i.e., Eq. (19)]. To distinguish the impacts of the spatial variability of recharge and topography, we calculated the subsurface flows based on Eqs. (19) and (5) for f = 1 and f = 2, respectively. Figures 9a and 10a show the differences between the subsurface flows calculated from Eqs. (19) and (5) for Tulpehocken and Walnut, respectively, for f = 1 and f = 2. They highlight the impacts of the combined effects of spatial variability of recharge and topography [Eq. (19)] versus the effects of spatial variability of topography alone [Eq. (5)]. Similar to Fig. 8, Figs. 9a and 10a show that Eq. (19) results in higher subsurface flows than Eq. (5), and the difference is larger for larger values of f. The exception in the Walnut Creek (Fig. 10a) when the groundwater table approaches the surface [i.e., 1 − (z/D) approaches 1] is due to the numerical error discussed earlier. That is, the negative value of ΔQb for f = 2 in Fig. 10a is caused by the same numerical error as discussed for Fig. 8f. Both figures also show that there is a mean groundwater table depth for each watershed at which the combined impacts of spatial variability on Qb are maximized. Such a maximum is physically meaningful because when the mean groundwater table is close to the surface, the impacts of spatial variability of the topography and recharge should become negligible. Therefore, the difference between Eqs. (19) and (5) becomes smaller. When the mean groundwater table is deep, the subsurface flow should be much less sensitive to the spatial variability of recharge and topography; hence, the impacts of topography and spatial variability of recharge become much smaller.

Figures 9b and 10b show the impacts of topography alone on Eqs. (19) and (5) at both sites. For the same mean groundwater table depth, the impact is generally higher for Eq. (19) than for Eq. (5). Our results suggest that for humid climate with a relatively shallow groundwater table [e.g., 1 − (z/D) between 0.5 and 1], the combined impacts of spatial variability of recharge and topography are important for hilly watersheds and must be taken into account. For arid climate with deep groundwater table [e.g., 1 − (z/D) < 0.5], simpler formulations for Qb may work. Conceptually, this is because interactions between the groundwater table and the land surface are enhanced in hilly watersheds with a relatively shallow groundwater table such that spatial variability of topography, recharge, their combinations, and/or other factors (e.g., vegetation cover) can significantly influence Qb.

2) Impact of ω

From the above analysis, it can be seen that parameter ζ in the function of ω controls the degree of spatial variability of recharge, and hence affects the magnitude of the subsurface flow. However, the precise form of ω has no effect on the mathematical derivations of the new topographic index and subsurface flow formulation. Empirical evidence suggests that it should be a nonincreasing function of z (Woods et al. 1997, note the difference in the definition of z in that paper).

From Eq. (7), when z is close to D, ω would be close to 0. This is problematic because when n = 1 and ω approaches 0, the denominator in Eq. (20) becomes 0 and gH is thus undefined. Consequently, numerical instability could be introduced into the baseflow calculation based on Eq. (19). To overcome this problem, Woods (1997) proposed a new form of ω as follows:
i1525-7541-9-6-1151-e22
where pmin and pmax are two parameters between 0 and 1 that give more flexibility to the shape of ω. Equation (7) is a special case of Eq. (22) when pmin = 0 and pmax = 1.0. Figure 6c shows the comparison of subsurface flows based on the new formulation with different values of ζ similar to Fig. 6b, but using Eq. (22) with pmin = 0.01 and pmax = 1.0. As can be seen, the difference between Figs. 6b and 6c is not significant. Since the new subsurface formulation is developed for potential use in land surface models and hydrologic models, we prefer to introduce as few parameters as possible and neglect pmin and pmax in the new subsurface flow formulation. However, we recommend using a small value (e.g., 0.01) for pmin and keeping pmax to be 1.0 to avoid the unstable behavior of Eq. (20) in real applications.

4. Conclusions and discussion

In this study, a new subsurface flow formulation is proposed and compared to the subsurface flows derived from observations at three watersheds. Furthermore, the sensitivity of the new formulation is investigated at three sites. The new subsurface flow formulation incorporates spatial variability of both topography and recharge. Compared to the ARNO parameterization, the new formulation is more physically based. Compared to Eqs. (5) and (3), the new formulation is more appropriate for applications (e.g., in hilly watersheds under humid climate) where the spatial variability of recharge is significant and cannot be ignored. It is also shown through theoretical derivation and case studies that Eqs. (5) and (3) and the parameterization proposed by Woods et al. (1997) are all special cases of the new subsurface flow formulation [i.e., Eq. (19)]. The application of the new formulation is straightforward. Following the procedures outlined in section 2b(2), the method can be applied to watersheds at the hillslope scale to large river basins or domains typically used by land surface models.

The subsurface flows calculated using the new formulation compare well with values derived from observations at both the Tulpehocken Creek and Walnut Creek watersheds (Fig. 4). Although we presented the observations at the Maimai hillslope, the available data points are too few to draw any meaningful conclusions on the comparisons between the observations and the new formulation at that site. The new formulation has positive impacts on simulating subsurface flow in wet basins [i.e., 1 − (z/D) > 0.5], where we see notable differences between the new formulation [i.e., Eq. (19)] and the TOPMODEL formulation with a power-law transmissivity [i.e., Eq. (5)]. For relatively dry basins, Eq. (5) is found to be a good approximation. Sensitivity studies show that when the spatial variability of topography or recharge, or both is increased, the subsurface flows increase at all three sites. This is probably due to enhancement of interactions between the groundwater table and the land surface that reduce the flow path. Our results also show that the combined affects on Qb are more pronounced than any individual effect (e.g., spatial variability of recharge and topography) acting alone (see Figs. 5, 8, 9 and 10). Therefore, under certain situations, it is important to consider the integrated effects of the multiple factors that affect subsurface flow, rather than to treat the individual factors separately.

Sensitivity analysis also shows that when n is large, the impact of spatial variability of recharge, controlled by the parameter ζ, on subsurface flow becomes less significant for the same topography. For areas that have larger spatial variability of topography (i.e., large f ), subsurface flow becomes more sensitive to ζ, and the difference between the new formulation [i.e., Eq. (19)] and the TOPMODEL formulation with a power-law transmissivity also becomes much larger. Furthermore, our results (Figs. 9a and 10a) show that there is a mean groundwater table depth at which the combined impacts on Qb are maximized because when the groundwater table is either very shallow or very deep, the impacts of spatial variability of the topography and recharge become negligible.

In summary, an important conclusion of this study is that the spatial variability of recharge alone, and/or in combination with the spatial variability of topography can substantially alter the behaviors of subsurface flows. This suggests that in macroscale hydrologic models or land surface models, subgrid variations of recharge and topography can make significant contributions to the grid mean subsurface flow and must be accounted for in regions with large surface heterogeneity. This is particularly true for regions with a humid climate and a relatively shallow groundwater table where the combined impacts of spatial variability of recharge and topography are shown to be more important. For regions with an arid climate and a relatively deep groundwater table, simpler formulations, such as the TOPMODEL formulation with a power-law transmissivity, for Qb can work well, and the impacts of subgrid variations of recharge and topography may be ignored.

Acknowledgments

The authors are grateful for the generosity of R. A. Woods, who provided his computer code for calculating the H index, and other relevant data for the Maimai site; and Y.-K. Zhang and K. Schilling, who provided the baseflow data and other relevant information for the Walnut Creek watershed in Iowa. We would also like to thank the three anonymous reviewers for their valuable suggestions, which helped us improve the presentation of the materials. This research is partially sponsored by the University of Pittsburgh and by the U.S. Department of Energy’s Office of Science Biological and Environmental Research under a bilateral agreement with the China Meteorological Administration on regional climate research. PNNL is operated for the U.S. DOE by Battelle Memorial Institute under Contract DE-AC06-76RLO1830.

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APPENDIX A

Derivation of the New Subsurface Flow Parameterization

Combining Eqs. (10) and (11), we obtain
i1525-7541-9-6-1151-ea1
Let Li = ai(t)/sinβi, and c′ = (P/ksx,iD)1/n, we have
i1525-7541-9-6-1151-ea2
Note that as shown in Eq. (7), ω is a function of the overall wetness zi/D and is always between 0 and 1, zi/D is also between 0 and 1 as zi varies between 0 and D. Therefore, the absolute value of (1 − ω)zi/D is between 0 and 1, and is much smaller than 1. Thus, [1 − (1 − ω)zi/D]1/n can be approximated by its first-order Taylor series expansion. Equation (A2) can be rewritten as
i1525-7541-9-6-1151-ea3
Let c = c′(n − 1 + ω)/n, then
i1525-7541-9-6-1151-ea4
Let g = c′(1 − ω)/n, we obtain
i1525-7541-9-6-1151-ea5
which is Eq. (13).
Let j be the collection of points upslope of any location i, and k be the collection of points upslope of any location j, and so on, until there are no points upslope. Assume it occurs at index k, then 1 − zk/D = 0. Based on Eq. (A5), we have
i1525-7541-9-6-1151-ea6
The recursive form can be written as
i1525-7541-9-6-1151-ea7
which is Eq. (14) and
i1525-7541-9-6-1151-ea8
which is Eq. (15), where Hi = mean(Hj), j is the collection of all the points upslope of i. If there is no points upslope of i, then Hi = 0, and Hi = L1/ni.
Let ai be the total drainage area upslope of i, and ai(t) be the dynamic contributing area upslope of i, the subsurface flow at point i is given by
i1525-7541-9-6-1151-ea9

It is worth mentioning that when n = 1, the derivation shown above reduces to the derivation shown in Woods et al. (1997).

APPENDIX B

Relationship between gH and z in the New Subsurface Flow Parameterization

Based on the quasi-steady-state assumption and Eq. (19), if we apply Eq. (8) to the whole catchment, we would obtain
i1525-7541-9-6-1151-eb1
Since
i1525-7541-9-6-1151-eb2
we have
i1525-7541-9-6-1151-eb3
To simplify Eq. (B3), we carry out a second-order Taylor expansion about 0 for the term [1 − (1 − ω)z/D)](1/n), which yields
i1525-7541-9-6-1151-eqb1
Rearranging the above expression, we obtain
i1525-7541-9-6-1151-eb4

Equation (B4) is an important relationship for use in calculation of subsurface flow using the new formulation described in this paper. It can be seen that the term gH is determined by the mean groundwater table depth z. Thus, the term gH can be used to represent the wetness of the studied area. It can be verified that when n = 1, Eq. (B4) becomes Eq. (12) in Woods et al. (1997).

APPENDIX C

Calculating the H Index

Let i be the index of the pixel on the DEM of the area of interest. Let Ai, wi, and si be the upslope contributing area (including the pixel itself), the width of the pixel, and the local slope, respectively. Assuming that pixel j is adjacent to and upslope of i, and fij represents the proportion of water from pixel j that flow toward pixel i. The methods of determining Ai, wi, si, and fij are described in Quinn et al. (1991) and Costa-Cabral and Burges (1994). The algorithm for calculating Hi and Hi at each pixel i for a given value of g through Eq. (15) is as follows:

  • Mark all pixels as unprocessed.
  • Find an element i that has no unprocessed pixels upslope of itself. If no such pixel exists, output all values of Hi and Hi and stop. (step 1)
  • With Li = (Ai/wi)/si and ΔAi is the area of pixel i.
  • Assign j(i) = the set of indices for elements next to and upslope of element i.
    • If j(i) is empty, then
      i1525-7541-9-6-1151-eqc1
      • otherwise,
        i1525-7541-9-6-1151-eqc2
        Mark pixed i as processed and return to step 1.

This algorithm can be easily implemented in most existing methods for calculating topographic indices.

APPENDIX D

List of Variables and Their Meanings

  • Qb  Subsurface flow for a watershed
  • K  A constant to describe the relationship between subsurface flow and storage
  • S  A storage term
  • P  A constant to describe the relationship between subsurface flow and storage
  • T  Lateral transmissivity
  • T0  Saturated lateral transmissivity
  • z  Local storage deficit of depth to water table
  • m  Scaling parameter describing the transmissivity profile with depth
  • λ  The topographic index introduced by Kirkby (1975)
  • a  The contributing area to a point per unit contour length
  • β  The local surface slope angle
  • A  The drainage area of a watershed
  • z  The mean storage deficit or mean depth to water table over a catchment
  • Λ1  The spatial average of the logarithmic topographic index defined as Λ1 = A−1A λ ds
  • M  The maximum storage deficit
  • n  Power-law coefficient, a nondimensional parameter between 1 and infinity
  • Λ2  The spatial average of the topographic index to the power 1/n over a watershed defined as Λ2 = A−1A (a/tanβ)1/n ds
  • ri  The spatially variable recharge rate in a watershed
  • P  The average net rainfall rate at the ground surface
  • zi  The depth from the surface to the water table at point i in a watershed
  • D  The soil depth
  • ω  The function describing how the spatial variability of recharge depends on catchment wetness
  • ζ  An empirical constant in ω
  • ri  The spatially average recharge rate per unit area from all points upslope of a unit contour width at i
  • zi   The average zi over all points upslope of location i
  • qi  Subsurface flow per unit contour length at location i
  • ksx,i  The saturated lateral hydraulic conductivity at location i
  • ksx  The effective lateral saturated hydraulic conductivity over an area
  • ks  The effective vertical saturated hydraulic conductivity over an area
  • α  An anisotropic factor between ksx and ks
  • βi  The local surface slope angle at location i
  • ai(t)  The contributing area to point i per unit contour length at time t
  • ai  The total drainage area upslope of i
    i1525-7541-9-6-1151-eqd1
  • Hi = L1/ni(1 + gHi)  The new topographic index at point i that incorporates spatial variability of recharge and topography upslope of i
  • Hi = mean(Hj),  where j is the collection of all the points upslope of i
  • H  The average of Hi over a watershed with a drainage area of A
  • pmin, pmax  Parameters between 0 and 1 that determines the shape of ω
  • f  A factor to modify DEMs for sensitivity analyses

Fig. 1.
Fig. 1.

The DEM of (a) Maimai hillslope, (b) Tulpehocken Creek Watershed, and (c) Walnut Creek Watershed.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM936.1

Fig. 2.
Fig. 2.

The histograms of baseflow at (a) the Maimai hillslope, (b) the Tulpehocken Creek watershed, and (c) the Walnut Creek Watershed.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM936.1

Fig. 3.
Fig. 3.

Subsurface flow calculated based on Eq. (19): (a) with different values of ζ for n = 1 and (b) with different values of the power-law coefficient (n) for the Maimai hillslope. The corresponding flows based on the TOPMODEL formulations with a power-law transmissivity profile, Eq. (5), (dash–dotted line), an exponential transmissivity profile, Eq. (3), (dotted line), and the observations obtained based on the procedure in section 3a(1) are also included.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM936.1

Fig. 4.
Fig. 4.

(a) The subsurface flow for the Tulpehocken Creek Watershed calculated based on the new formulation (i.e., solid line) with n = 3.5 and ζ = 5. Also shown are the corresponding subsurface flows based on the power-law formulation with n = 3.5 (dashed line), and the exponential formulation (dot–dashed line). Both Dgwt and Qobsb (circles) are calculated from observations are included for comparison. (b) As in (a), but for the Walnut Creek Watershed with the subsurface flow calculated from the new formulation (i.e., solid line) with n = 3.5 and ζ = 30.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM936.1

Fig. 5.
Fig. 5.

Comparison of subsurface flows calculated from Eq. (19) with different values of the power-law coefficient (i.e., n) for (a) Tulpehocken Creek Watershed and (b) Walnut Creek Watershed. The corresponding subsurface flows based on Eqs. (5) and (3) are also shown. In the figure, the solid and dashed lines represent the subsurface flows from Eqs. (19) and (5), respectively, for different n values.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM936.1

Fig. 6.
Fig. 6.

Comparison of subsurface flows calculated from Eq. (19) for the Maimai hillslope with different values of ζ for (a) n = 1, (b) n = 3, and (c) as in (b), but for the case with pmin = 0.01 and pmax = 1.0.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM936.1

Fig. 7.
Fig. 7.

Comparison of subsurface flows calculated based on Eq. (19) with different values of ζ for (a) Tulpehocken Creek Watershed and (b) Walnut Creek Watershed.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM936.1

Fig. 8.
Fig. 8.

Comparison of subsurface flows calculated from Eq. (19) based on different synthetic DEMs for the three sites: (a)–(d), Maimai hillslope with n = 1 and ζ = 1.5, n = 1 and ζ = 5, n = 3 and ζ = 1.5 , and n = 3 and ζ = 5, respectively; (e) Tulpehocken Creek Watershed with n = 3.5 and ζ = 5; and (f) Walnut Creek Watershed with n = 3 and ζ = 30. The corresponding subsurface flows based on Eq. (5) are also included. Note that f = 1 represents a case with the original DEM.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM936.1

Fig. 9.
Fig. 9.

(a) Difference between subsurface flow values calculated from Eq. (19) (i.e., new) and Eq. (5) (i.e., power law) for the Tulpehocken Creek Watershed. (b) Difference between subsurface flow values calculated from different synthetic DEMs for the Tulpehocken Creek Watershed. Note that f = 1 represents the case with the original DEM.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM936.1

Fig. 10.
Fig. 10.

Same as Fig. 9, but for the Walnut Creek Watershed.

Citation: Journal of Hydrometeorology 9, 6; 10.1175/2008JHM936.1

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