A simple model of saturation excess runoff generation based on geolnorphology, steady state soil moisture

Transcription

1 WATER RESOURCES RESEARCH, VOL. 37, NO. 1, PAGES , JANUARY 2001 A simple model of saturation excess runoff generation based on geolnorphology, steady state soil moisture Garry Willgoose and Hemantha Perera Department of Civil, Surveying, and Environmental Engineering, University of Newcastle, Callaghan New South Wales, Australia Abstract. A simple conceptual model of saturation excess runoff generation, the Hydrogeomorphic Steady State model (HGSS) is developed using widely observed geomorphic relationships: the saturation excess threshold, the area-slope relationship, and the cumulative area diagram. HGSS provides theoretical insight into the link between runoff generation by saturation excess and the geomorphology that governs it. Current models for saturation excess runoff generation require detailed analysis of catchment digital elevation maps (DEMs), and their prediction accuracy depends on the DEM grid resolution and the analysis techniques used on the DEM data. HGSS can predict the saturated area at steady state without site-specific DEM analysis. In its simplest form, HGSS requires only two parameters, each a combination of geomorphology parameters, together with soil and recharge properties. Effectively, HGSS predicts an antecedent recharge-dependent runoff coefficient. A scaling solution indicates that the percentage of the catchment saturate decreases with the square root of catchment area over a broad range of catchment areas and that the proportion of the catchment saturated has a log-log linear relationship with recharge. HGSS is tested by comparing its predictions with two DEM rainfall-runoff models, the steady state TOPMODEL and a kinematic wave saturation excess model DISTFW-EXT, with HGSS predictions matching for steady state conditions. 1. Introduction ant, with universal parameters over a large range of scales, so The hydrology and the geomorphology of a catchment inthat we are able to present a scale-invariant runoff generation model. The model yields valuable insights into the fundamenteract. The hydrology responds to the shape and slopes of the tal geomorphological basis of saturation excess runoff generacatchment. On the other hand, erosion (responding to hydrol- tion and its scale dependence. ogy) shapes the catchment over geologic time. The geomor- There are also good pragmatic reasons for wanting a simple phology of the catchment determines the flow paths within the saturation excess model. Many saturation excess runoff genercatchment. This interaction results in the catchment and flow paths being organized with simple laws governing their form [e.g., Shreve, 1966; Rodriguez-Iturbe and Valdes, 1979]. ation models use the topographic index or wetness index [e.g., Beven and Kirkby, 1979; O'Loughlin, 1981; Beven and Wood, 1983; Sivapalan et al., 1987]. To calculate this index requires This paper develops a simple model of saturation excess the measurement of the spatial distribution of slope and drainrunoff generation [Dunne and Black, 1970; Beven and Kirkby, age area within catchments. Digital elevation models (DEMs) 1979]. This type of runoff generation occurs when the infiltrated water saturates the total depth of soil mantle. A relatively impermeable layer underneath impedes the ability of the water to percolate vertically, so it must drain down slope within the soil. Our model explicitly incorporates recent knowledge about are a common source of these data, so modeling of saturation excess runoff involves analysis of a DEM. DEM analysis can be problematic. First, Gyasi-Agyei et al. [1995] and Walker and Willgoose [1999] showed that common DEM elevation errors limit the accuracy of the topographic index derived using them. Second, Zhang and Montgomery [1994] and Quinn et al. [1995] catchment organization by using geomorphology relationships. showed that the derived topographic index varies with DEM The model analytically predicts the percentage of catchment grid resolution and analysis procedure, with the topographic saturated by using these fundamental geomorphology relationships, which have been shown by the authors [Willgoose et al., 1991b; Willgoose, 1994; Perera and Willgoose, 1998] and others [Rodriguez-Iturbe et al., 1992; La Barber and Roth, 1994] to be directly linked to catchment forming processes. These geomorphology relationships have also been shown to be scale invariindex increasing as the grid resolution increased [Bruneau et al., 1995]. Finally, Tarboton [1997] showed that the topographic index value varies when calculated with different DEM analysis techniques. This paper presents a conceptual rainfall-runoff model for saturation excess runoff generation which reduces the need for site-specific DEM analysis and which is based on recent in- Now at Department of Land and Water Conservation (New South sights into the geomorphology of catchments. The conceptual Wales), Grafton, New South Wales, Australia. model, Hydrogeomorphic Steady State (HGSS) predicts the Copyright 2001 by the American Geophysical Union. saturated area at steady state using statistics of the catchment geomorphology. HGSS is tested by comparing its predictions Paper number 2000WR /01/2000WR $ with two other DEM-based models.

2 148 WILLGOOSE AND PERERA: MODEL OF SATURATION EXCESS RUNOFF GENERATION Slope Elimination ( B even/o ' Loughhn h /steady state saturation criteria (wetness index) J o Parameterized by slope L andrainage area J r Relationship between mean slope and drainage area Drainage Area Elimination (Steady state saturation oparametenzedbyi LdrainageareaaloneJ Cumulative area distribution curve HGSS Formulation Percentage of catchment saturated at steady state for catchment ß Independent of detailed catchment geomorphology i.e. catchment slope and drainage area ß Parameterized by statistics of the geomorphology Figure 1. A schematic representation of the HGSS model for predicting saturation-excess saturated area at steady state. 2. HGSS Model 2.1. HGSS Model Conceptualization Three geomorphology concepts are used in the derivation of HGSS: (1) the wetness index, (2) the relationship between the slope at a catchment outlet and its drainage area, and (3) the cumulative area distribution. The key steps in HGSS are the elimination of the slope term from the wetness index using the geomorphological relationship between drainage area and mean slope and the conversion of the wetness index into a percentage of catchment saturated using the cumulative area distribution (Figure 1). The wetness index l i [Beven and Kirkby, 1979; O'Loughlin, 1981] for a pixel in a gridded DEM can be written as 1 i = Ai/Si, (1) where A i and S i are the area draining through a pixel and the slope at the pixel i for which Xi is defined. For a given point in the catchment with known drainage area and slope this relationship can be used to determine if that point in the catchment is saturated (Xi >- T) or not (Xi < T), where T is a threshold, a function of rainfall and soil transmissivity. Flint [1974] and Tarboton et al. [1989] observed a log-log linear relationship between catchment area and slope for large areas (typically above ---1 km2). Willgoose et al. [1989, 1991b] and Willgoose [1994] presented a general analytical solution for the mean slope with area based on the balance between erosion and tectonics. The general form of this relationship is, = f(ai, G), (2) where is the mean slope at the outlet for all catchments with area draining to it A and G is a vector of geomorphology parameters. The first step in the derivation of HGSS is to substitute the mean slope in (2) for the slope in (1), eliminating the slope dependency of the wetness index and expressing it in terms of area and geomorphology parameters ' i ---- Ai/f(Ai, G). (3) This slope elimination step in HGSS is schematically represented in Figure 1. Note that this step assumes that there is a unique value of catchment outlet slope for a given drainage area: the mean slope of all catchments of area Ai. This point will be discussed in section 2.2. The key insight from (3) is that for a point in the catchment with known drainage area this relationship can be used to determine if that point in the catchment is saturated (, >- T) or not (X < T). We no longer need to explicitly determine the slope at that point. The consequence of this is that if we know the percentage of the catchmenthat has drainage area greater than this value we can determine the percentage of a catchment saturated. The second step in the derivation of HGSS uses the cumulative area distribution, sometimes called the area-mass diagram [Rodriguez-Iturbe et al., 1992; La Barbera and Roth, 1994; Moglen and Bras, 1994; Perera and Willgoose, 1998]. This curve gives the proportion or percentage of a catchment which has drainage area greater than or equal to a specified value of drainage area (Figure 1). In this way, the topographic index based on area in (3) can be converted into a percentage of the catchment saturated. The details and implications follow Testing the Slope Elimination Step The first step in HGSS used a generic area-slope relationship. For a catchment dominated by fluvial erosion this can be written as [Willgooset al., 1991b] A i = C, (4) where C and 0 are parameters of geomorphology and i is the mean slope at the outlet of a catchment with area A (not to be

3 ß WILLGOOSE AND PERERA: MODEL OF SATURATION EXCESS RUNOFF GENERATION 149 confused with the mean slope of a catchment with area A i). We will use a series of catchments complying with this equation in different ways to test this transformation. The reason for this testing is that (4) provides the mean slope at the outlet of many catchments with a given catchment area. However, the slope of field catchments for a given catchment area is scat- tered around this mean. The reasons for and characteristics of, 104 ' 1ooo A 100 (a) Drainage area (b) Transformed topographic index tual. t pograph index ß this scatter are not fully understood but, in part at least, consist of DEM error and spatial heterogeneity within catchments. ß o The HGSS model was applied to three catchments for which we have gridded DEMs. Two of these catchments were syn-,,,,,,,i,,,,,\, 1 thetic catchments simulated by SIBERIA [Willgoos et al., a], while the third was a high-accuracy DEM based on (a) Drainage Area (pixels) ground surveying for a catchment in the Pokolbin region of (b,c) Topographic Index Australia [Willgoose, 1994; Walker and Willgoose, 1999]. The Figure 3. Cumulative distribution for topographic index and SIBERIA catchments allow us to explore the implications of drainage area together with the transformed distribution using the scatter around the area-slope relationship arising from (5) for the Pokolbin catchment. spatial heterogeneity of catchment properties separately from other more complicated effects observed in the field. The three catchments were the following: (1) The first SIBERIA catchment was generated assuming spatially homogeneous erosion In the HGSS slope elimination step the cumulative distribution of drainage area was transformed to the cumulative disand runoff parameters and with only a fluvial erosion process tribution of the wetness index using the relationship between operating. This catchment has an area-slope relationship that falls on the log-log straight line in (4) [Willgooset al., 1991b]. (2) The second SIBERIA catchment was the same as the first drainage area and slope. Substituting the average slope from (4) into the wetness index in (1) yielded the wetness index parameterized by area except that the catchment erodibility was allowed to vary ran- A i ß domly in space. This catchment has an area-slope relationship = =, Ai C A[O C where the slope for a given area is scattered around the log-log (5) straight line in (4) [Willgoose, 1994]. (3) The Pokolbin natural catchment has a diffusion-dominated region, so the mean areawhere the only approximation is that we have substituted the average slope for a given drainage area (from the area-slope slope relationship does not follow a log-log linear relationship relationship (4)) for the actual value. The distributions ofœ on for small areas (Figure 2), and the slopes are scattered around the mean area-slope line [Willgoose, 1994]. We first look at the validity of the slope elimination from the and œ on A are then plotted on the same axes, and if all values of A transformed using (5) coincide with the distribution of œ with, the slope elimination step is exact. For the wetness index. If there is no scatter around the mean area- homogeneous SIBERIA catchment this step is exact because slope relationship then this step is exact. This section demonstrates that the scatter around the area-slope relationship does not significantly impact on the accuracy of HGSS. without the scatter the actual and average slopes for a pixel are identical. To test this step, we use the Pokolbin field catchment. Its To calculate the exact cumulative distribution for wetness drainage area-slope relationship show significant scatter (Figindex for the catchment, the wetness index for each pixel was ure 2). In Figure 2, drainage areas >10 pixels (with 20 m x derived using (1). The proportion of the area of the catchment 20 m grid) are dominated by fluvial erosion, while drainage œ which has the wetness index greater than or equal to a given value of index A was then computed. areas <10 pixels are dominated by diffusive transport [Willgoose, 1994]. For the analysis here the geomorphological parameters 0 and C 1 were estimated only for the fluvial transport region and were 0.35 and 0.177, respectively. These values were used in (5). Fluvial H G S S Pixel Data Figure 3 shows the cumulative area distribution, the exact ß o Aggregated ß ß Data cumulative distribution of the topographic index, and the "ii:i.' ß Mean Slope transformed topographic index distribution using (5). The transformed distribution is a good match to the cumulative '. area distribution on topographic index. For <10 pixels (i.e., the diffusive-dominant region) the transformed distribution deviates slightly. This is not unexpected since the area-slope curve deviates from the fluvial parameters in Figure 2 for these drainage areas. This analysis indicates that the elimination of Wetness Index Saturated slope is not significantly affected by the scatter in area-slope '... '... '... I relationship nor by minor deviation from log-log linearity in Area (m2/m width) the area-slope relationship. Figure 2. Area-slope relationship of the Pokolbin catch- An alternative interpretation of the HGSS approximation is ment. Dots are the raw data from a 20 m DEM, and circles are provided in Figure 2. The sloping line labeled "Wetness Index" the average of 20 adjacent points to more clearly show the is the criteria for a pixel to be saturated based on the wetness mean trend [after Willgoose, 1994]. The mean slope line and index definition in (1). All pixels to the right of the line are the HGSS and wetness index lines are explained in the text. saturated. All those to the left are unsaturated. The number of..

4 150 WILLGOOSE AND PERERA: MODEL OF SATURATION EXCESS RUNOFF GENERATION Table 1. Parameters for the Three Test Catchments Used for Prediction Parameter of the Saturated Area Value Synthetic Synthetic Parameter Homoge- Hetero- Type neous geneous Pokolbin Kh, m/h soil d, m soil w, m DEM resolution C geomorphology geomorphology saturated pixels is determined by analyzing all of the pixels in the catchment. On the other hand, the vertical line labeled "HGSS" (it cuts the mean slope line at the same point as the wetness index line) is the criteria for saturation in our simple model. In HGSS the number of pixels to the right of the curve is determined by use of the cumulative area diagram. Any difference between the models results from a difference in the The HGSS model was further evaluated by comparing its variation of saturated areas for different recharge rates at steady state with the DISTFW-EXT model and the topographic index. DISTFW-EXT is extension of an existing DEMbased, kinematic wave, HortonJan infiltration excess runoff DISTFW model [Willgoose and Kuczera, 1995] to incorporate saturation excess runoff following the approach of Moore et al. [1991]. The details of this extension are described elsewhere [Perera, 1998]. The three models were applied to the test catchments using the parameters in Table For HGSS the threshold drainage area for saturation was calculated for recharge rates between 0.1 and 10 mm/h using a simple case of HGSS described in section 4. The measured cumulative area diagram was used. 2. For DISTFW-EXT, saturated areas at steady state were simulated for the same recharge rates. Effectively, these model results are the same as for the wetness index, but results from this model will be central to testing a dynamic extension of HGSS to be described in a forthcoming paper, so they are presented here for completeness (H. Perera and G. Willgoose, manuscript in preparation, 2000). 3. Using wetness index, the slope and the drainage area of each pixel in the DEM for the three catchments were computed, and the index Ai/S i was computed for each pixel i. Using the soil parameters Kh and d in Table 1 and recharge rate as used for HGSS, the threshold index for saturation (A/S)t was computed for the three catchments. The areas (number of pixels) which have the topographic index greater than or equal to the threshold index values (i.e., area saturated) were then computed. The wetness index can have different variants depending on the modeled groundwater physics. A derivation of the wetness index used follows. Figure 4 shows a typical cross section of a part of a catchment having a constant depth of soil d overlying a relatively impermeable layer. The drainage area per unit width at a section Z is A i/w, where in Figure 4 w is unit width. The subsurface flow per unit width through Z in the saturated region q i is in balance with the constant recharge input R. If the infiltration capacity of the soil exceeds the recharge rate so that HortonJan infiltration excess overland flow cannot occur, then at steady state, qi = (AiR)/w. (6) If the spatially homogeneousoil has transmissivity T and the hydraulic gradient of the water table is assumed to be equal to the surface slope, then the capacity of the subsurface discharge per unit width through the section Z, q... is given by Darcy's number of points in the upper triangle between the wetness index and HGSS lines (pixels saturated by HGSS but not the qmax-- rxi, (7) wetness index) and the lower triangle (saturated by wetness where S i is the surface slope at Z. Saturation excess occurs index and not HGSS). If the number of points is equal, then when the subsurface flow exceeds the capacity of the soil to the models are identical; if the points in the upper triangle transmit water. The condition for saturation at point Z is exceed the lower triangle, then HGSS saturates too much and vice versa. Our validation shows that these two triangles con- At-> (TwS,)/R, (8) tain approximately equal numbers of points for our data sets. where At is the threshold area per unit width at which satura- Note that in the model comparison above the mean slope line tion occurs. used to determine the intersection between the saturation cri- The saturated areas predicted by the three models are comteria is the straight line labeled "fluvial line" not the downward pared in Figures 5-7. They show that saturated areas predicted concave line, which will be discussed in section 4. by the HGSS compared well with those for DISTFW-EXT and the topographic index for the three test catchments. HGSS 3. Evaluation of the HGSS Model predictions for the synthetic catchments deviate from DIS- TFW-EXT for higher recharge rates where the threshold drainage area for saturation is very small. This results from law slight deviations from log-log linearity in the area-slope relationship for small areas in the simulated landform. For the Pokolbin catchment the HGSS predictions match the other two models throughout the range of recharge rates used. The threshold drainage area for saturation computed for the 10 la er a Drainage Area A Slope S echarge Figure 4. Typical long section of a part of a catchment. q

5 WILLGOOSE AND PERERA: MODEL OF SATURATION EXCESS RUNOFF GENERATION , 100o o HGSS, x DISTFW-EXT o o Topographic index o 100 HGSS DISTFW-EXT Topographic index 100,,,,,,,,I,,,,,,,, Recharge Rate (mm/hr) Figure $. Comparisons of the predicted saturated area for HGSS, DISTFW-EXT, and the topographic index using the synthetic homogeneous catchment I I I t I I i i [ i i i i i i i i 1 lo Recharge Rate (mm/hr) Figure 7. Comparisons of the predicted saturated area saturated for HGSS, DISTFW-EXT, and the topographic index using the Pokolbin natural catchment. be presented in a forthcoming paper by the authors. In this mm/h recharge rate is 2 pixels (1600 m 2) for the Pokolbin section we derive an analytical expression for HGSS based on catchment. Although this area seems to be outside the region simple approximations which will highlight the geomorphic dominated by fluvial erosion for which the parameters of the dependence of HGSS. area-slope relationship were determined, the HGSS predic- Slope Si can be eliminated from (8) by using the mean tions of saturated area still match the results well, which is area-slope relationship (equation(4)), and rearranging yields consistent with the results above for the transformed topographic index distribution. At-> C2(1/R) 1/(1+ø), (9) That the Pokolbin site (Figure 7) is better fit by HGSS than the log-log linear heterogeneou s site (Figure 6) is somewhat C 2 = (TwCO /( +ø). (10) surprisingiven the clear deviation from log-log linearity in the At the point on the hillslope when the soil profile is fully average slope for small areas in the Pokolbin data. The better saturated the inequality in (9) is an equality. Equation (9) fit for Pokolbin can only result from the statistical charactershows that recharge rate has a log-10g linear relationship with istics of the scatter in the slopes around the mean. Since the threshold drainage area for saturation. For different constant reasons for and generic characteristics of this scatter are poorly recharge rates the distribution of the threshold area for a given understood, we can only postulate that the performance will be catchment can be plotted (Figure 8a). equally good in other catchments. We do, however, believe We can relate the proportion of the catchment saturated to that this technique will be generally applicable because there is this threshold area. Perera and Ve21lgoose [1998] described the nothing to suggest that the Pokolbin DEM is unusual. behavior of the cumulative area diagram (CAD) of a catch- 4. A Simple Geomorphology Scaling Model for Wetness Index As a steady state model, HGSS is an approximation of the transient condition of saturation, such as modeled by TOP- MODEL [Beven and Kirkby, 1979]. Moreover, HGSS is an essential basis for a geomorphology-based conceptualization of saturation excess runoff generation for transient conditions to R* 1000 HGSS DISTFW-EXT Topographic index o o o o o x log(drainage Area, A) (b) Area saturated Recharge rate (mm/hr) Figure 6. Comparisons of the predicted saturated area for HGSS, DISTFW-EXT, and the topographic index using the synthetic heterogeneous catchment. log(drainage Area, A) Figure 8. The prediction of saturated area within a catchment for a given recharge rate using the steady state conceptual model HGSS. (a) Predicted threshold drainage area and (b) predicted saturated area for the same threshold drainage area using the cumulative area distribution.

6 152 WILLGOOSE AND PERERA: MODEL OF SATURATION EXCESS RUNOFF GENERATION Table 2. Comparison of Analytically Computed Exponent to With That Predicted by HGSS and DISTFW-EXT Computed Predicted Predicted by Using (15) by HGSS DISTFW-EXT Synthetic homogeneous Synthetic heterogeneous Pokolbin ment. They showed that for the region where the fluvial erosion dominates, the CAD asymptotically obeys a power law assess runoff generation. After obtaining the threshold drainage area for saturation, we can use the cumulative area distri- CAD(A) oc A0, (11) bution of a catchmen to predict the percentage of the catchment saturated for that threshold contributing area (Figure 8). where CAD is the number of pixels with drainage area greater than A (in pixels) and has been observed by a number of authors to be about -0.5 [Rodriguez-Iturbet al., 1992; La Barbera and Roth, 1994; Moglen and Bras, 1994]. If the area is normalized using the total catchment area, then the cumulative Finally, the percentage of the catchment saturated decreases as a power of the catchment area provided only that the parameters in (9) or the recharge do not vary systematically in space. For the typical value of (k = -0.5 the runoff coefficient decreases with the square root of the catchment area. This scaling area diagram gives the proportion of the catchment saturated of the percentage of catchment saturated will only apply for for a given threshold area that range of catchment areas for which the scale-invariant p = (C3/Ac)(At) rk, (12) log-log linear relationship in (4) and (12) apply. The only restriction on this result is that the catchment area where A c is the total area of the catchment and.4' =.4/.4c is needs to be larger than the area needed for saturation (i.e., the nondimensional area. If (9) and (12) are combined by part of the catchment must be saturated) such that the persetting.4 - At, then the proportion of catchment saturated is centage of the catchment saturated is not calculated from that part of the cumulative area diagram affected by boundary effects [Perera and Willgoose, 1998], the drop off in the CAD p =Ac +. (13) figure at the extreme right-hand end (Figure 3). The relationship between saturated area at steady state and the recharge rate is This simple model ignores the break in log-log linearity in the area-slope relationship that occurs at small area. It is not A satocr "', (14) possible to provide a general analytical solution to the mean relationship in this case for all parameter values. For instance, where the exponent to is expressed in terms of fundamental geomorphology parameters the mean line in Figure 2 resulted from solving an equation for the equilibrium geomorphology of catchment where tectonic o, = + 0)]. uplift of the catchment U is exactly balanced by fluvial erosion (parameters/3, m, and n ) and diffusive mass transport The values of and 0 for the three test catchments are known, so to test this result, the value of to was computed from (15) and compared with the DISTFW-EXT model simulations (parameter D) [ Willgoose, 1994] UA -- [31A m l S n ' -[- DS. (16) (Figures 5-7). Table 2 shows that the predictions of (15) com- When the erosion processes are dominated by the first term on pare well with the fitted values. This suggests that the exponent the right-hand side, the catchment is dominated by fluvial of the relationship between saturated area at steady state and erosion and exhibits a concave up longitudinal profile. When the recharge rate can be computed without site-specific DEM the catchment is dominated by the second term (typically for data of a catchment using regional parameters of 4> and 0, so small catchment areas because of the area dependence of the that to can also be regionalized. first term), then the catchment is dominated by diffusive pro- The geomorphological parameters C, C 3, 0, and have cesses (e.g., creep) and exhibits a concave down longitudinal been observed by the authors to be constants for many subprofile. Equation (4) can be derived from (16) when D = 0. catchments in a given region. Thus for a known constant re- For nonzero D, Willgooset al. [1991] noted, however, that the charge rate the threshold drainage area for saturation at steady state of a catchment can be obtained for known parameters, T, description of the curved transition region (where both terms C and 0, using (9) and the proportion of the catchment saton the right-hand side of (16) are equally dominant) was inurated from (13). Furthermore, there are only two indepen- sensitive to the exact choice of n. Moreover, Willgoose [1994], dent properties, the coefficient and the exponent, a parsimo- when fitting the mean curve in Figure 2, showed that the value nious result that suggests calibration of this relationship to of n could not be accurately estimated from DEM data even catchment saturation data should be relatively straight for- though 0, which is function of n, could be accurately estiward. This method allows us to determine the saturated area mated. If a value of n = 2 is adopted, which is close to the from the distribution of drainage area. By eliminating the vertical domain this new conceptualization allows us to use only a planar geomorphic property, the drainage area.4, to theoretically accepted value [Henderson, 1966], then the slope can be eliminated between (8), (12), and (16) to yield a scaling relationship similar to (13):

7 WILLGOOSE AND PERERA: MODEL OF SATURATION EXCESS RUNOFF GENERATION 153 C3 I 13-- T2 ( U / R) 11/2(4'/1+ø), (17) P = Ac- where 0 = (m - 1)/n = (m - 1)/2, and if we define a nondimensional recharge rate R' = R/T and rearrange the expression, area less C3 [(U- R'D)] l/2(4'/l+ø)(r] 7) 4'/(l+ø) P = Ac 7 j3. (18) As for (13) this result scales with catchment area. For D = 0 this relationship simplifies to (13). However, we now find that 6. Discussion the variation of saturated area with recharge is no longer log-log linear. As R' increases and more of the diffusive por- HGSS can be used to predict the percentage of catchment tion of the catchment saturates, the deviation from linearity saturated for catchments without using site-specific DEM analincreases. The relative simplicity of the recharge-saturation ysis. There is good evidence that the geomorphological paramrelationship is only complicated by the addition of an extra eters of the area-slope relationship can be derived from reparameter that becomes important for high recharge rates. gional DEM analysis because they have been widely observed Both (13) and (18) demonstrate that the runoff can be modto have universal behavior in the field [Tarboton eta!., 1989] eled by a runoff coefficient (assuming 100% runoff in the and theoretically because of their link with the discharge and saturated region and 0% in the unsaturated region) that de- sedimentransport characteristics of catchments [Willgooset creases with a power of catchment area and is dependent on al., 1991b; Willgoose, 1994]. Likewise, there is good evidence antecedent recharge and thus rainfall conditions in a simple, that the cumulative area distribution can be subjected to repredictable fashion. gional analysis because a large number of field catchments show similar behavior [Rodriguez-Iturbe et al., 1992; La Barbara and Roth, 1994]. Furthermore, an analytical solution of the 5. Spatial Distribution of Water Table cumulative area distribution was derived by Perera and Will- Thickness goose [1998] based on widely accepted catchment geomorphic statistics that supports universal behavior. This suggests that The catchment base flow is used to predict the initial soil HGSS can be parameterized from regional geomorphology moisture distribution prior to a rainfall event in many soil analysis. moisture storage-based models that simulate saturation excess It is possible to conceive of a number of ways HGSS can be runoff [e.g., Beven and Kirkby, 1979; Beven and Wood, 1983] by used in practice. We imagine embedding this model inside matching the outlet discharge of the catchment with catchment existing conceptual rainfall runoff models, which currently ususoil moisture. These initial conditions assume steady state con- ally involve some form of infiltration excess runoff generation ditions. HGSS can be used to predict the distribution of the water table thickness prior to a rainfall event. The distribution of the thickness of the water table and the relationship between catchment average soil moisture and saturation depend to a great extent on the vertical profile of horizontal conductivity. To exemplify how HGSS can be used to predict this distribution, we examine the simplest case of conductivity constant with depth. At steady state the catchment outflow prior to a rainfall event Q o is assumed to be in equilibrium with the recharge rate Ro. The relationship between Qo (m3/s) and R o (mm/h) is Q0 = (RoAcat)/3.6 X 106, (19) where A ca t (m 2) is the total area of the catchment. At steady state the total recharge upstream of a pixel within a catchment is equal to the flow through that pixel. Thus, at pixel i, if the total soil depth is not saturated, RoAi = KhtiW S 0 -< t -< d, (20) where Ai, gh, ti, Wi, and S i are the drainage area, the saturated hydraulic conductivity, the thickness of water table, the width of flow, and slope at the pixel i, respectively. For a gridded DEM, w i is the grid resolution. Substituting for slope using (4) and rearranging the terms, the thickness of the water table at the pixel is R0 -- Ai +ø 0 -< t i -< d. (21) ti- ClKhWi Equation (21) gives the relationship between the thickness of the water table and the drainage area of a pixel at steady state when the total depth of soil layer is not saturated. The drainage area of each pixel can be calculated using DEM data. The threshold drainage areaao for saturation of the full soil profile d at steady state for a catchment outflow Q o can be calculated. The distribution of the water table thickness for the drainage thanao can be computed using (21). All pixels which have drainage area greater than A o are fully saturated. model and which are unable at the current time to model saturation excess runoff. The geomorphology parameters could be determined at site from these maps or determined using site-specific data. For instance, a number of high- resolution, high-accuracy study sites distributed over a region might be used to derive contour maps of geomorphology parameters for input to the model. This would circumvent many of the problems with the relatively low accuracy DEMs that are currently available, outlined in section 1. Alternatively, given that the two or three parameters of HGSS (depending on the formulation) are no more complicated than existing infiltration excess models typically used (e.g., Philip model), it may be possible to simply use HGSS as a more physically based infiltration model to be calibrated to at-station rainfall-runoff data, replacing the infiltration excess model at sites where it is believed to be inappropriate. Another use of HGSS would be as the basis of a simple model for spatially average evaporation for large areas (e.g., climate model grid cells). By being able to predict both the percentage of a catchment saturated from long-term rainfall and the distribution of soil saturation in the remaining partially saturated areas it may be possible to derive a simple evaporation model based on saturation deficit that would be more computationally efficient and physically defensible than cur- rent bucket models. HGSS is most applicable to a catchment where the timescales of change in water table levels are very much longer than the rainfall, so that steady state can be assumed. Dynamic variations in the water table potentially impact on HGSS in two

8 . 154 WILLGOOSE AND PERERA: MODEL OF SATURATION EXCESS RUNOFF GENERATION ways. First, even if the water table does not vary significantly that predicts the percentage of the catchment saturated given during a rainfall event, how do prior rainfall'events impact on soil transmissivity and steady state recharge. The parameters of the thickness of the water table immediately prior to, and thus the drainage area-slope relationship and cumulative area disduring the current rainfall event? This process will determine tribution can be derived on a regional basis, thus eliminating how the effective recharge in HGSS is defined (i.e., the re, the need for site-specific digital terrain analysis. For instance, charge rate that would be equivalento a steady state constant Perera and Willgoo se [1998] derived an analytical solution for recharge) in the scaling models of (13) and (18). seeond, in the the form of the cumulative area distribution based on geomorcase where.the Water table variesignificantly during the rain, phic insights and suggest. that the cumulative area distribution fall event, in what way does HGSS need to be extended to of a catchment can be analytically derived with regional geoallow for the effects of a nonsteady state, dynamic water table? morphic parameters, without local DEM analysis. For the first case, where the water table does not change HGSS links three important geomorphi concepts to deterduring the rainfall event, it is clear that the long periods of mine the saturation excess runoff generation at steady state. constant rainfall for which the catchment can reach steady They are (1) topographic index, (2) drainage area-slope relastate condition do not occur. This suggests that for natural tionship, and (3) cumulative area distribution. Using these catchments a strict steady' state will not normally be achieved. relationships, we derived a scaling of the runoff coefficient with For instance, Barling ½t al. [1994] observed that the time to the catchment area that depended only on geomorphology, steady state for their natural catchment exceeded 30 days. The recharge, and soils. The parsimonious form of the runoff cotime for a point within a catchment to reach Steady state efficient (two or three parameters depending on the geomordepends on its S0il properties, geomorphic parameters, initial phology) suggests that this result could be feasibly incorposoil moisture condition, and recharge rate and position within rated as a component of existing conceptual rainfall-runoff the catchment. The time taken for our three test catchments to models. Typically, these models do not incorporate DEM analattain steady state under variou s recharge conditions, as sim- yses and thus can only model saturation excess runoff generulated by D ISTFW-EXT, were also very long. For instance, for ation crudely, if at all, so HGSS would significantly improve the a 10 mm/h recharge it took the Pokolbin catchment hours physical basis of these models. to reach steady state, Using the soil parameters fitted by Moore Finally, the HGSS model highlights the geomorphologic unet al. [199!] to their Wagga catchment. We hypothesize that derpinnings of the saturation excess runoff generation mechthe recharge rate in HGSS can be conceptualized as some anism. HGSS complements the runoff-routing model provided average or effective recharge over a time period prior to the by the Geomorphic Instantaneous Unit Hydrograph (GIUH) rainfall event. To parame(erize this averaging, we need to [Rodriguez-Iturbe and Valdes, 1979]. Together these models understand the response time of the perched aquifer in the provide a powerful description of the underlyin geomorphosoils of catchments. Thus it will be importanto study the logic basis of surface water event hydrolofty in catchments. dynamics of the saturated area under different rainfall and Subject to the steady state assumptions of the model, we initial soil moisture conditions. For the second case, where the believe that HGSS provides a significant step forward in our water table changes during the rainfall event, a forthcoming understanding of the fundamental geomorphic basis of saturapaper by the authors will present a dynamic extension of HGSS tion excess runoff generation and may be used for analysis of that maintains its underlying geomorphologic basis and sim- the effect of changes in geomorphology on hydrolofty. The plicity [Perera, 1998; Perera and Willgoose, manuscript in prep- scaling solutions may also provide the basis for a parsimonious aration, 2000]. runoff coefficient that could be used in either subcatchmentbased, non-dem, conceptual rainfall-runoff models or bucket 7. Conclusions soil moisture accounting models used in global climate models. The conceptual model, HGSS, was developed using the to- Acknowledgement. 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