Click Here
WATER RESOURCES RESEARCH, VOL. 46, W03511, doi:10.1029/2009WR008500, 2010
for
Full Article
Development and application of a catchment similarity index for subsurface flow Steve W. Lyon1 and Peter A. Troch2 Received 7 August 2009; revised 16 October 2009; accepted 27 October 2009; published 11 March 2010.
[1] In this study, we develop a similarity parameter to describe shallow subsurface hydrological response of small catchments on the basis of the hillslope‐Péclet number. This new similarity parameter, named the catchment‐Péclet (caPe) number, provides a theoretical framework to compare the relative hydrologic response derived from shallow subsurface flow of small catchments on the basis of geomorphic properties. Using 400,000 synthetically derived catchments to model catchment‐scale characteristic response functions (CRFs), we see good agreement between the synthetic and theoretical relationships relating the caPe number to the first two dimensionless moments of the CRF of small catchments. Working with real‐world data, however, requires the estimation of hydrologic parameters and delineation of hillslopes to apply the caPe number. Allowing for uncertainty in the estimation of hydrologic parameters and in the definition of the extent of the channel network, the caPe number is able to recreate the observed moments of an approximate catchment‐scale CRF for four small catchments ranging in size from 0.025 to 880 ha in two distinct climatic and geologic settings. By using physics to underpin the link between landscape and hydrological response, the caPe number creates a functional relationship between hydraulic theory and a catchment’s pedogeomorphological structure. While this study is limited to small headwater catchments, it lays the groundwork for a catchment‐scale similarity parameter that could be expanded to larger scales where channel network structure and storage become more important. Citation: Lyon, S. W., and P. A. Troch (2010), Development and application of a catchment similarity index for subsurface flow, Water Resour. Res., 46, W03511, doi:10.1029/2009WR008500.
1. Introduction [2] The geomorphologic and pedologic structure of the landscape is a dominant control over the hydrological response of a catchment. This concept is not new to scientific hydrology. For example, the classical work of Beven and Kirkby [1979] showed how geomorphometric parameters can be used to describe the hydrological behavior at a given position within the landscape, while Rodríguez‐Iturbe and Valdés [1979] showed how the shape of a catchment unit hydrograph can be explained from the structure of the channel network. More recent work on the scale‐dependent controls of catchment behavior [D’Odorico and Rigon, 2003] shows that the catchment travel time distribution is dominated by the hillslopes. Hydrologic science has thus identified landscape structure as a dominant control over catchment hydrological response and hillslopes as a dominant control over catchment travel time distribution. [3] This is primarily true for small catchments where channel network effects are negligible. Recent commentary 1 Physical Geography and Quaternary Geology, Stockholm University, Stockholm, Sweden. 2 Hydrology and Water Resources, University of Arizona, Tucson, Arizona, USA.
Copyright 2010 by the American Geophysical Union. 0043‐1397/10/2009WR008500$09.00
by Bishop et al. [2008] highlights the disproportionate amount of total stream length contained in these small, headwater catchments (a region of hydrology they refer to as Aqua Incognita). This holds for many distinct geomorphologic landscapes. While Bishop et al. [2008] refer specifically to their experience in Sweden, Nadeau and Rains [2007] estimate that 53% of the total known stream length in the United States (excluding Alaska) is contained in headwater catchments. Here, headwater refers to streams that are first order [Strahler, 1952] when viewed at the 1:100,000 scale. Nadeau and Rains [2007] suggest that hillslopes, headwater streams, and downstream waters are best described as individual elements of integrated hydrological systems. This makes the hillslope a good candidate for a structural element of a landscape capable of describing small catchment hydrological response. [4] From this structural element, we should be able to derive similarity parameters that can be scaled up to the small catchment scale and translated to various climatic settings. Although considerable efforts have been put into studying hydrologic similarity [Sivapalan et al., 1987, 1990], definitive conclusions about similarity in behavior of landscapes, based on similarities in the dominant hillslope processes, have not yet been accepted in the field of hydrology. The upscaling of hydrologic processes has traditionally been [Dooge, 1986; Blöschl and Sivapalan, 1995] and still is [Woods, 2005; Harman and Sivapalan, 2009a,
W03511
1 of 13
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
2009b] a problem. This is partly due to a lack of analytical relationships between the flow processes and quantitative descriptors of landscape attributes [Aryal et al., 2005]. While attempts have been made in the past by geomorphologists [e.g., Morisawa, 1962, 1985; Gardiner, 1979] to establish regression relations between flow and morphometric indices (e.g., relief ratio, drainage density), it is possible to develop similarity indices on the basis of analytical solutions to the governing dynamic equations [Brutsaert, 2005]. The development of similarity indices in such a manner has been stymied by the complexity (due to their highly nonlinear nature) of these governing dynamic equations. For example, there are no general analytical solutions to the full, three‐ dimensional Richards equation that governs hillslope matrix flow processes [Paniconi and Wood, 1993; Bronstert, 1994]. The difficulties associated with such three‐dimensional approaches can be overcome by employing low‐dimensional hillslope models [Troch et al., 2003; Paniconi et al., 2003; Hilberts et al., 2004]. These models, called hillslope‐storage dynamics models, are able to treat geometric complexity in a simple way, resulting in a significant reduction in model complexity allowing them to serve as a basis for landscape hydrologic similarity. [5] For example dimensionless Péclet (Pe) numbers exist in many disciplines and at many physical scales (which has lead to much confusion in their use and interpretation). In chemical engineering, the Pe number is associated with molecular properties of the fluid in question, namely the molecular viscosity and the molecular (usually thermal, but not exclusively) diffusivity. In fluid dynamics, the Pe number is a dimensionless number relating the rate of advection of a flow to its rate of diffusion. While Pe numbers have traditionally been used to describe solute transport and not necessarily flow, the concept of a Pe number has been applied in a more general sense as a ratio of advective (gravity) forces to diffusion (pressure) forces to describe hydrologic processes at hillslope and catchment scales [e.g., Troch et al., 2004; Berne et al., 2005]. This general interpretation is particularly true for larger Pe numbers when simpler computational models can be adopted (such as a kinematic approach to model subsurface flow). To help avoid confusion across disciplines and scales, we apply a naming convention were the intended scale of applicability is imbedded within the Pe number’s name. As such, we refer to the hillslope‐Péclet (hsPe) numbers and the catchment‐ Péclet (caPe) numbers for the remainder of this article. [6] The dimensionless hillslope‐Péclet (hsPe) number can be taken in a general sense as a ratio of advective (gravity) forces to diffusion (pressure) forces at hillslope scales [e.g., Troch et al., 2004; Berne et al., 2005] expressed for an unconfined sloping aquifer as [Brutsaert, 1994; Troch et al., 2004] hsPe ¼
UL 2K
ð1Þ
where L is the hillslope length, U is a characteristic advective velocity and K is the complex‐hillslope diffusion coefficient. Troch et al. [2004], on the basis of an analytical solution of the hillslope‐storage Boussinesq (hsB) equation [Troch et al., 2003], showed that the hsPe number for advective‐ diffusive flow along complex hillslopes can be expressed in
W03511
terms of quantifiable geomorphic properties, such as slope, depth, and convergence rate: hsPe ¼
L ac L tan 2pD 2
ð2Þ
where a is the hillslope bedrock angle, pD is the average saturated depth, D is soil depth, p is a linearization parameter, and ac is the hillslope plan shape convergence rate. Starting from the Laplace domain solution derived by Troch et al. [2004] to the linearized hsB equation, Berne et al. [2005] used moment generating functions to relate analytically the hillslope hsPe number to the moments of the characteristic response function (CRF) of subsurface flow. The CRF is defined as the free drainage hydrograph, normalized by the total outflow volume, following an initial steady state storage profile corresponding to a constant recharge rate. The first moment of the CRF is the mean response time for the hillslope. Berne et al. [2005] showed that the hsPe number is an efficient similarity parameter to describe the hillslope subsurface flow response, and used laboratory data from a scaled hillslope model to test and validate the derived relationships. The hsPe number has since been successfully applied by Lyon and Troch [2007] to “real‐world” data (i.e., hydrological response determined from hillslope‐scale field experiments) by adopting effective parameters (namely, lateral hydraulic conductivity and drainable porosity) to capture the subsurface heterogeneity at the hillslope scale for making the CRF dimensionless. [7] In this study, we will develop a similarity parameter to describe shallow subsurface hydrological response of small catchments on the basis of the hsPe number. This new similarity parameter, named the catchment‐Péclet (caPe) numbers, provides a theoretical framework to compare the relative hydrologic response derived from shallow subsurface flow of small, headwater catchments on the basis of geomorphic properties. In addition to developing the caPe number, we will demonstrate its ability to describe catchment hydrologic response for four small catchments ranging in size from 0.025 ha to 880 ha and coming from two distinct climatic and geologic settings. This similarity parameter may provide some insight in the realm of Aqua Incognita mapped out by Bishop et al. [2008].
2. Development of the Catchment‐Péclet Number [8] Let us define a small catchment recession response as one that is dominated by hillslope subsurface flow and storage such that channel network effects are negligible and that there exists no significant riparian zone that delays flow. It is difficult to define an exact area threshold for what can be considered a “small” catchment (e.g., Bishop et al. [2008] assume small catchments are those less than 15 km2 in area on the basis of national‐scale surveys of Sweden). The main requirement, however, is a minimal influence of channel network and channel storage on hydrologic response. [9] Such small catchments can be thought of as being composed of h hillslopes of which hc are convergent (zero‐ order catchments or source draining), hd are divergent and hp are parallel draining, such that h = hc + hd + hp. Each individual hillslope has a characteristic response (Figure 1a) to an initial steady state storage profile corresponding to a
2 of 13
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
W03511
Figure 1. Schematic representation of (a) resulting catchment hydrograph as the sum of the individual hillslope responses and (b) how the characteristic response functions (CRFs) of the individual hillslopes combine to define the catchment CRF. constant recharge rate (i.e., a unique CRF). Berne et al. [2005] have shown that the moments of the linear CRF can be derived from the hsPe number (given in equation (2)) and the characteristic diffusive time constant (t K): K ¼
Lf 2k
L 2pD
1 cos
ð3Þ
where f is the drainable porosity and k is the hydraulic conductivity. Note this time constant is, in essence, similar to the storage coefficient used in storage outflow models (e.g., linear tank models) and has been shown to be relatively invariant for drought flows from larger catchments [Brutsaert, 2008]. The hsPe number and t K have also been demonstrated as predictors of the nonlinear CRF [Harman and Sivapalan, 2009b]. Given the geometric and hydraulic parameters for a catchment containing h hillslopes, it is
possible to “reconstruct” the resulting composite outflow CRF (Figure 1a). Since flow processes in the channel network and the riparian area do not induce significant delays and dampening of the individual CRFs, the resulting hydrograph “observed” at the outlet of the small catchment is the sum of the hillslope responses. If one were to relax this assumption, it would be necessary to account for such dampening through, for example, a convolution of the hillslope responses along the channel network. We restrict ourselves to consider only small catchments in this present study and leave the incorporation of channel network effects for future research. [10] After normalizing the observed composite hydrograph by the total volume drained from the catchment, one obtains the CRF for the catchment. It is clear that the dimensionless moments of this catchment CRF, m*n,c (where n is the 1st, 2nd,…, nth moment), are some functions of the
3 of 13
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
set of dimensionless moments for the individual hillslopes, {m*n,1, m*n,2,…, m*n,h} (Figure 1b), and, through inversion of the analytical relationship between the hsPe number and the CRF dimensionless moments derived by Berne et al. [2005], thus function of the hsPe numbers. For example, the first moment (center of gravity) shown by the vertical dashed lines in Figure 1b for the catchment hydrograph is a function of the centers of gravity of the individual hillslope hydrographs. [11] Let us denote qi(t) as the normalized CRF for the ith hillslope: Qi ðtÞ Vi
qi ðtÞ ¼
ð4Þ
where Qi(t) is the outflow hydrograph and Vi is the total volume of water for the ith hillslope recessing from an initial steady state storage profile. In general, we can say that the normalized CRF for the catchment (qc) is the volume‐ weighted sum of normalized CRF for each hillslope given by qc ðtÞ ¼
h X
vi qi ðtÞ
ð5Þ
i¼1
where vi is the fraction of hillslope i contributing to the catchment hydrograph or the ratio of Vi to the total volume of water draining from the catchment. [12] The first dimensional moment (mean response time) of the catchment CRF is given by Z 1;c ¼
1
0
tqc ðtÞdt ¼
h Z X
1
0
i¼1
h X
vi tqi ðtÞdt ¼
vi 1;i
ð6Þ
i¼1
and m1,i are the first dimensional moments of the individual hillslopes. Similarly, the second raw moment of the catchment CRF is given by Z m2;c ¼
1 0
t 2 qc ðtÞdt ¼
h Z X i¼1
1
0
vi t2 qi ðtÞdt ¼
h X
vi m2;i
ð7Þ
i¼1
where m2,i gives the second raw moment of each individual hillslope. This leads to a second dimensional (central) moment of the form 2;c ¼
h X i¼1
vi m2;i
h X
!2 vi 1;i
ð8Þ
i¼1
[13] We limit ourselves to expressions of the first and second moments in this study. Similar expressions to those above can be developed for higher‐order moments. Assuming that there is one characteristic diffusive time constant for each individual hillslope, all moments can be converted in a dimensionless context and give analytical expressions for the dimensionless moments of the catchment response. Using the analytical relationship outlined by Berne et al. [2005], we now have dimensionless moments of the catchment CRF that can be related to a caPe number. [14] But, how do we develop the caPe number? Let us consider the simple case of a catchment composed of three
W03511
hillslopes. It should be noted that the delineation of this small catchment into three hillslopes will be scale‐dependent. That is, using a digital elevation model (DEM) of higher resolution may lead to the delineation of more than three hillslopes. Also, there is no robust geomorphological or hydrological definition of what constitutes a hillslope and several separate hillslopes delineated from high‐resolution DEMs can be easily grouped together into one larger area which still can be considered, from a hydrological point of view, a hillslope. Returning to our example catchment, one can measure the hydrograph after the catchment was brought into steady state (after a long rainy period of almost constant intensity). As discussed above, this response function is the simple summation of the hillslope response functions, since we neglect channel and riparian area delays in flow. After normalization by volume and making the response function dimensionless using the characteristic diffusive time constant, the dimensionless moments of the CRF of our catchment will point to a specific caPe number relevant for the catchment. This is similar to Berne et al. [2005] using the relationship between the hsPe number and the dimensionless moments of the CRF such that all moments (first, second, third, and so on) will “predict” the same hsPe number. This caPe number will be some combination of the individual hsPe numbers. In the following paragraph we will show that this caPe number can be obtained by volume weighting the individual hsPe numbers: caPe ¼
h X
vi hsPei
ð9Þ
i¼1
where hsPei is the hsPe number (equation (2)) of the individual hillslopes comprising a given catchment. Note that “volume weighting” here is semantic as vi is a ratio of water volumes. To prove (9) we need to demonstrate that the caPe number predicted from the dimensionless moments of the catchment CRF is identical to the volume‐weighted average of the individual hsPe numbers. Here we choose to do this numerically, even though an analytical derivation of this proof may be possible. [15] We use synthetically generated hillslopes to construct composite small catchments. A set of 1018 hillslopes were randomly generated spanning a range of convergence rates from 0.025 m−1 to −0.025 m−1 and relief from 1 m to 5 m for fixed hillslope lengths of 100 m. Relief (R) here refers to difference in elevation from the hillslope outlet to hillslope divide and is related to the slope of a hillslope (a) such that R = tan(a)L. A subset of 25 hillslopes was selected to characterize the range of reliefs and convergence rates (ac) of the 1018 random hillslopes such that R = 1 m, 2 m, 3 m, 4 m or 5 m and ac = 0.025 m−1, 0.0125 m−1, 0.0 m−1, −0.0125 m−1 or −0.02 5 m−1. CRFs for each of the 25 synthetic hillslopes were modeled using the nonlinear hillslope‐storage Boussinesq (hsB) model developed by Troch et al. [2003]. The CRF was obtained by bringing the storage profile in each hillslope to steady state using a constant recharge of 10 mm d−1 then allowing each hillslope to recess. The modeled recession provides (by definition) the CRF for a given hillslope. The dimensional moments (here, only first and second are considered) for each of the CRFs for each of the individual hillslopes were determined. In addition, the diffusive time constant (t K) and the hsPe number [Berne et
4 of 13
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
W03511
Figure 2. First and second dimensionless moments as a function of (a) volume‐weighted and (b) area‐ weighted caPe numbers for synthetic catchments comprising randomly generated 5, 10, 15,…, 400 hillslope configurations. The influence of hillslope resolution on volume‐weighted caPe numbers is shown for (c) randomly merging 50% of the total number of hillslopes and (d) merging all hillslopes into one catchment. Curves give the theoretical relationship between the first and second dimensionless moments of the CRF and the hsPe number by Berne et al. [2005]. al., 2005] was determined for each individual hillslope using equation (3) and equation (2), respectively. [16] The synthetic hillslopes were used to generate composite catchments consisting of 5 to 400 hillslopes at intervals of 5 hillslopes. This gave a total of 80 composite catchment types (catchments consisting of 5, 10, 15,…, 400 hillslopes). Within each catchment type, 5000 random catchment realizations were generated from the synthetic hillslopes by randomly selecting from the subset of 25 within fixed relief values. By fixing the relief value, it is ensured that all the hillslopes in a given catchment realization are similar in terms of relief (as one would expect in real‐world catchments). The 400,000 total catchments were, thus, representations of various geomorphic compositions. The dimensional first and second moments of each catchment were determined using volume‐weighted average of the individual hillslope moments (equation (6) for the first
moment and equation (8) for the second moment). In a similar manner, the volume‐weighted average diffusive time constant was determined for each catchment. These catchment diffusive time constants were used to convert the catchment dimensional moments into catchment dimensionless moments. [17] A caPe number was determined for each synthetic catchment taking the volume‐weighted average of the individual hsPe numbers (equation (9)). We can relate these caPe numbers to the first and second dimensionless moments of the catchment CRFs (Figure 2). There is good agreement with the theoretical relationship between the hsPe number and dimensionless moments presented by Berne et al. [2005] to that seen for the synthetic small catchments (Figure 2a). This synthetic approach demonstrates how a volume‐ weighted average of individual hsPe numbers does a good job in capturing the shallow subsurface hydrologic response
5 of 13
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
W03511
Figure 3. Site map showing Upper Sabino and Marshall Gulch catchments located in the Santa Catalina Mountains, United States. The stream channel shown is determined using At = 50,000 m2. The catchments are located at approximately 32°25′N latitude and 110°45′W longitude. of a catchment lending support to the use of equation (9) as a definition of the caPe number. Note that equation (9) is inherently dependent on the resolution at which hillslopes are represented. This dependency can be demonstrated using the synthetic catchments generated. For example, by randomly merging together 50% of hillslopes considered in each synthetic catchment, there is drastic change in the resulting caPe numbers and, thus, poor representation of the shallow subsurface hydrologic response of each catchment (Figure 2c). Continuing along the line, we can merge together 100% of the hillslopes considered in each synthetic catchment (Figure 2d) such that each catchment is represented as one large hillslope. This would be the same as defining the caPe number as the ratio of averages or taking a bulk or effective approach. Using such an approach leads to quite different caPe numbers which do not agree well with the dimensionless moments presented by Berne et al. [2005] (Figure 2d). This supports the use of equation (9) (an average of ratios) as a definition of the caPe number as opposed to a bulk or effective representation at the catchment scale (a ratio of averages).
3. Application of the Catchment‐Péclet Number [18] To test the above theory and demonstrate its use in a similarity framework, we test the caPe number using observed small catchment hydrologic responses that approximate CRFs. This is done for four small catchments ranging in size from 0.025 ha to 880 ha and spanning two distinct climatic and geologic settings (i.e., Santa Catalina Mountains, Arizona, United States, and Maimai, New Zealand). The moments of the hydrologic responses for each small catchment are compared to the ones predicted by the caPe number determined from the individual geometric characteristics of the hillslopes that make up each catchment. [19] It is often not possible to know a priori the volume of water expected to drain from all the hillslopes in a catch-
ment as they relax from a steady state storage profile. Keeping to a more practical development of the caPe number as a similarity parameter, we test the use of an area‐ weighted caPe number to represent the dimensionless moments of a catchment CRF (Figure 2b). The fraction of the total catchment volume of water draining from each hillslope (vi) in equation (9) is replaced with the area of each hillslope relative to the total area of the catchment. Using this area‐weighted average approach, there is scatter around the theoretical relationship of Berne et al. [2005] for caPe numbers lower than about 6 (indicated with a vertical line in Figure 2b). This scatter is due to the influence of convergence rate on the total volume of water draining from each synthetic hillslope. However, for caPe numbers greater than 6 this influence is minimal. Accepting this area‐weighting methodology, we can now look at how the caPe number performs in a real‐world setting using observed CRFs. 3.1. Observed Catchment Response: Santa Catalina Mountains, Arizona, United States [20] Lyon et al. [2008] report on the hydrologic response of the 880 ha Upper Sabino catchment located in the Santa Catalina Mountains northeast of downtown Tucson, Arizona (United States) (Figure 3) during a series of extreme rainfall events. Using hydrometric and isotopic data, Lyon et al. [2008] were able to estimate the event‐scale transit time distribution (TTD) for the catchment and the event water recession hydrograph. This event water recession hydrograph was determined from the recession limb of the hydrograph after the cessation of rainfall such that it is, primarily, influenced by shallow subsurface flows. On the basis of similarity between the event‐scale TTD and the event water recession and the extreme wet conditions present in the catchment, it is likely the event water recession hydrograph for this event can be used to approximate the CRF for the catchment [Lyon et al., 2008]. We use the three event water recession
6 of 13
W03511
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
Figure 4. Event water recession using three different hydrograph separation techniques (solid line for two‐component separation, dotted line for TRANSEP [Weiler et al., 2003] using dispersion distribution for transport, and dashed line for TRANSEP [Weiler et al., 2003] using exponential distribution for transport) for an event approximating the CRF of the Upper Sabino catchment (modified from Lyon et al. [2008]).
hydrographs reported by Lyon et al. [2008] as the starting point of a real‐world application of the caPe number (Figure 4). Note that there are three event water recession hydrographs for the storm event due to the use of three different hydrograph separation techniques (see Lyon et al. [2008] for details). Lyon et al. [2009] demonstrated that spatial differences in event water isotopic compositions of dD versus d18O make it difficult to accurately separate the runoff event hydrograph. Thus, the values reported by Lyon et al. [2008] may underrepresent rainfall in up to about 25% of the total area of the Upper Sabino catchment and should be considered at best approximations of the CRF for the catchment. The first and second dimensional moments of these recession hydrograph are taken to represent the CRF of the Upper Sabino catchment (Table 1). [21] Concurrent to the hydrometric and isotopic observations at Upper Sabino catchment for the event reported on by
Lyon et al. [2008], streamflow was monitored at a smaller, nested catchment (Figure 3). This catchment, hereafter referred to as the Marshall Gulch catchment, covers 150 ha in the southwestern portion of the Upper Sabino catchment. Isotopic data are not available for the Marshall Gulch catchment during the event considered by Lyon et al. [2008]. If we assume that the separation between event and preevent water observed at Upper Sabino catchment is the same as that expected at Marshall Gulch catchment, we can use the instantaneous separations from Lyon et al. [2008] to separate the hydrograph for Marshall Gulch and obtain an estimate of the CRF for this smaller nested catchment (Table 1). While this may not be considered a purely separate observation of a small‐catchment‐scale CRF, it provides some additional support to the assumption that Upper Sabino can be considered a small catchment where in‐channel processes have minimal influence over hydrologic response.
Table 1. Dimensional Moments for the Event Water Recession Hydrographs Reported by Lyon et al. [2008] Using Three Different Hydrograph Separation Techniques for Both Upper Sabino and Marshall Gulch Catchments Upper Sabino Catchment Hydrograph Separation Method Two‐component hydrograph separation TRANSEP [Weiler et al., 2003] using exponential distribution transport transfer function TRANSEP [Weiler et al., 2003] using dispersion distribution transport transfer function
Marshall Gulch Catchment
First Moment (min)
Second Moment (min2)
First Moment (min)
Second Moment (min2)
98 133
7,753 13,390
91 131
7,500 13,701
99
8,043
101
8,370
7 of 13
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
Figure 5. The Brutsaert‐Nieber recession analysis and resulting intercept value fitting a lower envelope excluding 10% of the data for the 2006–2007 monsoon period of the Upper Sabino (circles, observed recession; solid line, fit power function) and Marshall Gulch (crosses, observed recession; dashed line, fit power function) catchments. [22] It is necessary to determine several geometric properties and hydraulic parameters at the support scale of a catchment to estimate the characteristic diffusive time constant (equation (3)) needed to evaluate the dimensionless moments for the Upper Sabino and Marshall Gulch catchments. Following a procedure similar to that used by Lyon and Troch [2007], we performed a Brutsaert‐Nieber recession analysis [Brutsaert and Nieber, 1977] using observed flow data from the 2006–2007 monsoon seasons (July– October) for both Upper Sabino and Marshall Gulch catchments (Figure 5). This allows drainable porosity (f) for the catchment to be defined as a function of hydraulic conductivity (k), channel length (Lc), and catchment area (A) as defined by Lyon and Troch [2007]. [23] Chief et al. [2008] report a range of saturated hydraulic conductivities from about 2 × 10−4 m s−1 to about 2 × 10−5 m s−1 for soils across the Santa Catalina Mountains (including sites within the Upper Sabino catchment) on the basis of 66 cm3 undisturbed soil core samples. Ferré et al. [2006] used a modeled hydraulic conductivity of around 1 × 10−6 m s−1 to describe an ephemeral stream reach at the base of these mountains. Newman et al. [1998] reported a range of hydraulic conductivity values their semiarid field site near Los Alamos, New Mexico, United States, which has similar soil texture (sandy loam) and dominant vegetation type (pines) to those observed at the Upper Sabino and Marshall Gulch catchments [Lyon et al., 2008]. Measured saturated hydraulic conductivities ranged from 5.7 × 10−9 m s−1 to 7.5 × 10−7 m s−1 for the A and Bw horizons, 2.5 × 10−10 m s−1 for the Bt horizon, and 1.3 × 10−9 to 7.1 × 10−9 m s−1 for the CB horizon [Newman et al., 1998]. Obviously, a range of possible hydraulic conductivities can be found in semiarid soils both locally and regionally. As such, we use these literature reported values to help constrain the possible catchment‐ scale representation of hydraulic conductivity values of our study catchments. Catchment area (A), channel length
W03511
(Lc) and hillslope angle (a) were determined for both Upper Sabino and Marshall Gulch using topographic analysis on a 10 m digital elevation model [Bogaart and Troch, 2006]. Soil depth (D) of 1.5 m was selected from reported values as a representative value on the basis of several in‐field investigations for both catchments (M. Guardiola‐ Claramonte, personal communication, 2007). For the linearization parameter p, we adopt the method of Lyon and Troch [2007] and use a theoretical value of 0.30 that can be derived from exact solutions to the Boussinesq equation [Brutsaert, 1994]. Catchment hillslope length (L) was taken as the hillslope length assuming the drainage area for each catchment is one continuous hillslope wrapped around the channel network [Bogaart and Troch, 2006]. [24] To estimate the caPe number for both the Upper Sabino and Marshall Gulch catchment, we used the area‐ weighted average of the hsPe numbers (equation (2)) of the individual hillslopes making up each catchment (Table 2). For high caPe numbers such as those evaluated for Upper Sabino and Marshall Gulch, the influence of using area weighting versus volume weighting is minimal (Figure 2b). Hillslopes were delineated and defined using the techniques given by Bogaart and Troch [2006]. The number of hillslopes delineated for either catchment depends on the definition of the channel network. Using an accumulated area threshold (At) on a flow routing map to define the channel network, we selected a threshold such that the resulting channel network matches well with field observations of the actual channel network (At = 50,000 m2). This resulted in 195 hillslopes for the Upper Sabino catchment and 42 hillslopes for the Marshall Gulch catchment. To address the influence of this method for defining the channel network, we also conducted hillslope delineation assuming an accumulated area threshold using twice (At = 100,000 m2) and half (At = 25,000 m2) this selected value. This resulted in 356 and 115 hillslopes for At = 25,000 m2 and At = 100,000 m2, respectively, for the Upper Sabino catchment and resulted in 86 and 23 hillslopes for At = 25,000 m2 and At = 100,000 m2, respectively, for the Marshall Gulch catchment. This allows us to calculate a range of caPe numbers for each catchment. Soil depth (D) and the linearization parameter (p) were taken as the same values used to determine the catchment characteristic diffusive time constant. Hillslope area (Ah), angle (a) and length (L) of each individual hillslope in both catchments was determined using topographic analysis on a 10 m digital elevation model [Bogaart and Troch, 2006]. The hillslope convergence rate (ac) for each individual hillslope was determined assuming exponential (ac ≠ 0) or uniform (ac = 0) width functions. When using an exponential width function, hillslope outlet width (Lh) was fixed and ac was adjusted while preserving its relation to hillslope area, Ah = Lh(exp(acL) 1)/ac [Troch et al., 2003; Berne et al., 2005]. [25] The first and second dimensionless moment and the caPe numbers for both Upper Sabino and Marshall Gulch catchments provide two real‐world small catchment tests for the utility of the caPe number (Figure 6). These are compared to the theoretical relationship given by Berne et al. [2005]. The center of each cross in Figure 6 represents the caPe number determined when delineating hillslopes for a channel network defined using a threshold accumulated area of 50,000 m2 and a value of hydraulic conductivity (k = 9.3 × 10−8 m s−1) simultaneously fitted to both the
8 of 13
W03511
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
Table 2. Variation in Catchment Average Geometric Properties and caPe Numbers Based on the Accumulated Area Threshold At Used to Delineate Channel Network for Upper Sabino and Marshall Gulch Catchments Upper Sabino Catchment
Marshall Gulch Catchment
Property
At = 25,000 m2
At = 50,000 m2
At = 100,000 m2
At = 25,000 m2
At = 50,000 m2
At = 100,000 m2
Channel length (m) Number of hillslopes Catchment average hillslope length (m) caPe number
7,411 356 640
5,895 195 660
4,672 115 760
1,236 86 500
805 42 580
634 23 760
109
141
190
104
153
203
theoretical first and second moment. This fitted hydraulic conductivity translates into an extremely low drainable porosity (6.6 × 10−7) for these soils. While this fit hydraulic conductivity value is lower than the range reported by Chief et al. [2008] for this area, it falls within the range of hydraulic conductivities reported by Newman et al. [1998]. For comparison, the dimensionless first moment estimated adopting hydraulic conductivity of k = 1 × 10−6 m s−1 [Ferré et al., 2006] leads to a first moment 3 times larger than that estimated fitting hydraulic conductive. Adopting the highest values from Chief et al. [2008] of k = 2 × 10−4 m s−1 increases the first moment by 47 times. Clearly, there is sensitivity to the value used to define hydraulic conductivity. [26] Note the upper and lower bounds of each cross are the maximum and minimum dimensionless first and second moments defined using three CRFs estimated for both catchments on the basis of the hydrograph separation made by Lyon et al. [2008]. The left and right bounds of each cross are the caPe numbers determined for channels defined using At = 25,000 m2 and At = 100,000 m2, respectively, in each catchment.
Figure 6. (a) Estimated first and second dimensionless moments from observed data related to the caPe number for Upper Sabino and Marshall Gulch catchments (black markers) in Santa Catalina Mountains, United States, and Maimai catchments 1 and 2 (gray markers) in New Zealand. Upper and lower bounds of the black markers refer to the range of moments dependent on the CRF selected from Figure 4 for Upper Sabino catchment. The left and right bounds of the black markers show the range of caPe numbers determined by changing the threshold accumulated area used to determine channel network and thus to delineate hillslopes for Upper Sabino and Marshall Gulch catchments. (b) Zooming in on observed data.
3.2. Observed Catchment Response: Maimai, New Zealand [27] In a second real‐world test of the caPe number, we use observations from the Maimai experimental hillslopes (Figure 7). The Maimai hillslopes and research catchments, located on the west coast of the South Island of New Zealand, have been host to numerous hydrological studies of the past 30 years (see McGlynn et al. [2002] for a thorough description of the various subcatchments and studies). Here we focus on runoff data collected at the base of the trenched experimental hillslopes using a trough system reported on by Woods and Rowe [1996]. Recently, Lyon and Troch [2007] used this data as an approximation of the CRF for each of six individual hillslopes in testing the applicability of the hsPe number. As the caPe number presented in the current study is based on this hsPe number, it is only natural to return to this Maimai data set of Woods and Rowe [1996] as a test for the applicability of the caPe number for characterizing the hydrologic response of small catchments. [28] As such, we combine the hydrologic response for the individual hillslopes at Maimai to create two individual small catchments. This grouping into two small catchments occurs quite naturally owing to the differences in hydrologic response between hillslopes 1 through 4 compared to hillslopes 6 through 7 [Lyon and Troch, 2007]. We can treat the observed outflow from troughs 1 through 20 as one individual small catchment (Maimai catchment 1) and the observed outflow from troughs 22 through 30 as a second
9 of 13
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
W03511
the small‐catchment representation of the Maimai hillslopes taken in this study. [29] To determine the caPe numbers for Maimai catchments 1 and 2, we use the area‐weighted average of the corresponding hsPe numbers reported by Lyon and Troch [2007] (Table 3). This is a clear demonstration of the utility of generating the caPe number from the individual hsPe numbers. Due to the clear definition of hillslopes in the Maimai catchments 1 and 2, there are no estimates of ranges of possible caPe numbers. In addition, since the trenched hillslopes allows for a direct observation of shallow subsurface recession, there is no estimated range of CRF moments due to variations in hydrograph separations. The combination of this area‐weighted caPe number and the above discussed dimensionless moments of the CRF for Maimai catchments 1 and 2 provide two more real‐world small catchment tests for the utility of the caPe number (Figure 6).
4. Discussion
Figure 7. Site map showing Maimai catchments 1 and 2 located in New Zealand. The catchments are located at approximately 45°05′S latitude and 171°48′E longitude.
individual small catchment (Maimai catchment 2) (Figure 7 and Table 3). The total area for these two small catchments are 0.17 ha and 0.025 ha for Maimai catchment 1 and Maimai catchment 2, respectively. Summing the observed outflow from these troughs over the same event considered to approximate the CRF as used by Lyon and Troch [2007], we obtain an event for each small catchment that approximates the CRF. Note that these small catchments perfectly match our previously stated definition of “small” as they have no effect of channel since, for this trenched setup, there is no channel network. Also note that here the small catchment response for Maimai catchments 1 and 2 is defined by summing the observed outflow from the two catchments and not by taking some averaging of the individual hillslope responses. Following the methodology from above and that presented by Lyon and Troch [2007], the first and second dimensional moments for Maimai catchments 1 and 2 can be obtained (Table 3). These dimensional moments can be made dimensionless using a characteristic diffusive time constant (equation (3)) on the basis of several geometric properties and hydraulic parameters. Lyon and Troch [2007] cover this procedure extensively (and it is similar to that reported above for the sites in the Santa Catalina Mountains) for the Maimai sites. The same analysis holds for determining the geometric properties and hydraulic parameters and the resulting characteristic diffusive time constant for
[30] The study builds on the previous work developing the hsPe number [Berne et al., 2005]. Berne et al. [2005] linked the geometry of a hillslope analytically to the dimensionless moments of its characteristic response function. Recent work [Lyon and Troch, 2007] has shown the applicability of such a relationship as a possible similarity parameter in a real‐world setting. In this study, we extend the application of a hsPe number as a similarity parameter to describe subsurface flow at the small‐catchment scale. In order to accomplish this, a catchment is first broken down into a number of hillslopes. The caPe number is determined by taking the weighted average of the individual hsPe number of each hillslope. This caPe number works as a similarity parameter as it is able to reproduce the dimensionless moments of the composite characteristic response function of the catchment. Synthetic analysis (Figure 2) shows that this weighted average approach works well to determine a caPe number as long as the relative relief of the hillslopes making up a composite catchment is similar. The impact of this assumption from a practical point of view is likely to be minimal in many real‐life small catchments where there exist similar geomorphology and formation processes. Thus, one should not expect (relatively speaking) huge variation in hillslope relief at (by definition) the small‐ catchment scale. [31] As it is proposed in this study, the caPe number does not explicitly consider variations in rainfall or vegetation patterns with respect to their possible influence on the CRF at the small‐catchment scale. In general, rainfall and vegetation Table 3. Summary of Catchment Hydrologic Responses and caPe Numbers for Maimai Catchments 1 and 2 Property Trough numbers from Woods and Rowe [1996] Hillslope numbers from Lyon and Troch [2007] Area (m2) First moment (h) Second moment (h2) caPe number
10 of 13
Maimai Catchment 1
Maimai Catchment 2
1–20
22–30
1–4
6–7
1,674 6.7 73.2 87
246 4.7 40.0 48
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
patterns, along with riparian and channel network structure, may exert a control on the travel time of water through river basins. Variations in the patterns of rainfall and vegetation could be leading to differences with respect to the storage and release of water within individual hillslopes. This could impart nonlinearity on hydrological responses or the travel time of water moving through a catchment stemming from differences in water entering and exiting the hillslope storage volumes under different initial and/or internal conditions. This is because of the nonlinear dependence of the output fluxes on the storage. Counter to this nonlinearity, the caPe approach presented in this study is not investigating catchment‐scale transit times and their distributions, per se, but considering an end‐member case of a characteristic response function (CRF). As such, these nonlinearity effects are minimal or nonexistent. In addition, the hillslope‐scale parameterizations used in this study can be considered representative of a prescribed state of the ecosystem during a given season. By working at the event scale and approximate steady state storage conditions (i.e., the CRF), intra‐annual variations of climatic inputs or vegetation patterns may be negligible. [32] The magnitude of the caPe number is influenced by the geomorphology of the catchment (Figure 6). There is a direct connection between the (delineated) extent of the catchment channel network determined using topographic analysis and the number of hillslopes represented in the catchment. As we increase the extent of the channel network (lower the threshold accumulated area used to delineate the channel network), there is a corresponding increase in the number of hillslopes in a catchment (Table 2). Holding all other catchment properties (e.g., soil depth, average slope) constant, this corresponds to a decrease in the caPe number owing (primarily) to shorter hillslope lengths and more first‐ order stream reaches. The opposite holds for a decrease in the extent of the channel network. While quantitative methods exist to determine the extent of the channel network from topographic analysis using the relationship between slope and accumulated area [e.g., Tarboton et al., 1991; Stock and Dietrich, 2003], visual inspection comparing actual to delineated channel network extent still remains a popular and effective method. Regardless, it is possible to define a range of caPe numbers (Figure 6) that takes into account variations in the definition of what is a hillslope if data is not available for determine the actual extent of the network. [33] The use of the caPe number as a similarity parameter to predict the CRF suffers from the uncertainty associated with defining hydrologic parameters (the bane of modern scientific hydrology). Estimates on a catchment’s hydrologic parameters directly influence the definition of the diffusive time constant (t K) and, thus, the moments of the CRF. The caPe number can be defined regardless of hydrologic parameters. This allows for the relative comparison of catchment responses where as the estimation of hydrologic parameters allows for the absolute gauging of catchment response. As such, the caPe number has promise as a classification scheme for grouping similar small catchments on the basis of the hydrological response. The uncertainty of knowing the hydrologic properties (specifically, drainable porosity and hydraulic conductivity) of a catchment could be accounted for by accepting a range of values. The accuracy and appropriateness of such measurements is (always)
W03511
questionable. In this study, we remove one degree of freedom by defining drainable porosity and hydraulic conductivity is established using a Brutsaert‐Nieber analysis (similar to the methodology adopted by Lyon and Troch [2007]). [34] The applicability of this commonly used technique has come under investigation in sloping aquifers by recent work of Rupp and Selker [2006]. Their work, based on numerical analysis, suggests that this technique may not be appropriate for settings where slope is an important driver of flow or hydraulic conductivity varies greatly with depth. They present an alternative empirical late time solution to the Boussinesq equation for a sloping aquifer with decreasing hydraulic conductivity with soil depth [see Rupp and Selker, 2006, Figure 3, parameter set xix]. Adopting their empirical relation to the observed relationship between dQ/dt and Q at, for example, the Upper Sabino catchment (Figure 5), there would be a corresponding 1/5 decrease in drainable porosity as a function of hydraulic conductivity (analysis not shown) leading to a fivefold increase in the first dimensionless moment and a 25‐fold increase in the second dimensionless moment of the CRF for Upper Sabino catchment. Regardless of the explanation used to interpret a dQ/dt versus Q plot, we can relate drainable porosity to hydraulic conductivity at the catchment scale. [35] The approach presented in this work and previous work from Troch et al. [2004], Berne et al. [2005], Lyon and Troch [2007] builds on a linearized Bousinessq assumption similar to that used by Brutsaert [1994]. Rupp and Selker [2006] have commented on shortcomings with the use of such a linearization parameter (p). Traditionally, this parameter is accepted to be within a narrow range of values for an initially saturated horizontal aquifer ranging from 1/3 to 1/2 [Brutsaert, 1994]. However, it is possible to develop similarity approaches without the linearization of the Bousinessq equation. Recent work by Harman and Sivapalan [2009b] shows the development of a characteristic aquifer thickness based on dimensional similarity. This thickness is proportional to both the hillslope number of Brutsaert [1994] and hsPe number of Berne et al. [2005]. The linearization assumption used to develop the hsPe (and therefore caPe) number does not appear to have a significant influence over the representation of the nonlinear CRF or its moments. [36] Direct observation of the CRF at the small‐catchment scale is difficult. It requires a combination of near steady state storage conditions and a monitoring network capable of isolating the shallow subsurface hydrologic response for the catchment. The four small‐catchment hydrologic responses used in this study to approximate small catchment CRFs draw upon to quite unique data sets. The first from the Santa Catalina Mountains, Arizona, United States, reported on by Lyon et al. [2008] was the result of an unprecedented rainfall event in the area co‐occurring with hydrometric and isotopic observations. Even with this data, there is difficulty in accurately obtaining the “true” event water recession hydrograph [Lyon et al., 2009]. For the second set of small catchments in Maimai, New Zealand, Woods and Rowe [1996] needed to trench a considerable length of hillslope to measure subsurface runoff. From this perspective, it is difficult to directly validate the caPe number using real‐ world data. However, the sites considered in this study demonstrate the strength of the caPe number as a method for intercatchment comparisons. While differences between the caPe numbers at Upper Sabino compared to Marshall Gulch
11 of 13
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
catchments is rather small, there is a large difference between the Santa Catalina Mountain catchments when compared to the Maimai, New Zealand catchments. Such a classification scheme allows for identification of likely small catchment hydrologic responses based on geometry of the composite hillslopes. [37] Perron et al. [2009] have recently demonstrated that a characteristic length scale for hillslopes based on a Pe number derived from an advection‐dispersion erosion model can be used to form a basis for landscape evolution models. As such, there is a strong physical meaning from geomorphology/hydrology for the development of the catchment Pe number as a similarity parameter. The caPe number can allow for comparisons to be made on a theoretical basis using the CRF as a common denominator regardless of the difficulty associated with observation of a “real” CRF. In other words, the CRF can be viewed as an end‐member of the possible hydrologic responses of a small catchment. This makes it a good basis for the intercomparison of small catchments. In addition, Harman and Sivapalan [2009b] lay out a framework using regime analysis of Robinson and Sivapalan [1997] that allows for extension of similarity parameters such as the hsPe and caPe number when observations or calibration data are inadequate. So, while this study applies the caPe number to a limited number of observed, approximate small catchment CRFs, it should be possible to extend such analysis to conditions and hydrological responses which result from temporally variable recharge leading to subsurface lateral hillslope flow.
5. Concluding Remarks [38] Using the area‐weighted average of the hsPe number, we are able to define a small catchment‐scale similarity parameter. This caPe number is shown to be linked to the dimensionless moments of the characteristic response function (CRF) of a small catchment using the same theoretical relationship as that given by Berne et al. [2005]. Using 400,000 synthetically derived small catchments, we have demonstrated that there is good agreement between the caPe number and the first two dimensionless moments of the CRF. In addition, the caPe number is able (allowing for uncertainty in the definition of hydrological parameters and channel extent) to recreate the observed moments of the catchment‐scale CRF for four small catchments ranging in size from 0.025 ha to 880 ha and spanning two distinct climatic and geologic settings. The ability of the caPe number to represent and intercompare small catchment shallow subsurface hydrologic response makes it a valuable addition to the hydrologist’s toolbox. This is especially true in light of the vastness of Aqua Incognita mapped out by Bishop et al. [2008]. [39] Acknowledgments. The authors are grateful for support from the National Science Foundation’s Science and Technology Center (STC) program through the Center for Sustainability of semi‐Arid Hydrology and Riparian Areas (SAHRA) under agreement EAR‐9876800 from the Hydrology and Water Resources Department at the University of Arizona.
References Aryal, S. K., E. M. O’Loughlin, and R. G. Mein (2005), A similarity approach to determine response times to steady‐state saturation in landscapes, Adv. Water Resour., 28(2), 99–115, doi:10.1016/j.advwatres.2004.10.008.
W03511
Berne, A., R. Uijlenhoet, and P. A. Troch (2005), Travel time distributions of subsurface flow along complex hillslopes with exponential width functions, Water Resour. Res., 41, W09410, doi:10.1029/2004WR003629. Beven, K. J., and M. J. Kirkby (1979), Towards a simple, physically based, variable contributing area model of catchment hydrology, Int. Assoc. Hydrol. Sci. Bull., 24, 43–69. Bishop, K., I. Buffam, M. Erlandsson, J. Fölster, H. Laudon, J. Seibert, and J. Temnerud (2008), Aqua Incognita: The unknown headwaters, Hydrol. Processes, 22, 1239–1242, doi:10.1002/hyp.7049. Blöschl, G., and M. Sivapalan (1995), Scale issues in hydrological modeling—A review, Hydrol. Processes, 9, 251–290, doi:10.1002/ hyp.3360090305. Bogaart, P. W., and P. A. Troch (2006), Curvature distribution within hillslopes and catchments and its effect on the hydrological response, Hydrol. Earth Syst. Sci., 10, 925–936. Bronstert, A. (1994), 1‐, 2‐, and 3‐dimensional modeling of the water dynamics of agricultural sites using a physically based modeling system “Hillflow”, in Advances in Water Resources Technology and Management, edited by G. Tsakiris and M. A. Santos, pp. 77–84, Balkema, Rotterdam, Netherlands. Brutsaert, W. (1994), The unit response of groundwater outflow from a hillslope, Water Resour. Res., 30(10), 2759–2763, doi:10.1029/94WR01396. Brutsaert, W. (2005), Hydrology: An Introduction, 605 pp., Cambridge Univ. Press, New York. Brutsaert, W. (2008), Long‐term groundwater storage trends estimated from streamflow records: Climatic perspective, Water Resour. Res., 44, W02409, doi:10.1029/2007WR006518. Brutsaert, W., and J. L. Nieber (1977), Regionalized drought flow hydrographs from a mature glaciated plateau, Water Resour. Res., 13(3), 637–643, doi:10.1029/WR013i003p00637. Chief, K., T. P. A. Ferré, and B. Nijssen (2008), Correlation between air permeability and saturated hydraulic conductivity in unburned and burned desert soils, Soil Sci. Soc. Am. J., 72(6), 1501–1509, doi:10.2136/ sssaj2006.0416. D’Odorico, P., and R. Rigon (2003), Hillslope and channel contributions to the hydrologic response, Water Resour. Res., 39(5), 1113, doi:10.1029/ 2002WR001708. Dooge, J. C. I. (1986), Looking for hydrologic laws, Water Resour. Res., 22(9S), 46S–58S, doi:10.1029/WR022i09Sp0046S. Ferré, T. P. A., A. C. Hinnell, and J. B. Blainey (2006), Examining the potential for inferring hydraulic properties using surface‐based electrical resistivity during infiltration, Leading Edge, 25(6), 720–723, doi:10.1190/1.2210055. Gardiner, V. (1979), Estimation of drainage density from topological variables, Water Resour. Res., 15(4), 909–917, doi:10.1029/ WR015i004p00909. Harman, C. J., and M. Sivapalan (2009a), Effects of hydraulic conductivity variability on hillslope‐scale shallow subsurface flow response and storage‐discharge relations, Water Resour. Res., 45, W01421, doi:10.1029/ 2008WR007228. Harman, C. J., and M. Sivapalan (2009b), A similarity framework to assess controls on shallow subsurface flow dynamics, Water Resour. Res., 45, W01417, doi:10.1029/2008WR007067. Hilberts, A. G. J., E. E. van Loon, P. A. Troch, and C. Paniconi (2004), The hillslope‐storage Boussinesq model for non‐constant bedrock slope, J. Hydrol., 291, 160–173, doi:10.1016/j.jhydrol.2003.12.043. Lyon, S. W., and P. A. Troch (2007), Hillslope subsurface flow similarity: Real‐world tests of the hillslope Péclet number, Water Resour. Res., 43, W07450, doi:10.1029/2006WR005323. Lyon, S. W., S. L. E. Desilets, and P. A. Troch (2008), Characterizing the response of a catchment to an extreme rainfall event using hydrometric and isotopic data, Water Resour. Res., 44, W06413, doi:10.1029/ 2007WR006259. Lyon, S. W., S. L. E. Desilets, and P. A. Troch (2009), A tale of two isotopes: Differences in hydrograph separation for a runoff event when using dD vs. d 18 O, Hydrol. Processes, 23, 2095–2101, doi:10.1002/ hyp.7326. McGlynn, B. L., J. J. McDonnell, and D. D. Brammer (2002), A review of the evolving perceptual model of hillslope flowpaths at the Maimai catchments, N. Z. J. Hydrol., 257, 1–26, doi:10.1016/S0022-1694(01) 00559-5. Morisawa, M. E. (1962), Quantitative geomorphology of some watersheds in the Appalachian Plateau, Geol. Soc. Am. Bull., 73, 1025–1046, doi:10.1130/0016-7606(1962)73[1025:QGOSWI]2.0.CO;2. Morisawa, M. E. (1985), Rivers: Forms and Processes, Longman, New York.
12 of 13
W03511
LYON AND TROCH: CATCHMENT SIMILARITY INDEX FOR SUBSURFACE FLOW
Nadeau, T. L., and M. C. Rains (2007), Hydrological connectivity between headwater streams and downstream waters: How science can inform policy, J. Am. Water Resour. Assoc., 43, 86–103, doi:10.1111/j.17521688.2007.00010.x. Newman, B. D., A. R. Campbell, and B. P. Wilcox (1998), Lateral subsurface flow pathways in a semiarid ponderosa pine hillslope, Water Resour. Res., 34(12), 3485–3496, doi:10.1029/98WR02684. Paniconi, C., and E. Wood (1993), A detailed model for simulation of catchment scale subsurface hydrologic processes, Water Resour. Res., 29(6), 1601–1620, doi:10.1029/92WR02333. Paniconi, C., P. A. Troch, E. E. van Loon, and A. Hilberts (2003), The hillslope‐storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 2. Numerical testing with 3D Richards equation model, Water Resour. Res., 39(11), 1317, doi:10.1029/ 2002WR001730. Perron, J. T., J. W. Kirchner, and W. E. Dietrich (2009), Formation of evenly spaced ridges and valleys, Nature, 460, 502–505, doi:10.1038/ nature08174. Robinson, J. S., and M. Sivapalan (1997), Temporal scale and hydrological regimes: Implications for flood frequency scaling, Water Resour. Res., 33(12), 2981–2999, doi:10.1029/97WR01964. Rodríguez‐Iturbe, I., and J. B. Valdés (1979), The geomorphologic structure of hydrologic response, Water Resour. Res., 15(6), 1409–1420, doi:10.1029/WR015i006p01409. Rupp, D. E., and J. S. Selker (2006), On the use of the Boussinesq equation for interpreting recession hydrographs from sloping aquifers, Water Resour. Res., 42, W12421, doi:10.1029/2006WR005080. Sivapalan, M., K. Beven, and E. F. Wood (1987), On hydrologic similarity: 1. A scaled model of storm runoff production, Water Resour. Res., 23(11), 2266–2278. Sivapalan, M., E. F. Wood, and K. J. Beven (1990), On hydrologic similarity: 3. A dimensionless flood frequency model using a generalized geomorphologic unit‐hydrograph and partial area runoff generation, Water Resour. Res., 26(1), 43–58.
W03511
Stock, J., and W. E. Dietrich (2003), Valley incision by debris flows: Evidence of a topographic signature, Water Resour. Res., 39(4), 1089, doi:10.1029/2001WR001057. Strahler, A. N. (1952), Hypsometric (area‐altitude) analysis of erosional topography, Geol. Soc. Am. Bull., 63, 1117–1142, doi:10.1130/00167606(1952)63[1117:HAAOET]2.0.CO;2. Tarboton, D. G., R. L. Bras, and I. Rodriguez‐Iturbe (1991), On the extraction of channel networks from digital elevation data, Hydrol. Processes, 5, 81–100, doi:10.1002/hyp.3360050107. Troch, P. A., C. Paniconi, and E. E. van Loon (2003), Hillslope‐storage Boussinesq model for subsurface flow and variable source areas along complex hillslopes: 1. Formulation and characteristic response, Water Resour. Res., 39(11), 1316, doi:10.1029/2002WR001728. Troch, P. A., A. H. van Loon, and A. G. J. Hilberts (2004), Analytical solution of the linearized hillslope‐storage Boussinesq equation for exponential hillslope width functions, Water Resour. Res., 40, W08601, doi:10.1029/2003WR002850. Weiler, M., B. L. McGlynn, K. J. McGuire, and J. J. McDonnell (2003), How does rainfall become runoff? A combined tracer and runoff transfer function approach, Water Resour. Res., 39(11), 1315, doi:10.1029/ 2003WR002331. Woods, R. (2005), Hydrologic concepts of variability and scale, in Encyclopedia of Hydrological Sciences, edited by M. G. Anderson, pp. 23–40, John Wiley, Chichester, U. K. Woods, R., and L. Rowe (1996), The changing spatial variability of subsurface flow across a hillside, J. Hydrol., 35, 51–86. S. W. Lyon, Physical Geography and Quaternary Geology, Stockholm University, SE‐106 91 Stockholm, Sweden. (
[email protected]) P. A. Troch, Hydrology and Water Resources, University of Arizona, JW Harshbarger 320A, Tucson, AZ 85721, USA. (
[email protected])
13 of 13
