Space-time variability of low streamflows in river networks - CiteSeerX

WATER RESOURCES RESEARCH, VOL. 36, NO. 9, PAGES 2679 –2690, SEPTEMBER 2000

Space-time variability of low streamflows in river networks Peter R. Furey Department of Geological Sciences, Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder

Vijay K. Gupta Department of Civil and Environmental Engineering, Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder

Abstract. A low-flow equation is derived from Darcy’s law and a mass balance equation for base flow from a hillside by assuming that groundwater recharge is constant over short time periods. This low-flow equation reduces to many of the existing mechanistic low-flow equations, but in its general form looks like a linear regression model and captures spatial variability in base flow among hillsides in a basin. As a result, it unifies the mechanics and statistics of low flows in river networks. This equation is tested against low-flow data from the Flint River basin in Georgia and the Gasconade River basin in Missouri, both of which are approximately 7000 km2 in size.

1.

Introduction

Low flows are generated largely by subsurface runoff from hills in river basins. Past efforts to understand low flows have used either purely statistical or purely mechanistic approaches. The former are derived from multiple regression analyses which correlate low-flow discharge with the physiographic characteristics of a basin or collection of basins [e.g., Carpenter and Hayes, 1996]. The later approaches use a simple physical model of phreatic groundwater movement through and out of a hill which is scaled deterministically to encompass a river basin [e.g., Troch et al., 1993]. These later approaches do not model low-flow fluctuations and thus are unable to characterize temporal and spatial variability inherent in low-flow data. By contrast, statistical equations are site-specific, and their physical foundations, if any, remain unclear. These issues point to a great need to develop a low-flow theory which connects low-flow statistics and physics. The development of such a theory is the main focus of this paper. It parallels a newly emerging approach to understanding space-time variability in peak flows [Gupta and Waymire, 1998]. The physical basis of our theory is a mass balance equation for base flow through a hillside [Gelhar and Wilson, 1974]. This equation is simplified, as supported by data, to equate the mean base flow from a hillside over a short time interval s with initial base flow from a hillside. Extensions to a river basin from a single hillside can be accomplished by viewing a river basin as a collection of hillsides along a channel network. An expression for base flow or “low flow” from a basin is derived by assuming that saturated hydraulic conductivity and mean hydraulic head vary randomly in space among hillsides, while all other hillside parameters are held constant. Under this assumption a low-flow equation is developed for a basin from the base flow equation for a hillside using the Central Limit Theorem. Like other low-flow equations, this equation expresses low flow as a function of saturated hydraulic conductivity, drainage density squared, and drainage basin area. UnCopyright 2000 by the American Geophysical Union. Paper number 2000WR900136. 0043-1397/00/2000WR900136$09.00

like other equations, it is not based on a steady state solution because low flows never reach steady state. Moreover, it is expressed as a linear regression model and therefore provides a framework for a statistical treatment of low-flow data. Our low-flow equation is developed for a river network and can be used for a spatial analysis of low-flow data. By contrast, the current regional approaches to low-flow analysis are not restricted to a network. It is shown that our equation should only be applied to unnested basins and should not be applied to quantiles, as is typical in existing regional low-flow studies. The reason for this second constraint is that quantiles, by definition, do not account for the time at which data are recorded. Thus corresponding quantile data from different stream gauges in a river network are not necessarily recorded at the same time, and unnesting this type of data can produce negative flow values. In the present study, streamflow data for different gauges are used that are recorded at the same time. Tests of our low-flow equation are carried out using data from two basins in the United States with drainage areas that are approximately 7000 km2. Results show that theoretical findings are supported by data. Section 2 reviews both the steady state and transient state physically based equations that have been developed and tested against low-flow data. It also gives a comparison between these equations and empirically based statistical regression equations. Section 3 introduces our low-flow equation, which, unlike other low-flow equations in the literature, is shown to explain some of the temporal and spatial variability in low-flow data. Section 4 describes the data used to test this low-flow equation, and section 5 presents and discusses analytical test results. It is shown that the equation is broadly consistent with data and that theoretical findings are qualitatively correct. Finally, section 6 summarizes the results of this research and presents some ideas for future research.

2.

Background

Various approaches to low-flow analysis have been developed using either (1) mechanistic steady state equations, (2) regression equations, or (3) mechanistic transient state equations. We will briefly review the highlights of each of these

2679

2680

FUREY AND GUPTA: SPACE-TIME VARIABILITY OF LOW STREAMFLOWS

Q ⫽ wal ⫽

Figure 1. A cross-sectional profile of an idealized hill.

approaches and point out important similarities and differences among them. Furey [1996] gives a more detailed discussion of the review below. The mechanistic theories, and the theoretical developments given in section 3, come from a model of one-dimensional groundwater flow from a hillside depicted in Figure 1. This figure displays the cross-sectional profile of an unconfined aquifer resting on a horizontal impermeable boundary. The aquifer is bordered on each side by a stream and is placed in a coordinate system where the origin is beside the left edge of the aquifer near the left-hand stream. Let the distance x from the origin to the right side of the aquifer be 2a, and the drainage divide be in the middle at x ⫽ a. To keep this model simple, assume that the aquifer has a depth l measured into the page and that flow lines do not change with respect to l. This parameter is later called the hillslope length. Also, assume that the aquifer consists of an isotropic and homogeneous material. We will use this model to introduce some notation and basic concepts for the review and for later on in section 3. 2.1.

Steady State Equations

A steady state expression for base flow through the hillside depicted in Figure 1 can be derived following Jacob [1943] and Gelhar and Wilson [1974]. Denote steady state recharge per unit area as w. Given Darcy’s law and the Dupuit-Forchheimer approximation, steady state discharge for a hillside per unit length of stream at a distance x from the stream, q( x), can be expressed as q共 x兲 ⫽ ⫺kh共 x兲

dh共 x兲 ⫽ w共 x ⫺ a兲 dx

x ⬍ a.

(1)

Here k is saturated hydraulic conductivity, and h( x) is the hydraulic head at the surface of the water table measured a distance x from the stream. Integrating this equation gives h共 x兲 ⫺ h共0兲 ⫽

wx共2a ⫺ x兲 2kb

(2)

where h(0) is the hydraulic head at x ⫽ 0, and b ⬅ b( x) ⫽ (h( x) ⫹ h(0))/ 2 is a constant which approximates the mean saturated depth of the aquifer. To write this equation in terms of base flow discharge, define the mean hydraulic head across the hillside from 0 to a as h៮ ⬅ 1/a 兰 a0 h( x) dx, and substitute (2) into this expression. Evaluating the integral shows that base flow discharge from a hillside is

3kb⌬hl a

(3)

where ⌬h ⬅ h៮ ⫺ h(0) and l is the length of the hillslope measured along the bordering stream. Steady state base flow equations such as (3) have been scaled deterministically to produce low-flow equations for drainage basins. To understand how, let m denote the number of firstorder streams in a basin or, equivalently, basin magnitude. Then, following Shreve [1967], the total number of links in the basin is (2m ⫺ 1), and the total number of hillsides is n ⫽ 2(2m ⫺ 1) since each link has two hills draining into it. Furthermore, the total channel length above a junction in the basin is L ⫽ l(2m ⫺ 1), drainage basin area is A B ⫽ 2al(2m ⫺ 1) ⫽ nal, and drainage density can be defined as D ⫽ L/A B ⫽ 1/ 2a [Horton, 1945]. To scale an equation like (3) from a hill to a drainage basin, base flow from a basin is represented as Q B ⫽ nQ. Substituting this expression and the relations for A B and D above into (3) gives a base flow equation for a basin that can be written in a generic form as Q B ⫽ Ckb⌬hD 2A B

(4)

where C is a constant and ⌬h represents hydraulic head difference. Several versions of this equation have been published in the literature; for example, see Dingman [1978], Zecharias and Brutsaert [1988], and Troch et al. [1993] to mention a few. The value of C and the definitions of ⌬h and b vary from one version to another depending on specific model assumptions. In some versions, ⌬h ⫽ h(a) ⫺ h(0) [Dingman, 1978], while in others ⌬h ⫽ h៮ ⫺ h(0) as in (3). In some versions the parameter b is assumed to be a fixed constant [Dingman, 1978], while in others it is taken to be equal to h៮ [Troch et al., 1993]. Carlston [1963] formulated a low-flow expression from Jacob’s [1943] equation as well. However, as Dingman [1978] observed, there is a conceptual error in this formulation because steady state discharge per unit length is equated to hydraulic transmissivity rather than to recharge rate per unit length. Many observations from field studies are found to be consistent with (4), but not all. Browne [1981] investigated the relationship between low flows and subsurface geology and observed that changes in subsurface geology, from granite to slate to sandstone, significantly influenced low-flow discharge. A similar observation was made by Trainer [1969]. These results can be understood from (4) which shows that discharge depends on saturated hydraulic conductivity, a parameter that varies with aquifer material. Gregory and Walling [1968] studied the relationship between low flows and active drainage density D a and found that for minimum daily flows Q B /A B ⬀ D 2.2 in a close agreement with (4). However, Gregory and Walling also found that for six river basins in Devon, England, the relationship between Q B /A B and D a depended on the timescale by which minimum flow was defined. It was observed that Q B /A B ⬀ D 0.8 when Q B /A B was the minimum daily mean flow per a area for October 1964. Yet it was also observed that Q B /A B ⬀ D ⫺2.5 when Q B /A B was the mean monthly minimum flow per a area for a year. Carlston’s results for minimum monthly flows averaged over 6 years showed that Q B /A B ⬀ D ⫺2 for 13 basins in the eastern United States. Similar results were obtained by Trainer [1969] for discharge per unit length of stream using yearly mean values of estimated daily base flow. In summary, many field observations agree with (4) expressed above. At the same time, many field observations disagree with relation Q B

FUREY AND GUPTA: SPACE-TIME VARIABILITY OF LOW STREAMFLOWS

2681

Table 1. Regression Equations for Low Flows Presented in an Abbreviated Form to Depict the Relationship Between Discharge and Drainage Area Equation Q B ⫽ c 1A Q B ⫽ c 1 f(

b B

)A Bb

Q B ⫽ c 1 A Bb ⫺ c 2 Q B ⫽ c 1 f( )A Bb ⫺ c 2

Exponent Values

Location

Reference

0.955 ⱕ b ⱕ 1.256 0.890 ⱕ b ⱕ 1.087 0.993 ⱕ b ⱕ 1.03 0.967 ⱕ b ⱕ 1.05 0.99 ⱕ b ⱕ 1.07 b ⫽ 0.89 0.728 ⱕ b ⱕ 1.202 0.876 ⱕ b ⱕ 0.925 0.918 ⱕ b ⱕ 1.165

MD, PA MD, PA IN KT TN Devon, England VA MD, DE VA

Carpenter and Hayes [1996] Carpenter and Hayes [1996] Arihood and Glatfelter [1991] Ruhl and Martin [1991] Bingham [1986] Browne [1981] Hayes [1991] Carpenter and Hayes [1996] Hayes [1991]

⬀ D 2 A B , expressed by the equation. Unfortunately, explanations for these anomalous results have not been offered in the literature and remain unclear. 2.2.

Regression Equations

Interestingly, the steady state low-flow equation derived above is similar in form to statistically based regression equations developed from low-flow data. Table 1 presents several regression equations in an abbreviated form and the references from which they are obtained. Low-flow discharge Q B is evaluated in these equations as a flow average from 3 to 30 days with a recurrence interval of 2, 10, or 20 years. In each equation, Q B depends on drainage area A B , raised to the power b, and in two of the four equations presented, depends on a function f( ), which is composed of the product of parameters such as soil type, rock type, or a flow index. This function does not include A B , drainage density D, or hydraulic conductivity, k. The parameters c 1 and c 2 are constants. According to these regression equations, Q B scales linearly with A B since values of b are around one. This relationship is found in other regression equations [e.g., see Vogel and Kroll, 1992], and some regression equations explicitly assume that this relationship holds true [e.g., see Pearson, 1995]. This same relationship is observed to hold in the physically based equation discussed above. Thus there is some broad consistency between the physically based and statistical regression equations with respect to their functional dependence on drainage area A B . 2.3.

Transient State Equations

Transient state low-flow equations have been developed from an integral representation of base flow involving an equation of continuity and a parametric form of a storage-discharge relationship. Both linear and nonlinear versions of parametric equations have been used; for example, see Brutsaert and Nieber [1977] for a nonlinear analysis, and Gelhar and Wilson [1974], Vogel and Kroll [1992], and Brutsaert and Lopez [1998] for a linear analysis. A linear storage-discharge relationship can be derived using the well-known Darcy’s equation for saturated porous media, and many studies of low-flow data support this relationship [e.g., Vogel and Kroll, 1992]. By contrast, the physical basis of a nonlinear relationship remains unclear. Yet empirical analyses of Brutsaert and Nieber [1977] and more recently Wittenberg [1999] and Wittenberg and Sivapalan [1999] suggest that the issue of nonlinearity cannot be dismissed. Explaining the nonlinearities often observed in low-flow data from physical considerations remains an important unsolved puzzle. The literature review above shows that drainage area is a

common variable in both the physically based and statistically based low-flow equations, and Vogel and Kroll [1992] have attempted to connect these two types of equations. They first developed a low-flow equation for a basin founded on a linear storage-discharge relationship. This equation was then multiplied by an error term to produce a regression model, and based on this model, 7-day-mean low-flow data were investigated using a multivariate ordinary least squares regression. The physical basis for the error term in this model was not explained. In section 3 below, a low-flow equation is developed that is founded on the physics of subsurface flow and in which two physical parameters are treated probabilistically, as motivated by empirical observations. This equation is a linear regression model, and its error term has a physical basis. With this equation, temporal and spatial statistical variability in lowflow data can be investigated in terms of the physics of low flows.

3. 3.1.

Theory Derivation of a Low-Flow Equation

Using Darcy’s law, Gelhar and Wilson [1974] expressed base flow as a linear reservoir equation and illustrated how this equation could be used to model temporal changes in water table height in aquifers in eastern Massachusetts. Their equation furnishes the starting point of our developments and is expressed as Sy

d⌬h共t兲 ⫽ w共t兲 ⫺ ␣ ⌬h共t兲. dt

(5)

Here S y is the specific yield, defined as the volume of water released from an aquifer per unit decline in hydraulic head per area, t is time, ⌬h(t) is the mean height of the water table as measured above the bordering stream, w(t) is recharge rate per unit area, and ␣ ⬅ 3kb/a 2 is an aquifer response constant where k, b, and a are the same variables as those defined in (3). Base flow discharge per unit area is represented above as ␣ ⌬h(t). Thus base flow Q(t) is linearly related to hydraulic head as Q(t) ⫽ ␣ ⌬h(t) A where A ⫽ al is hillside area. The term ⌬h(t) A can be thought of as aquifer storage S(t), hence Q(t) ⫽ ␣ S(t) formally represents a linear storage-discharge relationship. Equation (5) can be expressed in terms of Q(t) as dQ共t兲 ⫽ ␤ 共R共t兲 ⫺ Q共t兲兲 dt

(6)

where recharge R(t) ⫽ w(t)al and ␤ ⫽ ␣ /S y ⫽ 3kb/S y a 2 . The general solution of this equation can be written as

2682

FUREY AND GUPTA: SPACE-TIME VARIABILITY OF LOW STREAMFLOWS

Q共t 0, ␩ 兲 ⫽ ⫹

3kbA 共⌬h共t 0兲 exp 关⫺␤␩ 兴 a2

1 S yA

冕

t 0⫹ ␩

exp 关⫺␤ 共t 0 ⫹ ␩ ⫺ ␶ 兲兴R共 ␶ 兲 d ␶ )

(7)

៮ 共t 0, s兲 ⫽ Q

t0

where t 0 is a starting time, Q(t 0 , ␩ ) is base flow at time t 0 ⫹ ␩ , ⌬h(t 0 ) ⬅ ⌬h(0, t 0 ) is hydraulic head at time t 0 (see Figure 1), and the integral term is a convolution. Low flow equations are often formulated by assuming that hydraulic conditions are approximately steady state. For instance, if it is assumed in (7) that recharge is constant, then Q(t 0 , ␩ ) becomes independent of time as ␩ 3 ⬁, and thus base flow Q ⫽ 3kb⌬hl/a is steady state where hydraulic head is ⌬h ⫽ w/S y ␤ ⫽ wa 2 /3kb. This expression is identical to (3) from which a low-flow equation can be formulated as discussed in section 2. However, recharge R( ␶ ) is not constant over a long enough period of time for hydraulic conditions to become approximately steady state. An analysis of (7) using a timevarying recharge rate is outside the scope of this study. Instead, we consider a short-time approximation to (7) for developing a low-flow equation. Recharge R( ␶ ) ⱖ 0 can be expressed as the sum of a mean, ␮(␶), and a perturbation, ␧(␶), as R( ␶ ) ⫽ ␮ ( ␶ ) ⫹ ␧( ␶ ) where ␧(␶) is a stochastic process with mean zero. We assume that R( ␶ ) does not significantly change over short periods of time and, as a result, treat it as a constant ␮. Given that recharge is constant over ␩, base flow discharge in (7) becomes Q共t0 , ␩兲 ⫽

3kbA a2

冋冉

⌬h共t0 兲 ⫺

冊

册

␮ ␮ exp 关⫺␤␩兴 ⫹ . ␤Sy A ␤Sy A

(8)

This equation describes base flow at an instant in time, t 0 ⫹ ␩ . However, most streamflow data sets consist of averages over a time interval of length s. We call these averages s time mean values. As a result, to make this equation compatible with data, it is averaged over the time period (t 0 , t 0 ⫹ s) to produce an expression for s time mean base flow from a hillside. From (8), s time mean base flow can be computed as ៮ 共t 0, s兲 ⫽ Q

䡠

冉

1 s

冕

s

Q共t 0, ␩ 兲 d ␩ ⫽

3kbA a2

冊

册

0

1 ⫺ exp 关⫺␤ s兴 ␤s

⫹

冋冉

⌬h共t 0兲 ⫺

␮ ␤ S yA

␮ . ␤ S yA

冊 (9)

The exponential term in this equation can be represented as an infinite series as exp [⫺ ␤ s] ⫽ 1 ⫺ ␤ s ⫹ ( ␤ s) 2 / 2 ⫺ 䡠 䡠 䡠 and substituted into (9) to get ៮ 共t 0, s兲 ⫽ Q

䡠

冢

3kbA a2

1⫺

assumption consequently places an upper bound on s. As discussed later, it can be examined with estimates of ␤ obtained from data and, in this study, holds well for s ⫽ 1 day. Evaluating (10) under this assumption gives

冉

冤冉

⌬h共t 0兲 ⫺

␮ ␤ S yA

冊

共 ␤ s兲 2 ⫺... 1 ⫺ ␤s ⫹ 2 ␤s

冊冣

冥

␮ ⫹ . ␤ S yA

3kbA ⌬h共t 0兲 ⫽ Q共t 0兲 a2

(11)

where the second relation follows since Q(t 0 ) ⫽ (3kbA/ a 2 )⌬h(t 0 ). This expression shows that mean base flow over time (t 0 , t 0 ⫹ s) is for small s approximately proportional to the hydraulic head at time t 0 . To formulate an expression for s time low flow from a basin, assume that the travel time of channel water from each hillside to the basin’s outlet is so small in comparison with the travel time of base flow through a hillside that it is, effectively, instantaneous. Then, low-flow discharge from the basin at time n t 0 ⫹ ␩ can be expressed as Q B (t 0 , ␩ ) ⬅ ¥ i⫽1 Q i (t 0 , ␩ ) where n is the number of hillsides upstream of the basin’s outlet, and Q i (t 0 , ␩ ) is base flow discharge from the ith contributing hillside. If both sides of this expression are averaged over time s, then an s time mean low-flow discharge from ៮ B (t 0 , s) ⬅ 1/s 兰 s0 Q B (t 0 , ␩ ) d ␩ can be expressed as a basin Q

冘 n

៮ B共t 0, s兲 ⫽ Q

冘冉 n

៮ i共t 0, s兲 ⫽ Q

i⫽1

i⫽1

3kbA ⌬h共t 0兲 a2

冊

(12) i

where the right-hand terms follow from (11). This equation only holds for small values of s. To evaluate (12), certain assumptions about the spatial variability of parameters in the equation must be made. Unfortunately, the magnitude of the spatial variability of saturated thickness b and hydraulic head ⌬h(t) for a fixed t, among hills in a river basin, is currently unknown to the best of our knowledge. However, the spatial variability of hydraulic conductivity k has been investigated in the literature. With reference to a number of studies, Gelhar [1993] mentions that it is common for hydraulic conductivity to be extremely variable in space even within a given soil type. This variability can be up to many orders of magnitude. Therefore, as a first step, we assume that the spatial variability of b, a, and A in a basin is negligible relative to the spatial variability of k and ⌬h(t) and treat b, a, and A as constants. Under this assumption, (12) simplifies to ៮ B共t 0, s兲 ⫽ Q

3bA a2

冘 n

冘 n

2

共k⌬h共t 0兲兲 i ⫽ 12bD A

i⫽1

i⫽1

共k⌬h共t 0兲兲 i (13)

where the second relation follows from D ⫽ 1/ 2a as discussed in section 2. Let the random variables K i and H i denote spatial variability in k and ⌬h(t 0 ) among hillsides. Then, (13) becomes a stochastic equation given by

冘 n

៮ B ⫽ 12bD 2A Q (10)

To develop a testable low-flow equation from this expression, assume that ␤ s ⬍ 1 so that 1 ⫺ ␤ s ⫹ ( ␤ s) 2 / 2 ⫺ 䡠 䡠 䡠 ⬇ 1 ⫺ ␤ s. Notice that for a given value of ␤ there is a maximum value of s under which this assumption holds, and it is not appropriate to assume that recharge is constant if s is too large. This

K iH i

(14)

i⫽1

៮ B is now a random variable which denotes spatial where Q variability in s time mean low flow among basins composed of n hillsides. According to this expression, basins with the same number of hillsides can have different s time low-flow values because of differences in k⌬h(t 0 ) between them. Next, assume that the product K i H i is independent and identically distrib-

FUREY AND GUPTA: SPACE-TIME VARIABILITY OF LOW STREAMFLOWS

uted (iid) among hillsides and that n is large. Then, the Central Limit Theorem [Ross, 1993] can be used to approximate the summation in (14) to get ៮ B ⫽ 12bD 2AnE关KH兴 ⫹ 12bD 2A 冑n Var关KH兴 N. Q

(15)

Here E[KH] and Var[KH] are the expectation and variance of KH, and N ⬅ N(0, 1) is a standard normal random variable which reflects spatial variability in K i H i among different hillsides in a basin. Defining the drainage area of a basin A B ⫽ nA, (15) becomes ៮ B ⫽ 12bD 2E关KH兴A B ⫹ 冑A B ␴ N Q (16) where ␴ ⫽ 12bD 2 公Var[KH]A. Notice that (16) looks like a linear regression model but is not because the coefficient of the random variable N is 公A B ␴ which depends on basin area. However, this equation can be written as a linear regression model given by ៮B Q

冑A B

⫽ 12bD 2E关KH兴 冑A B ⫹ ␴ N.

(17)

This is the low-flow equation we set out to derive. The appearance of 公A B on the left-hand side makes it a “nonstandard” regression equation. A comparison of (16) with (17) in terms of their correlation structure is given in section 3.4. 3.2.

Physical Significance of the Low-Flow Equation

Equation (17) unifies the physics of base flow generation with the statistical variability of low flows in a river network. The first term in (17) indicates that low flow is a function of saturated hydraulic conductivity, drainage density squared, and drainage area. If one ignores the spatial variability in k and ⌬h(t 0 ) among hillsides, then ␴ ⫽ 0 and (17) is identical to others. However, (17) includes an extra term which represents spatial variability that is not considered in the physically based equations in section 2. This second term makes (17) a linear regression model which is similar to the regression equations in Table 1. However, (17) is different from the current statisti៮ B / 公A B is recally based regression equations insofar as Q gressed onto 公A B . Moreover, as explained later, the definition of regression analysis requires that the error terms N j for different j data points be iid. This assumption does not hold in nested basins, and therefore (17) should only be applied to basins that are not nested. By contrast, some of the low-flow data used to produce the regression equations in Table 1, excluding the equation given by Browne [1981], come from nested basins. Equation (17) indicates that a least squares regression of ៮ B / 公A B onto 公A B for several stream gauges or subbasins in Q a larger “parent” basin will exhibit a high linear correlation if Var[KH] is small. Theoretically, if k⌬h(t 0 ) does not vary in ៮ B / 公A B onto space, then Var[KH] ⫽ 0, and regressing Q 公A B will produce a linear correlation of one. If k⌬h(t 0 ) varies in space, then the correlation will be less than one. Equation (17) shows that the regression slope 12bD 2 E[KH] will change in time because of changes in ⌬h(t 0 ). Changes in ⌬h(t 0 ) can be understood from (7). By letting t 0 ⫽ 0 and duration ␩ ⫽ t 0 , (7) can be expressed in terms of hydraulic head as

1 S yA

冕

0

t0

exp 关⫺␤ 共t 0 ⫺ ␶ 兲兴R共 ␶ 兲 d ␶ .

Given that t 0 ⬎⬎ 0, this equation reduces to ⌬h(t 0 ) ⬇ 1/S y A 兰 t00 exp [⫺ ␤ (t 0 ⫺ ␶ )]R( ␶ ) d ␶ and shows that regression slopes will change due to changes in ⌬h(t 0 ) as a result of changes in recharge R( ␶ ) ⱖ 0. It follows from (17) that high regression slopes correspond to high values of E[KH] and thus large low flows, while low regression slopes correspond to low values of E[KH] and thus small low flows. Equation (17) also ៮ B / 公A B ] ⫽ 12bD 2 E[KH] 公A B and therefore shows that E[Q indicates that for any regression plot the intercept of the regressed line should be zero. In (17) the Gaussian random variable N reflects the spatial variability of K i H i among different hillsides in a basin. A realization of N therefore depends on the value of k⌬h(t 0 ) for each hillside in a given basin. Equation (18) shows that ⌬h(t 0 ) is a function of k through ␤ and groundwater recharge R( ␶ ). This feature indicates that the scatter of points about a fitted line in a regression plot, where each point denotes a stream gauge or subbasin within a “parent” basin, depends strongly on differences in k among the points. Since k is constant in time, this means that the pattern of scatter in different regression plots should be similar regardless of differences in regression slopes. Finally, (17) reveals that a first-order estimate of basin transˆ , can be obtained from data. As discussed missivity, denoted T later, drainage density D and a first-order estimate of the mean ˆ , can be obtained from digital elevation of ⌬h(t 0 ) for a basin, h ˆ can be evaluated as maps. With this data, T ˆ⫽ T

␥ bE关KH兴 ⫽ ˆ ˆ h 12D 2h

(19)

where ␥ ⫽ 12bD 2 E[KH] is a regression slope value. 3.3.

Effect of Nested Basins on Low-Flow Analysis

Equation (17) indicates that 12bD 2 E[KH] can be evaluated with low-flow data from a collection of basins by regressing ៮ B / 公A B onto 公A B . However, this regression requires that Q basins are not nested within one another. To understand this constraint, suppose that there are three basins denoted 1, 2, and 3, with drainage areas A 1 , A 2 , and A 3 and that basins 1 and 2 are nested in basin 3 such that A 3 ⫽ A 1 ⫹ A 2 . According to (14), low-flow s time discharge for the three basins is

冘 n1

៮ 1 ⫽ 12bD 2A Q

K iH i

i⫽1

冘 n2

៮ 2 ⫽ 12bD 2A Q

冉冘 n1

៮ 3 ⫽ 12bD 2A Q

K iH i

(20)

i⫽n1⫹1

冘 n2

K iH i ⫹

i⫽1

i⫽n1⫹1

K iH i

冊

where n 1 and n 2 are the number of hillsides in basins 1 and 2. Now assume that n 1 and n 2 are large so that each of the summations in (20) can be approximated using the Central Limit Theorem. Applying this theorem to the equations above gives

冑A 1 ␴ N 1 ៮ 2 ⫽ 12bD 2E关KH兴A 2 ⫹ 冑A 2 ␴ N 2 Q ៮ 3 ⫽ 12bD 2E关KH兴A 3 ⫹ 冑A 1 ␴ N 1 ⫹ 冑A 2 ␴ N 2. Q ៮ 1 ⫽ 12bD 2E关KH兴A 1 ⫹ Q

⌬h共t 0兲 ⫽ ⌬h共0兲 exp 关⫺␤ t 0兴 ⫹

2683

(18)

(21)

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FUREY AND GUPTA: SPACE-TIME VARIABILITY OF LOW STREAMFLOWS

In these equations, N 1 and N 2 are standard normal random variables for basins 1 and 2 and are iid. It can be shown from ៮ 1, Q ៮ 3 ] ⫽ A 1 ␴ 2 Var[N 1 ], Cov[Q ៮ 2, Q ៮ 3] ⫽ (21) that Cov[Q 2 ៮ 1, Q ៮ 2 ] ⫽ 0. These results show A 2 ␴ Var[N 2 ], and Cov[Q that low flows from basins 1 and 3 and basins 2 and 3 cannot be used together in a linear regression analysis because they are correlated. Yet, low flows from basins 1 and 2 can be used together because they are uncorrelated. Consequently, data from stream gauges of nested basins must be unnested before applying a linear regression analysis. Unnesting data is described later in section 4.4. ៮ B and AB 3.4. Comparing Correlations Between Q ៮ B/公AB and 公AB With Correlations Between Q ៮ B is linearly related to A B ; howEquation (16) shows how Q ever, it is not a linear regression equation because 公A B ap៮ B / 公A B pears in the error term. By contrast, (17) shows how Q is linearly related to 公A B and is a linear regression equation. Yet the dependent variable includes the independent variable 公A B , and thus it is not a standard regression equation. In ៮ B is regressed onto A B , but (16) many low-flow studies, Q suggests this is incorrect. This discrepancy prompts the issue of ៮ B and A B from (16), how the correlation coefficient for Q ៮ B/ denoted R 1 , compares with the correlation coefficient for Q 公A B and 公A B from (17), denoted R 2 . At first glance, one ៮ B/ expects that R 2 will be greater than R 1 , simply because Q 公A B includes 公A B . However, as shown below, the opposite can be true. Letting ␥ ⫽ 12bD 2 E[KH], (16) and (17) are expressed as ៮ B ⫽ ␥ A B ⫹ 公A B ␴ N and Q ៮ B / 公A B ⫽ ␥ 公A B ⫹ ␴ N. Q Since A B is independent of N, the correlation coefficients R 1 and R 2 can be derived from these relations as R1 ⫽

␥ 共Var关A B兴兲 1/ 2 共 ␥ 2 Var关A B兴 ⫹ E关A B兴 ␴ 2兲 1/ 2

R2 ⫽

␥ 共Var关 冑A B兴兲 1/ 2

共 ␥ 2 Var关 冑A B兴 ⫹ ␴ 2兲 1/ 2

(22)

.

(23)

Solving for ␴2 in (22) and substituting the result into (23) gives the relation R2 ⫽

冉

共Var关 冑A B兴兲 1/ 2 Var关A B兴 1 Var关 冑A B兴 ⫹ ⫺1 E关A B兴 R 12

冉

冊冊

1/ 2

(24)

which simplifies to R2 ⫽

冉

1 1⫺c⫹

c R 12

冊

1/ 2

(25)

by letting Var[A B ]/E[A B ] ⫽ c Var[ 公A B ] where c ⬎ 0 is a constant. Here R 2 ⫽ R 1 for c ⫽ 1. If c ⬎ 1, then, counter to intuition, R 2 ⬍ R 1 . This relationship is illustrated in Figure 2 for c ⫽ 7.13. As discussed in section 5, our low-flow analysis shows that c ⬎ 1 and thus the correlation coefficient for ៮ B / 公A B and 公A B in (17) is lower than the correlation coefQ ៮ B and A B in (16). ficient for Q A related issue called “spurious correlation” is examined by Pearson [1897] and Kenney [1982]. The correlation coefficient for two parameters y and x, R y, x , can be significantly different than the correlation coefficient for y/x and x, R ( y/x), x . However, without realizing this possibility, R ( y/x), x can be and often

Figure 2. A plot of equation (25) for c ⫽ 7.13. is misinterpreted as equivalent to R y, x . Similarly, we have examined the differences between R y, x and R ( y/ 公x), 公x where ៮ B and x ⫽ A B . The former coefficient is represented by y⫽Q R 1 , and the latter coefficient is represented by R 2 . 3.5.

Discussion

Many assumptions have been made to develop (17). The most critical of these seems to be the assumption that K i H i , the products of the random variables K i and H i , are iid among hillsides in a basin, and this assumption is made to apply the Central Limit Theorem. This assumption can be loosened to allow K i H i to be weakly dependent among hillsides, and still the Central Limit Theorem can be applied. However, there is no indication how well either of these assumptions hold. There may be a strong dependence between K i H i among hillsides, in which case another limit theorem would need to be invoked. Under this situation the relationship between low flow and drainage area shown in (17) would change. The assumption that b, a, and A are constant in space is made to keep our low-flow equation simple. Given the extreme variability often exhibited by k, these parameters probably are less variable relative to k.

4.

Data

To test our theory given by (17), low-flow discharge values are investigated for the Flint River basin above Lake Blackshear in Georgia, and the Gasconade River basin in Missouri (see Figures 3 and 4). Measurements of drainage basin area and 1-day mean streamflow were obtained for these basins from the United States Geological Survey (USGS), and lowflow values were evaluated from streamflow measurements for 3 months, August, September, and October. Low-flow values are examined for the years 1966 to 1969 for the Flint basin, and for the years 1968 to 1971 for the Gasconade basin. During these years, nine gauges are active in the Flint, and eight gauges are active in the Gasconade. The assumption that exp [⫺ ␤ s] ⬇ 1 ⫺ ␤ s used to derive (17) is honored by setting s to the lowest possible value. In this case, s ⫽ 1 day, the resolution of the streamflow data, and thus 1-day mean lowflow values are used to test (17). None of the gauges used exhibit streamflow values of zero, and it is assumed that base flow is the dominant, if not the only, source of the low-flow values. The Flint and Gasconade basins are investigated for three reasons. First, both basins consist of many stream gauges that are active during the same years, and none of these gauges are significantly affected by an upstream reservoir or dam or by domestic water supply use. Second, gauges in each basin collectively cover a wide range of drainage areas, and as a result,

FUREY AND GUPTA: SPACE-TIME VARIABILITY OF LOW STREAMFLOWS

2685

Figure 4. Drainage network of the Gasconade River basin generated from a digital elevation map. Stream gauge locations are shown. The geographic coordinate of the outlet is 38⬚40⬘N and 91⬚34⬘W.

Figure 3. Drainage network of the Flint River basin generated from a digital elevation map. Stream gauge locations are shown. The geographic coordinate of the outlet is 32⬚03⬘N and 83⬚59⬘W.

the spatial scale in the present context is well represented. Third, most of the hills covering both basins are of gentle to moderate relief [Hammond, 1970] indicating that the topography of both basins is fairly amenable to the DupuitForchheimer flow approximation assumed in equation (7) from which (17) is derived. 4.1.

Stream Gauges and Precipitation

Tables 2 and 3 present four columns of gauge characteristics particular to each basin. The first column lists the last five digits of the USGS station identification number. The second column displays the measurement accuracy of each gauge as indicated by USGS records: good, 95% of the daily discharge measurements are within 10% of the true value; fair, 95% of the measurements are within 15% of the true value; and poor, measurements may exceed 15% of the true value. The accuracies corresponding to good, fair, and poor are approximate and disregard any estimated discharge values. The third column presents the drainage area of each gauge, and in this column the area of a gauge may be a subset of the area of another gauge. The fourth column displays, where applicable, the unnested drainage area of a gauge, and the reason for including this column will be discussed later. The difference between columns three and four can be illustrated with gauge 44500 in Table 2. In the third column, gauge 44500 has a

drainage area of 704 km2, and gauge 44300, nested within it, has a drainage area of 44.5 km2. Therefore the unnested drainage area of gauge 44500 is 704 km2 ⫺ 44.5 km2 ⫽ 659.5 km2 as shown in the fourth column. The gauges presented in each table are ordered in terms of drainage area, and data regarding columns one, two, and three are given by Stokes and McFarlane [1995] and Reed et al. [1994]. Stream gauge locations are shown in Figures 3 and 4. Precipitation data from Barnston [1993] indicate that August, September, and October have the lowest total monthly precipitation in the Flint basin. This is also true in the Gasconade basin when the winter months (December, January, and February) are excluded. As a result, overland flow in these basins is deemed to be lowest during August, September, and October, and low-flow discharge values during these months are considered representative of base flow from hillsides.

Table 2. Characteristics of Stream Gauges for the Flint River Basin

Gauge

Accuracy

Drainage Area, km2

44300 49900 49000 44700 46500 44500 46180 47500 49500

fair good good good good good good good good

44.5 116.5 241.8 261 482 704 3159 4790 7508

Drainage Area, km2 Unnested 䡠䡠䡠 䡠䡠䡠 䡠䡠䡠 䡠䡠䡠 䡠䡠䡠 659.5 2194 1149 2476.2

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FUREY AND GUPTA: SPACE-TIME VARIABILITY OF LOW STREAMFLOWS

Table 3. Characteristics of Stream Gauges for the Gasconade River Basin

Gauge

Accuracy

Drainage Area, km2

31500 28700 32000 27800 30000 28000 28500 33500

fair good good poor fair fair good good

16.7 20.1 518 1046 1450 3236 4350 7353

4.2.

Drainage Area, km2 Unnested 䡠䡠䡠 䡠䡠䡠 501.3 䡠䡠䡠 1429.9 2190 1114 1035

Aquifers, Hydrogeology, Soils, and Vegetation

The Flint basin trends north-south and is encompassed by two geologic provinces. The Piedmont province encompasses about two thirds of the basin to the north and consists of “indurated and metamorphosed sedimentary rocks and crystalline igneous rocks” [Cederstrom et al., 1979, p. 2]. Nearsurface material in this province consists of saprolite. The Coastal Plain province to the south consists of “soft unconsolidated sand, gravel, and clay, and consolidated or semiconsolidated limestone” [Cederstrom et al. 1979, p. 2] and is separated from the Piedmont by the Fall Line (See Figure 3). Aquifers throughout the Flint basin are unconfined, and base flow is approximately 25 to 90% of streamflow in unconsolidated areas such as the Piedmont and is approximately 40 to 95% of streamflow in the Coastal Plain province [Cederstrom et al., 1979]. Thus, during low-flow conditions, base flow is likely the dominant source of streamflow in the Flint basin. The Gasconade basin trends northeast-southwest and is encompassed by the Roubidoux, Gasconade, Eminence, and Potosi aquifers. These aquifers are unconfined, composed primarily of dolomite, partly consist of cavern systems which can transmit groundwater quite rapidly, and partly consist of springs [Taylor, 1978]. The presence of caverns and springs indicates that the hydrogeology of the basin is complicated. Nevertheless, base flow is likely the dominant source of streamflow during low-flow conditions in this basin. The saturated hydraulic conductivity for the Flint basin should be within 5 ⫻ 10⫺9 to 5 ⫻ 10⫺4 m/s based on conductivity values for fractured igneous and metamorphic rocks given by Freeze and Cherry [1979]. Given that the mean saturated depth b is of the order of 50 m, the transmissivity should then range from 2.5 ⫻ 10⫺7 to 2.5 ⫻ 10⫺2 m2/s. Geochemical modeling of a basin near the Flint by Rose [1996] suggests that the transmissivity is at the low end of this range. Stricker [1983] has shown that transmissivity is approximately 10⫺3 m2/s for aquifers in the northern part of the Coastal Plain province in Georgia. However, only the southern end of the Flint basin used in this study is situated in this region. The saturated hydraulic conductivity for the Gasconade basin should be within 1 ⫻ 10⫺9 to 5 ⫻ 10⫺6 m/s based on conductivity values for dolomite given by Freeze and Cherry [1979]. For b ⫽ 50 m the transmissivity should range from 5.0 ⫻ 10⫺8 to 2.5 ⫻ 10⫺4 m2/s. The presence of caverns and springs in the Gasconade suggests that transmissivity might be higher in this basin than in the Flint. Similar types of soil and vegetation cover the Flint and Gasconade basins. According to the Soil Conservation Service [1970], most of the soils covering the two basins are Order

Ultisol and Suborder Udult. A vegetation map produced by Ku ¨chler [1970] indicates that Oak-hickory-pine forests cover the Flint basin, and Oak-hickory and Oak-hickory-pine forests cover most of the Gasconade basin. 4.3.

Geomorphology

Drainage density and mean hill height for the Flint and Gasconade basins were estimated from digital elevation maps using a software program developed in our research group. An upper bound and lower bound for drainage density D was evaluated from each basin map, and an estimate of D for each basin was determined by calculating the integer value of the average of these two bounds. For both the Flint and Gasconade basins, D was estimated to be 1/500 m. A mean hill height in each basin was determined by evaluating the mean elevation drop from pixels at local topographic highs down to pixels representing river channels. The mean hill height in the Flint and Gasconade basins was estimated to be 15 and 25 m, respectively. This result suggests that a first-order estimate of the ˆ ⫽ 10 m. A more detailed mean of ⌬h(t 0 ) for a basin is h ˆ were estimated is given by Furey explanation of how D and h [1996]. 4.4.

Defining Low Flows and Unnesting Data

Stream gauges 49500 and 33500 have the largest drainage areas in the Flint and Gasconade basins. Consequently, relative to the other stream gauges, these well represent the hydrologic conditions in these basins. As a result, these gauges are viewed as “reference gauges” and used with precipitation data to determine the day in a month where streamflow is predominantly or completely base flow. Precipitation data from rain gauges located in or near the Flint and Gasconade basins are used that are catalogued on two CD-ROMs by the National Climatic Data Center [1995]. Data are also used from one rain gauge in the Flint basin that was obtained from the South Carolina Department of Natural Resources. Nine rain gauges occupy the Flint basin, and 12 occupy the Gasconade basin. For a given month and year the days where all rain gauges in a basin record no rain are considered rainless days. Among these, the day with the lowest streamflow in the reference gauge is determined. Then, the streamflow values for all gauges in the basin on this day are defined to be the low-flow values for that month and year. The motivation for defining low flows in this way arises because, as discussed in section 3.3, data need to be unnested. Low-flow values are commonly expressed as quantiles in regional low-flow studies. However, if a quantile approach is used to define low flow, then unnesting low-flow data can produce flow values that are negative. To illustrate how data are unnested, consider gauge 44500. The drainage area of gauge 44500 is unnested by subtracting from it the drainage area of gauge 44300. This gives gauge 44500 an unnested drainage area of 659.5 km2 as shown in Table 2. Likewise, a measured low flow from gauge 44500 is unnested by subtracting from it the corresponding low flow from gauge 44300. Note that in formulating (17) it is assumed that the travel time of channel water from each hillside in a basin to the basin’s outlet is small relative to the travel time of water through each hillside. Under this assumption it follows that a measured low flow for gauge 44500 is equivalent to low flow from gauge 44300 plus low flow from the rest of the basin that drains into gauge 44500. Thus, low-flow values can be subtracted from one another to produce unnested flow values.

FUREY AND GUPTA: SPACE-TIME VARIABILITY OF LOW STREAMFLOWS

5.

2687

Analysis and Results

The relationship between 1-day low flows during August, September, and October, and drainage area is investigated for the Flint basin from 1966 to 1969 and for the Gasconade basin ៮ B / 公A B , from 1968 to 1971. Following (17), realizations of Q 公 公 denoted q៮ B / A B , are plotted against A B for each gauge in the two basins. A least squares linear regression is then applied to plotted data for each basin to study the physical meaning of the regression parameters generated for each plot. ៮ B Onto AB and Q ៮ B/公AB Onto 公AB 5.1. Regressing Q for Nested and Unnested Basins Figure 5 displays four linear regression plots for the Flint basin using low-flow values for September 1966. The plots shown are (1) q៮ B versus A B for nested basins, (2) q៮ B / 公A B versus 公A B for nested basins, (3) q៮ B versus A B for unnested basins, and (4) q៮ B / 公A B versus 公A B for unnested basins. Each plot depicts a least squares fitted line and shows the slope and y intercept of the regressed line, the coefficient of determination R 2 , and the standard error of prediction defined as SE ⫽ s( y) 公(1 ⫺ R 2 ) where s( y) is the sample standard deviation of the dependent variable. Plot 1 is the common approach to low-flow regression analysis, plot 4 is the approach suggested by (17), and plots 2 and 3 are intermediate steps from plot 1 to plot 4. The value of R 2 decreases from 0.93 in plot 1 to 0.56 in plot 2, and from 0.77 in plot 3 to 0.33 in plot 4. These results ៮ B and A B is indicate that the correlation coefficient for Q ៮ B / 公A B and greater than the correlation coefficient for Q 公A B . This observation is found for all months and years investigated for the Flint and Gasconade basins and agrees with (25) when c ⬎ 1. Consider plots 3 and 4 in Figure 5. The parameter R 1 is defined by (22) as the correlation coefficient ៮ B and A B . However, R in plot 3 is a sample correlation for Q ៮ B and A B . According to (22), R is an estimate coefficient for Q of R 1 based on an estimate of E[A B ] and Var[A B ] using a small number of realizations of A B . Likewise, R 2 is defined by ៮ B / 公A B and 公A B . (23) as the correlation coefficient for Q However, R in plot 4 is a sample correlation coefficient for ៮ B / 公A B and 公A B . Thus values of R in plots 3 and 4 are Q estimates of R 1 and R 2 , respectively. These estimates can be used to show that the changes in R observed in the plots agree with (25) for c ⬎ 1. Substituting in values of R from plots 3 and 4 into (25) shows that c ⫽ 7.13. Figure 2 shows a plot of R 2 versus R 1 for this estimated value. The value of R 2 also decreases from 0.93 in plot 1 to 0.77 in plot 3, and from 0.56 in plot 2 to 0.33 in plot 4. This result makes sense because low flows from nested basins are positively correlated by virtue of their nestedness, as shown in section 3.3. By unnesting data, correlation due to nestedness is removed, and R 2 is reduced. This result suggests that the correlation between low flow and drainage area may be weaker than has been reported in previous low-flow studies because many of these studies have included nested basins. Moreover, it shows that subsurface variability due to k and ⌬h(t 0 ) may impact low flows more than has been reported. Finally, the value of R 2 decreases from 0.93 in plot 1 to 0.33 in plot 4. This large decrease clearly reflects the cumulative affect of unnesting data and dividing by 公A B . Values of SE in Figure 5 can be compared to one another by dividing them by ␮ ( y), the mean of the dependent variable. Values of normalized standard error, NSE ⬅ ( s ( y ) / ␮ ( y)) 公(1 ⫺ R 2 ), are 0.386, 0.637, 0.608, and 0.779 for plots

Figure 5. Flint basin during September 1966: q៮ B versus A B and q៮ B / 公A B versus 公A B for nested and unnested basins. 1 to 4, and their differences clearly arise from the corresponding differences in R 2 values. The trends in the regression statistics observed in Figure 5 were reproduced in a computer simulation. It was assumed that K i H i in (14) is independent and identically distributed among hillsides in a basin as a lognormal random variable. n Then, for a basin of n hillsides a realization of ¥ i⫽1 K i H i was simulated on computer. A low-flow value q៮ B was calculated for a basin by defining hillside area A ⫽ 1 so that A B ⫽ n and by defining 12bD 2 to be a constant. Following this approach, values of q៮ B were generated for a group of unnested subbasins of different sizes. Similarly, q៮ B was generated for a group of nested subbasins by including the same realizations of K i H i in two or more basins within the group. Using these artificial ៮ B onto A B values of q៮ B and A B , we observed that regressing Q ៮ B / 公A B onto 公A B for nested and unnested and regressing Q basins produced statistical trends similar to those shown in Figure 5 for real data. ៮ B/公AB Onto 公AB for Unnested Basins: 5.2. Regressing Q Observed Theoretical and Physical Consistencies Figure 6 displays plots of q៮ B / 公A B versus 公A B for the Flint and Gasconade basins during September. Each plot displays a least squares fitted line and the same regression statistics as shown in the plots in Figure 5. Similar plots were generated for the Flint and Gasconade basins during August and October but are not shown here. Regression results for each month are summarized in Tables 4 and 5. For each regression parameter there are four values in a given month corresponding to the 4 years investigated. Tables 4 and 5 show that q៮ B / 公A B is only moderately correlated to 公A B . For the Flint basin, R 2 ranges from 0.23 to 0.59 and has a mean of 0.37. For the Gasconade basin, R 2 ranges from 0.14 to 0.63 and has a mean of 0.25. Interestingly, R 2 is usually higher in the Flint basin than in the Gasconade basin. According to (17) this feature suggests that the spatial variability of k⌬h(t 0 ) among hillsides is greater in the Gasconade basin than in the Flint. This result makes sense physically. The dolomite that encompasses the Gasconade basin partly consists of caverns formed by water-rock dissolution. In

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FUREY AND GUPTA: SPACE-TIME VARIABILITY OF LOW STREAMFLOWS

Table 5. Slope, y Intercept, R 2 , and SE Values for the ៮ B / 公A B Onto Gasconade Basin Calculated by Regressing Q 公A B August

September

October

Year

2.40 1.60 1.62 2.15 0.59 0.77 1.05 0.46 0.30 0.29 0.23 0.14 5.60 3.90 4.60 8.08

1.98 1.64 1.81 1.94 0.84 1.34 0.96 0.79 0.22 0.15 0.31 0.17 5.77 5.88 4.16 6.49

1.79 1.46 4.95 1.85 0.55 1.42 ⫺0.52 0.82 0.25 0.14 0.63 0.17 4.74 5.68 5.85 6.20

1968 1969 1970 1971 1968 1969 1970 1971 1968 1969 1970 1971 1968 1969 1970 1971

Slope, ⫻10⫺6 m2/km s

y intercept, ⫻10⫺5 m2/s

R2

SE, ⫻10⫺5 m2/s

Figure 6. The ratio q៮ B / 公A B versus 公A B during September for (left) the Flint basin and (right) the Gasconade basin. Units of q៮ B / 公A B and 公A B are m2/s and km. addition, there are many springs within the basin. Although these features are not accounted for in the formulation of (17), it seems that they should make the spatial variability of k⌬h(t 0 ) relatively high and thus produce the low R 2 values obtained. On the other hand, the Flint basin is relatively more uniform, hydrologically, and this is reflected in the correlation values for the Flint basin which are higher than in the Gasconade. As discussed in section 3.2, regression slope values change in time. These changes do not appear to be artifacts of statistical error as shown in Figure 7. The first plot in this figure shows regression slope values for the Flint basin versus nested lowTable 4. Slope, y Intercept, R 2 , and SE Values for the ៮ B / 公A B Onto 公A B Flint Basin Calculated by Regressing Q Slope, ⫻10⫺6 m2/km s

⫺5

y intercept, ⫻10

R2

SE, ⫻10⫺5 m2/s

2

m /s

August

September

October

Year

4.58 4.05 2.89 3.47 0.73 1.53 1.71 5.25 0.44 0.40 0.26 0.59 7.92 7.64 7.49 4.41

3.76 5.01 2.79 2.36 1.16 0.95 1.02 2.96 0.33 0.49 0.23 0.34 8.30 7.79 7.84 5.06

4.68 3.58 2.81 2.92 0.57 3.71 1.50 2.14 0.41 0.39 0.24 0.34 8.54 6.87 7.66 6.25

1966 1967 1968 1969 1966 1967 1968 1969 1966 1967 1968 1969 1966 1967 1968 1969

flow discharge for the “reference gauge” in this basin, stream gauge 49500. In this plot there are 12 points, one for each month and year investigated, August, September, and October over the period 1966 –1969. The second plot in the figure shows regression slope values for the Gasconade basin versus nested low-flow discharge for its reference gauge, stream gauge 33500. There are also 12 points in this plot, one for each month and year investigated, August, September, and October over the period 1968 –1971. Both plots show that as slope values increase, the nested low-flow discharge recorded by the reference gauge increases. This feature supports our theory that changes in regression slope have a physical basis. Following (17), it means that an increase in the hydraulic head throughout each basin causes an increase in low-flow discharge. Equation (17) shows that the expected y intercept value is zero. To test this theoretical result, the y intercept value was represented as b 0 , and null and alternative hypotheses were defined as H 0 ⬊ b 0 ⫽ 0 and H a ⬊ b 0 ⫽ 0. The 95% confidence interval for b 0 was calculated for each regression plot based on the standard error of the intercept following Draper and Smith [1968]. Table 6 shows the results of this test and indicates that for all plots, b 0 ⫽ 0 lies within the confidence interval. Therefore the null hypothesis cannot be rejected, and the expected y intercept value appears to be zero. Values of SE do not change too much among plots as shown in Tables 4 and 5. This is true even in the Gasconade basin Table 6. Ninety-Five Percent Confidence Intervals for the y Intercept Values Presented in Tables 4 and 5 August

September

October

Year

[⫺1.61, [⫺1.41, [⫺1.21, [⫺0.57,

1.75] 1.71] 1.55] 1.62]

[⫺1.49, [⫺1.64, [⫺1.32, [⫺0.69,

Flint 1.72] 1.83] 1.52] 1.28]

[⫺1.71, [⫺1.02, [⫺1.24, [⫺1.01,

1.83] 1.77] 1.54] 1.43]

1966 1967 1968 1969

[⫺1.16, [⫺0.76, [⫺0.85, [⫺1.55,

1.28] 0.92] 1.06] 1.64]

[⫺1.11, [⫺1.03, [⫺0.82, [⫺1.22,

Gasconade 1.27] 1.30] 1.01] 1.38]

[⫺0.94, [⫺0.97, [⫺1.80, [⫺1.16,

1.05] 1.26] 1.70] 1.33]

1968 1969 1970 1971

Interval values are reported in units of ⫻10⫺4 m2/s.

FUREY AND GUPTA: SPACE-TIME VARIABILITY OF LOW STREAMFLOWS

2689

slope is ␥ ⬇ 3.0 ⫻ 10 ⫺6 m2/s km or 3.0 ⫻ 10⫺9 m/s. As ˆ are D 2 ⬇ discussed in section 4, estimates of D 2 and h 2 ˆ ⬇ 10 m. Following (19), these values 1/ 250,000 m and h ˆ ⫽ 6.3 ⫻ 10 ⫺6 indicate that estimated basin transmissivity is T 2 m /s, which is within the range of transmissivity values discussed in section 4.2. An estimate of ␤ s from the regression slope shows that the assumption used to derive (17), that exp [⫺ ␤ s] ⬇ 1 ⫺ ␤ s, is ˆ ⫽ satisfied. Using (19), ␤ ⫽ 3kb/S y a 2 can be estimated as ␤ ˆ ⫽ ␥ /S y h ˆ where the regression slope ␥ ⫽ 3bE[KH]/S y a 2 h ˆ can be eval12bD 2 E[KH] ⫽ (3b/a 2 )E[KH]. As a result, ␤ uated from data. According to Freeze and Cherry [1979], S y usually ranges from 0.01 to 0.30. Using S y ⫽ 0.01, s ⫽ 1 day, ˆ discussed above shows that exp [⫺ ␤ ˆ s] ⫽ and values of ␥ and h ˆ s ⫽ 0.997408, indicating that the 0.997411 and 1 ⫺ ␤ assumption holds. The assumption is even better using S y ⫽ 0.30. Note that base flow recession constants have been found to range from 0.93 to 0.995 [Klaassen and Pilgrim, 1975] in close agreement with the value estimated here.

6.

Figure 7. Regression slope versus nested low-flow discharge for gauges 49500 and 33500 in the Flint and Gasconade basins. where, during October 1970, slope and R 2 values are abnormally high, 4.95 and 0.63, respectively. This feature is captured in Figure 6 which shows that the pattern of scatter about the regressed line is similar among the plots for the Flint basin and the Gasconade basin. As discussed in section 3.2, the pattern of scatter in a regression plot for a “parent” basin should be similar at different values of t 0 , regardless of the value of the regression slope. The relatively constant values of SE and the similar patterns of scatter observed in the regression plots for the Flint and Gasconade basins support this theoretical insight. For each of the months and years investigated, the low-flow of gauge 49000 is much greater than other gauges with similar drainage areas. Likewise, the low flow of gauge 49500 is much greater, most of the time. These features are shown in Figure 6 at 公A B ⫽ 15.5 km for 49000 and at 公A B ⫽ 49.8 km for 49500. Interestingly, the drainage area of 49000 and the unnested area of 49500 are located beside one another (see Figure 3). Consequently, there appears to be something different about the hydrology of the areas drained by these gauges; however, the cause of this difference is speculative. The areas of gauges 49000 and 49500 are located just south of the Fall Line in the Coastal Plain province. One possibility is that the Fall Line itself causes the low-flow discharge to be excessive. Another possibility is that the large low-flow discharges of these gauges are due to the unique geologic setting of the Coastal Plain province. Recall that the Coastal Plain consists of sand and gravel, unlike the Piedmont province. However, gauge 49900 is also located in the Coastal Plain, but does not appear anomalous. An estimate of transmissivity for the Flint and Gasconade basins based on (19) falls within the range of transmissivity values expected for these basins. For both basins the regression

Conclusion

We have developed a low-flow equation for a basin from a transient-state equation for base flow from a hillside. This low-flow equation, like others, expresses low-flow discharge as a function of saturated hydraulic conductivity, drainage density squared, and basin drainage area. However, unlike other lowflow equations, this equation is formally a linear regression model. As a result, it unifies the physics of base flow with the statistics of low flows in river networks and explains some of the temporal and spatial statistical variability observed in low flows. ៮ B / 公A B should be Our low-flow equation indicates that Q regressed onto 公A B to investigate low flows. We have shown analytically that only data from unnested basins should be used in a low-flow regression analysis. Otherwise, different data points in a regression plot can possess errors from the fitted line that are correlated to one another. If low-flow data need to be unnested to satisfy this condition, then it should not be expressed in terms of quantiles because unnesting quantile data can produce negative flow values. To test our equation, we have used low-flow data from different stream gauges that are recorded at the same time. Data analysis shows that regression slope values change both within a given year and among years. This observation is explained by our equation to come from changes in the hydraulic head throughout a basin. For each linear regression the pattern of scatter that data exhibits about the fitted line is similar, and this observation is explained to come from differences in saturated hydraulic conductivity among basins. The effective transmissivity in a river basin has been evaluated using a mean regression slope value, an estimate of drainage density, and a first-order estimate of the mean hillside hydraulic head. This value of basin transmissivity is consistent with field measurements of transmissivity. Future research efforts should focus on the applicability of (17) to other gauged basins. In particular, (17) should be examined in a basin that is well gauged and which includes detailed measurements of subsurface hydrologic and geologic conditions. In such a basin, the results of a linear regression analysis following (17) can be better investigated. We anticipate that the pattern of scatter about a fitted line can be directly correlated to the spatial pattern of subsurface geology in a basin. Future research efforts should also be focused on

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understanding the transient features of low flows in river basins at longer timescales than considered here and how they relate to spatial variability within river basins. Acknowledgments. We gratefully acknowledge various discussions with and comments from Dave Dawdy and Hari Rajaram on this work. This research was partially supported by grants from NSF and NASA. The first author was also supported by a Graduate Research Traineeship grant from NSF.

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