# Effects of excess rainfall on the temporal variability of observed peak-discharge power laws

**ABSTRACT** Few studies have been conducted to determine the empirical relationship between peak discharge and spatial scale within a single river basin. Only one study has determined this empirical relationship during single rainfall–runoff events. The study was conducted on the Goodwin Creek Experimental Watershed (GCEW) in Mississippi and shows that during single events peak discharge Q(A) and drainage area A are correlated as Q(A) = αAθ and that α and θ change between events. These observations are the first of their kind and to understand them from a physical standpoint we examined streamflow and rainfall data from 148 events in the basin.A time series of excess rainfall was estimated for each event in GCEW by assuming that a threshold infiltration rate partitions rainfall into infiltration and runoff. We evaluated this threshold iteratively using conservation of mass as a criterion and found that threshold values are consistent physically with independent measurements of near-surface soil moisture. We then estimated the excess rainfall duration for each event and placed events into groups of different durations. For many groups, data show that α is linearly related to excess rainfall depth and that the event-to-event variability in Q(A) is controlled mainly by variability in α through changes in . The exponent θ appears to be independent of for all groups, but mean values of θ tend to increase as the duration increases from group to group. This later result provides the first observational support for past theoretical results, all of which have been obtained under idealized conditions. Moreover, this result provides an avenue for predicting peak discharges at multiple spatial scales in the basin.

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**ABSTRACT:**We propose an extended study of recent flood-triggering storms and resulting hydrological responses for catchments in the Pyrenean foothills up to the Aude region. For hydrometeorological sciences, it appears relevant to characterize flash floods and the storm that triggered them over various temporal and spatial scales. There are very few studies of extreme storm-caused floods in the literature covering the Mediterranean and highlighting, for example, the quickness and seasonality of this natural phenomenon. The present analysis is based on statistics that clarify the dependence between the spatial and temporal distributions of rainfall at catchment scale, catchment morphology and runoff response. Given the specific space and time scales of rainfall cell development, we show that the combined use of radar and a rain gauge network appears pertinent. Rainfall depth and intensity are found to be lower for catchments in the Pyrenean foothills than for the nearby Corbières or Montagne Noire regions. We highlight various hydrological behaviours and show that an increase in initial soil saturation tends to foster quicker catchment flood response times, of around 3 to 10 h. The hydrometeorological data set characterized in this paper constitutes a wealth of information to constrain a physics-based distributed model for regionalization purposes in the case of flash floods. Moreover, the use of diagnostic indices for rainfall distribution over catchment drainage networks highlights a unimodal trend in spatial temporal storm distributions for the entire flood dataset. Finally, it appears that floods in mountainous Pyrenean catchments are generally triggered by rainfall near the catchment outlet, where the topography is lower.Atmospheric Research 01/2014; 137:14–24. · 2.20 Impact Factor - SourceAvailable from: Ricardo Mantilla[Show abstract] [Hide abstract]

**ABSTRACT:**Several studies revealed that peak discharges (Q) observed in a nested drainage network following a runoff-generating rainfall event exhibit power law scaling with respect to drainage area (A) as Q(A) = αAθ. However, multiple aspects of how rainfall-runoff process controls the value of the intercept (α) and the scaling exponent (θ) are not fully understood. We use the rainfall-runoff model CUENCAS and apply it to three different river basins in Iowa to investigate how the interplay among rainfall intensity, duration, hillslope overland flow velocity, channel flow velocity, and the drainage network structure affects these parameters. We show that, for a given catchment: (1) rainfall duration and hillslope overland flow velocity play a dominant role in controlling θ, followed by channel flow velocity and rainfall intensity; (2) α is systematically controlled by the interplay among rainfall intensity, duration, hillslope overland flow velocity, and channel flow velocity, which highlights that it is the combined effect of these factors that controls the exact values of α and θ; and (3) a scale break occurs when runoff generated on hillslopes runs off into the drainage network very rapidly and the scale at which the break happens is determined by the interplay among rainfall duration, hillslope overland flow velocity, and channel flow velocity.Advances in Water Resources 02/2014; 64:9–20. · 2.41 Impact Factor - SourceAvailable from: Tibebu B. Ayalew[Show abstract] [Hide abstract]

**ABSTRACT:**We have conducted extensive hydrologic simulation experiments in order to investigate how the flood scaling parameters in the power-law relationship Q(A)=αAθQ(A)=αAθ, between peak-discharges resulting from a single rainfall-runoff event Q(A)Q(A) and upstream area AA, change as a function of rainfall, runoff coefficient (CrCr) that we use as a proxy for catchment antecedent moisture state, hillslope overland flow velocity (vhvh), and channel flow velocity (vcvc), all of which are variable in space. We use a physically-based distributed numerical framework that is based on an accurate representation of the drainage network and apply it to the Cedar River basin (A=16,861km2A=16,861km2), which is located in Eastern Iowa, USA. Our work is motivated by seminal empirical studies that show that the flood scaling parameters αα and θθ change from event to event. Uncovering the underlying physical mechanism behind the event-to-event variability of αα and θθ in terms of catchment physical processes and rainfall properties would significantly improve our ability to predict peak-discharge in ungauged basins (PUB). The simulation results demonstrate how both αα and θθ are systematically controlled by the interplay among rainfall duration TT, spatially averaged rainfall intensity E[I]E[I], as well as E[Cr]E[Cr], E[vh]E[vh], and vcvc. Specifically, we found that the value of θθ generally decreases with increasing values of E[I]E[I], E[Cr]E[Cr], and E[vh]E[vh], whereas its value generally increases with increasing TT. Moreover, while αα is primarily controlled by E[I]E[I], it increases with increasing E[Cr]E[Cr] and E[vh]E[vh]. These results highlight the fact that the flood scaling parameters are able to be estimated from the aforementioned catchment rainfall and physical variables, which can be measured either directly or indirectly.Advances in Water Resources 01/2014; · 2.41 Impact Factor

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Effects of excess rainfall on the temporal variability

of observed peak-discharge power laws

Peter R. Fureya,b,*, Vijay K. Guptac

aCooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO, USA

bColorado Research Associates (CoRa)/Northwest Research Associates (NWRA), 3380 Mitchell Lane, Boulder, CO 80301, USA

cDepartment of Civil and Environmental Engineering, Cooperative Institute for Research in Environmental Sciences,

University of Colorado, Boulder, CO, USA

Received 4 October 2004; received in revised form 10 March 2005; accepted 21 March 2005

Available online 20 June 2005

Abstract

Few studies have been conducted to determine the empirical relationship between peak discharge and spatial scale within a single

river basin. Only one study has determined this empirical relationship during single rainfall–runoff events. The study was conducted

on the Goodwin Creek Experimental Watershed (GCEW) in Mississippi and shows that during single events peak discharge Q(A)

and drainage area A are correlated as Q(A) = aAhand that a and h change between events. These observations are the first of their

kind and to understand them from a physical standpoint we examined streamflow and rainfall data from 148 events in the basin.

A time series of excess rainfall was estimated for each event in GCEW by assuming that a threshold infiltration rate partitions

rainfall into infiltration and runoff. We evaluated this threshold iteratively using conservation of mass as a criterion and found that

threshold values are consistent physically with independent measurements of near-surface soil moisture. We then estimated the

excess rainfall duration for each event and placed events into groups of different durations. For many groups, data show that a

is linearly related to excess rainfall depth ^ rdand that the event-to-event variability in Q(A) is controlled mainly by variability in

a through changes in ^ rd. The exponent h appears to be independent of ^ rdfor all groups, but mean values of h tend to increase

as the duration increases from group to group. This later result provides the first observational support for past theoretical results,

all of which have been obtained under idealized conditions. Moreover, this result provides an avenue for predicting peak discharges

at multiple spatial scales in the basin.

? 2005 Elsevier Ltd. All rights reserved.

Keywords: Rainfall–runoff relationship; Streamflow; Scaling; Floods

1. Introduction

For nearly 60 years, flood peaks in separate unnested

basins over large geographic regions have been analyzed

using purely statistical techniques (e.g., [13]). Studies

using statistical regression show that drainage basin area

describes most of the variability in flood peaks, and it is

widely observed that a power law relates flood peak

quantiles to drainage areas in homogeneous regions

[10,3]. Statistics that are related to drainage area by a

power law are known as ‘‘scaling statistics’’ because they

are invariant under a change in drainage area, the scale

parameter. An important fundamental research topic is

to understand the physical basis of observed scaling

statistics.

It is difficult to use flood quantile data to understand

the physical basis of scaling statistics. The difficulty

arises because quantiles of the same return period can

be associated with different rainfall events. However, a

promising alternative approach is to analyze the statis-

0309-1708/$ - see front matter ? 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2005.03.014

*Corresponding author.

E-mail addresses: furey@cires.colorado.edu, furey@cora.nwra.com

(P.R. Furey), guptav@cires.colorado.edu (V.K. Gupta).

Advances in Water Resources 28 (2005) 1240–1253

www.elsevier.com/locate/advwatres

Page 2

tics of flood peaks during single rainfall–runoff events at

multiple spatial scales throughout a single river basin.

We take this approach to examine peak-discharge data

from 148 events in Goodwin Creek Experimental Wa-

tershed (GCEW), Mississippi (MS). The events were se-

lected on the basis of an objective set of streamflow and

rainfall criteria using a computer algorithm. We focus

our study on understanding the physical causes for the

temporal variability of observed peak-discharge power

laws in GCEW.

Our current understanding of peak-discharge scaling

statistics during single events on nested river basins is

based mainly on results from analytical and numerical

solutions of equations derived under idealized condi-

tions [9,11,16,17,23,22,18]. Such results suggest that

rainfall and the river network structure of a basin

greatly influence the relationship between flood peaks

Q(A) and drainage area A. For example, in an event

with a constant rainfall rate and without infiltration,

Q(A) = aA when rainfall duration is long enough to sat-

urate the basin, and Q(A) = aAhwhen rainfall duration

is instantaneous and there is no flow attenuation in the

channels of the river network. The network structure

of a basin can be described by the width function, de-

fined as the number of links in a network that are a dis-

tance x from the network?s outlet. For the case of

instantaneous rainfall, when Q(A) = aAh, the parameter

h is related to the maxima of width functions sampled at

the end of complete Horton–Strahler streams within a

basin. A transition zone lies between these two extreme

cases of rainfall.

Only two empirical studies have been made in which

the spatial variability of peak discharge is examined

throughout a single river basin. One is a quantile-based

study of peaks within the Walnut Gulch basin, AZ [7],

and the other is both a quantile and an event-based

study of peak discharge in GCEW [19]. The studies sug-

gest that Q(A) and A are related physically as

Q(A) = aAhover some finite range of scales. A funda-

mental open problem in hydrologic sciences and engi-

neering is to understand the observed values of the

statistical parameters a and h in terms of rainfall and

the physical processes that transform rainfall to peak

discharge. To date, there is no empirical study that

examines the effect of excess rainfall duration on these

power-law parameters. Based on theoretical results, it

is expected that (1) values of h increase to 1 as the dura-

tion of excess rainfall increases among a group of events;

and (2) if the total amount of excess rainfall is the same

for a group of events, then a will decrease as excess rain-

fall duration increases among the events.

The broad goal of this paper is to develop an empir-

ical understanding of how and why peak discharge

changes with spatial scale during single rainfall–runoff

events in river basins. To pursue this goal, we examine

peak discharges from 148 events in GCEW, the same ba-

sin studied by Ogden and Dawdy [19] where ordinary

least-squares linear regression shows that Q(A) = aAh

on average for an event. A discussion about event-based

studies of peak discharge is given in Section 2 that moti-

vates the central question addressed in this paper—Why

do values of a and h determined from linear regression

change among events in GCEW? In Section 3, we esti-

mate the time series of excess rainfall for each event,

and estimates are shown to be physically consistent with

soil moisture data. Data are examined in Section 4 to

answer our main question and results show that two the-

oretical predictions are supported by the data examined.

A summary of the findings is given in Section 5.

2. Peak discharge at multiple spatial scales in a basin

2.1. Theoretical results

Gupta et al. [9] and Gupta and Waymire [11] exam-

ined the relationship between peak discharge and spatial

scale in idealized river networks where there was no sur-

face infiltration of rainfall or attenuation during stream-

flow routing. In Gupta et al. [9], instantaneous rainfall

was idealized as a random cascade and applied to a river

basin with a network idealized as a Peano tree. They

showed numerically and analytically that Q(A) and A

under this scenario are related as Q(A) / Ahwhere the

exponent h can be predicted from both rainfall spatial

variability and the maxima of width functions sampled

at the end of complete Horton–Strahler streams within

the basin. By assuming that rainfall occurs at a constant

rate uniformally distributed over a mean Shreve tree,

Gupta and Waymire [11] derived an equation for mean

peak discharge as a function of scale and rainfall dura-

tion. If basin scale is fixed, then as rainfall duration

T ! 0 the equation becomes a power law with a scaling

exponent 0 < h < 1. This exponent can be determined

from the maxima of width functions in the basin. As

T ! tc, where tcis the time to basin saturation, it be-

comes a power law with h = 1. A region in which the

exponent changes lies between these two limits. Besides

rainfall duration T, hillslope and channel residence time

can potentially affect the relationship between the mean

peak discharge and scale. As shown in [5], decreasing

hillslope velocity relative to channel velocity can have

the same effect on streamflow as increasing the duration

of excess rainfall.

In [9] and [11], there is no flow attenuation because

routing of water in a network is translational. Using

numerical simulations, Menabde et al. [16] investigated

how attenuation affects peak-discharge scaling. Simu-

lated streamflow was forced by instantaneous excess

rainfall and routed linearly. For a network idealized as

a Peano tree, they found that the scaling exponent of

peak discharges is lower than the scaling exponent of

P.R. Furey, V.K. Gupta / Advances in Water Resources 28 (2005) 1240–1253

1241

Page 3

the maxima of the width functions because of flow atten-

uation. Extending this work, Menabde and Sivapalan

[17] examined how peak discharges scale in a basin

with a network idealized as a Mandelbrot–Vicsek tree.

Numerical simulations were forced with spatially and

temporally variable rainfall, and streamflow was routed

using a Chezy expression for friction. The hydraulic

geometry of network links was defined to change with

spatial scale based on two empirical expressions. For

spatially uniform rainfall, the first-order features of their

results are comparable to those found in the analytical

expression introduced in [11]. However, as found in

[16], the peak-discharge scaling exponent was less than

the width function exponent in the limit as T ! 0. By

contrast, the authors of [15] have shown that stream-

flows simulated over real river networks, forced with

instantaneous spatially uniform excess rainfall, and rou-

ted dynamically similar to [17] can produce a flood scal-

ing exponent that exceeds the exponent for the width

function maxima.

The theoretical results for single rainfall–runoff

events discussed above show that Q(A) = aAhfor instan-

taneous and long-duration rainfall events. Before pro-

ceeding further, we examine whether this expression is

dimensionally consistent. Consider two basins with

areas A1 and A25 A1. For the same rainfall–runoff

event, the ratio of the peak discharges from these basins

is QðA2Þ=QðA1Þ ¼ Ah

QðA2Þ ¼ Qð1ÞAh

aAh, the coefficient a = Q(1) and Ahis implicitly norma-

lized by a unit area raised to the power h. It is not obvi-

ous at first glance, but Q(A) = aAhis dimensionally

consistent.

The theoretical studies discussed above indicate that

the network structure in real river networks has a great

impact on how peak discharges change with spatial scale

when rainfall is or is approximately spatially uniform. A

recent short review of prior research is given in [8].

When rainfall is spatially variable, it can reduce or mask

network effects [17,22]. As discussed below, Ogden and

Dawdy [19] observed that rainfall in GCEW tends to fall

into the former category and that Q(A) = aAhon aver-

age for an event in the basin. These observations suggest

that peak discharges in GCEW are strongly influenced

by the basin?s network structure.

2=Ah

1¼ ðA2=A1Þh. If A1= 1, then

2. Therefore, in the expression Q(A) =

2.2. Observations in two experimental watersheds

There are very few river basins in the world with a

large number of active stream gauges that encompass

a wide range of drainage areas. Only a few of these ba-

sins consist of a dense network of rainfall gauges that

are sampled simultaneously with the stream gauges. As

a result, there are only a couple of empirical studies in

which the relationship between peak discharge and spa-

tial scale within a single basin is examined.

Goodrich et al. [7] and Ogden and Dawdy [19] exam-

ined the relationships between annual peak-discharge

quantiles and drainage area A within single basins. In

[7], peak quantiles from the Walnut Gulch basin, AZ

were correlated as a power law to A across one range

of areas, and as a different power law across another

range. The observed that power-law exponents de-

pended marginally on the return period of the data,

which suggested the presence of statistical simple scaling

in peak discharges [11]. Also, for a fixed return period, it

was unclear how the two separate relationships were

connected because of a gap in data. In [19], peak quan-

tiles from GCEW were correlated as a power law to A

and, like the observations in [7], the exponent was rela-

tively insensitive to return period.

Ogden and Dawdy [19] also examined the relation-

ship between peak discharge Q(A) and drainage area A

for 279 rainfall–runoff events in GCEW that occurred

from 1981 to 1996. They observed that Q(A) and A were

correlated as Q(A) = aAhwith R2> 0.865 for 226 events.

Fig. 1 shows an example of this relationship for four

events in GCEW. Values of a and h determined from lin-

ear regression change between events in the figure. For

each event, the regression line can be interpreted as a

power law relationship between mean peak discharge

and drainage area.

In their study, Ogden and Dawdy [19] analyzed the

spatial variability of rainfall in GCEW to assess its influ-

ence on the observed relation Q(A) = aAh. A correlation

analysis of rainfall, using 1 and 3 h accumulation data,

indicated that rainfall at both time scales was highly cor-

related in space when rainfall was detected at all gauges.

The correlation length scale of rainfall greatly exceeded

the length scale of the basin. They concluded that rain-

fall could be considered uniform over these time scales

during the events studied. This conclusion indicates that

the observed event-to-event variability of a and h was

not caused by spatial variability in rainfall.

The results of [19] produce two interesting questions

regarding GCEW. What physical processes are responsi-

ble for the observed changes in a and h among events?

How are mean peak discharges and drainage areas re-

lated physically by a power law as data suggests? We ad-

dress this first question by examining streamflow and

rainfall data for 148 rainfall–runoff events. A second

paper by the authors addresses both questions analyti-

cally and using numerical simulations of streamflows [6].

3. Using data to select events and estimate excess rainfall

GCEW has a drainage area of 21.2 km2and is located

near Batesville, MS. Streamflow is measured in flumes

located at 14 nested subbasins [2]. Rainfall is measured

at 31 locations in or near the basin using standard tip-

ping-bucket and weighing rain gauges [20]. All stream-

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P.R. Furey, V.K. Gupta / Advances in Water Resources 28 (2005) 1240–1253

Page 4

flow and rainfall gauges are operated by the US Depart-

ment of Agriculture, Agricultural Research Service

(ARS), National Sedimentation Laboratory [2]. Drain-

age areas of the stream gauges range from 0.172 km2

for gauge 9 to 21.2 km2for gauge 1 at the outlet. One

streamflow gauge, gauge 10, is not used in this study be-

cause problems with its measurements have been ob-

served [19].

The rain gauges record the amount of rainfall that

accumulates during an event. Data values for a gauge

should increase monotonically during an event, but they

sometimes decrease over a short period of time within an

event. A likely source for this data error is the process

used to record and store the data; for most rain gauges,

data was digitized from recording charts. To account for

this discrepancy, we assumed that accumulated rainfall

is zero when a decrease occurs in the rainfall data during

an event.

Perennial streamflow occurs in the lower reaches of

GCEW, but most streamflow is ephemeral in the basin.

Streamflow quickly exits the basin during a rainstorm

with pre-storm streamflow levels returning within one

to three days [1,2]. The dominant runoff process in

GCEW is Hortonian overland flow. Groundwater levels

taken beside the main channel of the basin show that the

water table is several meters deep while channels

throughout the basin are incised 2–3 m on average.

Base flow at the outlet tends to be less than 0.05 m3/s

[19].

3.1. Peak discharge events

Peak discharge events in GCEW were chosen where

(1) the hydrograph at the outlet (gauge 1) has a single

peak, (2) all rain gauge values are zero during the second

day before the peak, and (3) all rain and streamflow

gauge values are positive at least once during the day

leading up to the peak. Using a computer algorithm,

we identified 148 events from 1981 through 1995 that

meet these criteria. Fig. 2 shows the hydrograph over

this period and the peaks associated with the chosen

events. There are no events in 1995 partly because one

rain gauge was inactive during most of the year. For a

majority of the events, the time series of rainfall rate

exhibits a single distinct peak.

Our criteria for selecting events require both stream-

flow and rainfall data. The second criterion constrains

the type of antecedent condition that can occur prior

to a rainfall–runoff event. We felt that this constraint

was important for taking a first step at understanding

a and h physically. By contrast, Ogden and Dawdy

[19] selected events using criteria that are based only

on streamflow data. They chose events where (1) dis-

charge at gauge 1 exceeded 0.87 m3/s, (2) runoff was ob-

servedatallstreamflow

regression, as presented in Fig. 1, showed that

R2P 0.865. The criteria used in [19] neglect small rain-

fall–runoff events. As shown in Fig. 2, our criteria in-

clude events of all sizes.

gauges,and (3)linear

Fig. 1. Peak discharge Q(A) versus drainage area A for four events in GCEW. The regression intercept and slope are loga and h.

P.R. Furey, V.K. Gupta / Advances in Water Resources 28 (2005) 1240–1253

1243

Page 5

In GCEW, the active channels are located within dee-

per historical channels and often are bounded by mini-

floodplains. Bankfull discharge for the active channels

has a return period of about two years, and the 2-year

return period at the outlet of GCEW is 78 m3/s based

on streamflow data from 1982 to 1993 [2]. A streamflow

rating relation given in [2] shows that historical bankfull

discharge at the outlet of GCEW is 339 m3/s; to obtain

this value, the top of the flume at gauge 1 was used

for bankfull depth. A comparison of these values for ac-

tive and historical bankfull discharge (78 and 339 m3/s)

to the hydrograph in Fig. 2 shows that all peak-dis-

charge values in this study are confined within the his-

torical channel banks and most are confined within the

active banks.

3.2. Estimating infiltration and excess rainfall time series

A time series of excess rainfall is needed to under-

stand why a and h change between events and to test

theoretical results. In GCEW, each rain gauge records

accumulated rainfall. Rain gauge measurements occur

at unequal intervals, ranging from 1 to over 500 min be-

tween measurements. They are generally more frequent

as storm intensity increases and do not coincide in time

among the 31 rain gauges. Therefore, to produce a time

series of excess rainfall for the basin, we first produced

an interpolated time series of accumulated rainfall for

each gauge. Interpolated values were obtained from a

line connecting adjacent data points and defined to

occur every 5 min. We then generated a time series of

mean 5-min rainfall rates by taking the difference be-

tween consecutive values of accumulated rainfall. Finally,

we constructed a spatially averaged time series of mean

5-min rates for the entire GCEW using the time series of

all 31 gauges. This was possible using only the interpo-

lated time series because interpolated values coincided

in time.

During each rainfall–runoff event, we assumed that

infiltration rate Ijis related to rainfall as

Ij¼ Pj

Ij¼ b

where j is a 5-min time step, Pjis rainfall rate, and b is a

constant that represents an upper bound or threshold on

infiltration rate. We also assumed that most of the water

which infiltrates the subsurface during an event does not

also exit the subsurface and contribute to streamflow.

This assumption seems reasonable given that Hortonian

overland flow is the dominant runoff process as dis-

cussed at the beginning of Section 3. Under this assump-

tion, conservation of mass means that total infiltration

for an event can be approximated as It= Pt? qtwhere

Ptis total rainfall and qtis total streamflow discharge

minus total base flow. To evaluate qt, base flow was as-

sumed to be constant during an event and total base

flow was estimated as the discharge at the start of an

event times the event duration. As mentioned in Section

3, most streamflow is ephemeral in GCEW. Conse-

quently, base flow is often zero prior to an event and

qtequals the total discharge.

Using observed values of Ptand estimates of qtfrom

gauge 1 at the outlet of GCEW we evaluated b itera-

tively to satisfy conservation of mass. This approach

to determining b is comparable to the Phi–Index Meth-

od explained in [14]. For this calculation, we measured

Ptand qtstarting one day before the streamflow peak

at the outlet of GCEW because rainfall begins at or after

this time for most events. These totals were evaluated

over two days because storm discharge levels return to

pre-storm levels within this period [2]. The 2-day stream-

flow totals were calculated from an interpolated stream-

flow time series, like the one generated for rainfall.

Magnitudes of qt/Ptfor the events examined agree with

those found in [19]. Fig. 3 presents the rainfall time ser-

ies of two events and the corresponding values of b. The

time series of excess rainfall for the event is shown by the

values above b. For most events in this study, the time

series of excess rainfall has a single distinct peak like

the time series shown in the first plot.

for Pj< b

for PjP b

ð1Þ

Fig. 2. Hydrograph from October 1981 to the end of 1995 at the outlet of GCEW, stream gauge 1. Circled peaks denote the events used to test

theory.

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P.R. Furey, V.K. Gupta / Advances in Water Resources 28 (2005) 1240–1253

Page 6

3.3. Consistency between infiltration estimates and soil

moisture measurements

Our investigation relies on values of the infiltration

threshold b in (1) that, as discussed above, were deter-

mined iteratively using streamflow and rainfall data.

To assess whether the values obtained were physically

reasonable, we compared them to independent measure-

ments of soil moisture. Fig. 4 shows a plot of b versus

time-of-year and a plot of measured soil moisture values

versus time-of-year. The soil moisture data come from

the Soil Climate Analysis Network (SCAN) operated

by the US Department of Agriculture, Natural Re-

sources Conservation Service. The Pasture site in SCAN

is located near the center of GCEW (34?150N, 89?520W)

and the Timber site is just outside of GCEW near the

outlet (34?140N, 89?540W). The soil moisture data were

taken from January 1999 to June 2004 and do not coin-

cide in time with the streamflow and rainfall data used

to evaluate b.

The plots show that b and the observed soil moisture

values vary seasonally. High values of b tend to occur

during summer (June–September) the time of year when

soil moisture is at its lowest. Conversely, low values of b

tend to occur during winter (December–March) when

soil moisture tends to be high and near saturation. Val-

ues of b are about 3 mm/h during the winter and agree in

magnitude with values of saturated hydraulic conductiv-

ity for GCEW as reported in [12]. Similar features ap-

pear when we plot estimated mean infiltration rate

versus time-of-year. These results indicate that infiltra-

tion rates decrease as soil moisture increases, which is

qualitatively consistent with predictions by the Green-

and-Ampt and Philip infiltration models [4]. Thus, our

estimates of b are physically consistent with observed

values of soil moisture, observed values of saturated

hydraulic conductivity, and physical models of infiltra-

tion. This result indicates that our estimates of excess

rainfall are physically reasonable.

It is important recognize that the spatial scale for

both the observed values of soil moisture and the phys-

ical models of infiltration is a ‘‘point’’. By contrast, the

spatial scale of b is the entire GCEW. Therefore, the soil

moisture observations and infiltration models cannot be

directly equated to b since b is influenced by the spatial

variability of surface properties such as vegetation and

subsurface properties such as hydraulic conductivity.

One can only hope to find a qualitative consistency be-

tween them as discussed above.

3.4. Evaluating excess rainfall duration

An objective method for evaluating excess rainfall

duration is needed to understand whether regression

slopes depend on duration as suggested theoretically.

The method must characterize how duration changes

among events, but it does not need to produce estimates

Fig. 3. The rainfall time series and infiltration threshold b for two

events. Rainfall values above the threshold show the excess rainfall

time series.

Fig. 4. Infiltration threshold b and soil moisture versus time of year.

Soil moisture was measured 5 cm below the surface from January 1999

to June 2004 at two separate locations—the Pasture site in GCEW and

the Timber site just outside of GCEW. The general relationship

between b and time is a mirror image of the general relationship

between soil moisture and time. This feature shows that our estimates

of excess rainfall are physically reasonable.

P.R. Furey, V.K. Gupta / Advances in Water Resources 28 (2005) 1240–1253

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of duration magnitude. Unfortunately, evaluating the

duration of excess rainfall for a single rainfall–runoff

event is not always simple because, during an event,

the time series of excess rainfall rate can exhibit intermit-

tency and have large fluctuations.

Two idealized time series of excess rainfall suggest

that excess rainfall duration can be assessed using the ra-

tio rd/rpwhere rdis the total depth of excess rainfall and

rpis the excess rainfall peak. First, consider a situation

where the excess rainfall rate is constant for a fixed time,

meaning that the time series looks like a rectangular

pulse. Here, rd/rpequals the duration measured directly

from the time series, from when excess rainfall begins to

when it ends. Second, consider a situation where the ex-

cess rainfall time series is a Gaussian density function. In

this case, one can define duration to be proportional to

the standard deviation, r ¼ rd=

distinguish between different duration values among

events.

We choose to compare values of a and h determined

from regression to values of rd/rpfor two reasons. First,

measuring excess rainfall duration directly from a time

series with intermittancy can be ambiguous. For exam-

ple, the duration of excess rainfall is 650 min in the sec-

ond plot of Fig. 3 when using the entire time series.

However, it is possible that the first cluster of positive

excess rainfall values shown in the plot does not contrib-

ute significantly to peak discharge. When the first cluster

is neglected, a duration of 225 min is obtained which is

65% smaller than the first measurement. By contrast,

rd/rpis 70 min using the entire time series, while it is

57 min when the first cluster is neglected. This second

value is 19% smaller than the first, and the difference be-

tween these results is minimal by contrast with the differ-

ence between the direct measurements of duration.

Thus, rd/rpis less sensitive to the ambiguity of determin-

ing the period in an excess rainfall time series over which

to assess duration. Second, any assessment of duration

depends on the precision of the excess rainfall estimates.

However, it can be illustrated that measurements of rd/rp

are less sensitive to precision than direct measurements

of duration. It means that rainfall from a single storm

will have approximately the same rd/rpvalue if deter-

mined from excess rainfall estimates of two different pre-

cisions. To summarize, rd/rpappears to be a more robust

characterization of excess rainfall duration than a direct

measurement, and, as a result, we use this ratio to exa-

mine how duration influences a and h.

The excess rainfall time series generated for an event

was used to produce estimates of rdand rp, which we de-

note^ rdand^ rp. We evaluated^ rpas the maximum value of

the time series, which for all events occurs before the

streamflow peak at the outlet. To evaluate ^ rd, we

summed all positive excess-rainfall values including

any values that occurred after the streamflow peak at

the outlet. We do not analyze the estimation error asso-

ffiffiffiffiffiffi

2p

p

rp, and use rd/rpto

ciated with these parameters in this paper, but recognize

that it could affect our results and interpretations of

physical processes. In Section 4 below, duration for an

event is represented by ^ r ¼ ^ rd=^ rp and changes in ^ r

are comparable to changes in T discussed in Section

2.1.

4. Examining the relationships between peak discharge

and excess rainfall

To get an empirical relationship between Q(A) and A,

we plotted logQ(A) versus logA for each of the 148

events in GCEW. Ordinary least-squares (OLS) regres-

sion gives the equation E[logQ(A)] = hlogA + loga,

which represents the equation for the line fitted to the

plots in Fig. 1. A discharge data point is expressed as

logQ(A) = hlogA + loga + ? where ? is the amount that

the point deviates from the line. The amount of devia-

tion is random for any given value of logA. For the data

analysis discussed below, we write the regression and

data point equations as exp(E[logQ(A)]) = aAhand

Q(A) = aAhexp(?).

We assumed that it is appropriate statistically to use

OLS regression analysis to examine the peak discharge

data. However, we note that the peak-discharge data

set is nested and using OLS regression on nested data

is not an optimal technique because errors about the fit-

ted line may be correlated in space (e.g., [21]). Alterna-

tively, generalized least-squares regression could be

used but this approach requires an estimation of the spa-

tial correlation structure of the errors. Accurately evalu-

ating this correlation structure in a basin during an

event remains an open problem.

As explained in Section 3.1, we selected rainfall–run-

off events based on specific features in rainfall and

streamflow data, and not based on R2regression values

as done by Ogden and Dawdy [19]. However, values of

R2indicate how much confidence one should have in a

and h. Consequently, we investigated the relationship

between R2, a, h, and estimates of excess rainfall depth

and duration. To examine seasonality, we focused some

of our analysis on events that occur between October 1

and March 31. Over this time period, infiltration rates

tend to be low and soil moisture conditions tend to be

near saturation.

4.1. R2, rainfall spatial variability, and total excess

rainfall

The first plot in Fig. 5 shows the relation between R2

for an event obtained from OLS regression (see Fig. 1)

and the corresponding spatial coefficient of variation

(CV) of total rainfall. The CV for an event was obtained

using the total rainfall depths recorded by each rain

gauge. The CV is large when the spatial variability of

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Page 8

total rainfall is large. The second plot in Fig. 5 shows the

relation between R2and the total depth of excess rainfall

^ rd. A value of ^ rdis an estimated spatial average of excess

rainfall depth for an event in GCEW. A spatial CV of

total excess rainfall could not be evaluated because we

did not estimate excess rainfall separately for each

gauged subbasin in GCEW. Values of R2range from

about 0.15 to 0.99, and over 85% of the values are greater

than 0.7, even though the criteria used to select events

do not depend on R2. Both plots in the figure show that

low values of R2occur over a wide range of CV values

but tend to occur when ^ rd is small. Thus, low values

of R2are more correlated to ^ rdthan to the CV of total

rainfall. This result suggests that the partitioning of

rainfall into infiltration and excess rainfall components

has an important influence on the value of R2. This issue

is examined in greater detail in Section 4.3.

As shown in Fig. 5, the spatial variability of rainfall

appears to be large for a number of events that we exam-

ined. By comparison, Ogden and Dawdy [19] analyzed

hourly and 3-h rainfall accumulation data and found

that the correlation distance was nearly 40 km, approx-

imately three times the maximum distance across

GCEW. They remarked that rainfall was approximately

spatially uniform based on this result. However, as dis-

cussed in Section 3.1, Ogden and Dawdy [19] neglected

small rainfall–runoff events in their study, and by con-

trast, our assessment of spatial rainfall variability in-

cludes small events.

4.2. a and h versus excess rainfall

Fig. 6 shows plots of a obtained from regression (see

Fig. 1) versus excess rainfall depth ^ rd. Each plot is con-

ditioned on a narrow range of ^ r values (representing ex-

cess-rainfall durations) that we call a ^ r group. A circle

denotes an event and a grey circle is an event occurring

between October 1 and March 31. Circle diameter repre-

sents the CV of rainfall introduced in Section 4.1 multi-

plied by 3.5. This factor is introduced to make events

with different CV values graphically distinguishable.

Thus, given two events with circle diameters A and B,

the ratio (B ? A)/A equals the corresponding ratio cal-

culated using CV values for the events. However,

B ? A is larger by a factor of 3.5 than the corresponding

difference between CV values.

A linear trend between a and^ rdis observed for many ^ r

groups suggesting that, to a first-order, a ¼ c1^ rdwhere c1

is a constant. It means that ^ rdaccounts for most of the

variability in a among rainfall–runoff events. It also im-

plies that ^ rdaccounts for most of the variability in Q(1)

because exp(E[logQ(1)]) = a. The linear trend is weakest

in the second plot, among those plots with more than two

events. Here, events that deviate from linearity have rela-

tively large CV values, as indicated by the large circles. It

appears, qualitatively, that c1does not change systemat-

ically as excess rainfall duration increases. However, the

second row of plots show that c1decreases as ^ r increases

and the significance of this feature is discussed below.

The plots in Fig. 1 show that exp(E[logQ(A)]) changes

between events because of changes in a and/or h. They

also indicate that peak discharge at a given gauge,

Q(A), changes between events mainly because of changes

in a and/or h; the event-to-event change in exp(?) is

relatively small. Determining which of the parameters

contributes to most of the event-to-event temporal vari-

ability in exp(E[logQ(A)]) and in Q(A) for a given gauge

is central to a physical understanding of a and h.

To examine the relationship between a and Q(A) for a

given gauge, Fig. 7 shows plots of a versus QG1where

QG1is peak discharge at gauge 1, the outlet of GCEW.

Each plot is conditioned on a ^ r group and circle diam-

eters reflect the CV of rainfall. The figure shows that a

is linearly related to QG1 such that, to a first order,

QG1= c2a where c2is a constant. Substituting in a ¼

c1^ rd observed in Fig. 6 indicates that QG1¼ c2c1^ rd for

each ^ r group. Qualitatively, it appears that c2does not

change systematically as excess rainfall duration in-

creases. However, like c1, the second row of plots show

that c2decreases as ^ r increases.

Theoretical results indicate that if rdis the same for a

group of rainfall–runoff events, then a will decrease as r

increases among the events. As discussed above, Figs. 6

and 7 indicate that a ¼ c1^ rd and QG1= c2a to a first

order. Also, c1appears to decrease as ^ r increases in the

second row of Fig. 6, and a similar feature is observed

Fig. 5. Values of R2versus the spatial CV of total rainfall and^ rd. Grey

circles are events that occurred between October 1 and March 31.

P.R. Furey, V.K. Gupta / Advances in Water Resources 28 (2005) 1240–1253

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in the second row of Fig. 7. However, neither c1nor c2

appears to decrease systematically across all the plots

in either figure, as theory predicts. Consequently, to fur-

ther examine the theoretical prediction, Fig. 8 shows a

versus ^ r given a range of ^ rd values. There are only a

few events where ^ rd> 6 cm and these are not shown

the figure. The second and fourth plots in Fig. 8 show

that a decreases as ^ r increases. This trend is particularly

compelling in the second plot and provides support for

the theoretical prediction.

To further examine the relative influence of a and h on

Q(A) for a given gauge, we compared the event-to-event

temporal variability of Q(A) for gauges 1 and 14 to the

variability in a and Ah. The drainage areas of gauges 1

and 14 are 21.1 and 1.70 km2, respectively. In log space,

among the gauges in GCEW, the drainage area of gauge

14 is closest to the mean drainage area. With ^ r ¼ 22–

38 min, the CV of Q(A), a, and Ahis 1.15, 1.11, and

0.63 for gauge 1 and 1.63, 1.11, and 0.08 for gauge 14.

In both cases, the CVs of Q(A) and a are much higher

than the CV of Ah. These relationships are typical of

all plots except the first where a few extreme values of

h produce a high CV for Ah. Overall, the CV values indi-

cate that the event-to-event variability in Q(A) for gauges

1 and 14 depend largely on a. Conversely, variability in

Q(A) depends marginally on h; this result is found in

[19] for gauge 1.

The observations provided by Figs. 6 and 7 suggest

that h is relatively independent of ^ rdfor a given ^ r group.

This relationship is tested directly in Figs. 9 and 10. Fig.

9 shows plots of h versus ^ rdfor each ^ r group for events

that occur between October 1 and March 31. Fig. 10

Fig. 6. Coefficient a versus ^ rd. Plots are conditioned on a narrow range of ^ r values. Circle diameter represents the spatial CV of total rainfall

multiplied by 3.5. Large circles denote events where the spatial CV of total rainfall is relatively large. Grey circles are events that occurred between

October 1 and March 31. For graphical clarity, the x-axis range for the first six plots is smaller than the range for the second six plots.

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P.R. Furey, V.K. Gupta / Advances in Water Resources 28 (2005) 1240–1253

Page 10

shows the same plots for events between April 1 to Sep-

tember 30. Both figures present a space–time analysis of

data because h characterizes how peak discharge

changes with spatial scale. They illustrate that there is

no clear relationship between h and ^ rdbecause, in many

plots, multiple values of h are associated with a single

value of ^ rd. However, each plot also shows the number

of plotted events n and?h ¼Pn

a sample mean of h when n > 1. For both figures,?h tends

to increase as ^ r increases.

Theoretical results indicate that h 6 1 will increase to

1 as r increases among a group of rainfall–runoff events.

This result appears to hold on average as shown in Fig.

11. The top plot shows the relationship between h and ^ r

for all events. The bottom plot shows the relationship

between?h and corresponding values of ? r ¼Pn

evaluated for each plot in Figs. 9 and 10. Here, ? r repre-

i¼1hi=n which represents

i¼1^ ri=n

sents a sample mean of ^ r when n > 1. Grey circles corre-

spond to the plots in Fig. 9, while the other circles

correspond to the plots in Fig. 10. The trend shown in

Fig. 11 provides the first empirical support for the theo-

retical result, which has been obtained only under ideal-

ized conditions (see Section 2.1). Interestingly, the

increasing trend is more clearly shown for the points

corresponding to Fig. 9, which represents events be-

tween October 1 and March 31. As discussed below, a

likely cause for this feature is that the spatial variability

of excess rainfall is low during this time of year relative

to the rest of the year.

4.3. Spatial variability of excess rainfall

Figs. 5, 9, and 10 suggest that the spatial variability of

excess rainfall is relatively large for short-duration

Fig. 7. Coefficient a versus peak flow at the outlet QG1. Circle diameter represents the spatial CV of total rainfall multiplied by 3.5. Grey circles are

events that occurred between October 1 and March 31.

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events where the total excess rainfall is small. As dis-

cussed in Section 4.1, Fig. 5 shows that events where

R2< 0.7 are correlated with low values of ^ rd. These

events are also presented in Figs. 9 and 10 as circles di-

vided with a diagonal line. Most of these events occur in

the first plot, which shows the shortest-duration events

of all those examined. Figs. 9 and 10 also show that

for a given ^ r group the variability in h tends to be great-

est when ^ rdis small. Of all ^ r groups, the greatest vari-

ability in h occurs in the first plot when ^ r is smallest.

Moreover, as shown in Fig. 10, values of h can exceed

1 even though past theoretical results predict that

h 6 1 for spatially uniform excess rainfall (see Section

2.1). To summarize, short-duration events where the to-

tal excess rainfall is small tend to have low values of R2,

the variability of h for these events tends to be relatively

large, and values of h for these events can exceed unity.

A likely cause for these observations is that the spatial

variability of excess rainfall is relatively large for these

events. Indeed, the spatial CV of total rainfall for many

of these events is also large.

Figs. 5, 9, and 10 also suggest that the spatial vari-

ability of excess rainfall is relatively low for events that

occur between October 1 and March 31 when soil mois-

ture tends to be high. Fig. 5 shows that events between

October 1 and March 31 have fewer low values of R2

than events outside of this time period. In particular,

R2ranges from about 0.35–0.99 for these events, and

only six of them have R2values less than 0.7. Further-

more, the variability of h is much lower in Fig. 9 than

in Fig. 10, and the spatial CV of total rainfall tends to

be smaller for events in Fig. 9 than for those in Fig.

10. Accordingly, R2values tend to be high, the variabil-

ity in h is relatively low, and the spatial CV of total rain-

fall is relatively low between October 1 and March 31

when soil moisture tends to be high. A likely cause for

these observations is that the spatial variability of excess

rainfall is small for events that occur during this time

period compared to events that occur outside of it.

Direct measurements or estimated values of excess

rainfall throughout GCEW are needed to confirm

whether the spatial variability of excess rainfall causes

the features discussed above. Estimates of excess rainfall

times series for each subbasin in GCEW could be pro-

duced using the approach described in Section 3.2. How-

ever, it would require a time series of interpolated rainfall

fields for each subbasin in GCEW. We have not pursued

this research here, but it remains open for the future. In

addition, direct measurements or estimated values of

rainfall and soil moisture throughout GCEW are needed

to determine the primary source for any spatial variabil-

ity in excess rainfall. Spatially variable rainfall fields im-

posed on spatially uniform soil moisture fields will lead

to spatially variable excess rainfall. The converse is also

true and it is not clear how the space–time patterns of

rainfall and soil moisture interact in GCEW.

5. Summary and future research

Our long-term goal is to understand how the spatial

statistics of peak discharges in river basins can be related

to physical processes in a scale invariant manner. With

this goal in mind, we investigated how observed peak-

discharge power laws for individual rainfall–runoff

events are related to physical processes. In the first paper

on this topic, Ogden and Dawdy [19] observed that the

exponent h and the coefficient a in peak-discharge power

Fig. 8. Coefficient a versus ^ r. Plots are conditioned on a narrow range of ^ rdvalues. Circle diameter represents the spatial CV of total rainfall

multiplied by 3.5. Grey circles are events that occurred between October 1 and March 31.

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Page 12

laws in GCEW change between events. To understand

these observations physically, we examined streamflow

and rainfall data from 148 rainfall–runoff events in the

basin. We also tested two theoretical predictions: (1) val-

ues of h 6 1 increase to 1 as the duration of excess rain-

fall r increases among a group of events; and (2) if the

total amount of excess rainfall rdis the same for a group

of events, then a will decrease as r increases among the

events. As summarized below, results show that the tem-

poral variability of the power laws observed in GCEW

depends strongly on the duration and amount of excess

rainfall.

To obtain our results, the time series of excess rainfall

for each rainfall–runoff event was estimated by assuming

that a threshold infiltration rate b partitions rainfall into

infiltration and runoff. The base flow contribution to

streamflow in GCEW is not a significant factor, which

simplified our analysis. We evaluated the threshold iter-

atively using conservation of mass as a criterion and

found that threshold values are consistent physically

with independent measurements of near-surface soil

moisture. We then estimated the excess rainfall duration

for each event and, to analyze the power law parameters

a and h, placed events into groups of different durations.

Data show that a is linearly related to excess rainfall

depth ^ rdand that the event-to-event variability in Q(A)

is controlled mainly by variability in a through changes

in ^ rd. Data provide some support for the theoretical pre-

diction that if ^ rdis the same for a collection of events

then a decreases as the duration of excess rainfall in-

creases among the events. The exponent h appears to

be independent of ^ rdand mean values of h tend to in-

crease as the duration increases. This empirical result

is the first of its kind and supports past theoretical

Fig. 9. Exponent h versus ^ rdfor events that occurred between October 1 and March 31. Circle diameter represents the spatial CV of total rainfall

multiplied by 3.5. Circles divided with a diagonal line denote events where R2< 0.7.

P.R. Furey, V.K. Gupta / Advances in Water Resources 28 (2005) 1240–1253

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results, all of which have been obtained under idealized

runoff conditions in idealized river networks [8].

Estimating the time series of excess rainfall for an

event in a basin is inherently difficult because of the

space–time variability of soil moisture and infiltration

processes that govern runoff generation from rainfall.

It is made more difficult when stream gauging data are

limited or nonexistent in space in a basin. However,

for GCEW, this problem tends to be reduced from

October 1 to March 31. During this time, soil moisture

data show that the land surface in GCEW tends to be

near saturation and this is also observed in estimates

of the infiltration threshold b. Therefore, during this

time of year, the excess rainfall and rainfall time series

are similar and this suggests that one could estimate a

and?h from measured rainfall alone. This finding also

suggests the possibility of using radar rainfall data to

the west of GCEW, where rainstorms tend to originate,

to estimate excess rainfall depth and duration prior the

arrival of rainfall in the basin. Given such estimates,

one could then estimate a and?h and predict peak dis-

charge throughout GCEW before a storm arrives. This

idea is clearly speculative but serves to motivate future

research in ungauged basins.

New theoretical advances suggest that Horton–Strah-

ler order is a more appropriate scale index than drainage

area for a scaling analysis of peak discharge in real ba-

sins [15]. By using Horton–Strahler order, correlations

among peak-discharge data can be addressed to give

estimates for scaling parameters that are more valid

than those obtained using statistical regression [15].

Moreover, in a Hortonian analysis of peak discharge,

spatial statistical means and probability distributions

of peak discharges for an event are computed. This is

Fig. 10. Exponent h versus ^ rdfor events that occurred between April 1 and September 30. Circle diameter represents the spatial CV of total rainfall

multiplied by 3.5. Circles divided with a diagonal line denote events where R2< 0.7.

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Page 14

an important feature of the Hortonian framework be-

cause there is a great need for suitable statistical scale-

invariant measures of peak discharge for single events.

In this context, it is very interesting to note that a pat-

tern of physical dependence between h and excess rain-

fall duration is observed only for mean values of h

rather than for individual values (see Fig. 11). Unfortu-

nately, the location of streamflow gauges in GCEW does

not allow for a scaling analysis based on Horton–Strah-

ler order. An important open problem is to understand

precisely how the results of ordinary least-squares

regression, as used in this study, can be linked with a

Horton–Strahler scaling analysis.

Acknowledgements

We thank the National Science Foundation for sup-

porting this research and Ricardo Mantilla, Jerry Tuttle,

and Ron Bingner for many helpful discussions during

the preparation of this paper.

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Fig. 11. Exponent h versus ^ r (top plot) and?h versus ? r (bottom plot).

Grey circles correspond to Fig. 9 while the other circles correspond to

Fig. 10. In the top plot, circle diameter represents the spatial CV of

total rainfall multiplied by 3.5. The long tick marks at the top of the

bottom plot denote the bounds of the ^ r groups in Figs. 9 and 10.

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