prediction of statistical scaling in peak flows for rainfall-runoff events: a new framework for...

8/11/2019 Prediction of statistical scaling in peak flows for rainfall-runoff events: a new framework for testing physical hypotheses.

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98

Vijay K. Gupta

I N T R O D U C T I O N

Many practical problems need prediction of peak flows or floods at a wide range of

space and time scales in drainage basins. Practical interest in floods comes from a

variety of sources in addition to engineering design. For example, the variation of

flows in terms of their magnitude and frequency, or regime, is a primary factor that

controls channel form and process and the nature of aquatic and riparian ecosystems. It

causes stream channels and ecosystems to co-exist in a constant state of flux. More

over, the formation of a channel network and its ecosystems are a direct result of the

culminated history of flow events and their future condition depends upon the

characteristic flow regime of a basin (Poff

et al,

1997; Beschta

et al.,

2000; Wolman

& Miller, 1960). This paper explains some basic elements of a physical theory of

scaling in spatial flood statistics on nested channel networks that has been developing

since Gupta

et al.

(1996) published the first paper about it.

Results in Gupta etal. (1996) have been generalized in many directions on idealized

mean self-similar channel networks (Menabde

et al.,

2 001 ; Men abde & Sivapalan,

2 1 ;

Troutm an & Over, 200 1). How ever, there is a pressing need to further generalize

and test the scaling theory of floods on real channel networks. Without this key step, it

will not be possible to apply this new theory to real world situations and substantially

advance the existing technology for flood prediction in poorly gauged and ungauged

basins. Such a generalization will need to incorporate the presence of natural spatial

variability in the topological, hydraulic and the hydrological properties of real channel

networks, and space-time variability of rainfall, soil moisture and runoff generation.

Moreover, nested networks with a large number of streamgauges are required for

testing the predictions of the scaling theory. Unfortunately, such basins are quite

limited in the USA and the rest of the world. These issues make the problem of

generalization of the scaling theory from idealized to real networks a major scientific

challenge. Fortunately, a large number of streamgauges and raingauges are available

on two experimental basins of the US Department of Agriculture (UDSA), the Walnut

Gulch basin (Arizona) and the Goodwin Creek basin (Mississippi). We are using these

two basins to develop and test the scaling theory of floods on real networks (Mantilla

et al., 2004; Fury & Gupta, 2004). In this paper, I use the topographic and the stream

flow data from the Goodwin Creek basin to illustrate some key elements of the scaling

theory of floods for individual rainfall-runoff events.

In its simplest form, statistical simple scaling connects two (joint) probability

distributions at any two arbitrary scales by a power law. For simplicity, let us take

drainage area as a spatial scale parameter. Equality in distributions implies that the

mean and higher-order finite moments of flood peaks, or the quantiles, can be

represented as power laws with respect to drainage area (Gupta & Waymire, 1998a,b).

Power laws describe similarity or similitude. For example, power laws have served as

the foundation for dimensional analysis and dynamic similitude in fluid mechanics

(Barenblatt, 1996, Chap. 1). Contemporary scientific literature dealing with

geometrical self-similarity and fractals is based in power laws (Barenblatt, 1996;

Mandelbrot, 1982). The concept of self-similarity and its generalizations are having a

great impact on hydrology (Sposito, 1998), geosciences (Turcotte, 1997), and other

sciences such as fluid mechanics (Barenblatt, 1996).


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Prediction of statistical scaling in peak flows for rainfall-runoff e vents 99

Three types of empirical statistical scaling analyses for floods have been

published. The first considers scaling in observed peak flows for individual rainfall-

runoff events on nested channel networks (Ogden & Dawdy, 2003). The second

considers scaling in annual peak flow statistics on nested channel networks (Goodrich

et al., 1997), and the third considers scaling in annual peak flow statistics in un-nested

basins within a hom ogen eous region (C athcart, 20 01 ; Gup ta & Da wdy , 1995; Smith,

1992). Analyses of peak flows for individual rainfall-runoff events on nested channel

networks provide a natural starting point for developing a physical understanding of

statistical scaling. In this paper, I have chosen to illustrate how channel network

geometry involving scaling in width function maxima enters into the scaling theory of

floods for individual rainfall-runoff events. Scaling in width function maxima is

illustrated here for the Goodwin Creek basin. It is empirically observed for many real

networks that have been analysed by Veitzer & Gupta (2001), who also found scaling

in the maxima of width functions for a new class of channel network models called

Random Self-Similar Networks (RSN) (Veitzer & Gupta, 2000). As il lustrated here,

scaling in width function maxima provides a natural starting point for understanding

how space-time variable physical processes transform rainfall to floods that exhibit

statistical scaling on real networks.

The scaling theory of floods predicts empirically observed statistical scaling

parameters under a set of assumptions about physical processes and channel network

geometry. Prediction allows us to test different physical assumptions and hypotheses

within a rigorous ma thematical framework without calibrating and fitt ing m odel

parameters. This is the main conceptual issue that I wish to illustrate in this paper

through a simple example. Klemes (1997) has thoughtfully discussed the issue of

testing a model using data for advancing physical understanding versus curve fitting a

model to data. Testability in scaling theory serves as a major point of departure of this

approach from a large number of hydrological models that are fitted to data; see Singh

(1995) for many examples of such hydrological models. Research on the scaling theory

of floods was instigated by the long-standing need to develop a fundamental physical

basis for purely statistical approaches to regionalization of floods in ungauged and

poorly gauged basins (Cathcart, 200 1; Gupta et al., 1994; Smith, 1992). Almost all the

basins in the world are either unga uged or poorly gau ged. A m ajor long-term objective

of this theory is to provide a process-based predictive understanding of flood scaling

statistics in nested and un-nested ungauged basins ranging from the time scales of

individual events to seasonal, annual, inter-annual and longer t ime scales. Gupta &

Waymire (1998a) provided a self-contained exposition of the developments until 1996,

and Gupta (2004) published a brief review of further progress until recently. In

recognition of the global importance of the basic problem of prediction in ungauged

basins (PU B) to hydrological sciences and engineering, and to closely related sciences,

IAHS has launched a new decadal initiative on PUB (Sivapalan

et al.,

2003).

The scaling theory of floods addresses three key problem s in river basin hyd rology

that have been articulated to varying degrees in the literature: (a) the problem of scale;

(b) the problem of l inking observed space-time statistical variabili ty with underlying

physical processes at multiple scales. Following Gupta (2004), I discuss here how the

scaling theory tackles the problem of a very large number of dynamic parameters that

are needed to model spatially variable physical processe s in rainfall-runoff relationships;


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100 Vijay K. Gupta

and (c) the problem that major gaps exist between data needs and data availability for

developing and testing the scaling theory. The significance of these three problems

individually and collectively is not limited to floods but applies to many hydrological

variables, e.g. rainfall and vapotranspiration. Collectively, these three problems can

be regarded as a central challenge for hydrology. I will focus on explaining these key

issues through a simple example from the scaling theory of floods, and refer to the

publishe d literature for ma ny technical details of the theory (Og den Da wd y, 2003;

Menabde et al, 200 1; Menabde Sivapalan, 20 01; Troutman Over, 20 01; Vei tzer

Gupta, 20 01; Reggiani et al, 200 1). This pape r has the dual objective of prov iding a

clear road map for understanding a newly growing body of literature on the scaling

theory of floods (Gupta, 2004).

This paper is organized as follows. The next section begins with a spatially

discrete equation of mass balance at the hillslope-link scale in a basin, and the scale

issues that arise in representing physical processes in this equation. This is followed by

a brief discussion of the parametric complexity that ensues due to spatial variability of

the physical processes that transform rainfall to

runoff.

The next section explains some

elements of the scaling theory through an idealized example and illustrates how model

predictions can be tested against data. Our approach is based on predicting the empir

ically observed spatial statistical scaling in peak flows for rainfall-runoff events.

Deviations between model predictions and empirical scaling relationships provide a

new framework for testing physical assumptions and hypotheses. However,

computation of empirical scaling relationships requires stream flow data at several

locations in a nested basin, which are rarely available in the USA or globally. I briefly

discuss our approach for solving this problem in a separate section. Scaling provides a

spatially distributed metric, based in similarity, for testing model predictions of peak

flows. It should be contrasted with the minimum mean square error between predicted

and observed flow hydrographs at a basin outlet, which represents a spatially inte

grated metric that is widely used in calibrating rainfall-runoff models. I close with a

brief discussion of future research.

SC LE PR OBLEM S IN FOR M U L TIN G M THEM TIC L THEOR Y OF

FLOODS

The mathematical formulation begins with a discrete mass-balance or continuity

equation and uses the geomorphic decomposition of a drainage basin as a collection of

channel links and hillslopes as shown in Fig. 1. Shreve (1966) introduced the concept

of a l ink in his well-known theory of channel networks known as the random model.

Each link of a river network is surrounded by two hillslopes, one on each side of it,

which are shown in the same colour in Fig. 1. Precipitation on each hillslope produces

runoff into its adjoining link, and it involves the physical processes of infiltration,

vapotranspiration, and overland and subsurface flows. A channel network covers

spatial scales ranging from 10

2

m to 10

6

m (Dooge, 1988; Gupta W aym ire, 1998a),

and provides a natural partitioning of a landscape into hillslopes and links. The

mathematical formulation underlying the scaling theory for floods is based on this

partitioning. This theory is designed to understand statistical scaling in tenus of


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Prediction of statistical scaling in peak flows for rainfall-runoff events

101

F ig .

A system of hillslopes and links for a drainage basin.

physical processes and network geometry across multiple spatial scales of a channel

network. Empirical evidence shows that breaks can appear in statistical scaling

relationships for annual peak flows in nested basins (Goodrich et al., 1997) as well as

in un-nested basins in homogeneous regions (Cathcart, 2001). Although, no breaks in

peak flow scaling have been observed in individual rainfall-runoff events (Ogden &

Dawdy, 2003), the theory provides a physical framework to understand scaling breaks

in floods (Gupta, 2004).

Follow ing G upta & W aym ire (199 8a), assume that stream flow in a netwo rk, T, is

observed in multiples of a characteristic time scale At at t = nAt, n =0 ,

1,2...

One can

think of

At

as the time it takes for water to flow through a link with some

mean

velocity, v, i.e.At = //v. For simplicity, Gupta & Waymire (1998a) assumed thatAt was

the same for all the links. Many results were derived using this assumption for idealized

channel networks (Gupta

et al.,

1996; Menabde

et al.,

20 01 ; M enabde & Sivapalan,

2001;

Troutman & Over, 2001), but it is too restrictive for natural channel networks in

which both link lengths and link velocities vary amongst links. Therefore, following

Mantilla

et al.

(2004), we will take time to be continuous rather than discrete by letting

At

>

in the continuity equation given in Gupta & Waymire (1998a). Let q e,t), a s t ,

t>0 be a spa ce-t im e field representing river disch arge, or the volum etric flow p er unit

time, at time t at the outlet of a link e. Each link, e represents the outlet of a sub

network draining into it. Let R e,t)c e) denote the volume of runoff into a link e from

adjacent hills at time t, where c(e) denotes the area of hills draining into the link e, and

R e,t)

is the runoff intensity measured in the units of length per unit time. Finally, let


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Vijay K. Gupta

S(e,t) denote the total volume of runoff stored in a link eat time t, and let dS/dt denote

the rate of change of total volume at time t. The equation of continuity for a link

representing a control volume for the hillslope-link system can be written as:

The sum on the right hand side of Equation (1) consists of discharges from all

those l in ks / th at jo in l ink e at its top. The formulation of the continuity equation does

not assume the channel network is binary. However, it assumes that no loops are

present in the network.

The topology and geometry of a network enters explicitly into the solution of

Equation (1) through the summation term as illustrated below through an example. A

momentum balance equation is required for expressing the link storage,

S(e,t),

as a

function of discharge,q(e,t); see Reggiani et al. (2001) for a formal derivation of these

equations at the hillslope-link scale. The runoff intensity R{e,t) in Equation (1) can be

determined from a water balance equation for a hillslope draining into the link e. It

includes overland storage, saturated and unsaturated subsurface storage, precipitation,

vapotranspiration (ET) and runoff (Gupta W aym ire, 1998a). Spatial hy dr au lic-

geometric relationships are required for completing the mathematical formulation

(M enabde Sivapalan, 20 01 ; M antilla et al, 2004).

The above formulation assumes that the integral dynamical equations governing

runoff generation from a hillslope and water transport through a link are available.

Both these scales are much larger than the laboratory scales that are typically of the

order of 0.1-1.0 m. Individual hydrological processes at the laboratory scale, e.g. flow

through saturated and unsaturated columns of a porous medium, are fairly well under

stood. By contrast, the spatial scale for a single hillslope is of the order of 1-10

2

m at

which spatial variability becomes important, and new physical elements, e.g. hillslope

topography and vegetation, influence the process of runoff generation. The key scale

problem here is to spatially integrate laboratory-scale equations over a hillslope, and

use these integrated equations for specifying hillslope runoff R(e,t) in Equation (1).

Some basic wo rk has been published on deriving an aggregate behaviour of infiltration

on a spatially variable hillslope (Chen et al, 1994). Duffy (199 6) has investigated an

integral representation of runoff-generation from a hillslope as a dynamic system.

However, much more basic theoretical and experimental work remains to be done on

this important problem.

Momentum equations of fluid mechanics for open-channel flows at the hydro-

dynamic scale are well established. They can be used as the starting point for obtaining

aggregate flow dynamics for a link. Reggiani et al. (2001) have developed a new

framework to write a link-based momentum balance equation, but specific model

assumptions are needed to complete the mathematical specification. For example, one

problem is to model spatially variable channel boundary roughness within a channel

link. Kean (2003) has investigated this problem for modelling boundary sheer stress

and flow field near the banks of streams. This type of a theoretical development is

required for deriving an average momentum balance equation for a link. In addition,

hydraulic-geometric relationships are needed to extend the momentum balance

equation to all the links in a network, because velocity, width, depth, slope, and

dS(e,t)

dt

q(e, t)

+

q

(f, t)

+

R(e, t)c(e), es

,t

> 0

(1)


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Prediction of statistical scaling in peak flows for rainfall-runoff events 103

friction vary spatially over a channel network. For example, Ibbit et al. (1998) give a

unique set of field measured spatial hydraulic-geometric relationships on a channel

network in Ne w Zealand that clearly exhibit pow er law relationships for the me ans but

stochastic variations around the means. A derivation of hydraulic-geometric relation

ships from first principles involving a combination of deterministic and stochastic

elements remains a major open scale problem.

ST TISTIC L SC LIN G N D D Y N MIC P R METR IC C OM PLEX ITY

DUE TO SP TI L V RI BILITY

Spatial variability in physical parameters describing runoff-generation processes arises

due to differences in geometric, hydraulic and biophysical properties among individual

channel links and hillslopes in a drainage basin. The number of hillslopes and links

increases with the area of a drainage basin representing spatial scale. Therefore, the

number of different values that the physical parameters can take also increases with

scale. Gupta (2004) calls this the number of degrees of freedom NDOF) to underscore

the idea that hydrological complexity in a river basin can be compared to the

complexity of a statistical-mechanical system that has a very largeNDOF (Lienhard,

1964;

Gu pta W aym ire, 1983). To illustrate that theNDOF in our physical system is

very large, let us take the example from Gupta (2004). He considered an idealized

bucket-type representation of runoff-generation and transport processes for a single

hillslope-link system following M enabd e Sivapalan

(2001,

pp. 1003-1004). Four

physical parameters are required to specify runoff generation for each hillslope. These

are,

the (j) index gove rning infiltration and H ortonia n runoff gene ration, a charac teristic

time scale,

T

e

,

governing vapotranspiration, a characteristic time scale,

T

s

,

governing

subsurface flow, and threshold soil-moisture storage, So. There are three parameters

governing link dynamics: the friction coefficient given by Chezy's resistance

coefficient, C; the link wid th,W and link depth, D.

The values of the seven physical parameters vary spatially among hillslopes and

links throughout a basin. For illustration assume that a typical hillslope has an area of

0.05 km

2

. Therefore, a 1 la n

2

basin can be partitioned into

1/0.05

= 20 hillslopes and

10 links, because each link is drained by two hillslopes. The number of different

spatial values to the seven dynamic parameters is: (20 x 4) + (10 x 3) = 110.

Therefore, in a basin of 1k m

2

,NDOF =110. This simple calculation generalizes to an

arbitrary size basin of drainage area,A , and leads to the formula:

NDOF=UQA (2)

It shows that the NDOF increases linearly with the spatial scale of a basin, A. For

example, in a small-size basin of 100 km

2

, NDOF = 11 000, and in a m edium-size

basin of 5000 km

2

,NDOF is about half a million.

In order to solve Equation (1) either analytically or numerically, statistical and

other simplifying assumptions are necessary to reduce the NDOF. Unfortunately,

assumptions regarding spatial variabili ty of these seven parameters cannot be tested

directly against empirical observations because these physical parameters cannot be

measured across hundreds and thousands of hillslopes. Remote sensing can provide


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104

Vijay K. Gupta

some of the spatially variable information, especially as it pertains to channel network

geometry, precipitation and ET. Still, most of the parametric information regarding the

processes of runoff generation and transport for spatially variable hillslopes and links

is not available. The simple calculation given in Equation (2) illustrates the funda

mental difficulty that underlies distributed rainfall-runoff modelling due to dynamic

parametric complexity. In the absence of a theoretical framework, hydrologists and

engineers have routinely and widely used calibration of dynamic parameters to reduce

large

NDOF.

But calibration is an

ad hoc

approach and is unsuitable for developing a

foundational understanding of space-time rainfall-runoff relationships (Klemes, 1997;

Woolhiser, 1996). Statistical scaling provides a new theoretical framework to reduce

the large

NDOF

of dynamic parameters as explained below.

In contrast to the space-time complexity that physical processes exhibit at the

hillslope-link scale, empirical observations show that the aggregated behaviour of peak

flow statistics in rainfall-runoff events exhibits statistical scale invariance or scaling at

successively larger spatial scales of a drainage network (Ogden & Dawdy, 2003).

Scaling is an asymptotic property, which holds in the limit of large area. This result

was obtained analytically for the idealized Peano network model by Gupta

et al.

(1996). Troutman & Over (2001) generalized it to the class of mean self-similar

networks. Menabde

et al.

(2001) considered attenuation in flow by taking

dS/dt

> 0 in

Equation (1) on a Peano network, and found using numerical calculations that scaling

holds asymptotically for large areas. Because scaling is an asymptotic property, the

scaling parameters must remain insensitive to many physical details of the system at

the hillslope-link scale. This is the key idea that allows us to reduce the largeNDOF of

dynamic parameters in the present context. This situation is somewhat similar to the

Central Limit Theorem (CLT) of probability theory, in which the convergence of a

random sum to a normal distribution does not depend on all the details of the

individual probability distributions of the summands, but on only the common mean

and variance of individual summands. However, demonstration of asymptotic scaling

in peak flows on real channel networks under both idealized and realistic flow

conditions remains a very important open theoretical problem.

Spatial statistical scaling in peak flows refers to similitude or similarity across

multiple scales and is represented by log-log linear equations between the mean, or

higher order finite statistical moments, and the drainage area, which serves as the scale

parameter. The slopes and the intercepts of log-log linear relationships are called

scaling parameters.

They represent integrated

basin-scale

behaviour. Scaling can be

defined in terms of quantiles instead of moments. This is particularly important for

extrem e events wh ose prob ability distributions often exhibit fat tails, and

a priori

one does not know the order above which the statistical moments are infinite. The

scaling of quantiles can be used to empirically estimate an upper bound for the order of

finite moments (Pavlopoulos & Gupta, 2003). Scaling parameters are analogous to

macroscopic laboratory-scale parameters, e.g. hydraulic conductivity, fluid viscosity

and density, in so far as they arise from microscopic statistical-dynamical equations

governing molecular motions with very large

NDOF.

The goal of the scaling theory of

floods on river networks is to derive scaling parameters from physical processes on

real channel networks, which is similar to the goal of deriving ma croscopic param eters

from microscopic statistical physics.


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Vijay K. Gupta

geometric structures of a network. Therefore, the maxima of the local width functions

and the peak flows in (5) represen t spatial random fields. Trou tman Ka rlinger

(1984) were the first to consider the spatial dependence of width function for the

random model of channel networks. Spatial dependence of floods and width functions

is a fundamental aspect of the scaling theory of floods. It is quite different from the

GIUH theories and rainfall-runoff models in which space is fixed at a basin outlet

(Rod riguez-Iturbe Rina ldo,19 97). I will illustrate this key idea in the exam ple given

below. The dependence of runoff on spatial scales of a channel network arises through

the indexing parameter, e, denoting a link. However, spatial scales for a channel

network are not unique and any one of several choices is possible. As a simple

example, consider the spatial scale given by the drainage area, Aie), draining into the

link e. Since each sub-basin draining into a link has a different area, it follows that the

field of peak flows, Q(e), and the runoff field, q(e,t), depends on the multiple spatial

scales of sub-basins within a basin. From here on we will denote the dependence of the

peak flow and maximum of the width function on drainage area and write, Q(e) =

Q(A(e)) = Q(A), and (e) = (A(e)) = (A). Therefore, Equation (5) can be rewritten

as:

Q(A) = Q(A),VA (6)

We computed width function maxima for the Goodwin Creek basin using a GIS

software called hidrosig. It has been developed by my research group to conduct state

of the art hydrological analyses and numerical simulations on a river basin after

partitioning it into hillslopes and links as shown in Fig. 1. The empirical estimation of

width function maxima needs an explanation of some key issues that arise in this

context. First, note that total drainage area,A(e), is the sum of all the hillslope areas

draining into a link e. Summing over all the hillslope areas, it follows that,

A(e) - c(2m{e) -1), where m(e) is the linkmagnitude representing the total number of

source links

draining into the link

e, (2m(e) -

1) is total number of links and

c

is the

mean hillslope area (Shreve, 1967). From our previous assumption that hillslope areas

draining into each link, c(e) = 1, it follows that c = 1. Magnitude and drainage area

vary stochastically from one link to the other throughout a network. Therefore, they are

not always the most convenient scale param eters.

As an alternative, I consider another well-known scale parameter, known as the

Horton order, denoted by ((e),or simply co. Ho rton order va ries from 1 to 4 on the

Goodwin Creek basin. We take all links of the same Horton order and compute the

mean magnitude, m , and the mean of the width function ma xim a,

0

(m), for thes e

links. Mean magnitude for a fixed order serves as the scale parameter in this

computation. Figure 2 shows the relationship between the empirical,

0

(m), and the

mean magnitude m. Each point represents a given Horton order, and the order

increases from 1 to 4 going from left to right. The lo g- log linearity holds w ell and a

regression equation fit to it can be written as:

log E[Q

0

(m)] =

y logm

+

b (7)

The Goodwin Creek data gives an estimate of the slope, y = 0.46 as shown in Fig. 2.

Technical issues arise in estimating the scaling parameter, y, because regression is not


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Prediction of statistical scaling in peak flows for rainfall-runoff events 107

1 1

1

1

1

Mean Magni tude (m)

Fig. 2 Scaling of mean width function maxima with respect to mean magnitude for

Goodwi n Creek basin, Mississ ippi.

quite suitable for this purpose. Mantilla et al. (2004) have explained some of these

estimation issues. I use regression only for illustration. Equations (6) and (7) lead to

two theoretical predictions: (a) that peak flows exhibit statistical scaling, and (b) the

flood scaling exponent 8 is given by:

DAT A NE E DS T O T E S T DIS T RIB UT E D PH YS ICAL AS S UMPT IO NS US ING

S CAL ING PRE DICT IO NS

Empirical tests of model predictions described above require that observed streamflow

hydrographs are available for individual rainfall-runoff events at many spatial

locations on a channel network. This type of streamflow information at several

locations is rarely available to test predictions of scaling theory. Goodwin Creek basin

is an exce ption bec ause it contains 14 streamflow gaug es in a 21 la n

2

area. This

allowed us to test the two scaling predictions given above. Empirical observations of

peak flows for individual rainfall-runoff events on the Goodwin Creek basin have

show n that they exhibit spatial statistical scaling (Ogd en Da wd y, 200 3). For

exam ple, Fig. 3 shows a plot of observed peak flows against drainage areas at 14 stteam-

flow gauges on the Goodwin Creek basin for the rainfall-runoff event of 12 March

1986. A fitted regression equation can be interpreted as scaling in mean peak flows

conditioned on area and can be written as:

Here, the mean peak flow at some reference drainage area is denoted by a. Equation

(9) shows that log-log linearity, or a power law, holds between mean peak flows and

the drainage areas predicted by Equations (6) and (7). Figure 3 gives an empirical value

of theflood scaling expone nt, 9 = 0.82. The standard regression analysis is used here

mainly for illustration. Interesting technical issues arise in estimating scaling exponent

without using regression that have been investigated elsewhere (Mantilla et ai, 2004).

= y = 0.46

(8)

io

g

[e

0

(^)]= e i o g^ + f

(9)


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108 Vijay K. Gupta

1 0 0 . 0 0 0

So 10 .0 00

1 .000

2 0 . 1 0 0

s

^ 0 . 0 1 0

0 . 0 0 1

0.1 1.0 10.0

Area

F ig .3 Scaling of observed peak flows for the event of 12 March 1986, Goodwin Creek.

The observed value of the flood-scaling exponent does not agree with the predicted

value of 0 . 4 6 given by Equation (8). This failure should not come as a big surprise to

the reader because the set of three physical assumptions used here for making this

prediction are highly idealized. Nonetheless, this example demonstrates how the

physical assumptions can be tested within the scaling framework.

I wish to make a major point regarding streamgauging with this example. Tests of

the scaling theory of floods in other basins in the USA and other countries will need

streamflow data at several locations on nested basins for individual rainfall-runo ff

events. Such data sets are rarely available either in the USA or other countries because

streamflow measurements requiring the establishment of standard USGS gauging

stations are too expensive or impractical. Under these circumstances, emplacement of

pressure gaug es at judic iously chose n field sites and conv ersion of the stage mea sure

ments to discharge with a rating curve generated from an appropriately designed flow

model and ancillary models, is a viable and reasonably accurate choice. In order to be

useful for generating discharge-rating curves, a flow model must be able to treat

curved channels w ith irregular boundaries without em ploying any em pirically adjusted

para me ters. Conseq uently, the mo del must explicitly calculate the effective flow

resistance as a function of stage from measurements of: (a) the physical roughness of

the river bed, (b) the physical roughness of the channel banks, (c) the physical

characteristics of the bank vegetation, (d) the physical roughness of the flood plain,

and (e) the physical characteristics of the flood plain vegetation. We are currently

developing and verifying flow, sediment transport, and geomorphic adjustment models

for this purpose u sing the method of Kean ( 2 0 0 3 ) .

In the long run, simultaneous observations of space-time rainfall, streamflows, and

vapotranspiration will be needed within individual mesoscale river basins for

conducting scaling analyses of floods and other hydrological variables. Such data sets

are typically unavailable in the USA or the rest of the world. Substantial progress on

these scaling and other problems in mesoscale basins requires that new comprehensive

field programmes be undertaken in the next decade. A recent report to the National

Science Foundation, USA (http://cires.colorado.edu/hydrology) articulated this key

hydrological issue. This issue is also fundamental for developing and testing new

theories and models for PUB (Sivapalanet al, 2 0 0 3 ) .
http://cires.colorado.edu/hydrologyhttp://cires.colorado.edu/hydrology


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Prediction of statistical scaling in peak flows for rainfall-runoff events

109

F U TU R E R ESEA R C H

How do we relax some of the physical idealizations that led to predicting the flood

scaling exponent in Equation (8)? This is a big question that I will touch upon very

briefly. The empirical results given by Ogden & Dawdy (2003) show that the mean

flood scaling exponents for over 200 individual rainfall-runoff events vary between

0.9 and 0.6 with a mean of about 0.8. Therefore, the flood scaling exponents are

significantly larger than 0.46 predicted by the width function maxima. By contrast, the

simulation results given by Menabde & Sivapalan (2001) and Menabde

et al.

(2001)

for idealized mean self-similar channel networks showed that incorporation of more

realistic link dynamics than contained in our three assumptions here, produced a flood

scaling exponent that is

below

the value predicted by the scaling of the width function

maxima. What are the physical reasons that cause the empirical flood scaling exponent

to be larger than the width function scaling exponent on the Goodwin Creek and the

Walnut Gulch basins? How can we understand the physical basis of variability in the

flood scaling exponents in individual rainfall-runoff events? My current work at the

University of Colorado is investigating this issue by incorporating space-time variable

rainfall into this mathematical framework for testing flood scaling predictions for the

Goodwin Creek basin (Furey & Gupta, 2004). Likewise, event-based flood scaling

analyses on the Walnut Gulch basin, Arizona, is giving new physical insights into how

the spatially variable friction as a hydraulic-geometric variable affects the flood scaling

exponent (Mantilla

et al,

2004).

Ac kno wle dge m ents Some of the ideas in this paper were presented at a conference on

Water and Environment

(Bhop al, India, 200 3), and published in the conference proc

eedings without peer reviews. V. P. Singh, the conference convener, kindly provided

comments on an earlier draft of this paper. Y. Alila, University of British Colombia,

Canada, sent me several key references, which were extremely useful, reviewed an

earlier draft of this paper and provided critical comments on it. Brent Troutman,

U S G S , Lakewood, Colorado, provided comments on an earlier draft. S. Veitzer,

P.

Furey, and Ricardo M antilla, me mb ers of my research grou p gave many insightful

comm ents. Discussions with Jim Smith and Jason Kean, USG S, Boulder, Colorado, on

this research have been very fruitful. These comments helped me to clarify important

concepts and they resulted in a substantial imp rovem ent of this paper. I am grateful for

all their input. This research was supported by grants from NSF and NASA.

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