Dissecting the effect of rainfall variability on the statistical structure of peak flows

Advances in Water Resources 32 (2009) 1508–1525

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Advances in Water Resources

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Dissecting the effect of rainfall variability on the statistical structure of peak flows

Pradeep V. Mandapaka a,*, Witold F. Krajewski a, Ricardo Mantilla a, Vijay K. Gupta b

a IIHR-Hydroscience and Engineering, The University of Iowa, Iowa City, IA 52242-1585, USAb Civil, Environmental and Architectural Engineering, Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO 80309, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 15 December 2008Received in revised form 13 July 2009Accepted 14 July 2009Available online 25 July 2009

Keywords:Peak flowsScalingRainfall variabilityChannel network

0309-1708/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.advwatres.2009.07.005

* Corresponding author. Tel.: +1 319 594 2481.E-mail address: [email protected]

This study examines the role of rainfall variability on the spatial scaling structure of peak flows using theWhitewater River basin in Kansas as an illustration. Specifically, we investigate the effect of rainfall onthe scatter, the scale break and the power law (peak flows vs. upstream areas) regression exponent.We illustrate why considering individual hydrographs at the outlet of a basin can lead to misleadinginterpretations of the effects of rainfall variability. We begin with the simple scenario of a basin receivingspatially uniform rainfall of varying intensities and durations and subsequently investigate the role ofstorm advection velocity, storm variability characterized by variance, spatial correlation and intermit-tency. Finally, we use a realistic space–time rainfall field obtained from a popular rainfall model thatcombines the aforementioned features. For each of these scenarios, we employ a recent formulation offlow velocity for a network of channels, assume idealized conditions of runoff generation and flowdynamics and calculate peak flow scaling exponents, which are then compared to the scaling exponentof the width function maxima. Our results show that the peak flow scaling exponent is always larger thanthe width function scaling exponent. The simulation scenarios are used to identify the smaller scalebasins, whose response is dominated by the rainfall variability and the larger scale basins, which are dri-ven by rainfall volume, river network aggregation and flow dynamics. The rainfall variability has a greaterimpact on peak flows at smaller scales. The effect of rainfall variability is reduced for larger scale basins asthe river network aggregates and smoothes out the storm variability. The results obtained from simplescenarios are used to make rigorous interpretations of the peak flow scaling structure that is obtainedfrom rainfall generated with the space–time rainfall model and realistic rainfall fields derived from NEX-RAD radar data.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Peak flows in a basin are difficult to predict because they resultfrom a complex interaction among rainfall and various processes inthe landscape. Hydrology literature is rife with models developedto predict hydrographs at the outlet or at specific locations (e.g.,[3,40,41]). Several studies have examined the sensitivity of thehydrologic response of a basin to the spatio-temporal variabilityof rainfall (e.g., [22,35,32]). However, as we illustrate with a simplesimulation experiment in Section 4, examining the basin responsein terms of outlet hydrograph can be misleading. On the otherhand, studies have also revealed that the peak flows from a basindisplay power-law behavior (or scaling or scale-invariance) withrespect to the drainage areas (e.g., [43,17,11,34,8,9]). The exponentof such a power law is widely known as the scaling exponent. Gup-ta et al. [19] have demonstrated that a physical understanding ofthe scaling behavior of the peak flows is crucial for building a uni-

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(P.V. Mandapaka).

fied geophysical theory of flood peaks. Such a theory would beinvaluable for the prediction of peak flows, particularly in unga-uged basins (e.g., [42]).

In the past two decades, numerous simulation and data-basedstudies were conducted to determine the physical basis of scale-invariance (e.g., [15,38,18,4,37,30,29,28,34,8,9]). A general consen-sus emerging from these studies is that the rainfall and the channelnetwork topology, both shown to be scale-invariant, play key rolesin determining the scaling exponents of the power laws in peakflows. While most of the aforementioned research is focused on an-nual peak flows, there has been a recent shift toward investigatingsingle-event peak flows (e.g., [18,34,8,26,9]). The physical mecha-nisms responsible for scale-invariance can be identified in a muchbetter manner for individual rainfall–runoff events and can be ex-tended to annual time scales by considering multiple events in ayear (e.g., [19]). Also, recent studies suggest that the scaling expo-nents of annual peak flows are related to those of single-event peakflows (e.g., [34,19]). Gupta [14] and Gupta et al. [19] offer a com-prehensive overview of the research pertaining to the scaling offlood peaks.

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The goal of our study is to clarify the role of rainfall variabilityon the scaling structure of peak flows. It is well known that rainfallis highly variable in space and time and that our observationalcapabilities result in rainfall estimates subject to considerableuncertainties (e.g., [5,6]). This study is limited to the effects of rain-fall variability, and the effect of rainfall estimation uncertainty istherefore outside of the scope of this paper. We separate the rain-fall variability into various components and study, via simulationexperiments, the sensitivity of peak flow scaling structure to eachof them. We then apply those results in order to understand thestatistical structure of peak flows obtained using rainfall from aspace–time model capable of simulating realistic rainfall events.Rigorous understanding of the role of rainfall on the scaling struc-ture of peak flows provides the basis for the scaling-based frame-work to predict the peak flows from real basins.

Following the introduction, Section 2 offers definitions of somebasic concepts and provides a short description of the literature re-lated to the single-event peak flow scaling structure. In Section 3,we describe the study area, the simulation framework and relevantassumptions. Section 4 compares the hydrograph-oriented andscaling-based approaches to studying the hydrologic response ofa basin. In Section 5, we show the scaling structure of peak flowsobtained from an actual rainfall event measured by NEXRADweather radar in Wichita, Kansas. Section 6 includes the presenta-tion of the results for the basic simulation scenarios that we con-sidered. The peak flow scaling structure obtained using therainfall from the space–time model is discussed in Section 7. InSection 8, we present an analysis of scatter seen in the scalingstructure of peak flows, followed by additional remarks in Section9. Section 10 summarizes and concludes the study.

2. Background

In this section, we briefly discuss key results in the literature re-lated to the statistical structure of single-event peak flows. We firstprovide definitions of some basic concepts and then proceed to adiscussion of simulation-based and data-based studies in theliterature.

2.1. Basic concepts

The Horton ratio RX is defined as a ratio of the averagesE½Xxþ1�=E½Xx�, where Xx is a generic random field indexed by Hor-ton order x, a stream ordering system developed by Horton [20]and later modified by Strahler [44,45]. For instance, the field Xcan be the upstream areas or width function maxima or peak flows.For more details on the Horton order and the Horton ratios, pleasesee Rodriguez-Iturbe and Rinaldo [39] and Peckham and Gupta[36].

The width function of a river network is a measure of the rivernetwork branching structure. There are basically two types ofwidth functions: topologic and geometric. Throughout this study,we employ the topologic width function, which is defined as thenumber of links which are s links upstream of the outlet of the ba-sin as a function of s (e.g., [47]). Under idealized conditions of run-off generation and constant flow velocity, the width functionrepresents the response of the river network to spatially uniforminstantaneous rainfall. The statistical structure of the width func-tion and its relation to the hydrologic response of the basin hasbeen the object of several recent studies (e.g., [47,31,23]). Veitzerand Gupta [47] showed that the width function maxima of the sim-ulated random self-similar channel networks follow distributionalsimple scaling. That is, the generalized Horton law in terms ofprobability distributions (e.g., [36]) holds for the width function

maxima, and the Horton ratios of width function maxima RH andupstream areas RA are related by a power law of the form:

RH ¼ RbA ð1Þ

where b is the scaling exponent of the width function maxima. Sim-ilarly, the Horton ratios for the peak flow RQ and upstream areas RA

are related by a power law

RQ ¼ R/A ð2Þ

when peak flow distributions exhibit statistical self-similarity,which has been shown to be the case under certain conditions offlow and rainfall (e.g., [26]). The exponent / in Eq. (2) is referredto as the peak flow scaling exponent.

A scale break is defined in our study as a transition point in thelog–log plot of peak flows vs. drainage areas. As discussed in Sec-tions 6 and 7, the scale break separates the smaller scale basin re-sponse dominated by the rainfall intensity from the larger scalebasins, whose response is dominated by river network characteris-tics and flow dynamics and is therefore rainfall volume driven.

2.2. Simulation-based studies

Gupta et al. [18] was the first study to focus on the effect ofrainfall and channel network on the scale-invariance of single-event peak flows from a deterministic Peano network. Using anumerical simulation framework, they showed that peak flows ex-hibit simple scaling for uniform rainfall, with the scaling exponentdependent on the fractal dimension of the channel network widthfunction maxima. For spatially variable rainfall, they reported thatthe peak flows display multi-scaling, with the exponent being afunction of the channel network and the spatial variability of therainfall. Troutman and Over [46] derived analytical expressionsfor channel networks and rainfall mass exponents for the generalclass of recursive replacement trees and instantaneous multifractalrainfall. Menabde et al. [29] focused on the attenuation due to stor-age in channel networks and its effect on the scaling exponents ofpeak flows from deterministic (Mandelbrot-Viscek and Peano net-works) and random self-similar networks with linear routing andfor spatially uniform rainfall. For the deterministic self-similar net-works (SSNs), the scaling exponent of peak flows is smaller thanthe one predicted for the width function maxima (i.e., ignoringthe attenuation due to storage in channel networks). Menabdeet al. [29] also showed that for random SSN with smaller bifurca-tion ratios, the peak flows scale asymptotically.

To better understand and predict the scaling behavior of peakflows, Menabde and Sivapalan [28] introduced a dynamic and spa-tially distributed hillslope-link rainfall–runoff model based on rep-resentative elementary watershed (REW) consisting of three maincomponents: a space–time model of rainfall, a hillslope model anda channel network model. The rainfall model can generate stormswhose spatial structure is characterized by a discrete random cas-cade. The hillslope model partitions the rainfall into Hortonianrunoff, subsurface flow and evaporation, which are assumed tobe zero during periods of rainfall. They further assumed that allof the surface runoff reaches the channel instantaneously. Thechannel network is a deterministic Mandelbrot-Viscek networkin which the hydraulic geometry properties at every link are ob-tained from observed empirical relationships. They investigatedthe effect of rainfall on the scaling structure of the peak flows,starting from a spatially uniform rainfall scenario and moving tothe individual storms based on discrete random cascade. They alsoextended the study to include continuous rainfall and annual floodpeaks. The results from event-based simulations with spatially uni-form rainfall and the rainfall based on the random cascade modeldemonstrated that the interplay between the catchment response

1510 P.V. Mandapaka et al. / Advances in Water Resources 32 (2009) 1508–1525

time and the storm duration controls the scaling exponent of peakflows.

Mantilla et al. [26] discussed the difficulties in generalizing thescaling theory to the real networks and tested whether the randomspatial variability of the real channel networks and their hydraulicgeometry properties, coupled with flow dynamics, produce Horto-nian scaling in peak flows. Based on the results from Veitzer andGupta [47], the value of the scaling exponent of the network widthfunction was computed for the 149 km2 Walnut Gulch basin in Ari-zona (e.g., [12]). The runoff rates were estimated from two verysmall gauged sub-basins within the Walnut Gulch, assuming thatrainfall was spatially uniform. For an instantaneously applied run-off rate, the system of ordinary differential equations describingthe runoff dynamics was solved for three different scenarios: (a)constant velocity, (b) constant Chezy and (c) spatially varying Che-zy constant. They showed that the scaling exponent of peak flowsis larger than the exponent of the width function maxima, whichcontradicted the results from the studies performed on the ideal-ized basins, where the flow scaling exponent is always smallerthan the exponent of the width function maxima. The contradic-tion is explained in terms of the relative roles of flow attenuationand flow aggregation in the river networks that were considered.

2.3. Data-based studies

Ogden and Dawdy [34] investigated the single-event and an-nual peak flows from the 21.2 km2 Goodwin Creek watershed inMississippi (e.g., [1]), where the Hortonian mechanism of runoffgeneration is dominant. They considered 279 events for whichflows were recorded at several interior gauging stations. The re-sults showed that the peak flows follow simple scaling but theexponents vary from event to event and depend on the runoff pro-duction efficiency. The mean of scaling exponents is 0.831 with astandard deviation of 0.10. Some events are then filtered out witha threshold on the correlation coefficient (0.93) between the loga-rithm of peak flows and the upstream areas. The mean of scalingexponents from the 226 remaining events is equal to 0.826, witha standard deviation of 0.047 and a mean correlation coefficientof 0.98.

Furey and Gupta [8] explained this event-to-event variability inthe peak flow power laws in Goodwin Creek watershed in terms ofvariability in the rainfall’s excess depth and the duration. To under-stand the physical origin of the observed peak flow scaling, Fureyand Gupta [9] proposed and applied a five-step framework to theGoodwin Creek watershed. Gupta et al. [19] provided further obser-vational evidence on scaling in single-event peak flows for the Wal-nut Gulch basin, Arizona. They reported two different sets of scalingexponents for smaller and larger scales with a scale break at around1 km2. They also noticed that for the events that cover almost theentire basin, the single-event scaling exponents are quite close tothe scaling exponents of the annual flood quantiles.

All the studies discussed in this section focused on the funda-mental question, ‘‘How is the peak flow scaling exponent linkedto the channel network characteristics such as width functionmaxima and variability in the rainfall?” In the studies that ad-dressed this question using numerical simulations under idealizedconditions, the complexity in the simulations increased from Gup-ta et al. [18] to Mantilla et al. [26]. The rainfall varied from spatiallyuniform to the complex cascade-based case, and the networks ran-ged from deterministic self-similar to random self-similar and ac-tual river networks with linear and nonlinear routing mechanisms(e.g., [18,47,46,29,28,26]). In the data-based analyses (e.g.,[34,8,19,9]), the variability in the scaling exponents was explainedin terms of variability in antecedent conditions and storm charac-teristics. However, the smaller size of the basins (21.2 km2 Good-win Creek and the 149 km2 Walnut Gulch basins) limited the

range of scales available to explore the effect of rainfall variabilityon the peak flow scaling structure. Regardless of the approach fol-lowed, these studies enhanced our understanding of the relation-ship between the statistical structure of flood peaks and thecharacteristics of rainfall and channel network. However, we needto further understand and generalize the role that rainfall plays inthe statistical structure of peak flows from actual river basinsacross a range of scales.

In this study, we perform a series of simulation experimentsstarting from a simple scenario of spatially uniform rainfall for afixed duration and moving to a complex scenario in which the rain-fall is obtained from a space–time rainfall model. We also investi-gate the sensitivity of the scaling behavior to linear and nonlinearchannel routing mechanisms. We selected the simulation frame-work instead of a data-based analysis since it allows complete free-dom to systematically explore various aspects of scale-invariance.Also, there are very few basins in the United States where stream-flow data necessary for rigorous scaling analyses are available. Oursimulation covers a range of scales from �0.1 to 1000 km2, thusaddressing the peak flow scaling for basin response times rangingfrom minutes to days.

3. Simulation framework

3.1. Study area

The Whitewater River basin (Fig. 1), with an area of 1100 km2,stretches between latitudes 37�460E and 38�090E and longitudes96�510W and 97�180W. The river network extraction was basedon the maximum gradient method, also known as the D8 algorithm(e.g., [33]). Mantilla and Gupta [25] compared the network ex-tracted from CUENCAS with those extracted from popular GIS soft-ware such as ArcInfo, GRASS and RiverTools and found no majordifferences when high-resolution DEMs were used. They showedthat a 30 m resolution DEM is sufficient to extract the drainagenetwork that is close to the terrain’s actual network. We use theone arc-second resolution (�30 m) digital elevation model (DEM)from USGS to extract the channel network. This results in some20,000 hillslopes and, thus, channel links for this basin. In Fig. 1,we show the extracted channel network with links of Horton or-ders 4–7.

Section 2 indicated that the width function maxima play animportant role in understanding the scaling structure of the peakflows. Fig. 2a and c shows the Horton plots for drainage areasand width function maxima of links of various orders for theWhitewater River basin, Kansas. If the channel network is self-sim-ilar, the averages of drainage areas and width function maxima dis-play linearity with respect to the corresponding Horton orders inthe log-linear domain (e.g., [45,36,10]). The log-linearity inFig. 2a and c confirm the statistical self-similarity of the upstreamareas and the width function maxima. In the regression analysis,we use the areas and width function maxima corresponding tothe Horton orders 2–6. The order 7 stream is not used in the Hortonregression due to sampling reasons: we have only one point corre-sponding to the order 7. Although, averages corresponding to order1 streams do not suffer from sampling issues, they are usually notconsidered in the regression (e.g., [36,25]) as they represent thefinest detail in a stream network, and therefore the correspondingbasins do not contain a ‘‘network”. The Horton ratios for the areasand width function maxima are then obtained by exponentiationof the slopes from the regression analysis. The scaling exponentof width function maxima obtained through Horton ratios in (1)is 0.49.

If the upstream areas and width function maxima display log-linearity, as shown in Fig. 2a and c, then E½Xx� ¼ E½X1� � ðRXÞx�1,

Fig. 1. A shaded relief map of the Whitewater River basin showing the hillslope and channel link structure of the CUANCAS model. The channel network with links of orders4–7 is shown.

Fig. 2. Statistical self-similarity of upstream areas and width function maxima in terms of Horton plots (left panels) and rescaled distributions (right panels). The ordinaryleast square regression is used to obtain the corresponding Horton ratios. The first order and seventh order links are not considered in fitting.


where X is either the upstream area or the width function maximaand RX is the corresponding Horton ratio. The rescaled upstreamareas and width function maxima are obtained by dividing eachvalue of Xx by E½X1� � ðRXÞx�1. The probability distribution of the

quantity Xx=½E½X1� � ðRXÞx�1� is called the rescaled probability dis-tribution. In Fig. 2b and d, we show the statistical self-similarityof areas and width function maxima in terms of their rescaledprobability distributions for orders 1–5. Although order 1 basins


were not considered in the regression analysis, it can be seen thattheir rescaled probability distribution collapses onto those of or-ders 2–5.

3.2. Hydrologic model

Because of the fundamental effect of the river network structureon peak flows, it is necessary to have a distributed hydrologic mod-el that can calculate hydrographs for all river network links in or-der to carry out a systematic investigation. In this study, we usedthe CUENCAS model, developed by Mantilla and Gupta [25], whichis based on hillslope-link decomposition of the landscape and massconservation equations (e.g., [16]). The model can be run with lin-ear routing with constant flow velocity throughout the channelnetwork or nonlinear routing with velocity that depends on thedischarge in each link and the corresponding upstream area. Forthe nonlinear case, the velocities are estimated using [24]:

VcðtÞ ¼ vR �qðtÞQR

� �k1

� AAR

� �k2

ð3Þ

where VcðtÞ is the velocity in the channel and A is the upstream areaof the corresponding channel. The coefficients k1 and k2 are thevelocity scaling exponents for discharge and upstream area, respec-tively, and vR; QR and AR, are reference velocity, discharge and area,whose values are taken in this study to be 1.0 m/s, 200 m3/s and1100 km2. These values are obtained from measurements during

Fig. 3. Sensitivity of the channel velocity (m/s) to the k1 and k2 in Eq. (3). The velocities arpanel) for the Horton orders 2–7. The area units are in km2.

the rainfall–runoff events in the Whitewater River basin. The aboveequation gives the instantaneous velocity as a function of dischargeqðtÞ in the channel link, which in turn gives rise to a nonlinear or-dinary differential equation that represents fluxes coming out ofthe channel link. Please see Eqs. (6) and (9)–(11) in Mantilla et al.[26] for more details.

Although the nonlinear routing mechanism is closer to reality,we also included the linear routing analysis in this study as it isa good starting point to investigate the effect of rainfall variabilityon the peak flow scaling structure. Throughout this study, we use avalue of 0.5 m/s for the Vc for the linear routing scenario and k1 andk2 of 0.3 and �0.1 for the nonlinear routing scenario, obtainedbased on field data from the region. In Fig. 3, we show the velocityobtained using (3) for the link that corresponds to the largest up-stream area of each Horton order for the Whitewater River basin.Throughout the study, we employed a rainfall grid of size40 � 40 km2 with a spatial resolution of 1 km. The temporal reso-lution and the duration of the event (simulated as well as radardata) are different for different events, as mentioned in the corre-sponding sections.

3.3. Assumptions

In all of our simulation scenarios, we assume (1) negligibleevaporation; (2) purely surface runoff (i.e., no infiltration and nosubsurface runoff); and (3) instantaneous flow of runoff into the

e shown only for the channels that correspond to the largest area (displayed on each


channel. Evaporation rate is often an order of magnitude lowerthan storm rainfall rate, and Hortonian runoff generation is oneof the main flood producing mechanisms. From the brief reviewof literature presented in Section 2, one can infer that the complex-ity in the simulation-based studies that were carried out to under-stand the scaling behavior of peak flows have steadily increasedsince the early 90s. For instance, one of the first studies was basedon the deterministic Peano network and uniform rainfall (e.g.,[18]). Some recent studies have used random self-similar networksto mimic the river network behavior (e.g., [47,24]). We continue onthis trajectory by introducing complexity one step at a time. There-fore, in this study, the complexity is in terms of rainfall variabilityand the river network structure, which is why we limited our anal-ysis to the Hortonian runoff generation mechanism. We under-stand that in reality, other runoff producing mechanisms are alsopossible in the selected study area. The hydrologic model we usedcan account for the saturation excess mechanism, for instance.However, including it in the study would only add additional var-iability, and it is difficult to separate the role of rainfall variabilityand the variability introduced by the saturation excess mechanism.The third assumption regarding the instantaneous flow to thechannel plays a key role in shaping the hydrologic response. Forsmaller basins (<�10 km2), it leads to overestimation of the peakflows as the hillslope travel times are comparable to the time spentin the channel network (e.g., [7]). But the error is smoothed out forlarger basins.

Therefore, the assumptions are reasonable in the context ofexploring the roles of rainfall and channel network on the scalingexponents of peak flows, i.e., floods, for individual rainfall–runoffevents. Relaxing these assumptions and including other detailssuch as saturation excess flood production, hillslope travel timesand channel hydraulic geometry will be part of our futurecommunications.

Fig. 4. Hydrographs at six locations in the Whitewater River basin obtaine

4. Hydrographs vs. a scaling-based framework

This section illustrates via simple simulation experiments theadvantages of the scaling-based analysis of hydrologic response.The hydrologic model CUENCAS is forced with two simple rainfallscenarios of changing intensity (60 mm/h and 10 min) and dura-tion (5 mm/h and 120 min), while keeping the total rainfall volumeconstant (1.1 � 107 m3). The simulated rainfall is spatially uniformover the basin for the given duration. We also assumed that therunoff is Hortonian and reaches the channel instantaneously. Thedischarges are normalized with respect to the peak flow corre-sponding to the rainfall scenario of 60 mm/h for 10 min. The timeof occurrence is then normalized with the time at which the nor-malized discharge corresponding to the scenario of 60 mm/h and10 min reaches 0.01. In Fig. 4, we show the normalized hydro-graphs at six different locations in the Whitewater River basin.Although we show the normalized hydrographs at only six loca-tions, we simulated hydrographs for all the interior sub-basins aswell as for the outlet of the Whitewater River basin (Fig. 1) by solv-ing the mass and momentum equations throughout the river net-work. Fig. 4 demonstrates that at smaller scales, the values offlow peaks differ greatly from each other and occur at different in-stances. However, the flow hydrographs are indistinguishable aswe move to the larger scales.

We then relax the spatial uniformity assumption and assumethat the rainfall is randomly distributed in space over the hillslopesof that same basin. We obtained 10 realizations of the rainfall fol-lowing a uniform distribution over the range of 20–100 mm/h withthe average intensity equal to 60 mm/h and the duration kept at10 min. That is, for each rainfall field of size 40 � 40 km2, we gen-erated 1600 random numbers following a uniform distributionwith a range of [20,100] and a mean of 60 mm/h. It should benoted that these fields do not possess any spatial correlation. In

d from a distributed hydrologic model for a spatially uniform rainfall.

Fig. 5. Hydrographs at six locations in the Whitewater River basin obtained from a distributed hydrologic model. The gray lines are the hydrographs for each of the 10 rainfallrealizations assumed to be random in space with the intensities following the uniform distribution U [20,100] mm/h for a duration of 10 min. The solid line representshydrographs for the spatially uniform rainfall of 60 mm/h for 10 min.


Fig. 5, we compare the normalized hydrographs obtained withthese 10 rainfall fields with the one obtained for the spatially uni-form case of Fig. 4. It is clear from Fig. 5 that for spatially randomrainfall, the variability in the hydrographs at smaller scales is high-er compared to those of larger scales. Therefore, to develop a com-prehensive understanding of river basin response, it is imperativethat we study the hydrographs throughout the basin across multi-ple scales.

In this context, the results from spatially uniform rainfall(Fig. 4) can be alternatively represented in the form of Fig. 6aand b. Similarly, the results from spatially variable rainfall(Fig. 5) for two of the simulated realizations are shown in Fig. 6cand d. This framework allows us to study the basin response acrossmultiple scales. Fig. 6 illustrates that our simulated peak flows dis-play scaling structure with respect to the drainage area, and thescaling regime depends on the intensity, duration and variabilityof the rainfall. Fig. 6 also demonstrates that the effect of rainfallvariability on the basin response is scale-dependent. While peakflows are sensitive to the intensity, duration and spatial distribu-tion of rainfall at small scales (�10 km2), the variability in rainfallis dampened at larger scales (�1000 km2) by the river network viaaggregation of flows.

5. Basin response to the radar-rainfall data

To investigate the statistical structure of peak flows for a rangeof scales, it is necessary to have information on the spatial-tempo-ral distribution of rainfall events. Such information can be conve-niently provided by the ground-based weather radar network.We obtained radar estimates of three rainfall events that occurredin 2007 over Whitewater River basin, Kansas. The spatial resolu-tion of the data is 1 km, and the temporal resolution is 15 min.We forced the hydrologic model CUENCAS with radar-rainfall esti-mates and obtained the hydrographs for all the interior sub-basins

and the outlet of the Whitewater River basin. We assumed a linearrouting mechanism with constant flow velocity throughout the riv-er network. Fig. 7 shows the peak flow structure for the 6th of May2007 event that lasted for approximately 25 h. Fig. 7 reveals thatpeak flows display scaling with a scaling exponent of 0.70. We ob-tained ordinary least squares fit to the peak flows, though in theHortonian regression framework, similar to the width functionanalysis presented in the previous section. That is, instead ofobtaining the scaling exponent by regression of peak flows withupstream areas, we used (2) to obtain the scaling exponent.

The scaling exponent of 0.70 is larger than the scaling exponentof the width function maxima. For the other two 2007 events thatwe analyzed, the scaling exponents were 0.68 and 0.77. From thestudies discussed in Section 2, we know that when a spatially uni-form rainfall is applied instantaneously, the peak flow scalingexponent is very close to that of the width function maxima. A realrainfall event is far from being spatially uniform and lasts for a cer-tain duration. Therefore, the scaling exponent is different from thatof the width function maxima. Fig. 7 shows that the scatter at smallscales is different from that of the spatially uniform or spatiallyrandom case presented in Fig. 3. Another conspicuous feature inFig. 7 is that the scale break is poorly defined, possibly becauseof the inherent space–time variability of the rainfall event suchas zero-rain intermittency and its spatio-temporal correlationstructure.

The studies discussed in Section 2, which were carried out un-der idealized conditions, cannot clearly explain the effect of vari-ous characteristics of rainfall that resulted in Fig. 7. It is also wellknown that remotely sensed rainfall products suffer from largeuncertainties (e.g., [21,6]) that propagate through the hydrologicmodels and contribute to the variability of the predicted peakflows across scales. Consequently, it becomes necessary to separatethe effects of uncertainties from the effects of variability of rainfallon the peak flow scaling structure. The natural variability of rainfall

Fig. 6. Scaling of peak flows with respect to the upstream areas of all the sub-basins in the Whitewater River basin. For (a) and (b), the rainfall is spatially uniform withintensity and duration indicated on the panels. For (c) and (d), the rainfall is random in space with the intensities following the uniform distribution U [20,100] mm/h, and theduration is equal to 10 min.

Fig. 7. Scaling of peak flows with respect to the upstream areas of all the sub-basinsin the Whitewater River basin, Kansas. The rainfall data is obtained from the KICTNEXRAD weather radar in Wichita, Kansas. The color scheme indicates the Hortonorders as in Fig. 6. (For interpretation of the references in color in this figure legend,the reader is referred to the web version of this article.)


itself has a great impact on the statistical structure of peak flows,and understanding its role is the main goal of this study. Hereafter,we avoid using rainfall data and follow a systematic simulation

framework to explore the effect of rainfall variability on the peakflows.

6. Simulation scenarios and results

We start our experiments with simple but less realistic modelsof rainfall and proceed to complex space–time models that yieldmore plausible rainfall. We obtain parameters of these modelsfrom analyzing real rainfall events.

6.1. Sensitivity to the intensity and duration of spatially uniformrainfall

We start with the simple scenario of a basin receiving spatiallyuniform rainfall of a certain intensity and duration. Fig. 8 showsthe peak flows vs. drainage areas for different rainfall intensitiesand durations with a linear channel routing mechanism. Threeimportant features of the peak flow scaling structure that areapparent in the plots are the scatter, the scale break and the scalingexponent. For a fixed rainfall intensity, the scatter decreases as theduration of the event increases. For each link, there is an upper lim-it for the peak flow that is not exceeded. This upper limit corre-sponds to the equilibrium discharge reached when the rainfallduration is larger than the concentration time. With an increasein the duration of rainfall, more hillslopes reach saturation, therebydecreasing the scatter. The peak flows for the links that reachedsteady-state correspond to the well known rational method

Fig. 8. Sensitivity of peak flow scaling structure to intensity and duration of spatially uniform rainfall and linear channel routing with a velocity of 0.5 m/s. The solid black linerepresents the ordinary least squares fit (equation on each panel) obtained in the Hortonian framework. The color scheme indicates the Horton orders as in Fig. 6. The solidred line indicates the scale break. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)


Q = cIA, where c is the runoff coefficient, I is the rainfall intensityand A is the upstream drainage area. In this study, since the infil-tration is assumed to be zero, the value of c is equal to 1.0. We ob-tain the scale break by comparing the peak flows obtained fromour simulations to those from the above rational method equation.A window of fixed size in the logarithmic domain is moved alongthe upstream area axis of each panel in Fig. 8. Within such a win-dow, we compute the following ratio

v ¼Pnp

i¼1Ip;i

npð4Þ

where Ip;i ¼ 1 if 0:9Qrat 6 Qp;i 6 Qrat , Qp;i is the peak flow for the linki, Qrat is the corresponding peak flow obtained from the rationalmethod and np is the total number of links in the network. If the ra-tio v is less than 0.75, the scale break is considered to be at the aver-age of the upstream areas within that window. We realize that thisdefinition of scale break is subjective. However, it serves the pur-pose of a qualitative comparison only. The scale break is indicatedby a red line in Fig. 8. As the duration of the rainfall increases, morelinks reach saturation and the scale break moves towards the larger

areas. Because of the large scatter, we do not estimate the scalebreak for the shortest duration of 5 min.

Fig. 8 shows that for a fixed duration, the scale break and scatterin the peak flow scaling structure remain unchanged with inten-sity. As in Section 5, we fit these peak flows in the Hortonianframework using (2). In Fig. 9, we show the Horton plots of peakflows for all of the intensities and durations. The Horton ratio ofpeak flows is estimated considering only the orders that lie onthe higher side of the scale break. Since the scale break is not obvi-ous in the Horton plots, we select the orders for regression basedon Fig. 8. The Horton ratio of peak flows obtained by exponentia-tion of the regression slope is shown in each panel of Fig. 9. TheHorton ratio of peak flows and the upstream area are plugged intoEq. (2) to obtain the scaling exponent of the peak flows. The coef-ficient of the power law is obtained so that the regression linepasses through the average of the peak flows corresponding tothe top three orders (Fig. 8). The regression equations in Fig. 8allow us to conclude that, for a fixed duration, the peak flows arelinearly related to the rainfall intensities when the routing mecha-nism in the channels is linear. This result is similar to the one ob-served by Furey and Gupta [8] over the Goodwin Creek Watershed.

Fig. 9. Horton plots of peak flows for different combinations of intensity and duration of spatially uniform rainfall applied throughout the basin. The solid line indicates theordinary least squares regression fit. The corresponding Horton ratios are also indicated on each panel.


For a fixed intensity, the scaling exponents range from 0.50 to 0.56as the duration changes from 5 to 360 min. We have also noticedthat the peak flow at the outlet of the basin changes linearly withthe duration. The scaling exponents for all of the cases are largerthan the width function scaling exponent of 0.49, which confirmsthe result of Mantilla et al. [26] for the Walnut Gulch watershed.

Fig. 10 shows the effect of the intensity and duration of spatiallyuniform rainfall when the channel routing mechanism is nonlinear.The parameters selected in this study for the nonlinear routingmechanism result in different velocities in different links, and astraightforward panel to panel comparison between Figs. 8 and10 is therefore not meaningful. We do not obtain the scale breakfor the shortest duration simulations for the same reason men-tioned in the linear routing case. Regression is not performed forthe longest duration of 360 min, as most of the links have reachedsaturation and the scale break is poorly defined. For the 120 minduration, the fitted regression equations reveal that the scalingexponent decreases as the rainfall intensity decreases. It is alsoclear from Fig. 10 that the relationship between flow peaks andrainfall intensities is nonlinear. The peak flow scaling exponentsfor all the cases of nonlinear routing are larger than the exponentof the width function maxima. They range from 0.55 for the short-est duration of 5 min to 0.66 when the rainfall intensity is 50 mm/h

and the duration is 120 min. For the longest duration of 360 min,the scaling exponent is close to 1.0.

6.2. Sensitivity to the advection velocity

A spatially uniform rainfall of intensity 30 mm/h and coveringapproximately half the size of the basin (40 � 20 km2) is movedfrom west to east at five different velocities (4, 8, 16, 32 and64 km/h). It should be noted that the spatial uniformity is onlywithin the 40 � 20 km2 area; the rainfall actually received by thebasin cannot be considered as uniform. The peak flows are fittedin the Hortonian framework, and the regression equations areobtained. Fig. 11 plots the scaling exponents vs. advection veloci-ties for linear and nonlinear routing mechanisms. As expected fromthe results shown in Fig. 10, there is no scale break for the smallestadvection velocity of 4 km/h for the nonlinear routing mechanism,and therefore we did not perform any regression analysis. For bothchannel routing mechanisms, the scaling exponent decreases withan increase in advection velocity. Our motive for plotting linearand nonlinear routing mechanisms in the same figure is not tocompare them point-to-point but to compare how the exponentsdecrease with advection velocity. Fig. 11 also shows the powerlaw fit for both routing mechanisms. The fitted equations reveal

Fig. 10. Sensitivity of the peak flow scaling structure to intensity and duration of spatially uniform rainfall and nonlinear channel routing. The solid black line represents theordinary least squares fit (equation on each panel) performed in the Hortonian framework. The color scheme indicates the Horton orders as in Fig. 6. The solid red lineindicates the scale break. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. Sensitivity of the scaling exponent of peak flows to the advection velocityof spatially uniform rainfall of intensity 30 mm/h and linear and nonlinear channelrouting mechanisms.


that the trend is the same for both routing mechanisms. Thedecreasing trend can be explained in terms of the effect of durationdiscussed in the previous subsection. With the increase in advec-tion velocity, the duration for which the block of rainfall stays overthe basin decreases, and therefore the scaling exponent alsodecreases.

6.3. Sensitivity to the spatial variability of rainfall

We have thus far assumed that the rainfall is spatially uniformthroughout the basin. In this subsection, we investigate the effectof spatial variability on the scaling exponents of peak flows.Though there are many ways in which spatial variability can becharacterized, we explore it in terms of variance, correlation struc-ture and zero-rainfall intermittency. Our experiments are designedso that we depart from the uniform rainfall scenarios in a gradual,simple manner to keep from losing the benefits of the recentlygained understanding of the effects of the uniform intensity andduration.

Fig. 12. Effect of space–time variability of rainfall on the peak flow scalingstructure. The rainfall is taken to be spatially uniform with the intensity of 25 mm/hfor 30 min over the western half of the basin. For the eastern half of the basin, therainfall is 50 mm/h for 120 min. The color scheme indicates the Horton orders as inFig. 6. (For interpretation of the references in color in this figure legend, the readeris referred to the web version of this article.)


6.3.1. Simple block structureWe relax the spatial uniformity of rainfall over the basin by

breaking it into two components: a block of uniform rainfall withan intensity of 25 mm/h for a duration of 30 min on the westernhalf of the basin and a rainfall of 50 mm/h for a duration of120 min over the eastern half of the basin. The channel network

Fig. 13. Sensitivity of the peak flow scaling structure to the variance of the rainfall field. Tindicated on each panel. The color scheme indicates the Horton orders as in Fig. 6. (For inthe web version of this article.)

routing is assumed to be linear with a velocity of 0.5 m/s through-out the network. Fig. 12 shows the scaling structure of peak flowsfor this scenario. The banded structure apparent in Fig. 12 is a directmanifestation of the different rainfall intensities received by thewestern and eastern regions of the basin. The scale break for thesetwo bands also occurs at different locations because of the differentrainfall durations over the western and eastern parts of the basin.We noticed a similar trend for the nonlinear routing in channels.

6.3.2. Gaussian uncorrelated fieldIn this scenario, we simulate Gaussian random fields with a

mean of 25.0 mm/h and the standard deviation ranging from 0.1to 6 mm/h. By gradually varying the variance, we gently departfrom the well-understood case of uniform intensity. The durationof the rainfall is fixed at 120 min. The peak flow scaling structurefor four different cases of standard deviation and linear routingmechanism is shown in Fig. 13. In the Hortonian regression, weused the orders 3–7. Table 1 lists the average rainfall, intercept,scaling exponent and the peak flow at the outlet of the basin forall the cases and for both routing mechanisms. Fig. 13 and Table1 reveal that the increasing variance has no significant effect onthe fitted regression equations. The main effect of the variance isto increase the scatter in the peak flow scaling structure. The scat-ter is averaged out by the basin at the larger scales, as seen fromthe peak flow values at the outlet (Table 1). The slight variationin the intercepts and outlet peak flows is expected given that weare using realizations of a random process. This is further evidentfrom the estimated values of the mean, which are different fromthe theoretical value of 25 mm/h.

he rainfall field is assumed to be Gaussian with a mean of 25 mm/h and the varianceterpretation of the references in color in this figure legend, the reader is referred to

Table 1Sensitivity of intercepts, scaling exponents and outlet peak flows to the variance of the Gaussian rainfall field with a mean intensity of 25 mm/h.

Nðl;r2Þ Mean (mm/h) Linear routing Nonlinear routing

Intercept Slope Outlet peak flow (m3/s) Intercept Slope Outlet peak flow (m3/s)

N(25,0.1) 24.99 12.02 0.54 587.54 12.79 0.63 1246.87N(25,1.0) 24.98 12.00 0.54 587.13 12.73 0.63 1245.98N(25,4.0) 25.02 12.00 0.54 587.01 12.74 0.63 1246.65N(25,9.0) 24.93 12.08 0.54 588.67 12.85 0.63 1228.01N(25,16.0) 25.09 11.95 0.54 580.44 12.70 0.64 1249.81N(25,25.0) 24.88 11.93 0.54 587.15 12.70 0.63 1243.16N(25,36.0) 24.88 11.97 0.54 589.58 12.64 0.64 1252.31

Fig. 14. Sensitivity of the peak flow scaling structure to the spatial correlation of the rainfall field. The rainfall field is assumed to be Gaussian with a mean of 25 mm/h and astandard deviation of 2.0 mm/h. The spatial structure of the rainfall field is characterized by an exponential correlation structure with the correlation distance indicated oneach panel. The color scheme indicates the Horton orders as in Fig. 6. (For interpretation of the references in color in this figure legend, the reader is referred to the webversion of this article.)

Table 2Sensitivity of intercepts and scaling exponents to the spatial correlation structure of the rainfall field. The rainfall field is assumed to be Gaussian with a mean intensity of 25 mm/h and a variance of 2.0 mm/h and is characterized by an exponential correlation function with the correlation distances indicated in the table.

Correlation distance (km) Mean (mm/h) Linear routing Nonlinear routing

Intercept Slope Outlet peak flow (m3/s) Intercept Slope Outlet peak flow (m3/s)

5.0 25.06 12.19 0.54 581.92 12.99 0.63 1232.8510.0 24.41 11.92 0.54 566.09 12.67 0.63 1192.2520.0 24.83 11.86 0.54 582.06 12.62 0.64 1222.6530.0 26.49 12.54 0.54 619.69 13.44 0.64 1340.3740.0 25.48 12.33 0.54 597.79 13.17 0.63 1279.2550.0 23.46 11.29 0.54 545.59 11.88 0.63 1120.85


6.3.3. Gaussian correlated fieldTo investigate the effect of the spatial correlation of the rainfall

field on the scaling exponents, we obtain Gaussian fields with amean of 25 mm/h and a standard deviation of 2 mm/h and thatis characterized by an exponential correlation structure

qðdÞ ¼ h0 � exp � dh1

� �h2" #

ð5Þ

where h0 characterizes the nugget effect and the small scale vari-ability, h1 is the correlation distance defined as the distance atwhich the correlation drops to 1/e and h2 is the shape factor thatcontrols the shape of the correlation function at the origin. We fixedthe nugget parameter and the shape factor at one and generated therandom fields with the correlation distances varying from 5 to50 km. Each field is then applied for 120 min over the basin, andthe peak flows are estimated for linear and nonlinear routing mech-anisms. Fig. 14 shows the scaling structure of peak flows for two ex-

treme cases of correlation distances and a linear routingmechanism. The effect of increasing correlation is to decrease thescatter in the scaling structure (Fig. 14). Table 2 shows that the lar-ger scale basin response is almost independent of the correlationstructure, although there is some variability in the intercepts andoutlet peak flows, which is mainly due to the fact that we are usingrealizations of a random process.

6.3.4. Spatial zero-rain intermittency: uncorrelated random fieldsTo investigate the effect of zero-rainfall intermittency, we sim-

ulated random rainfall with varying degrees of zero-rainfall inter-mittency and a duration of 120 min. The value of rainfall over eachpixel was drawn from uniform distribution U [10,30], and intermit-tency is introduced randomly but maintains an overall mean fixedat 20 mm/h. We considered four values of intermittencies: 0.0,0.05, 0.25 and 0.50 (corresponding rainy area fractions are 1.0,0.95, 0.75 and 0.50). These rainfall scenarios were supplied as input


to the CUENCAS model, and the peak flow scaling structure was ob-tained for linear routing mechanisms. The sensitivity of the peakflow scaling to the intermittent random fields is shown inFig. 15. With the increase in intermittency (or decrease in rainyarea), the scatter for the smaller scale basin peak flows increased.However, the effect of intermittency is reduced for the larger scalebasins, as evidenced by the linear regression equations shown ineach panel of Fig. 15 and also from the outlet peak flow valuesshown in Table 3. The simulations are repeated for the nonlinearrouting mechanism, and we found a similar pattern to the patternfound in linear routing, although the intercepts and scaling expo-nents differed (Table 3).

6.3.5. Spatial zero-rain intermittency: correlated random fieldsTo study the effect of intermittency in a more realistic manner,

we selected the spatial component of the rainfall model developedby Bell [2]. The model belongs to the class of meta-Gaussian mod-

Fig. 15. Sensitivity of the peak flow scaling structure to the zero-rainfall intermittency ineach pixel drawn from uniform distribution U [10,30] with a mean of 20 mm/h and a dinterpretation of the references in color in this figure legend, the reader is referred to th

Table 3Sensitivity of intercepts and scaling exponents to the intermittency in the uncorrelated randU [10,30], and the duration of the event is 120 min. The mean of the field is kept constan

Intermittency (%) Mean (mm/h) Linear routing

Intercept Slope Outlet

0 19.88 11.45 0.50 464.665 20.04 11.73 0.50 473.95

25 20.05 11.23 0.50 469.3750 19.84 11.91 0.50 485.78

els (e.g., [27,2,13]) and generates a two-dimensional isotropic, cor-related random field using spectral analysis. A nonlineartransformation and an external threshold are then applied to ob-tain a rainfall field with desired intermittency, average intensityand correlation structure. In Bell [2], the use of exponential trans-formation resulted in lognormally distributed rainfall. The param-eters for the model are the log-transformed (Gaussian) mean andvariance of the rainy area, the zero-rainfall intermittency factorand the spatial correlation structure. The parameters we selectedare 0 and 0.5 for the log-transformed mean and variance of therainy area and exponential correlation structure with a correlationdistance of 20 km. We observed that the realizations from themodel, besides having the desired spatial correlation structure,also displayed spatial scaling behavior (not shown). The durationof the event is fixed at 120 min. To keep the volume constant withchanging intermittency, we simulated a single realization with agiven correlation structure on a large (256 � 256) domain and se-

rainfall fields. The rainfall fields are distributed randomly in space with the value aturation of 120 min. The color scheme indicates the Horton orders as in Fig. 6. (Fore web version of this article.)

om fields. The value of rainfall over each pixel was drawn from a uniform distributiont for different intermittencies.

Nonlinear routing

peak flow (m3/s) Intercept Slope Outlet peak flow (m3/s)

12.26 0.58 898.4812.23 0.58 907.8811.58 0.58 907.2412.41 0.58 942.22

Fig. 16. Sensitivity of the peak flow scaling structure to the spatial intermittency of the rainfall field. The rainy portion of the field is assumed to follow lognormal distribution.We did not show the regression equations as the peak flow scaling structure is too noisy to perform Hortonian regression for the bottom two panels of the figure. The colorscheme indicates the Horton orders as in Fig. 6. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)


lected the portion that yielded the desired intermittency and vol-ume. The spatial structure of the field thus obtained will remainthe same as the larger one. Fig. 16 shows the scaling structure ofpeak flows for four different intermittency factors starting from 0to 0.50 for the linear channel routing mechanism. Unlike inFig. 15, significant scatter was observed even for higher order ba-sins (particularly for the bottom panels of Fig. 16) when the pixelsare correlated. The large scatter is due to the high probability ofconcentrated intermittent pixels present in correlated intermittentfields. Whereas, for uncorrelated intermittent fields, the river net-work efficiently aggregates the randomness in the fields. However,the overall behavior – of increasing scatter with increasing inter-mittency – is similar for both scenarios.

7. Simulations from the space–time rainfall model

The scenarios investigated so far have offered insight into theeffects of different characteristics of rainfall on the spatial scalingstructure of peak flows. We will now investigate the basin’s re-sponse to more realistic space–time rainfall events. Therefore, we

Table 4Characteristics of rainfall events simulated from the space–time rainfall model. A two para

Mean (mm/h) Standard deviation (mm/h) Coefficient of variation (mm

Storm 1 3.97 10.37 2.61Storm 2 1.41 5.97 4.23

have simulated a space–time rainfall event from a model devel-oped by Bell [2] and applied it over the basin. The spatial compo-nent of the model is described in the previous section. Thetemporal evolution of the rainfall is modeled as an autoregressiveprocess with parameters based on the correlation time of area-averaged rainfall. The parameters for the model are obtained byanalyzing several storms over the Midwest. In this study, we sim-ulate two different storm events with characteristics listed in Table4. Storm 2 is more variable than storm 1, as seen from the values ofthe coefficient of variation and correlation distance. Anotherimportant difference is that storm 1 lasts longer and has larger val-ues of mean and rainy fraction compared to storm 2. The values ofthe shape factor suggest that storm 2 is more correlated at verysmall scales than storm 1.

Fig. 17 shows the basin response to the two storms for the lin-ear routing mechanism. The peak flow scaling structure for thiscomplex scenario can now be explained using the results from Sec-tion 6. Fig. 17 considers three components: scale break, scatter andregression equations. Although the scale break is sharp and evidentfor the simulation scenarios of Section 6 under idealized condi-tions, it is not clearly seen in the peak flow scaling structure result-

meter exponential correlation function is used to characterize the spatial dependence.

/h) Correlation distance (km) Shape factor Rainy area (%) Duration (h)

15.40 0.73 45.84 205.52 0.92 17.51 4

Fig. 17. Response of the watershed to the simulated space–time rainfall events. The characteristics of the storms are listed in Table 3. The color scheme indicates the Hortonorders as in Fig. 6. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)


ing from the simulated realistic space–time rainfall event. Sincethe scaling exponent is not close to 1.0, basin saturation (for in-stance, top panels of Fig. 8) is not the reason for the absence ofscale break. The lack of sharp scale break for realistic rainfall sce-narios can be best explained by revisiting the idealized scenariosin Section 6. For instance, a combination of just two different inten-sities and durations has diffused the scale break in Fig. 12. Thespace–time rainfall fields are characterized by different intensities,durations, correlations and intermittencies and move with a cer-tain advection velocity. In a way, these fields are a combinationof all of the scenarios considered in Section 6. This explains the ab-sence of scale break for storm 1.

The scatter for storm 1 is smaller than that of storm 2. Two mainfactors responsible for the reduced scatter are the duration of thestorm and zero-rain intermittency. Results from Section 6.1 indi-cate that one consequence of longer duration events is the de-creased scatter that is most pronounced at smaller scales. InSection 6.3.4 that pertains to the effect of zero-rain intermittency,we note that the scatter in the peak flow structure increases rap-idly as the area of rainfall decreases. Though the rainy fraction ofstorm 1 is 46%, the advection of the storm eventually increasesthe effective wetted area of the basin, thereby decreasing the scat-ter seen at smaller scales. For the second storm, the large scatter is

Fig. 18. Probability distributions of rescaled areas, width functi

due to the increased intermittency combined with the shorterduration and large coefficient of variation.

The regression equation seen for storm 1 is obtained in a Horto-nian framework using the orders 2–7. For storm 2, peak flows cor-responding to the Horton orders of 4–7 are used in this storm’sregression. The scaling exponent for both storms is larger thanthe scaling exponent of the width function maxima. For the nonlin-ear routing scenario (not shown), we noticed a similar pattern withlarger scatter and higher values for scaling exponents than in thelinear routing case.

8. Analysis of scatter

The scatter in the peak flow scaling structure for the lower orderbasins can be explained in terms of peak flows reaching equilib-rium (basins reaching saturation), while the scatter for the higherorder basins can be explained in terms of aggregation and attenu-ation of flows. We illustrate this by analyzing the peak flows fortwo of the rainfall scenarios in Section 6.1 and Fig. 8. Specifically,we compare the probability distributions of the rescaled peakflows corresponding to spatially uniform rainfall of 5 mm/h anddurations of 5 and 120 min with the probability distributions of re-scaled areas and width function maxima. Fig. 18a illustrates that

on maxima and peak flows for order 1 and order 5 basins.


for a spatially uniform rainfall intensity of 5 mm/h and a durationof 120 min, the order 1 probability distributions of rescaled peakflows and areas are indistinguishable (negligible scatter in Fig. 8),but for a shorter duration of 5 min, the probability distributionsare very different (large scatter in Fig. 8). Since the width functionmaxima have the signature of the aggregation of flows in the chan-nel network, we compare the order 5 probability distributions ofrescaled peak flows and width function maxima in Fig. 18b. We in-tended to compare the probability distribution of peak flows withthat of width function maxima for higher order basins. However,for orders 6 and 7, there are not enough points to obtain the prob-ability distributions. Therefore, we limit the comparison to order 5basins. Unlike the order 1 distribution, the order 5 distribution isnot very sensitive to the duration of the rainfall event. For bothrainfall scenarios, the distributions of rescaled peak flows matchreasonably well with those of width function maxima (Fig. 18b).

9. Remarks

While our results are subject to the usual limitations of a simu-lation study, our experiments contain many realistic aspects. First,our river basin has a substantially larger size (scale) than manysmall experimental basins that are the basis for many hydrologicstudies. Second, using high-resolution DEM data, we extracted ariver network that closely approximates the actual drainage pat-tern of the selected basin. Third, our rainfall variability cases andrange of values, though simple, capture the key aspects of naturalrain systems.

We modeled rainfall space–time variability in terms of station-ary random fields with a certain intermittency and correlationstructure in space and time. Although the emphasis of this ap-proach is more on generation of space–time random fields andlacks a direct link to the physical aspects of the rainfall process,one can qualitatively relate the statistical parameters to the mete-orological aspects such as back-building thunderstorms, squalllines and convective systems. For instance, the size of the convec-tive system with respect to the size of the basin is an importantcharacteristic that controls the basin response. This informationis embedded in the correlation distance and shape factor in Eq.(5). There are other ways of characterizing the space–time variabil-ity of rainfall fields such as modeling of rainfall based on spatialcluster processes for rain cells, dynamic modeling of rainfall basedon partial differential equations for mass and momentum conser-vation and scaling-based modeling of rainfall space–time struc-ture. It would be interesting to employ these models and relatethe parameters of rainfall models to the characteristics of peakflow scaling structure.

Anthropogenic alteration of the landscape will have an impacton the extracted drainage network and subsequently on the aggre-gation of the flows affecting the larger scale basin response. Thehydraulic geometry of the channels is another key factor whichis strongly related to the channel–floodplain interactions and influ-ences the storage zones, movement of the flood waves and the tra-vel time within the channel. Therefore, it is expected to have aninfluence on the peak flow scaling structure. The CUENCAS frame-work allows for the inclusion of this feature by modifying the localvelocity law. However, investigation of this factor is beyond thescope of this work, and our strategy is to avoid any aspects inthe dynamics that can obscure the effect of rainfall variability.

Throughout this study, we have estimated the Horton ratios byordinary least squares regression of logarithms of arithmetic aver-ages with the Horton order (see Fig. 2). Based on a simulationstudy, Furey and Troutman [10] suggested the use of individualquantities (for example, areas or peak flows) instead of arithmeticaverages. Since the main focus of the study is on the role that rain-

fall variability plays in the scaling structure of peak flows, our useof arithmetic averages instead of individual quantities in the Hor-ton analysis would not affect the results. Also, in this study, wehave assumed that the runoff generation is Hortonian with no infil-tration so that we can focus on the role that rainfall plays in thestatistical structure of peak flows. The key issue is how to specifythe infiltration threshold for each of 20,000 hillslopes in CUENCAS,which differ due to spatial variability in soil and vegetation prop-erties. This problem of ‘‘dynamic parametric complexity” is a majorresearch problem (e.g., [14,9]) and not addressed in this study.

10. Summary and conclusions

In this study, we carried out a systematic investigation tounderstand the role that rainfall plays in the spatial structure ofpeak flows. Due to the lack of adequate field data, i.e., numerousstream gauges as well as highly accurate rainfall maps, we usedsimulations. Our simulation experiments consisted of simple sce-narios aimed at isolating the effects of rainfall variability on thepeak flow scaling structure. We demonstrated that rainfall vari-ability has a different impact on the magnitude of peak flows forbasins of different scales. We selected the Whitewater River basinin Kansas for this study and a distributed hillslope-link basedhydrological model to obtain the peak flows for each link withinthe basin. The channel network that was extracted is characterizedin terms of width function maxima. The width function maxima ofthe Whitewater River basin displayed scaling behavior with re-spect to the Horton orders. The scaling exponent of width functionmaxima was estimated to be 0.49.

We focused on three aspects of the peak flow scaling structurefor all the scenarios: scatter, scale break and the scaling exponents.The results showed that the peak flow scaling exponents for all thescenarios considered in this study are greater than the width func-tion scaling exponent. This result is in agreement with the hypoth-esis of Mantilla et al. [26] that in the river networks, the peak flowscaling exponent is governed by the competition between theattenuation and aggregation of the flows. For a fixed intensity,the scaling exponent increases with an increase in the rainfallduration, and for a fixed duration, the scaling exponent does notchange with intensity for linear channel routing and decreaseswith intensity for nonlinear channel routing. For the 2 h duration,the fitted regression equations reveal that the scaling exponent de-creases as the rainfall intensity decreases. Based on simulationswith spatially uniform rainfall of varying depths and a fixed dura-tion of 10 min on a deterministic Mandelbrot-Viscek network,Menabde and Sivapalan [28] reported that the scaling exponent in-creases as the rainfall depth decreases. For the Whitewater Riverbasin, and therefore in a real river network, we did not notice sucha trend for 30 and 5 min duration simulations and noticed a re-verse trend when the duration was 120 min.

For a constant volume of rainfall, the effect of spatial variability,as characterized by variance, spatial correlation and the spatialintermittency, is to increase the scatter in the peak flow scalingstructure. At larger scales, the effect of variability decreases, asseen from the regression equations and peak discharges at the out-let of the basin (Tables 1–3). Based on the simulations on a deter-ministic Mandelbrot-Viscek network, Menabde and Sivapalan [28]reported that the variability in the rainfall decreases the scalingexponent of peak flows on both sides of the scale break. We didnot observe such behavior in our simulations. For homogeneousrainfall fields and under idealized conditions of flow routing onhillslopes and in channels, we observed that the smaller scale basinresponse was dominated by the rainfall intensity (and spatial dis-tribution), while the hydrologic response of larger scale basins wasdriven by rainfall volume, river network topology and flow dynam-


ics. We expect that the heterogeneity in rainfall will have similarbehavior, at least for the larger scale basins as the river networkaggregates the heterogeneity. However, heterogeneity in rainfallwill have a larger impact for smaller scale basins.

The results obtained from the above simple scenarios enhancedour understanding of the peak flow scaling structure obtained fromsimulated space–time variable rainfall. Storm duration and advec-tion are the key factors that control the effective zero-rain inter-mittency, which in turn affects the scatter in the peak flows. Thepeak flow scaling structure for the realistic space–time rainfall sce-narios did not present a clear and sharp scale break. The scalebreak was masked due to the inherent space–time variability inthe realistic rainfall fields.

The results in this study also foster the development of a scal-ing-based predictive framework for peak flows using remotelysensed rainfall products over basins ranging from very small tovery large scales. A key question is, ‘‘What is the scale at which re-mote sensing products provide meaningful predictions?” Our re-sults suggest that the variability contributed by random errors ofremote sensing sensors, such as weather radars and satellites, arefiltered out by the drainage structure of river basins at some scales.Investigations of the above problem are underway and requiremodels of uncertainty such as those developed by Ciach et al. [6],Villarini et al. [49] and Villarini and Krajewski [48].

Acknowledgements

The authors recognize support from the National Science Foun-dation grant EAR-0450320, the Rose and Joseph Summers endow-ment to The University of Iowa and IIHR-Hydroscience andEngineering. We also acknowledge valuable discussions with Dr.Thomas Bell regarding the space–time model of rainfall.

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http://dx.doi.org/10.1029/2002WR001708

http://dx.doi.org/10.1029/2006WR005782

http://dx.doi.org/10.1029/2006WR005782

http://dx.doi.org/10.1029/2006WR005329

http://dx.doi.org/10.1029/2007WR006118

http://dx.doi.org/10.1029/2007WR006654

http://dx.doi.org/10.1029/2006GL029147

http://dx.doi.org/10.1029/2008WR006946

Dissecting the effect of rainfall variability on the statistical structure of peak flows

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