a conceptual investigation of process controls upon ﬂood frequency… · 2016. 1. 10. · i....

Hydrol. Earth Syst. Sci., 11, 1405–1416, 2007www.hydrol-earth-syst-sci.net/11/1405/2007/© Author(s) 2007. This work is licensedunder a Creative Commons License.

Hydrology andEarth System

Sciences

A conceptual investigation of process controls upon flood frequency:role of thresholds

I. Struthers1 and M. Sivapalan2,3

1School of Environmental Systems Engineering, The University of Western Australia, 35 Stirling Highway, Crawley WA6009, Australia2Centre for Water Research, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia3now at: Department of Geography and Department of Civil & Environmental Engineering University of Illinois atUrbana-Champaign, Urbana, Illinois, IL 61801, USA

Received: 11 October 2006 – Published in Hydrol. Earth Syst. Sci. Discuss.: 26 October 2006Revised: 15 May 2007 – Accepted: 8 June 2007 – Published: 6 July 2007

Abstract. Traditional statistical approaches to flood fre-quency inherently assume homogeneity and stationarity inthe flood generation process. This study illustrates the impactof heterogeneity associated with threshold non-linearitiesin the storage-discharge relationship associated with therainfall-runoff process upon flood frequency behaviour. For asimplified, non-threshold (i.e. homogeneous) scenario, floodfrequency can be characterised in terms of rainfall frequency,the characteristic response time of the catchment, and stormintermittency, modified by the relative strength of evapora-tion. The flood frequency curve is then a consistent transfor-mation of the rainfall frequency curve, and could be readilydescribed by traditional statistical methods. The introductionof storage thresholds, namely a field capacity storage anda catchment storage capacity, however, results in differentflood frequency “regions” associated with distinctly differ-ent rainfall-runoff response behaviour and different processcontrols. The return period associated with the transition be-tween these regions is directly related to the frequency ofthreshold exceedence. Where threshold exceedence is rela-tively rare, statistical extrapolation of flood frequency on thebasis of short historical flood records risks ignoring this het-erogeneity, and therefore significantly underestimating themagnitude of extreme flood peaks.

1 Introduction

Flood frequency analysis is concerned with describing theprobabilistic behaviour of flood peaks for a given location;specifically, it attempts to quantify the likelihood of a flood

Correspondence to:M. Sivapalan([email protected])

of a given magnitude occurring within a given time inter-val (Pattison et al., 1977). An understanding of flood fre-quency is of direct applicability for flood forecasting, but ismost commonly used for engineering design against risk; forthe design of flood-managing structures, such as reservoirs,drainage systems and levees, as well as flood-impacted struc-tures, such as roads, bridges and buildings.

Traditional methods of flood frequency analysis rely onstatistical fitting of historical flood records, where these areavailable. Where historical records are short, or the catch-ment is ungauged, flood frequency at high return periodsis estimated either by extrapolation of statistical fits, or byregionalisation methods, in which the flood frequency be-haviour of “similar” catchments are used to infer (usuallyinvolving some scaling relationship) the flood frequency be-haviour of an ungauged or data-poor catchment (e.g. Gupta etal., 1996). Central to traditional statistical methods for floodfrequency are the assumptions of homogeneity (i.e. that allflood peaks are independent, and drawn from the same prob-ability distribution) and stationarity (i.e. that the probabilitydistribution of flood peaks is temporally invariant). Land usechange, including the construction of dams and levees, havelong been acknowledged as violating the assumption of sta-tionarity, such that flood frequency must be re-determinedwhenever the properties of the rainfall to flood transforma-tion are altered due to changes in the catchment. More re-cently, non-stationarity due to the impact of climate variabil-ity and climate change have also been a subject of study (e.g.Kiem et al., 2003). The consequence of non-stationarity isthat the flood frequency curve varies with time, such that thereturn period associated with a flood of a given magnitudedoes not have a fixed value, and loses its traditional meaning.

Published by Copernicus Publications on behalf of the European Geosciences Union.

1406 I. Struthers and M. Sivapalan: Thresholds and flood frequency

Less attention has been given to an examination of the im-pact of heterogeneity upon altering flood frequency. Hetero-geneity is most commonly associated with a multiplicity ofmechanisms responsible for a flood peak, whether these re-late to heterogeneity in rainfall generation, or in the trans-formation of rainfall into a flood peak. Merz and Bloschl(2003) identified different storm types as resulting in dif-ferent flood responses, with different storm types being im-portant in describing different ranges of return period withinthe flood frequency curve. The consequence of heterogene-ity is that, while the flood frequency curve does not varywith time, the flood frequency curve consists of different“regions” associated with different mechanisms or combina-tions of mechanisms that cannot be readily fitted by a sim-ple probability distribution function. Where heterogeneity ornon-stationarity is significant, therefore, traditional statisticalmethods for flood frequency analysis are unsuited.

An alternative, derived flood frequency approach attemptsto apply our understanding of rainfall generation behaviour,and of the transformation process from rainfall into floodresponse, in order to understand the dynamics of flood fre-quency and its process controls (e.g. Eagleson, 1972). Un-like statistical methods, an understanding of process con-trols upon flood frequency permits theforecastingof floodfrequency subject to non-stationarity and heterogeneity, aswell as a rationalisation of non-stationarity and heterogene-ity in historic flood frequency behaviour – at least qualita-tively, if not quantitatively. The objective of this study is topresent an analysis of the process controls on flood frequencyfor a simplified representation of catchment hydrological re-sponse, and then to illustrate the impact of heterogeneity inthe hydrological response – specifically, that associated withthreshold non-linearities in the catchment storage-dischargerelationship – in altering flood frequency behaviour. The im-pacts of specific elements of spatial and temporal variabilityupon flood frequency are then considered. While Kusumas-tuti et al. (2006) considers the sensitivity of flood frequencyto threshold impacts for a specific climate using similar cli-mate and catchment parameterisations, this study aims todescribe climate and catchment process controls upon floodfrequency more generally, and to identify important param-eter groupings that can be used to generally characterise theflood frequency response, including heterogeneity therein, ofa given climate-catchment combination.

2 Methodology

This study couples a stochastic rainfall generator with aspatially-lumped rainfall-runoff model, which are run con-tinuously for 1000 years in order to obtain estimates of floodfrequency based on the maximum hourly flood intensity ineach individual year of simulation (i.e. annual maxima se-ries). Attenuation due to runoff routing in the river networkis not explicitly considered. Our analysis employs “essen-

tial” models, which contain only the basic elements of cli-mate forcing and catchment hydrological response; evapora-tion is essentially continuous, except during rainfall, whilethe essential properties of rainfall are its intermittency andits stochasticity. The fundamental “action” of the catchmentis to facilitate attenuation and loss, giving rise to a flood re-sponse signal which is filtered relative to the rainfall forcingsignal. The various simplifications employed in this studymake this methodology unsuitable for accurate flood predic-tion, but have the over-riding benefit – given the study ob-jectives – of permitting the derivation of a relatively clearexplanatory framework for the resulting flood frequency be-haviour in terms of dominant climate and landscape proper-ties.

2.1 Stochastic rainfall generator

Flood frequency is stochastic by its nature; in the absenceof substantial historical precipitation records, the use of astochastic rainfall generator is therefore logical for flood fre-quency examination. This study employs an event-basedrainfall model similar to that of Robinson and Sivapalan(1997). In an event-based model, each “event” consists ofa storm period of durationtr and average rainfall intensityi and an inter-storm period of durationtb. While Robinsonand Sivapalan (1997) considered storm intensity to be corre-lated to storm duration, in this study each of these propertiesis considered to be an independent, exponentially-distributedrandom variable, such that:

fX

(x|x

)=

1

xexp

(−

x

x

)(1)

wherex is the storm property variable (tr , i, or tb), andx is anaverage value for that variable, and noting thatE [x] terms incertain figures and expressions are equivalent tox. Within-storm patterns in rainfall intensity for a given storm are gen-erated using a stochastic disaggregation method, with identi-cal parameterisation and methodology as given in Robinsonand Sivapalan (1997). Seasonality in storm properties is ini-tially ignored.

2.2 Conceptual catchment model

The catchment is represented as a single bucket (e.g. Man-abe, 1969) of total storage capacity,Sb, and field capacitystorage,Sf c. At storages above field capacity the subsurfaceflow rate is linearly related to catchment storage, modulatedby a characteristic response time,tc, such that:

qss (t) =

{S(t)−Sf c

tcS (t) > Sf c

0 S (t) ≤ Sf c

(2)

Total storage capacity and field capacity storage can be di-rectly related to catchment average soil depth, average poros-ity and moisture content at field capacity, while the character-istic response time can be estimated by calibration using aslittle as 5–10 days of streamflow data (Atkinson et al., 2002).

Hydrol. Earth Syst. Sci., 11, 1405–1416, 2007 www.hydrol-earth-syst-sci.net/11/1405/2007/

I. Struthers and M. Sivapalan: Thresholds and flood frequency 1407

Table 1. Parameter values used in the construction of figures. V indicates parameters which can be freely varied. AG indicates parametersthat vary “as given” in the legend between individual curves, and N/A indicates variables which are not applicable to the particular figure.

Figure tr tb i tc Sb rA rB β Other

1a, b V V V N/A N/A N/A N/A N/A2a 10 100 1 AG ∞ 0 N/A 02b V V V V ∞ 0 N/A 03a, b, c, d V V V V ∞ AG AG 03e 10 100 1 100 ∞ V V 04a 10 100 1 100 V 2 5 04b V V V AG V AG AG 05 10 100 1 100 200 2 5 0

6 10 100 1 100 300 2 5 0

tra=6 htba=50 hia=0.4 mm/hrepa = 0.8 mm/hr

7a 10 100 1 100 V AG AG AG7b 10 100 1 100 200 2 5 AG

During a storm event, evaporation is considered to be neg-ligible, with rainfall infiltration constrained only by the stor-age capacity of the catchment. When the catchment is full,surface runoff occurs at a rate equal to the rainfall rate minusthe ongoing subsurface flow rate;

qs (t) =

{max[i (t) − qss (t) , 0] S (t) = Sb

0 S < Sb(3)

Thus, the equation for the rate of change of storage during astorm event is:

dS (t)

dt= i (t) − qss (t) − qs (t) (4)

So long as storages are above field capacity, subsurface flowis continuous during the inter-storm period, while surfacerunoff will cease. Evaporation is considered to occur duringthe inter-storm period at potential rates for storages greaterthan field capacity, and at a linearly-reduced rate below fieldcapacity;

e (t) =

{ep (t) S (t) ≥ Sf c

ep (t)S(t)Sf c

S (t) < Sf c(5)

such that the rate of change of storage during the inter-stormperiod can be expressed as:

dS (t)

dt= −e (t) − qss (t) (6)

Unlike storm properties, the potential evaporation rate is notconsidered to be stochastically variable; seasonality inep isalso initially ignored, such that the value is constant duringall interstorm periods.

3 Results and discussion

In attempting to summarise modelled flood frequency be-haviour, it is useful to consider certain “end-member” lim-iting cases, the behaviour of which is in some ways simpli-fied relative to the full complexity of the response. A triv-ial end-member is the case where storage is always belowfield capacity, such that no flood response occurs; obviouslythis case is of no consequence for flood frequency modelling.Two more meaningful end-member cases will be consideredin turn: (i) Case 1 (maximum): an impervious catchment(i.e.Sb=0), for which all rainfall is transformed into surfacerunoff, and (ii) Case 2 (baseline): attenuation without thresh-olds; specifically, a deep soil with no evaporation, such thatstorage is always greater than field capacity but less than thetotal storage capacity, for which subsurface runoff will becontinuously produced. Each of these end-member cases arethemselves homogeneous; once these homogeneous casesare understood, we can then generalise to consider hetero-geneity associated with evaporation below field capacity andintermittent saturation. In what follows, the expected valueof the annual flood peak is denoted byQav; numerical sub-script refers to a flood peak associated with a given returnperiod, e.g. the 1 in 50 year flood peak is denoted asQ50;and superscripts denote flood peaks associated with a givenend-member case. Parameterisations used in the constructionof specific figures are summarised in Table 1.

Figures presented in this paper utilise estimated values ofthe average annual flood peak, and flood peaks for speci-fied return periods. Each datapoint in Figs. 1a, b, 2a, 3a,b, c, d, 4a, and 7a is obtained by analysis of the output ofa 1000 year simulation, with different datapoints relating todifferent catchment or average climate parameter values, asdescribed in Table 1. The remaining figures present flood

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1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50.5

1

1.5

2

2.5

ln(n)

ln(Q

/E[i

])

Qav

*

Q10

*

Q50

*

Q100

*

Figure 1a. Rainfall frequency ( ) indices for the case without within-storm variability in

storm intensity. Shown are the expected value of the rainfall intensity peak, the 1 in 10 year

peak, 1 in 50 year peak and 1 in 100 year peak, for an assortment of rainfall

parameterisations.

maxTQ

27

Fig. 1a. Rainfall frequency (QmaxT

) indices for the case withoutwithin-storm variability in storm intensity. Shown are the expectedvalue of the rainfall intensity peak, the 1 in 10 year peak, 1 in 50year peak and 1 in 100 year peak, for an assortment of rainfall pa-rameterisations.

frequency curves, with each individual curve correspondingto a single simulation. As is always the case with flood fre-quency, uncertainty increases with increasing return period,such that replication with a different random number seed iscapable of producing different flood frequency curves – es-pecially at high return periods; while predictions may there-fore be quantitatively different for each 1000 year realisation,the qualitative features of the output will be the same. Forthis reason, results from only one realization are presentedin this paper. The primary motivation of this study is to un-derstand process controls upon flood frequency, such that amajority of the effort in data analysis involved the mostlyempirical identification of parameter groupings which couldbe used to “collapse” the full variety of flood response be-haviour into singular curves with understandable behaviour;parameter groupings used as the x-ordinate in many of thefigures in this study represent the end-result of such analysis.

3.1 Examination of “rainfall frequency” behaviour (Qmax)

Runoff and rainfall are intimately linked; the action of acatchment is essentially to filter and attenuate the rainfall sig-nal to produce a runoff signal. In attempting to understandprocess controls upon flood frequency, an understanding ofthe stochastic nature of rainfall is therefore crucial. In theextreme case of a catchment that insignificantly filters andattenuates rainfall (e.g. a small car park or an initially verywet catchment), the runoff peak will be approximately thesame as the rainfall intensity peak. In terms of the modelemployed in this study, this refers to the case where the totalstorage capacity,Sb=0, as well as the no evaporation caseswheretc is infinitely large (i.e. all rainfall is converted to sur-face runoff) or else extremely small (i.e. all rainfall is con-verted immediately to subsurface runoff). Flood frequencyanalysis, specifically the annual exceedence probability, isconcerned with peak intensities occurring in each year. Sinceall storm properties are considered independent, the magni-tude of the largest storm intensity in a given year will be de-pendent upon the properties of the probability distributionfunction of storm intensity as well as the number of storms

5 6 7 8 9 10 11

1

2

3

4

5

ln(ntr2)

ln(Q

/E[i

])

Qav

*

Q10

*

Q50

*

Q100

*

Figure 1b. Rainfall frequency ( ) indices including within-storm variability in storm

intensity. Shown are the expected value of the rainfall intensity peak, the 1 in 10 year peak, 1

in 50 year peak and 1 in 100 year peak, for an assortment of rainfall parameterisations.

maxTQ

28

Fig. 1b. Rainfall frequency (QmaxT

) indices including within-stormvariability in storm intensity. Shown are the expected value of therainfall intensity peak, the 1 in 10 year peak, 1 in 50 year peak and1 in 100 year peak, for an assortment of rainfall parameterisations.

(i.e. number of random realisations) in the year; the averagenumber of storms in a year is given by:

n =365× 24

tr + tb(7)

The exponential distribution for storm intensity has only oneparameter, the average storm intensityi, which acts as a scal-ing factor upon the distribution of storm intensity values; byscaling the rainfall intensity peaks byi, Fig. 1a shows that(for the case ignoring within-storm variability in storm in-tensity) the expected value of the scaled annual rainfall in-tensity peak, as well as scaled rainfall intensity peaks associ-ated with any given return period, will increase as the aver-age number of storm events in a year increases, as expected.Specifically, the average value of the annual rainfall intensitypeak obtained through simulations with the stochastic rain-fall model (without within-storm variability) was found to fitthe following empirical expression:

Qmaxav = 1.42in0.845 (8)

This figure is essentially a representation of the extremevalue distribution of average storm intensity; for the assumedexponential parent distribution function for storm intensity,the extreme value distribution will exactly fit a Gumbel dis-tribution (Sivapalan and Bloschl, 1998). The addition ofwithin-storm variability in storm intensity adds an additionaldependency upon the average storm duration itself (Fig. 1b);this is due to the fact that the number of bisections necessaryto resolve within-storm variability to a specified resolution(e.g., hourly) is directly functional upon storm duration. It iscrucial to note, however, that the relative impact of the aver-age storm duration upon resulting peak intensity behaviour,represented by the exponent value of 2, is not necessarily ageneral finding; alternative parameterisations and temporalresolutions for rainfall disaggregation may result in differentexponents, if not completely different functional forms, pos-sibly incorporating parameters relating to the disaggregationof rainfall to fine scales. Nonetheless, for the specific param-eterisation employed in this study, the average annual rainfall



1.01 1.1 1.5 2 5 10 20 50 100 50010

−2

10−1

100

101

102

103

Average Recurrence Interval (yrs)

No

rmal

ised

Dis

char

ge,

Q/E

[i]

tc=100000 hrs

tc=10000 hrs

tc=100 hrs

Precipitation

Figure 2a. The impact of the characteristic response time for subsurface flow upon baseline

flood frequency.

29

Fig. 2a. The impact of the characteristic response time for subsur-face flow upon baseline flood frequency.

intensity peak including the impact of within-storm variabil-ity was found to fit the following empirical expression:

Qmaxav = 0.031i

(nt2

r

)3.04(9)

Similar relationships, with different numerical coefficientsbut of the same functional form, can be derived relating topeaks for specified return periods, such as those presented inFigs. 1a and b. Of course, these empirical relationships arevalid for the specific assumptions made in relation to rainfallstochasticity; deviation of storm duration, inter-storm periodor average storm intensity from the assumed exponential dis-tributions, correlation between storm duration and averageintensity, and systematic long-term variability in storm prop-erties (associated with specific climate variability cycles, forexample), are among the possible complexities which maylead to actual rainfall frequency deviating from these pre-dictions. Seasonal variability in average storm propertieswill, of course, also have an impact; this will be consideredlater. Note that all simulations conducted hereinafter includewithin-storm variability in storm intensity.

3.2 Flood frequency ignoring thresholds and evaporation(Qbase)

We initially consider the simplified no evaporation case withan infinitely large storage capacity; for this scenario sub-surface flow is continuous and is the sole runoff genera-tion mechanism. We will herein refer to this scenario as the“baseline” case, i.e. the flood frequency response as impactedby the basic intermittency of storms, and attenuation by thecatchment. Figure 2a illustrates the impact that the charac-teristic response time for subsurface flow,tc, has upon floodfrequency for a particular climate for the baseline case. For

−40 −35 −30 −25 −20 −15 −10 −5 0 5 100

0.2

0.4

0.6

0.8

1

ln(1/ntc3)

QT*

Qav

* Q

10*

Q50

* Q

100*

Figure 2b. Baseline flood frequency indices for an assortment of storm and catchment

parameterisations.

30

Fig. 2b. Baseline flood frequency indices for an assortment of stormand catchment parameterisations.

the sake of comparison we also presentQmaxT in this figure

(representing essentially the rainfall peak). For extremelyslow subsurface flow, astc→∞, stochastic variability in theannual flood peak is negligible, making the flood frequencycurve essentially flat. From analysis conducted using a rangeof climate properties, this minimum value of flood peak forthe baseline case is found to be independent ofT and to beapproximately equal to:

Qmin= i

(tr

tr + tb

)(10)

Faster subsurface flow, associated with lower values oftc, causes a steepening of the flood frequency curve, asrunoff behaviour becomes more responsive to stochasticityin rainfall behaviour. As we have previously identified, astc→0, flood frequency will converge with rainfall frequency,Qmax

T (corresponding to the rainfall peak). We investigatedwhether the “baseline” behaviour for specifiedtc values be-tween falling these extremes can therefore be expressed com-pactly in terms ofQmin, Qmax

T and the specific value oftc.Figure 2b illustrates the end result of this exploration for awide range of average storm properties and values oftc. Notethat the y-ordinate in this and many subsequent figures is anormalised value of the natural logarithm of the flood peak,given by:

Q∗

T =ln

(Qbase

T

)− ln

(Qmin

)ln

(Qmax

T

)− ln

(Qmin

) (11)

whereT refers to the return period of interest. The expres-sion for the x-ordinate in Fig. 2b (and many subsequent plots)is dominated by the impact oftc, such that we can generaliseit to describe increasing values of x to correspond to fastersubsurface flow response time (i.e. lowertc). From Fig. 2bit is apparent that the scaled flood peak for the baseline casetakes the form of a simple logistic function with respect to thenatural logarithm of the parameter groupnt3

c . The resultingempirical functional form describing the curves presented inFig. 2b is therefore of the form:

Q∗

T =1

1 + a(nt3

c

)b(12)



−50 −40 −30 −20 −10 0 10 20−1.5

−1

−0.5

0

0.5

1

ln(1/ntc3)

Qav

*

Baseline (ep=0)

rA

=1,Sfc

=0

rA

=10,Sfc

=0

rA

=100,Sfc

=0

Figure 3a. Expected value of the annual flood peak, for the non-threshold case including

evaporation for different values of the dimensionless ratio, rA.

31

Fig. 3a. Expected value of the annual flood peak, for the non-threshold case including evaporation for different values of the di-mensionless ratio,rA.

where a and b are empirical coefficients that vary onlyslightly depending uponT (i.e. a=0.0519 andb=−0.174for T =10 years, compared toa=0.0534 andb=−0.163 forT =100 years). Allowing for the scatter associated with thefinite simulation length – especially in relation to large re-turn periods – these curves approximately overlap, such thatvariation in a and b with T is relatively small, i.e.Q∗

T isapproximately constant for a given value ofnt3

c . We can re-gardQ∗

T , which ranges in value from 0 to 1, as an index ofthe slope of the flood frequency curve relative to the rain-fall frequency curve. A value ofQ∗

T approaching 0 corre-sponds totc→∞, with flood peaks due to subsurface flowbeing relatively invariant from one year to another, approx-imately equal toQmin. As Q∗

T increases, flood peaks dueto subsurface flow will be increasingly impacted by event-scale variability in storm properties (and consequent event-scale variability in the antecedent condition), such that floodmagnitude increases significantly with increasing return pe-riod. AsQ∗

T approaches 1, attenuation by the catchment be-comes minimal, and the flood frequency curve has the sameslope as (and is therefore equivalent to) the rainfall frequencycurve. In summary, the slope and magnitude of the baselineflood frequency curve is functional upon the average valuesof storm intensity, storm duration, and number of storms peryear, which together determine the magnitude ofQmax

T , andthe characteristic response time for subsurface flow, whichdetermines the degree of attenuation associated with the rain-fall to runoff transformation.

3.3 Impact of evaporation on subsurface flow flood fre-quency

In terms of the rainfall to runoff transformation, evaporationacts essentially as a loss mechanism. If we consider evap-oration during rainfall to be negligible, evaporation duringinter-storm periods is capable of reducing the flood peak in-directly via its impact upon reducing antecedent storage priorto a given storm event. In the absence of a field capacitystorage threshold, fast-draining soils (i.e. lowtc) will tendto dry towards zero storage by the end of inter-storm periodseven without evaporation, such that the impact of evaporation

−50 −40 −30 −20 −10 0 10 20−1

−0.5

0

0.5

1

ln(1/ntc3)

Qav

*

Baseline (ep=0)

rA

=1,rB

=0

rA

=10,rB

=20

rA

=100,rB

=50

Figure 3b. Expected value of the annual flood peak, for the case with field capacity threshold

and evaporation for different combinations of the dimensionless ratios rA and rB. B

32

Fig. 3b. Expected value of the annual flood peak, for the case withfield capacity threshold and evaporation for different combinationsof the dimensionless ratiosrA andrB .

upon the antecedent condition and hence flood frequency willbe negligible. In contrast, for slow-draining soils, the impactof evaporation in reducing the antecedent moisture contentwill be significant, such that the evaporation will have a no-ticeable impact upon flood frequency. Figure 3a presents theexpected value of the annual flood peak in the absence of afield capacity threshold for various strengths of evaporationin contrast to the baseline (no evaporation) case. For val-ues of the x-ordinate greater than−20, which typically cor-responds to values oftc<200 h, evaporation has no impact,such that all curves overlap with the baseline curve. Only fortc>200 h is the soil sufficiently slow-responding that evapo-rative reduction in the antecedent condition causes a reduc-tion in the average annual flood peak. The magnitude of theevaporative impact obtained from the simulations was foundto be a function of the ratio of the per-event expected po-tential evaporation volume to the expected rainfall volume,

rA=ep tb

i tr. Where this ratio is significantly greater than 1, the

behaviour ofQ∗av for x values less than−20 is approximately

linear (on semi-log axes) and relatively unchanging. If theratio is equal to 0, the baseline expression describes its be-haviour. Where 0<rA<1, the behaviour falls between thesetwo “envelope” curves.

3.4 Impact of field capacity on subsurface flow flood fre-quency

For the no evaporation case, and for an infinite catchmentstorage capacity, the addition of a field capacity has no im-pact whatsoever upon flood frequency; it will merely modifythe storage corresponding to a given subsurface flow rate, asgiven by (2). This was already reflected in the results pre-sented in Figs. 2a and b. Only when combined with evap-oration is the field capacity threshold capable of impactingrunoff generation, and hence flood frequency. Specifically,the impact of evaporation in combination with field capacityis to permit the (intermittent) cessation of subsurface flow.Particularly in dry years, this will result in significant re-ductions in the magnitude of the annual flood peak; whereflow cessation lasts for the entire year, the annual flood peakis 0. The frequency and persistence of flow cessation was



−40 −30 −20 −10 0 10 20−0.5

0

0.5

1

ln(1/ntc3)

Qav

*

Baseline (ep=0)

rA

=0.6 (Sfc

large)

rA

=0.9 (Sfc

large)

rA

=1.0 (Sfc

large)

Figure 3c. Expected value of the annual flood peak for the case of extremely large field

capacity threshold and evaporation.

33

Fig. 3c. Expected value of the annual flood peak for the case ofextremely large field capacity threshold and evaporation.

−40 −30 −20 −10 0 10 20−0.5

0

0.5

1

ln(1/ntc3)

Q10

0*

Baseline (ep=0)

rA

=0.6 (Sfc

large)

rA

=0.9 (Sfc

large)

rA

=1.0 (Sfc

large)

Figure 3d. Predictions of the 100 year return period flood peak for the case of extremely large

field capacity threshold and evaporation for different values of the dimensionless ratio rA.

34

Fig. 3d. Predictions of the 100 year return period flood peak for thecase of extremely large field capacity threshold and evaporation fordifferent values of the dimensionless ratiorA.

again found to depend uponrA, as well asrB=Sf c

ep tb, the ra-

tio of field capacity storage to the event potential evapora-tion volume, which is a measure of the relative magnitudeof potential storage deficit below field capacity. The role ofevaporation is to reduce the antecedent storage to below thefield capacity threshold and its impact upon flood frequencyis maximised where these two ratios are large. Figure 3b il-lustrates how the expected value of the annual flood peak forsubsurface flow is modified from the baseline behaviour bythe introduction of a field capacity storage threshold, in com-bination with evaporation. ForrA≤1, behaviour is essentiallythe same as the case without the field capacity threshold (i.e.Fig. 3a) unlessrB>>1. AsrB increases, the impact of evap-oration below field capacity is manifested as a downwardtranslation of the curves relative to the “evaporation withoutfield capacity” case for the same value ofrA.

Figure 3c shows the impact of evaporation for the case ofan extremely large field capacity,rB>1×108; for this sce-nario, the ratioS/Sf c in (5), remains approximately equalto 1 throughout the simulation, such that evaporation occursat potential rates above and below field capacity. Contrast-ing this to Fig. 3b illustrates the potential importance of thepiecewise evaporation assumption in (5) upon resulting floodfrequency behaviour. In this scenario, flood frequency be-haviour is found to be highly sensitive to the value ofrA;for values significantly lower than 1, the volume of evapo-ration is low relative to the storm volume, and the impactof evaporation is not significant. AsrA increases towards 1,the evaporative volume is comparable to the storm volume,

1.01 1.1 1.5 2 5 10 20 50 100 50010

−4

10−3

10−2

10−1

100

101

102


No

rmal

ised

Dis

char

ge,

Q/E

[i]

No evap. rA

=1 , rB

=10rA

=2 , rB

=5 rA

=5 , rB

=2

Precipitation

Figure 3e. Impact of evaporation and field capacity threshold upon flood frequency for

different combinations of the dimensionless ratios rA and rB. B

35

Fig. 3e. Impact of evaporation and field capacity threshold uponflood frequency for different combinations of the dimensionless ra-tios rA andrB .

such that the runoff response is highly sensitive to stochas-tic variability in storm properties, and hence the antecedentstorage condition. The impact of evaporation is less signif-icant in wetter years associated with higher return periods;Fig. 3d shows the 100 year return period flood peak for thesame cases as Fig. 3c. For applications specifically interestedin extreme flood events only, therefore, accurate accountingof the complex impacts of evaporation upon flood frequencymay not be particularly crucial, especially whererA<1.

Figure 3e presents a partial summary of the impact ofevaporation upon flood frequency, showing the impact of in-creasing the value ofep alone (which is manifested as a pro-portional increase in the value ofrA and reduction inrB).This figure provides further insights into the summary resultspresented above in Figs. 3c and d; higher evaporation relativeto storm volume (i.e. higherrA) for a moderately large fieldcapacity results in an increased prevalence of flow cessation,with many years generating no flood response. But at largerreturn periods, the discrepancy between each curve reducesrapidly, until the impact of evaporation is minor for floodswith an average recurrence interval of 100 years or more. Ofcourse, the return period at which these curves converge isdependent upon the values ofrA andrB ; where these valuesare larger, convergence could occur at return periods greaterthan 100 years.

3.5 Impact of saturation

Analysis thus far has assumed an infinite catchment storagecapacity, such that saturation, and hence saturation excesssurface runoff, cannot occur. A finite storage capacity intro-duces the possibility of saturation excess, which is a fasterrunoff response mechanism (i.e. considered here to have an



1.01 1.1 1.5 2 5 10 20 50 100 50010

−3

10−2

10−1

100

101

102


No

rmal

ised

Dis

char

ge,

Q/E

[i]

Infinite Sb

Sb=300 mm

Sb=200 mm

Sb=150mm

Precipitation

Figure 4a. Impact of finite catchment storage capacity, and saturation excess, upon flood

frequency.

36

Fig. 4a. Impact of finite catchment storage capacity, and saturationexcess, upon flood frequency.

instantaneous concentration time) than subsurface flow.Figure 4a illustrates the impact of a finite catchment stor-

age capacity and resulting saturation excess upon flood fre-quency in contrast to a catchment with infinite storage ca-pacity, and hence no saturation excess. The onset of satu-ration excess results in a significant break in the flood fre-quency curve at a return period associated with the averagerecurrence interval between years in which saturation excessis triggered. Flood frequency at high return periods even-tually converges towards the rainfall frequency curve (sincesurface runoff is assumed to be converted instantaneously toa flood response; a finite response time for surface runoffwould result in peaks below the rainfall frequency values).The break in curve occurs at lower return periods as thecatchment storage capacity reduces. At return periods be-low this break, flood peaks are associated with subsurfaceflow only, such that this region of the flood frequency curveis not impacted by changes in the value ofSb. Similar breaksor “transitions” in the flood frequency curve have previouslybeen described by Sivapalan et al. (1990), associated witha transition from partial area saturation excess-dominated toinfiltration excess-dominated runoff. Where several runoffmechanisms exist, the flood frequency curve may thereforehave multiple breaks, each associated with a transition in thedominant runoff-producing mechanism.

Previous work by Struthers et al. (2007) formulated a di-mensionless index of the relative frequency of saturation ex-cess triggering for equivalent rainfall and runoff models asthose used here:

α =

itc

[exp

(trtc

)− 1

](Sb − Sf c

) [exp

(trtc

)−

(S0−Sf c

Sb−Sf c

)] (13)

−8 −6 −4 −2 0 2 40

0.2

0.4

0.6

0.8

1

ln(α)

Qav

**

tc=100 hrs,e

p=0

tc=100 hrs,r

A=1,r

B=10

tc=1000 hrs,e

p=0

tc=1000 hrs,r

A=1,r

B=10

tc=1000 hrs,r

A=10,r

B=1

tc=10000 hrs,e

p=0

tc=10000 hrs,r

A=1,r

B=10

Figure 4b. Example relationships between normalised values of the annual flood peak and the

index α for different combinations of the dimensionless ratios rA and rB. B

37

Fig. 4b. Example relationships between normalised values of theannual flood peak and the indexα for different combinations of thedimensionless ratiosrA andrB .

whereS0 is the expected value of antecedent storage, whichin this study is determined a posteriori from analysis of sim-ulation data. This index is a direct re-arrangement of thesingle-storm criteria for the triggering of saturation excess;for an individual storm with these average properties inci-dent upon a catchment with average antecedent storage, sat-uration excess will occur only ifα>1. When applied insteadto a population of stochastically varying storms, and hencestochastically varying antecedent conditions,α instead re-lates to the per-storm frequency of saturation excess occur-rence, with this relative frequency increasing asα increases.

In the case of a sufficiently large catchment storage capac-ity, even an extreme duration and intensity storm occurringin wet antecedent conditions will be insufficient to generatesaturation excess, and flood frequency behaviour is identicalto the subsurface flow-only case considered thus far. In theopposite extreme, where the catchment storage capacity issufficiently small that all storms trigger saturation excess, itis apparent from the summation of (2) and (3) that the an-nual flood peak will correspond to the annual rainfall inten-sity peak. Once more, it is therefore possible to normaliseflood frequency behaviour between the subsurface flow onlyvalue, which we will denote asQss , and the rainfall fre-quency value,Qmax

T , for a given return periodT :

Q∗∗

T =ln (QT ) − ln

(Qss

T

)ln

(Qmax

T

)− ln

(Qss

T

) (14)

Figure 4b presents example cases of the impact of intermit-tent triggering of saturation excess, which is functional onα, upon the expected value of the annual flood peak; for agiven curve, variation inα is obtained by altering the bucketstorage capacity,Sb (noting that this will have a consequentimpact in alteringS0). As expected, below some thresholdvalue ofα, wetting due to rainfall is insufficient to cause satu-ration, such that flood frequency is the same as the subsurfaceflow only case,Qss . As α increases, saturation excess mayoccur rarely, associated with long return periods only. Fur-ther increases inα, associated with a wetter climate and/or asmaller catchment storage capacity, results in saturation ex-cess occurring more often, but the likelihood of the timing ofsaturation excess corresponding with the timing of the peak



annual rainfall intensity is still small, and therefore the floodfrequency is still significantly belowQmax

T . Only with furtherincreases inα is the occurrence of saturation excess so com-mon that, in most years, its occurrence is concurrent with theannual rainfall intensity peak, and flood frequency is approx-imately equal to the rainfall frequency. Note, however, thatthe return period at which the break in the flood frequencycurve occurs does not necessarily correspond to the averagetime between saturation excess events; the importance of an-tecedent conditions upon saturation excess triggering meansthat its occurrence is often clustered in time, i.e. there is arelatively high probability of repeated triggering of satura-tion excess by storms immediately following its initial trig-gering. Due to clustering, in years when saturation excessoccurs there is a finite probability that it will occur multipletimes. As a consequence, the average time between satura-tion excess triggering (i.e. the average of the full distribution)is less than or equal to the return period associated with thebreak in the flood frequency curve (i.e. the average of the ex-treme value distribution). Nonetheless, increases inα abovethe lower threshold value are associated with a reduction inthe average time between saturation excess triggering (as im-plied from the results of Struthers et al., 2007), as well as areduction in the return period associated with the break in theflood frequency curve.

Unlike the description in relation to subsurface flow floodfrequency,Qss , the behaviour of flood frequency includingsaturation excess is still not firmly defined; while behaviouris relatively consistent for a given value oftc, the thresholdvalue ofα associated with the onset of saturation excess andthe degree of curvature changes with changing magnitudesof tc in a manner that is as yet not understood. The indexα

is therefore imperfect, but is nonetheless useful in describingthe basic process controls upon the frequency of saturationexcess occurrence, and its consequent impact upon flood fre-quency.

3.6 Summary

For the simplified rainfall and runoff response assumptionsused in this study, Fig. 5 presents a summary of the hetero-geneity in the resulting flood frequency in terms of four re-gions, corresponding to (i) evaporation- and field capacity-controlled flow reduction and cessation, (ii) storm intermit-tency and stochasticity-controlled subsurface flow, (iii) sat-uration excess surface flow limited by saturation triggeringfrequency, and (iv) rainfall-limited saturation excess surfaceflow. The transition between regions (i) and (ii) is usuallygradual, associated with a reduction in the impact of evapora-tion upon flood peaks in wetter rainfall years. In contrast, thetransition between regions (ii) and (iii) is abrupt, due to thesignificantly quicker response time associated with surfacerunoff compared to subsurface runoff, which results in orderof magnitude changes in the resulting flood peak. Region (iii)refers to the situation where saturation excess is occurring,

1.01 1.1 1.5 2 5 10 20 50 100 500 10

−2

10−1

100

101

102

103


Nor

mal

ised

Dis

char

ge (

Q/E

[i])

Peak Discharge

Precipitation

(i)

(ii)

(iii)

(iv)

Figure 5. Illustration of distinct regions of behaviour and process control within the modelled

flood frequency curve.

38

Fig. 5. Illustration of distinct regions of behaviour and process con-trol within the modelled flood frequency curve.

but does not necessarily occur at the same time as extremerainfall intensities in a given year, such that flood frequencypeaks for a given return period are still significantly belowthe corresponding rainfall intensity peak. Rainfall frequencyprovides an upper limit for flood frequency, such that region(iv) contains values of annual flood peak caused by saturationexcess which is concurrent with extreme rainfall intensities(i.e. intensities of the same order as the annual rainfall peakin a given year).

Each region essentially has distinct process controls and avery distinct curve shape, such that it would be difficult to fitbehaviour with a simple statistical model without introduc-ing significant inaccuracies. Given that statistical approachesare commonly used to obtain estimates of large return periodflood magnitudes by extrapolation from short historical floodrecords, which may not incorporate rare events such as sat-uration excess overland flow, the benefit of a derived floodfrequency approach over a statistical approach is readily ap-parent.

The framework developed in this study can be extendedto consider the impact of additional complexities in alteringthe idealised flood frequency behaviour presented in Fig. 5.Seasonality has been ignored thus far, in order to reduce thenumber of parameters in the model, but will undoubtedly im-pact upon the magnitude of the annual flood peak. By con-ceptualising the catchment using a single bucket, the modelassumes that overland flow is, in a sense, a binary process;occurring over the whole catchment, or not occurring at all.Of course, the more conventional and realistic approach tooverland flow involves the consideration of contributing ar-eas where, for a majority of time, some percentage of thecatchment, greater than 0% but less than 100%, is contribut-ing to saturation excess overland flow.



1.01 1.1 1.5 2 5 10 20 50 100 500 10

−3

10−2

10−1

100

101

102

103


Nor

mal

ised

Dis

char

ge (

Q/E

[i])

No seasonalitySeasonalityRescaled no seasonality

Figure 6. The impact of seasonality upon flood frequency. Also illustrated is the ability to

approximately account for simple seasonality implicitly using appropriately scaled average

parameter values without seasonality.

39

Fig. 6. The impact of seasonality upon flood frequency. Also illus-trated is the ability to approximately account for simple seasonal-ity implicitly using appropriately scaled average parameter valueswithout seasonality.

3.7 Impact of seasonality

The analysis thus far considered the impact of threshold non-linearities upon flood frequency in response to climate forc-ing with stochastic variability at within-storm and between-storm scales, but without deterministic variability associatedwith seasonality. It is beyond the scope of a conceptual studysuch as this to consider the full complexity of seasonal vari-ability in storm and inter-storm properties associated with,for example, seasonal differences in storm type (e.g., synop-tic versus cyclonic rainfall). We simply wish to illustrate theimpact that seasonal variability in storm properties, evapora-tion, and hence antecedent conditions, could have upon floodfrequency. Seasonality is incorporated in the rainfall genera-tor using a simple sinusoidal variability in average storm andinterstorm properties, of the form:

xt = x + xa sin

{2π

ωh

(t − t ′)

}(15)

wherext is the average value of the property in question (i.e.storm duration, interstorm period, potential evaporation rate,average storm intensity), for use in (1),x is the sinusoidalmean value,xa is the seasonal amplitude,t is the hour ofthe year, andt ′ is the phase lag in hours. As with the origi-nal (non-seasonal) methodology, the actual value of a givenrealisation of storm duration, interstorm period, and averagestorm intensity is considered to be stochastically variable (i.e.with xt as the mean value of the stochastic distribution attime t), while potential evaporation is non-variable (i.e. witha value equal toxt at timet). Our examination of seasonalityimpacts is restricted to the case of out-of-phase seasonalitybetween “wetting” and “drying” climate elements; i.e. sea-

sonality in storm duration and average storm intensity areconsidered to be perfectly in phase with one another, and ofopposing phase to interstorm period and potential evapora-tion seasonality. This “wet season-dry season” simplificationis reasonable for most non-tropical climates.

For flood frequency, which is concerned with annual ex-tremes rather than the complete stochasticity of flood re-sponse, we would expect seasonality to have two types ofimpact: a direct impact, in terms of longer or more intensestorms in a given season, leading to larger runoff response,and an indirect impact associated with seasonality in the an-tecedent condition resulting from seasonality in storm prop-erties. Seasonality in antecedent moisture will be lagged tosome degree relative to seasonality in storm properties, suchthat it probably will not simply be a case of applying themethodology previously described using “wet season” aver-age climate parameters in place of seasonally-averaged cli-mate parameters. Indeed, it was found that a given “wetseason-dry season” scenario could be reasonably replicatedby a “no seasonality” scenario in which the required param-eter values (i.e.tr , tb, i andep) were obtained by modifyingthe sinusoidal average values as appropriate (i.e. increasedfor wetting climate elements, decreased for drying climate el-ements) by approximately 50% of the seasonal amplitude. Inother words, a case in whichxa is non-zero for a given prop-erty can be approximately replicated by a case in whichxa isassigned a value of zero, andx is appropriately modified. Forexample, a climate with an 11 hr sinusoidal mean storm dura-tion with 4 hr seasonal variability in this value can be approx-imately replicated by assuming a 13 h average storm durationwithout seasonal variability; seasonality in average storm in-tensity can be similarly accounted for. Conversely, if poten-tial evaporation varies seasonally with a 0.1 mm/hr sinusoidalaverage with 0.04 mm/hr seasonal variability, this can be ap-proximately replicated by using a value of 0.08 mm/hr with-out seasonal variability; seasonality in the average interstormperiod can likewise be accounted for.

Invariably, the impact of seasonality is to increase the floodpeak associated with a given return period, relative to thesame average climate without seasonality. More specifically,seasonality will decreases the frequency of, or eliminate al-together, flow cessation over an entire year. It will also in-crease the frequency of saturation excess triggering, such thatthe break of slope in the flood frequency curve will occurat lower return periods. In effect, the impact of seasonal-ity is to translate the flood frequency curve to the left, suchthat floods of a given magnitude are associated with a lowerreturn period relative to the no seasonality case. Figure 6illustrates the impact of seasonality upon increasing the fre-quency of saturation excess (hence reducing the return periodassociated with the break in the flood frequency curve), re-ducing the frequency of or eliminating flow cessation, andincreasing the magnitude of flood peaks associated with sub-surface flow. Also illustrated is the ability of a non-seasonalsimulation with appropriately modified parameter values to



−8 −6 −4 −2 0 2 40

0.2

0.4

0.6

0.8

1

ln(α)

Qav

**

ep=0,β=0

rA

=1,rB

=10,β=0

ep=0,β=1

rA

=1,rB

=10,β=1

Figure 7a. Impact of variable soil depth and partial saturation upon the α relationship with

the normalised expected annual flood peak for different combinations of the dimensionless

ratios rA and rB and the parameter β of the Xinanjiang distribution. B

40

Fig. 7a. Impact of variable soil depth and partial saturation upon theα relationship with the normalised expected annual flood peak fordifferent combinations of the dimensionless ratiosrA and rB andthe parameterβ of the Xinanjiang distribution.

reasonably replicate the flood frequency of a seasonally-variable climate.

3.8 Impact of landscape spatial variability

A lumped representation of the catchment was employed inthis study so far, which ignores the impact of spatial variabil-ity in the timing and intensity of rainfall, as well as spatialorganisation in landscape properties. Issues of spatial scaleand spatial variability are considered in detail in other stud-ies (e.g. Bloschl and Sivapalan, 1997; Jothityangkoon andSivapalan, 2001); this study restricts itself to considering theimpact of simple spatial organisation in soil depth, and con-sequent spatial variability in threshold triggering, upon floodfrequency. Soil depths are considered to vary with distanceaway from the stream according to the Xinanjiang distribu-tion (Wood et al., 1992), where the depth of bucketi in aseries ofn buckets is given by:

Sb,i = Smax

[1 −

(1 −

i − 0.5

n

)1/β]

(16)

whereSmax is effectively a scaling coefficient for all soildepths andβ is a shape parameter. Buckets are consideredto be connected in series, from deepest to shallowest, withthe characteristic response time for subsurface flow in eachbucket,tc,i , being equal. The impact of spatial variability insoil depth is assessed by comparing the obtained flood fre-quency against the flood frequency for a lumped catchmentwith the same average soil depth, i.e.Sb=E

[Sb,i

], and with

an equivalent catchment-scale characteristic response timefor subsurface flow,tc=ntc,i . Note that the multiple bucketcase withβ=0 behaves identically to the single-bucket casewith equivalent average soil depth and catchment-scale char-acteristic response time for subsurface flow, such that thereis no discretisation artefact.

Figure 7a provides an example of the impact of spatialvariability in soil depth upon the triggering of saturation ex-cess; the value ofα is calculated for the equivalent single-bucket case, but response behaviour relates to a variable soildepth. As for Fig. 4a, variation inα is obtained by alter-ing the bucket storage capacity,Sb, and the onset of (partial)

1.01 1.1 1.5 2 5 10 20 50 100 50010

−4

10−3

10−2

10−1

100

101

102

103


No

rmal

ised

Dis

char

ge,

Q/E

[i]

β = 0 β = 0.1β = 1 β = 10

Figure 7b. The impact of variable soil depth and partial saturation upon flood frequency.

41

Fig. 7b. The impact of variable soil depth and partial saturationupon flood frequency.

saturation is associated with normalised flood peak valuesgreater than 0. The impact of variable soil depth is to createshallower-than-average buckets, which will saturate at lowervalues ofα compared to the single-bucket case; the largerthe value ofβ, the smaller the shallowest bucket depth is,and partial saturation excess onset will occur at progressivelylower values ofα. Figure 7b illustrates the impact of partialsaturation upon the flood frequency curve. For small valuesof β, which are associated with reductions in the near-streamsoil depth, but relatively constant soil depths away from thestream, the incidence of flow cessation is significantly re-duced, and (partial) saturation becomes more common. Asβ

becomes larger, partial saturation occurs in most or all years,such that flood magnitudes at low return periods are signifi-cantly increased. Conversely, extreme flood magnitudes as-sociated with large return periods are reduced by the impactof variable soil depth; in effect, partial saturation increasesthe efficiency of runoff generation and reduces the overallretention of water in the catchment at any point in time, suchthat full saturation (i.e. 100% of the catchment) rarely if everoccurs.

Variable soil depth therefore seems to simplify the be-haviour of the flood frequency curve, by reducing or elim-inating flow reduction and cessation due to evaporation (re-gion (i)), and allowing a smooth instead of abrupt transitionassociated with steadily increasing partial saturation insteadof binary saturation behaviour. It is crucial to note, how-ever, that this simplifying impact has been exaggerated bythe assumed organisation of soil depths within the catch-ment. By assuming that soil depth monotonically decreasestowards the stream, the impact of run-on of surface runoffas it moves down slope is neglected, which would otherwiseresult in more abrupt transitions in the extent of partial satu-ration.



4 Conclusions

In the absence of thresholds, runoff is produced by a sin-gle mechanism; in this study, subsurface flow. Deviation offlood frequency from rainfall frequency for this single mech-anism will be functional upon the characteristic responsetimescale associated with the runoff generation and routingprocesses (i.e. attenuation) and non-runoff losses which im-pact antecedent conditions and runoff recession (e.g., evapo-ration).

The addition of thresholds introduced a multiplicity ofintermittently-active runoff generation pathways; deviationof flood frequency from rainfall frequency will therefore beheterogeneous, with different ranges of return period associ-ated with different regimes of runoff pathway activity. Acti-vation of a given flow pathway is conditional upon thresh-old exceedence, such that the transition from one regimeto another will depend upon the frequency of threshold ex-ceedence. For storage-based thresholds, the relative fre-quency of threshold exceedance depends upon the value ofthe threshold itself, properties of rainfall forcing includingdeterministic and stochastic variability, and non-runoff losseswhich impact the antecedent condition. The degree of hetero-geneity in the flood frequency curve will therefore be max-imised where there is a wide discrepancy between the re-sponse timescales associated with each runoff mechanism,and where the per-storm frequency of threshold exceedenceis greater than 0 but less than 1.

Temporal variability in storm properties associated withseasonality always has the impact of increasing the magni-tude of the flood frequency response associated with a givenrunoff process, and is likely to increase the frequency ofthreshold exceedence, relative to a non-seasonal case withthe same average storm properties. Spatial variability inlandscape properties and climate properties results in spatialvariability in the local frequency of threshold exceedence,with the nature of landscape spatial variability being crucialin determining how this impacts upon catchment flood fre-quency. For decreasing soil depth towards the stream, forexample, partial saturation (i.e. threshold exceedance in atemporally-variable proportion of the total catchment area)results in a masking of the impacts of threshold behaviourupon the resulting flood frequency; nonetheless, flood fre-quency behaviour remains heterogeneous.

Acknowledgements.The authors wish to acknowledge fundingsupport provided through an Australian Research Council (ARC)Discovery Grant awarded to the second author. SESE Reference053.

Edited by: L. Pfister

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