Linking flood frequency to long-term water balance: Incorporating effects of seasonality

Authors


Abstract

[1] Derived flood frequency models can be used to study climate and land use change effects on the flood frequency curve. Intra-annual (i.e., within year) climate variability strongly impacts upon the flood frequency characteristics in two ways: in a direct way through the seasonal variability of storm characteristics and indirectly through the seasonality of rainfall and evapotranspiration which then affect the antecedent catchment conditions for individual storm events. In this paper we propose a quasi-analytical derived flood frequency model that is able to account for both types of seasonalities. The model treats individual events separately. It consists of a rainfall model with seasonally varying parameters. Increased flood peaks, as compared to block rainfall, due to random within-storm rainfall time patterns are represented by a factor that is a function of the ratio of storm duration and catchment response time. Event runoff coefficients are allowed to vary seasonally and include a random component. Their statistical characteristics are derived from long-term water balance simulations. The components of the derived flood frequency model are integrated in probability space to derive monthly flood frequency curves. These are then combined into annual flood frequency curves. Comparisons with Monte Carlo simulations using parameters that are typical of Austrian catchments indicate that the approximations used here are appropriate. We perform sensitivity analyses to explore the effects of the interaction of rainfall and antecedent soil moisture seasonalities on the flood frequency curve. When the two seasonalities are in phase, there is resonance, which increases the flood frequency curve dramatically. We are also able to isolate the contributions of individual months to the annual flood frequency curve. Monthly flood frequency curves cross over for the parameters chosen here, as extreme floods tend to mainly occur in summer while less extreme floods may occur throughout the year.

1. Introduction

[2] Floods are often generated by different processes in the same catchment. In parts of Western Australia, for example, floods with a return period less than 10 years are typically winter floods, whereas floods with a return period larger than 30 years tend to be summer floods despite generally drier soils in summer [Sivandran, 2002]. This arises due to different mechanisms of rain producing events (frontal events in winter, thunderstorms and tropical cyclones in summer), and their interaction with different flooding processes dominating in different times of the year. In Austria there also exist significant seasonal patterns of flood processes. In most of the country, the most extreme floods are summer floods, often produced by long-duration synoptic events, while smaller floods can occur throughout the year [Merz and Blöschl, 2003]. Differences in flood processes related to rain-fed (in summer and autumn) and snowmelt driven (spring) floods have received attention in North America and Europe [Waylen and Woo, 1982; Stedinger et al., 1992]. Hirschboeck [1987] performed a detailed analysis on causative mechanisms of floods in a number of catchments in Arizona based on surface and upper weather maps [Hirschboeck, 1988]. This scheme was updated by House and Hirschboeck [1997] and simplified into three event types (tropical, convective, and frontal events). The body of work on causative mechanisms allowed Hirschboeck [1987] and Alila and Mtiraoui [2002] to examine the flood statistics for each group of events and derive hydroclimatically defined mixed distributions in flood series. Merz and Blöschl [2003] found significantly different flood frequency statistics for long-rain floods, short-rain floods, flash floods, rain-on-snow floods, and snowmelt floods in Austria. The need to distinguish between flood frequency curves in different months of the year, e.g., between summer and winter, or between rain-fed or snowmelt driven floods, is becoming much more critical because changes in climate and land use cannot be fully investigated without explicitly incorporating changes in the associated intra-annual (e.g., seasonal) and interannual (i.e., between year) variabilities. This is the main motivation for this paper.

[3] Traditional flood frequency analysis depends upon the existence of long periods of flood records. A recent concern with such data-driven procedures is that changes in climate and/or land uses may affect the flood frequencies, yet may not be reflected by the short records we currently have [Franks and Kuczera, 2002; Sankarasubramanian and Lall, 2003], and the consequent nonstationarities in the data record may invalidate traditional flood frequency analysis. Also, unraveling land use and climate change effects from observed flood frequency curves is notoriously difficult [Kundzewicz, 2003].

[4] Therefore flood frequency estimation in the future is increasingly likely to be based on a combination of traditional data-driven procedures, with increased use of rainfall-runoff models that can capture the effects of both climate and land use changes [Blazkova and Beven, 1997]. Derived flood frequency procedures are amenable to this kind of investigation. The derived flood frequency approach consists of the following elements: (1) a statistical model of rainfall, usually expressed in the form of a joint probability distribution of rainfall intensity and duration, including, if necessary, a correction for the effects of catchment size; (2) a deterministic rainfall-runoff model which, in turn, contains three components, namely, a runoff generation model, a runoff routing model, and a method for the accounting of antecedent catchment wetness; and (3) a mathematical framework, or “methodology”, within which the above two elements are combined together to permit the “derivation” or estimation of the probability of exceedance of a given flood magnitude, thus leading to the “derived” flood frequency curve. In previous work two alternative methodologies have been adopted for deriving the flood frequency curve. The first approach is a Monte Carlo approach where rainfall time series are generated by a stochastic rainfall model and used to drive a continuous rainfall-runoff model [Ott and Linsley, 1972; Beven, 1986; Rahman et al., 2002]. From the runoff time series so generated the flood frequency curve is constructed. The second approach is a direct or analytical approach where the flood frequency curve is derived from the rainfall frequency curve using derived distribution theory [Eagleson, 1972; Sivapalan et al., 1990; Fiorentino and Iacobellis, 2001]. It is only feasible when the rainfall-runoff model and the stochastic models of rainfall and antecedent conditions are simple enough for the derivations to be analytically tractable but has the advantage that the effects of the various processes can be clearly distinguished in the final set of equations. The main contribution of the derived flood frequency approach has been the ability to understand the process controls of flood frequency behavior, especially the ability to focus attention on change of dominant processes with increasing return period [Sivapalan et al., 1990; Blöschl and Sivapalan, 1997], and therefore the ability to adapt flood estimation procedures to these dominant processes [Jothityangkoon and Sivapalan, 2001].

[5] Intra-annual variability in climate impacts upon the flood frequency curves in two ways: (1) seasonal and interannual variability of storm characteristics have a direct bearing on flood frequency distributions and (2) seasonality of rainfall and evapotranspiration affect the antecedent catchment conditions for individual storm events, and thus have an indirect effect on the magnitudes of the flood peaks. In spite of the widely acknowledged importance of seasonality of storm characteristics and of antecedent conditions on flood frequency these effects, to our knowledge, have never been explicitly included in any of the analytical derived flood frequency models presented in the literature. Clearly, this requires an explicit linking of derived flood frequency models with long-term water balance models, around a mathematical framework that enables the explicit inclusion of seasonality. This is the subject matter of this paper. Specifically, the aim of this paper is to explore the connection between the seasonality of floods and the seasonalities of climatic characteristics and the catchment state.

[6] The presentation of the paper will adopt the following outline. In section 2 we give examples of seasonal variability of flood frequency. We then give a mathematical presentation of the quasi-analytical derived flood frequency methodology in section 3, including a description of the way we derive annual flood frequency curves from monthly curves. In section 4 we describe the implementation of the continuous simulation or Monte Carlo approach with the same or equivalent model structures and parameters as the quasi-analytical approach, which we will use as a check of the quasi-analytical results. Section 5 is devoted to the application of the new derived flood frequency methodology to a typical catchment in Austria, its validation by comparing against results obtained using the Monte Carlo simulation approach, and the use of sensitivity analysis to gain insights into the effects of seasonality on monthly and annual flood frequency curves. Section 6 provides a discussion and the conclusions.

2. Examples of Seasonal Variability of Flood Frequency

[7] To further motivate the use of derived flood frequency models that represent the seasonalities of climate and soil moisture we illustrate the importance of seasonality through a number of examples taken from Austrian catchments. These examples also assist in guiding the development of our generalized flood frequency method.

[8] Figure 1 presents, for four Austrian catchments, plots of the month in which the annual maximum flood occurred for each year of record against the estimated return period for that flood. In the Teufelmühle catchment (Figure 1a) located in the north of Austria, we note that at low return periods the annual maximum floods could occur in any of the 12 months in a fairly uniform manner, whereas with increasing return period, the annual maximum floods tend to be produced preferentially in the December to March (winter) period. On the other hand, in the Wienerbruck catchment (Figure 1b) located in the Alps, there is a similar narrowing down in the timing of annual maximum floods with increasing return period, but this time around the July to September (summer) period. In the Anger catchment (Figure 1c) located in southeastern Austria, there is not the same degree of narrowing down in the timing of floods, with most floods occurring in the May to October period, regardless of return period. Finally, in the Rattendorf catchment (Figure 1d) located in southern Austria, the narrowing down in the timing of annual maximum floods shows a rather bimodal variation, with one set of floods occurring preferentially around June (spring) and the next around October (early winter). Clearly, these observed patterns must have both hydrological causes, and hydrological consequences, and yet analyses of these are precluded in traditional flood frequency analysis.

Figure 1.

Seasonality of flood frequency (date versus return period) for four example catchments in Austria: (a) Teufelmühle, (b) Wienerbruck, (c) Anger, and (d) Rattendorf. The mean catchment elevations and catchment areas are Teufelmühle, 751 m, 452 km2; Wienerbruck, 1013 m, 36 km2; Anger, 994 m, 408 km2; Rattendorf, 1380 m, 595 km2.

[9] We can speculate on possible physical causes of the observed seasonality in the timing of annual maximum flood peaks. Strong seasonal contrast in a number of contributing factors can have a significant effect on the extremes: rainfall, snowmelt, the underlying water balance reflected in the runoff coefficient and initial base flow in the streams. The first obvious cause could be the seasonality of rainfall inputs, especially the seasonality of extreme (flood producing) storms. Figure 2 presents the number of “extreme storms” as a function of the month for four regions in Austria that are loosely associated with the catchments of Figure 1 described above: Figure 2a shows Moldau (Teufelmühle), Figure 2b shows Donau between Enns and March (Wienerbruck), Figure 2c shows Raab (Anger), and Figure 2d shows Drau (Rattendorf). These were taken from Gutknecht and Watzinger [1996], where “extreme storms” are defined as those storms that have the property that d >equation image, where tr is rainfall duration in minutes and d is the total rainfall depth in mm over this duration. In the case of the Drau region and the associated Rattendorf catchment, presented in Figures 2d and 1d, the close association between the bimodal distribution of the timing of annual maximum floods (June and October) and the similarly bimodal distribution of extreme storms is unmistakable, and therefore the seasonality of extreme storms must be a primary determinant of seasonality of floods. In the Donau between Enns and March region and the Wienerbruck catchment, presented in Figures 2b and 1b, the most extreme storms appear to occur in summer which seems to be reflected also in the timing of annual maximum floods. The same appears to hold for the Raab region and the Anger catchment, where the extreme storms are mainly centered around June (Figure 2c) and the timing of annual maxima floods directly follow (Figure 1c). However, in the Moldau region and the Teufelmühle catchment the temporal distribution of extreme storms tends to be centered around July (Figure 2a), and yet annual maximum floods tend to be strongly centered around January at large return periods (Figure 1a). At least in the latter case, the observed seasonality is not directly attributable to seasonality of extreme storms. We must therefore seek alternative or additional explanations.

Figure 2.

Seasonality of extreme storms [from Gutknecht and Watzinger, 1996]: Number of extreme storms (expressed in percentages) versus month for four example regions in Austria, (a) Moldau, (b) Donau between Enns and March, (c) Raab, and (d) Drau. “Extreme storms” are defined as those storms that have the property that d >equation image, where tr is rainfall duration in minutes and d is the total rainfall depth in millimeters. Histograms in Figures 2a–2d are based on 48, 1795, 1512, and 3071 storms, respectively.

[10] Figure 3 presents the seasonal (monthly) variation of the runoff coefficient at event scale and the base flow for the four Austrian catchments considered before. In Figure 3a the base flow is defined as the minimum of daily runoff in each month, averaged over 30 years. Minimum runoff is an indicator of low flow and has here been used as a proxy for base flow. In Figure 3b the runoff coefficient has been estimated through the application of a variant of the HBV model [Merz and Blöschl, 2003, 2004] and is defined as the direct runoff divided by the sum of rainfall and melt input, averaged over each month and averaged over 30 years of record (thick lines). The thin lines in Figure 3b show the standard deviations between years and within each month of the runoff coefficients so defined. Both of these numbers show strong seasonal variation, and being measures of the antecedent condition of the catchment, they are likely to differently impact on the magnitude of the flood peaks that might result from storms that occur in different times of the year. The most dramatic effect is in Rattendorf (solid lines), where the runoff coefficient shows a bimodality that is in phase with, and therefore reinforces, the seasonality of extreme storms, thus contributing to the strong bimodality in the seasonal variation of annual maximum floods. The spring peak of the runoff coefficient appears to be associated with snowmelt while the autumn peak appears to be associated with heavy rainfall. In Teufelmühle (dashed lines) the apparent shift of the annual maximum flood peaks toward winter must be due to the high runoff coefficients obtained during the winter months, and not necessarily related to extreme storms. The high runoff coefficients in winter are a result of low evaporation and the presence of shallow snowpacks as indicated by the runoff simulations. Merz et al. [2004] have shown that the main effect of snowmelt on flood response in this type of catchments is in increasing antecedent soil moisture and hence runoff coefficients rather than through snowmelt contributions during the event. Wienerbruck is located at higher altitudes than Teufelmühle, so most of the snowmelt occurs in spring rather than in winter. The runoff coefficients in Figure 3b (dotted line) indeed peak in spring. The relatively flat response in Anger (dashed dotted lines) appears to be related to the rather flat variation of runoff coefficient and base flow during the year, and this suggests that the distribution of extreme storms is therefore the dominant control.

Figure 3.

Seasonal base flow and runoff coefficient for four example catchments in Austria: Teufelmühle (dashed lines), Wienerbruck (dotted lines), Anger (dash-dotted lines), and Rattendorf (solid lines). (a) Base flow (minimum of daily runoff in each month, averaged over 30 years). (b) Runoff coefficient estimated by a rainfall-runoff model, averaged over each month and averaged over 30 years (thick lines) and standard deviation between years and within month of the runoff coefficient (thin lines).

[11] Our basic hypothesis in this paper therefore is that the interactions between seasonality of rainfall characteristics and the seasonality of antecedent conditions can lead to the complex patterns of flood frequency behavior shown in Figure 1. We further hypothesize that these interactions can also be reflected in interesting regional/spatial patterns of flood frequency behavior. By way of illustration of the latter phenomenon, we present in Figure 4 regional patterns of flood frequency behavior for Austria [Merz et al., 1999]. Figure 4a presents the timing of all of the recorded annual maximum floods over a period of about 40 years regardless of return period, and presented as a map of arrows. On the other hand, Figure 4b presents the seasonality of only the three largest recorded floods (high return periods). The length of the arrows is a measure of the strength of the seasonality: r = 1 if the floods occur on the same day of the year; r = 0 if the floods occur uniformly around the year. The direction of each arrow gives the mean timing of the floods in terms of day of the year counted from 1 January. Very low seasonalities are indicated as disks. The arrows in Figure 4b are significantly longer than those in Figure 4a, which indicates a narrowing down in the timing of annual maximum floods with increasing return period similar to what we saw in Figure 1 for a few of the catchments. However, the observed timings of the largest floods also display strong regional variations, suggesting that the interactions between extreme storms and antecedent conditions are highly spatially heterogeneous, being dependent on the heterogeneity of the climatic and catchment characteristics governing the generation of floods. Figure 4 suggests that the flood processes may indeed change as one moves from moderate to extreme floods.

Figure 4.

Flood seasonality in Austria [from Merz et al., 1999]. The direction of the arrows indicates the average occurrence of floods in a year, and the length of the arrows is a measure of the strength of the seasonality. (a) All maximum annual floods. (b) Only the three biggest floods on record.

[12] A major objective of this paper therefore is to explore the connection between the seasonality of floods and the climatic and catchment properties that govern the generation of these floods. However, while we are motivated by these empirical observations, the objective of the paper is limited to presenting a quasi-analytical framework within which to investigate these interactions. At this time, we are not endeavoring to recreate, through a modeling exercise, the exact temporal patterns presented for the four individual catchments (Figure 1), nor the observed regional patterns (Figure 4). From this perspective the work presented here represents the initial steps of a much more long-term investigation.

3. Quasi-analytical Derived Flood Frequency Model

[13] The approach proposed here is a generalization of the derived flood frequency approach presented by Eagleson [1972] and subsequently adapted and extended by others [e.g., Sivapalan et al., 1990; Fiorentino and Iacobellis, 2001]. We will use the derived flood frequency approach to construct “monthly” flood frequency curves, for each month of the year, which we will then combine into “annual” flood frequency curves based on the “theory of mixed distributions” [Buishand and Demaré, 1990].

3.1. Details of the Stochastic Rainfall Model

[14] The stochastic rainfall model used here is an adaptation of the one used previously by Robinson and Sivapalan [1997a]. The model is adapted for conditions in Austria, and the parameters of the model are adjusted to mimic the observed rainfall record at Frankenfels station in the east of the country. The essential details of the model are presented next for completeness, with the parameter values adopted summarized in Table 1. Essentially, the model consists of discrete rainfall events whose arrival times, durations, average rainfall intensity and the within-storm intensity patterns are all random, governed by specified distributions. In addition, the parameters of these distributions are assumed to be seasonally dependent. At this stage, we have ignored spatial variability of the rainfall intensities.

Table 1. Parameters of the Rainfall and Runoff Models
ParameterSymbolEquationValue Used in Figure 8Value Used in Figure 9Value Used in Figure 10Value Used in Figures 11a and 12aValue Used in Figures 11b and 12bUnits
Weibull parameter of storm durationβr(1)0.70.70.70.70.7 
Weibull parameter of inter-storm periodβb(2)0.70.70.70.70.7 
Annual average storm durationδr(3)5.75.75.75.75.7hours
Amplitudes of the seasonal variation of storm durationαr(3)1.41.41.4; 01.41.4hours
Month of maximum storm durationτr(3)00000months
Annual average interstorm periodδb(4)43.043.043.043.043.0hours
Amplitudes of the seasonal variation of interstorm periodαb(4)5.65.65.6; 05.65.6hours
Month of maximum interstorm periodτb(4)00000months
Rainfall model parametera2(5)1.51.51.51.51.5 
Rainfall model parameterb1(5)0.010.010.010.010.01 
Rainfall model parameterb2(5)−0.55−0.55−0.55−0.55−0.55 
Parameter of mean rainfall intensityδa(6)1.051.051.051.051.05mm image
Amplitude of rainfall intensityαa(6)0.650.650.650.650.65mm image
Month of maximum rainfall intensityτa(6)66666months
Annual average base flowδq(9)0.0650.065000mm/h
Amplitude of the seasonal variation of base flowαq(9)0.0350.035000mm/h
Month of maximum base flowτq(9)6.56.5-6.56.5months
Annual average runoff coefficientδc(10)0.350.350.350.350.35 
Amplitude of the seasonal variations of the runoff coefficientαc(10)0.250.250.25; 00.250.25 
Month of maximum runoff coefficientsτc(10)3.53.5variable63months
Variance of the random component of the runoff coefficientσc2(12)00; 0.040.040.040.04 
Catchment response time for flood eventstc(13), (14)6.06.06.06.06.0hours
Mean of the rescaled χ ratio of flood peaksμη(16)-0.30.30.30.3 
Variance of the rescaled χ ratio of flood peaksση2(16)-0.040.040.040.04 

3.1.1. Storm Duration and Interstorm Period

[15] Guided by data, storm duration, tr, and interstorm period, tb, are assumed to follow the Weibull distribution (probability density function or pdf), with parameters that vary seasonally, i.e.,

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where βr and βb are the shape parameters of the Weibull distributions, and γr and γb are the scale parameters which can be expressed by the mean storm duration μr, and the mean interstorm period μb. These are allowed to vary sinusoidally with time of year, τ, as follows:

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where δr and δb are annual average storm duration and interstorm period, respectively, τr and τb are seasonal phase shifts, αr and αb are the amplitudes of the seasonal variations of tr and tb, τ describes the time of year, and ω is the total number of time units in a year (i.e., ω = 12 months if τ, τr and τb are in month units). αr αb, δr, δb, τr and τb have simply been estimated by fitting equations (3) and (4) to seasonal plots of mean storm duration and mean interstorm period derived from the Frankenfels rainfall time series. βr and βb were assumed to be constant throughout the year and were estimated from the distribution of storm duration and interstorm period of the rainfall data. The data also indicated that the observed distributions can be closely approximated by Weibull distributions.

3.1.2. Rainfall Intensities

[16] We describe the mean storm rainfall intensity i as a random variable stochastically dependent upon storm duration, tr, i.e., i and tr follow the joint pdf, image (i, tr), which is also seasonally varying due to the variation of γr with time of year, τ. The mean storm rainfall intensity is disaggregated further to hourly intensity patterns using stochastically generated mass curves [Huff, 1967], as shown later.

3.1.3. Average Storm Rainfall Intensity

[17] To parameterize the dependence between i and tr we assumed the conditional statistics of E[itr] and CV2 [itr] to be power functions of tr as follows:

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with the coefficient a1 being assumed to also vary seasonally, as follows, to account for variability of rainfall generating mechanisms within the year:

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The power functions given in equation (5) provide relationships between tr and the first two moments of fI (itr), the conditional distribution of i given tr, which is assumed to follow the gamma distribution:

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with the gamma parameters, κ and λ being functions of tr. By matching expressions for the statistics of the general gamma distribution with equation (5), we can estimate κ and λ as follows:

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The parameters b1, b2, a2, δa, αa and τa have been estimated by fitting equations (5) and (6) to the rainfall data at Frankenfels.

3.2. Details of the Simple Rainfall-Runoff Model

[18] The rainfall-runoff model used in the study is rather simple, as the main motivation is to present the generalized derived flood frequency framework, and validate it using results from the Monte Carlo approach. The rainfall-runoff model consists of three components: (1) a base flow component, which is assumed to be seasonally varying; (2) a quick flow component that utilizes a single runoff coefficient, whose variation within the year has a seasonal component (i.e., its mean) and a random component (i.e., deviation from the mean); and (3) a runoff routing component which consists of a linear reservoir with a response time of tc. The details of these three components are given below.

[19] 1. The first component is initial flow in the river (base flow). Base flow q0 is assumed to vary seasonally according to the sinusoidal expression

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No randomness is allowed in the case of the base flow variation.

[20] 2. The second component is quick flow runoff generation and the runoff coefficient. The volume of runoff generation during a storm is estimated according to a specified runoff coefficient, rc, (fixed for the event and thus nonlinearity is not considered), the mean of which is assumed to vary seasonally, i.e., with the time of occurrence of the storm, as follows:

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In addition to the seasonal variation, the runoff coefficient is also assumed to contain a random element. For simplicity, and on the basis of empirical results (see Figure 3), we assume that the variance of the random component remains constant throughout the year at a value equal to σc2. We assume further based on previous work of Gottschalk and Weingartner [1998] that rc is distributed according to the beta distribution:

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where B(u, v) is the incomplete beta function, and given the mean and standard deviation of runoff coefficient as above, the parameters u and v of the beta distribution can then be estimated from:

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In the model the runoff coefficients represent the lumped effect of a number of processes that vary seasonally including evaporation and snowmelt as well as random processes such as antecedent rainfall.

[21] 3. The final component is runoff routing. For routing purposes, we conceptualize the catchment as a linear reservoir with a response time tc. Effectively, this is equivalent to the convolution of the time series of quick flow generation with an exponential instantaneous unit hydrograph (IUH). For a single storm, the transformation of rainfall to runoff can be expressed through the convolution integral given below:

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where i(t) is the rainfall input time series, and q(t) is the resulting runoff time series. For response time tc, we used a typical value of medium sized catchments in Austria of 6 hours.

3.3. Effect of Within-Storm Patterns on Flood Peaks

[22] In this paper we are interested in the flood peak resulting from the above convolution for all the storms occurring in the year. For the Monte Carlo simulation approach (to be presented later), this is fairly straightforward, and only involves repeated application of equation (13) for the entire rainfall record.

[23] For the quasi-analytical approach, however, we require analytical or quasi-analytical expressions for the flood peak in terms of the average rainfall intensity, duration, also accounting for within-storm patterns. This is not so straightforward, and in this section we present an approximate approach that involves the implementation of a random correction factor χ to account for the random nature of within-storm patterns.

[24] To illustrate this, consider a single storm of average intensity, i, and of duration, tr. For a constant storm runoff coefficient, rc, the depth of (new) runoff generated is rc · i · tr. As a base case, let us assume that the rainfall intensity within the storm is also constant. Under these simplifying assumptions, by completing the integration in equation (13) we can easily derive an expression for the flood peak as follows:

display math

where for the simple rainfall pattern used in this base case, a correction factor is indeed not needed, and χ = 1. To come up with a priori estimates for χ for more complex within-storm patterns, we utilized synthetic time series of rainfall intensities, generated using the rainfall model presented earlier, with variable duration, average intensity but including within-storm temporal patterns. The within-storm temporal patterns are generated using Robinson and Sivapalan's [1997a] cascade method which is summarized in Appendix A. An example of a typical storm is presented in Figure 5 which, by visual inspection, is similar to observed storm patterns. As a check of the ability of the rainfall model to reproduce observed patterns of variability in the observed record at Frankenfels, we carried out a number of comparisons. As an example, cumulative mass curves are shown in Figure 6, including the mean and the 0.1 and 0.9 quantiles all of which mimic the corresponding patterns in the observed record. The cascade model in Appendix A is used to find values for χ and it is also used in the Monte Carlo approach discussed later but it is not directly used in the quasi-analytical derived flood frequency model.

Figure 5.

Typical simulated hyetograph (a major event in October).

Figure 6.

Mass curves (see equation (A1)). Thick lines are the 10 and 90% quantiles and mean of simulations. Dots are the 10 and 90% quantiles and mean from 7 years of hourly rainfall data at the Frankenfels rain gauge. Thin line is the mass curve of the simulated event in Figure 5.

[25] For finding values for χ, each of the storms so produced is routed through a linear store, as in equation (13), for three different specified response times tc. For each tc this kind of filtering produces a synthetic runoff record for each storm, which is analyzed to identify the single flood peak for each storm. Clearly, the required correction factor χ for estimation of the flood peak using equation (14) is a number generally greater than 1, and also random due to the random nature of the within-storm patterns. We also find that, on average, increasing tc leads to a reduction of the magnitude of the flood peak, and thus χ.

[26] In order to obtain a simple expression for χ we plotted it against tr/tc, the ratio of storm duration to response time, and as shown in Figure 7a, we find that it collapses into a single scatterplot regardless of the individual values of tr and tc used. However, there is considerable scatter about the mean curve, arising from the randomness of the within-storm patterns. To account for the scatter we fitted four curves: the lower and upper envelopes (χl and χu) of the scatterplots, and the mean and standard deviation (χm and σχ) for each value of tr/tc. Figure 7b presents the fitted curves, which are expressed analytically as follows:

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where χ′m = 0.17, σ′χ = 0.08, χ′l = 0.05, χ′u = 0.45. These parameters are a function of the models chosen for rainfall (i.e., within storm patterns) and the runoff response, but are independent of the actual event characteristics used in their estimation, namely tr and tc. Let us consider a new random variable η, which is a scaled version of the χ ratio defined above, with the property that 0 ≤ η ≤ 1, as follows:

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or

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The mean, μη, and variance, ση2, of the rescaled ratio of flood peaks, η, can then be estimated using the expressions given in equations (15a) to (15d). Upon doing this, remarkably, we find that the mean and variance collapse to being mere constants, and completely independent of tr/tc, that is,

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Figure 7.

Effect of time patterns (i.e., within storm temporal rainfall patterns) on peak flow, as expressed in terms of the χ ratio. Here tr is the storm duration, and tc is the catchment response time. (a) χ from Monte Carlo simulations (circles, tc = 2 hours; triangles, tc = 6 hours; squares, tc = 24 hours) with fitted mean χ indicated as a solid line. (b) Fitted upper and lower envelopes (dashed lines), mean from Monte Carlo simulations for a number of tr/tc classes (solid circles), fitted mean (thick solid line); 1 + standard deviation from Monte Carlo simulations (open circles), and 1 + fitted standard deviation (thin solid line).

[27] This is a highly convenient result, yielding μη = 0.3 and ση2 = 0.04. We further assume that η is also distributed according to the beta distribution:

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where B(ρ, ϑ) is the incomplete beta function. As in the case of the runoff coefficient, the parameters ρ and ϑ are then given by

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Note that the properties of the χ and η ratios presented above, i.e., μη = 0.3 and ση2 = 0.04, are a function of the nature of the within-storm patterns, i.e., the cascade model used to generate these within-storm patterns, and the rainfall-runoff model used for the runoff generation and routing. More complex models of both rainfall and runoff routing may lead to different properties of the χ ratio. For example, nonlinearity of the rainfall-runoff process will lead to higher χ ratios. More detailed investigation of the interactions of the rainfall model and the rainfall-runoff model can lead to significant improvements in the accuracy of flood estimates, and is left for further research.

3.4. Quasi-analytical Derived Flood Frequency Framework

3.4.1. Probability Distribution of the Population of Flood Peaks

[28] Flood frequency analysis concerns the estimation of annual peak discharges with specified probabilities of exceedance. If we let a random variable Y denote the peak discharge of all independent storms, and Q denote the annual maximum peak discharge, then the probability distribution of Q is the extreme value distribution associated with Y. Noting that the rainfall inputs i and tr are random variables (potentially also dependent on each other), we can derive the probability distribution of Y in terms of the probability density functions of i and tr, using derived distribution theory [Eagleson, 1972; Wood, 1976; Robinson and Sivapalan, 1997b]. Given the joint probability density function of i and tr as image (i, tr), the cumulative distribution of Y is given by:

display math

Here FY (qp) is the probability, for a given flood event, that the flood peak Y is less than or equal to qp. For the rainfall-runoff model given by equation (14), R is the region such that qpq0 + rciχ[1 − exp (−tr/tc)]. Then, using equation (20) for the simplifying case where χ and rc are allowed to remain constant for all storms, the above integral simplifies to

display math

where FI (.∣tr) is the conditional cumulative distribution of rainfall intensity, conditioned on tr.

3.4.2. Conversion to Extreme Value Distribution

[29] Assuming m independent floods in a year, the extreme value distribution is formed by selecting the largest of these m floods in each year, and deriving the distribution function of the truncated series. Using the theory of order statistics [Mood et al., 1974; Kottegoda and Rosso, 1997], the cdf of annual maximum floods, denoted by FQ (qp), is given by

display math

The same result can also be expressed in terms of a return period (in years):

display math

3.4.3. Seasonal Variability of Climate and Antecedent Conditions

[30] The above result is only valid if the rainfall and catchment properties do not vary systematically within the year. The effects of seasonal variability, if present, must somehow be factored in the derivation. Let us suppose that pdfs of i and tr vary seasonally, as do the runoff coefficient, rc, and initial base flow, q0. We will assume that these pdfs remain sufficiently stationary within individual months, so that the integral in equation (21) can still be carried out in the normal way for each month. For simplicity we will here assume that the χ ratio for rainfall temporal variability remains constant for all months. Thus, for any month j, j = 1,…12, we can rewrite the integral of equation (21) as follows:

display math

[31] At this point we generalize the above result by introducing random variability of both runoff coefficient and the χ ratio. Using the probability density functions derived before for runoff coefficient and the scaled ratio of flood peaks, η, we then have

display math

where it should be noted that χu and χl are empirical functions of tr/tc.

[32] Let us assume that the number of flood peaks in month j, is mj, j = 1, … 12. Then, equation (23) can be rewritten for each month of the year, as follows:

display math

[33] We will term the resulting flood frequencies monthly flood frequency curves, since they relate to probability of exceedances of the given flood peak qp within a single month, yet the return periods are in years, e.g., probability that the January flood peak exceeds qp in any given year. Provided seasonal variations of all controlling variables are known, it is straightforward to compute, using equations (25) and (26), the monthly flood frequency distributions for all 12 months.

3.4.4. Derivation of Annual Flood Frequency and Mixed Distributions

[34] For many engineering applications, we want to derive the annual flood frequency, i.e., the probability that the annual flood peak (maximum of the 12 monthly maxima) exceeds a given q, and the corresponding return period. Here this will be accomplished using the theory of mixed distributions, noting that the cdfs of the flood peaks for each month are different and the cdf of annual maxima is thus drawn from a mixture of these. Assuming the flood peaks in month j are independent from those in month j + 1 and considering the number of flood peaks in month j, mj, j = 1, … 12, again using order statistics [Mood et al., 1974; Kottegoda and Rosso, 1997; Buishand and Demaré, 1990], we can show that

display math
display math

[35] Note above that, essentially, mixing of the monthly flood frequency curves is equivalent to estimating the product of the monthly nonexceedance probabilities, under the product sign in equations (27) and (28). It can be shown to be invariant of the number of stationary periods used (i.e., in this case a month), provided that the number of events in a year are preserved. Equations (22) and (27) are based on the assumption of the independence of individual events. While this assumption is clearly appropriate for the annual case (equation (22)) it is less obviously so for the monthly case (equation (27)) and will be checked by Monte Carlo simulations.

4. Monte Carlo Simulation Approach

[36] The continuous or Monte Carlo simulation approach is used to check the integration and the χ ratio approximation adopted by the quasi-analytical derived flood frequency approach as well as the assumption of independence of monthly flood peaks. The steps involved in the Monte Carlo approach are summarized as follows.

[37] 1. Synthetically generate (where necessary multiple realizations of) continuous time series of rainfall intensities, using the rainfall model presented in section 3.1 and in Appendix A, using parameters presented in Table 1.

[38] 2. Apply the runoff coefficient, for each storm, incorporating the seasonality and randomness on a storm by storm basis as per the date that the storm occurred, multiply the rainfall time series by this runoff coefficient.

[39] 3. Convolute the resulting runoff generation time series with the exponential IUH presented in equation (13) with response time tc being assumed to remain constant with date, and add the seasonally varying base flow at the end.

[40] 4. Scan the resulting continuous time series and pick the largest flood in each month (forming the monthly maxima data series), and in the whole year (annual maxima data series), and construct the monthly and annual flood frequency curves.

5. Results

5.1. Test of the Generalized Derived Flood Frequency Approach

[41] As a test of the derived flood frequency approach we compare the estimated flood frequency curve for the case with no within-storm patterns, against the corresponding predictions by the Monte Carlo method. For both the rainfall and the catchment models, parameters that are typical of Austrian conditions have been used (Table 1). For this application, the runoff coefficient is assumed to vary seasonally, but no randomness is included. The results are presented in Figure 8. All flood peaks are expressed in terms of runoff depths (mm/h). Given the catchment response time of 6 hours assumed here, the average hourly values (Monte Carlo simulations) are comparable with the instantaneous values of the quasi-analytical approach. The 1 mm/h of runoff depth is equivalent to a specific peak discharge of 0.28 m3/s/km2. The annual flood frequency curve resulting from the quasi-analytical approach (thick solid line in Figure 8) is close to that resulting from the Monte Carlo approach (circles) suggesting that the generalized derived flood frequency approach presented in this paper captures the seasonality and that the integration is correct. Figure 8 also presents the monthly flood frequency curves (thin lines). The probability that a given flood peak discharge occurs in a particular month is smaller (larger T) than for it to occur at any time of the year. The monthly flood frequency curves hence plot to the right of the annual curve in Figure 8. For a flood peak of, say, 1.5 mm/h the annual return period is 1.4 (equivalent to a nonexceedance probability of 0.30) while the monthly return periods are 2.8, 3.0, 5.5 .. (equivalent to nonexceedance probabilities FYj of 0.64, 0.67, 0.82 …). The annual nonexceedance probability of 0.30 is simply the product of the monthly nonexceedance probabilities. The monthly flood frequency curves for the summer months plot at the top while those for the winter months plot at the bottom indicating that big floods in winter are very unlikely for the parameters chosen here (Table 1).

Figure 8.

Monthly (thin lines) and annual (thick line) quasi-analytical derived flood frequency curves and annual flood frequency curves from Monte Carlo simulations (circles). No time patterns are used, and runoff coefficient is seasonal but not random. Horizontal scale is linear using Gumbel reduced variates. For parameters, see Table 1.

5.2. Effect of Randomness of Within-Storm Patterns and Runoff Coefficient

[42] Figure 9 presents the estimated flood frequency curves when within-storm patterns are included (by both the quasi-analytical and Monte Carlo approaches). In the bottom line of Figure 9, the runoff coefficient is assumed to vary seasonally in a deterministic manner (no randomness is included). For small to medium return periods the Monte Carlo results are close to the results from the quasi-analytical approach. This suggests that the approximate analytical χ ratio approach adequately captures the effects of within-storm temporal patterns on the flood peaks. However, for return periods larger than about 30 years, the quasi-analytical approach seems to slightly underestimate flood peaks and the flood frequency curve is slightly less curvilinear than in the case of the Monte Carlo simulations. This suggests that the χ ratio approximation is somewhat more linear than the explicit time patterns and the probability of extreme events is slightly underestimated.

Figure 9.

Annual flood frequency curves using time patterns from the quasi-analytical (lines) and Monte Carlo (dots) methods. Bottom line shows the case with within-storm time patterns included, runoff coefficient seasonal but not random. Top line shows the case with storm time patterns included, runoff coefficient seasonal and random.

[43] As compared to Figure 8, the bottom line in Figure 9 gives almost twice the flood peak for the same return period. This is the effect of within-storm temporal patterns. It is clear that the inclusion of rainfall time patterns is critically important for realistic derived flood frequency predictions.

[44] We also present in Figure 9 the results of the case where the runoff coefficient varies both seasonally (in the mean) along with a random component (top line in Figure 9). With the addition of the random component, the flood frequency curve not only increases on average (shifts upward), but it also increases in steepness. When allowing for random runoff coefficients a small number of events occur where large runoff coefficients (wet antecedent moisture conditions) coincide with large rainfall depths to produce very large flood peaks.

5.3. Further Insights into the Effects of Seasonality

[45] With the derived flood frequency model developed above, we next carry out additional simulations to gain further insights into the effects of seasonality. There are two ways that seasonality enters into the flood frequency curve: seasonality in the rainfall inputs, especially “extreme rainfalls,” and the seasonality in the transformation between rainfall and runoff. The latter appears through the runoff coefficient and base flow in the present model, whose seasonality ultimately can be connected to seasonality of rainfall and evaporation. Figure 10 examines the effect of the interaction of these two types of seasonality on the flood frequency curve. Figure 10a shows the annual flood frequency curves for different phase shifts between seasonal variabilities of rainfall and runoff coefficient (thin lines). The phase shift is defined as Δτ = τc − τa with τr = τb = τa − 6. Figure 10a also includes the base case with no seasonality either in the rainfall or in the runoff coefficient (thick line with circles). Figure 10a shows that the flood magnitude is largest when the rainfall and runoff coefficient are perfectly in phase, i.e., the phase shift Δτ = 0, which means the highest runoff coefficients occur in the same month as highest rainfall intensity. This reflects a kind of resonance. The flood frequency curves decrease in magnitude when they become increasingly out of phase. It is interesting that any kind of seasonality (including out of phase) produces higher flood frequency curves than the case where no seasonality is included (thick line with circles in Figure 10). When seasonality is present a coincidence of large runoff coefficients (wet antecedent moisture conditions) with large rainfall depths is more likely than for the case without seasonality thus producing larger flood peaks. Note that the “resonance” highlighted above will be even more dramatic if the runoff generation process were to be nonlinear, with a runoff coefficient that increases with increased rainfall.

Figure 10.

Effect of phase shift. (a) Annual flood frequency curves for different phase shifts Δτ of the runoff coefficient rc as compared to rainfall (Δτ = τc − τa with τr = τb = τa − 6; thin lines) and annual flood frequency curve without any seasonality (i.e., all amplitudes are 0; thick line with circles). (b) Nonexceedance probabilities FYj for three flood peaks (2, 4, and 6 mm/h) as a function of phase shift Δτ. A phase shift of Δτ = 0 corresponds to highest runoff coefficients in the same month as the highest rainfall intensity.

[46] Figure 10b presents the same results of Figure 10a in an alternative manner to illustrate the role of the nonexceedance probabilities. In this case, we estimate the nonexceedance probabilities for three different specified values of the flood peak (2, 4 and 6 mm/h). Essentially, this amounts to slicing Figure 10a horizontally at the three specified flood levels, reading off the return period for each phase shift, and then estimating the corresponding nonexceedance probabilities (=1 − T−1). The results in Figure 10b show that the nonexceedance probability of a given flood peak is smallest for zero phase shift (case of resonance), and increases with increasing phase shift. For the largest of the flood peak values selected (6 mm/h) the exceedance probability varies from 0.01 (6 month phase shift) to 0.08 (no phase shift).

[47] Having determined the effect of seasonality of extreme storms and antecedent conditions on the magnitude of the annual and monthly flood frequency curves, we next investigate the question as to what are the relative contributions of individual months toward the annual flood frequency curve. In doing this we note at the outset that whereas in the observed record only one month produces the observed annual maximum flood in each year, the derived flood frequency approach can generate the relative contributions of the various months toward the annual maximum flood, i.e., the probability that the annual maximum flood of a given return period would occur in a specified month, although this cannot be verified in practice with the limited length of record currently available.

[48] Figure 11 presents the flood frequency curves for different months, with different phase shifts between seasonal variability of extreme storms and antecedent conditions. Figure 11a presents the case with zero phase shift, i.e., where maximum rainfall and maximum runoff coefficients occur in the same month. Figure 11b presents the case for a phase shift of 3 months (Δτ = −3) i.e., where maximum rainfall occurs three months after maximum runoff coefficients. The thick lines are the annual flood frequency curves and the thin lines are the monthly curves. The highest monthly curves in Figure 11a are for the summer months (June and July) when both highest rainfall and runoff coefficients occur. The two lines lie on top of each other as all model assumptions are strictly symmetrical around the mid of the year. The lowest monthly curves are for the winter months (December, January) where both rainfall intensities and runoff coefficients are low. For the case of a phase shift of 3 months (Δτ = −3, i.e., maximum rain intensities in summer, maximum runoff coefficients in spring) the monthly contributions are more complex. Figure 11b shows that the monthly flood frequency curves cross over, with the probability of exceedance changing depending upon the month. The high monthly curves occur in early summer while the low curves occur in early winter. The crossing over of the monthly curves is related to the interaction of rainfall seasonality and the seasonality of antecedent soil moisture. In months with relatively wet antecedent conditions but small rainfall intensities (March, for example) small floods are quite frequent, leading to a flat flood frequency curve. In contrast, in months with relatively dry antecedent conditions and large rainfall intensities (September, for example) small floods are less frequent but a number of large floods do occur, leading to a steep flood frequency curve.

Figure 11.

Flood frequency curves for different months (thin lines) and annual flood frequency curve (thick line). (a) Phase shift Δτ = 0 (i.e., maximum rain in summer, maximum runoff coefficient rc in summer). (b) Phase shift Δτ = −3 months (i.e., maximum rain in summer, maximum rc in spring).

[49] In order to shed more light on this interaction of climate and catchment state seasonalities, Figure 12 shows the contributions to the probability of nonexceedance from different months in terms of the return period, and for different values of the flood peak Qp. As in Figure 11, Figure 12a presents the case for zero phase shift between extreme rainfalls and antecedent conditions Δτ = 0 whereas Figure 12b presents the case for a three month phase shift (Δτ = −3). Figure 12 can be thought of as slices of Figure 11 for different flood peak values (0.5, 1, 2, 5 mm/h). Figure 12 mirrors the same phenomenon exhibited in Figure 11, namely, the occurrence of extreme floods in summer, and less extreme floods throughout the year.

Figure 12.

Probability contributions of months in terms of return period versus month for different flood peaks (0.5, 1, 2, and 5 mm/h). (a) Phase shift Δτ = 0. (b) Phase shift Δτ = −3 months as in Figure 11. The 1 mm/h of runoff depth is equivalent to a specific peak discharge of 0.28 m3/s/km2.

6. Discussion and Conclusions

[50] The paper has presented a derived flood frequency methodology that applies the theory of “mixed distributions” to account for the seasonalities of catchment state and climate forcing, and uses the χ ratio approach to incorporate the effects of within-storm patterns. Both of these advances remedy serious drawbacks in much of previous derived flood frequency research. The inclusion of seasonality and within storm time patterns change the shape of the flood frequency curve dramatically over the cases where these two aspects are ignored.

[51] The χ ratio approach is a parsimonious parameterization of the combined effect of within-storm rainfall characteristics and catchment response characteristics. Its simplicity, being a function of the ratio of storm duration and catchment response time only, makes the method amenable to analytical derivations. While the parameters given here are valid for the particular rainfall and runoff models used here, it is straightforward to derive the parameters for any rainfall and runoff model used in a particular context. For example, nonlinearity of the rainfall-runoff process will lead to higher χ ratios. More detailed investigation of the interactions of the rainfall model and the rainfall-runoff model can lead to significant improvements in the accuracy of flood estimates, and this is left for further research. The effect of the time patterns, as demonstrated here, is dramatic. As compared to the case of constant intensity rainfall, inclusion of time patterns results in almost twice the flood peak for the same return period. It is clear that the inclusion of rainfall time patterns is critically important for realistic derived flood frequency predictions and makes them more consistent with current rainfall-runoff models used in a design context.

[52] Analyses of the effects of seasonality on flood frequency indicated that indeed interannual climate variability strongly impacts upon the flood frequency characteristics in two ways, in a direct way through the seasonal variability of storm characteristics, and indirectly through the seasonality of rainfall and evapotranspiration which affect the antecedent catchment conditions for individual storm events. The relative phase shift between climate forcing and catchment state determines the magnitude and shape of the flood frequency curve. The highest impact of seasonality occurred when the seasonality in rainfall intensities was in phase with the seasonality of antecedent catchment wetness, i.e., when high rainfall intensities and large runoff coefficients occurred in the same season. This resonance increased the magnitude of the flood frequency curves by more than 50% over the case where climate forcing and catchment moisture state were out of phase. It is likely that the resonance highlighted above will be even more dramatic if the runoff generation processes were to be nonlinear, with a runoff coefficient that increases with increased rainfall. For certain combinations of the relative phase shift between climate forcing and catchment state, e.g., three months with maximum rain intensities in summer and maximum runoff coefficients in spring, the monthly flood frequency curves cross over. In months with relatively wet antecedent conditions but small rainfall intensities (March, for example) small floods are quite frequent, leading to a flat flood frequency curve. In contrast, in months with relatively dry antecedent conditions and large rainfall intensities (September, for example) small floods are less frequent but a number of large floods do occur, leading to a steep flood frequency curve. This is reminiscent of flood frequency behavior in different climates. In dry climates flood frequency curves tend to be steep while in wet climates they tend to be much flatter. This is the case in Austria. In western Austria, which is relatively wet, the CVs of the maximum annual floods are only about 0.3 while in eastern Austria which is relatively dry, the CVs of the flood frequency curves are of the order of 0.7. In different climates around the world this contrast may be even more pronounced [see, e.g., Iacobellis et al., 2002].

[53] It is interesting that any kind of seasonality (including out of phase) produces higher flood frequency curves than the case where no seasonality is included. When seasonality is present a coincidence of large runoff coefficients (wet antecedent moisture conditions) with large rainfall depths is more likely than for the case without seasonality thus producing larger flood peaks. A similar effect occurs when the runoff coefficient is allowed to vary randomly between events. When allowing for random runoff coefficients a small number of events occur where large runoff coefficients (wet antecedent moisture conditions) coincide with large rainfall depths producing very large flood peaks. The flood frequency curve not only increases on average (shifts upward), but it also increases in steepness. This is in line with the general experience that extreme floods are often produced by the combined effect of a number of factors, each of which is not particularly unusual [Snorasson et al., 2002]. This is also consistent with the assessment of Wood [1976] on the role of random variability of model parameters in a flood frequency context.

[54] The seasonal derived flood frequency approach presented here not only allows to isolate climate versus catchment seasonalities but also to examine the probability contributions of individual months. Whereas in the observed record only one month produces the observed annual maximum flood in each year, the derived flood frequency approach can generate the relative contributions of the various months toward the annual maximum flood, i.e., the probability that the annual maximum flood of a given return period would occur in a specified month, although this cannot be verified in practice with the limited length of record currently available. For the hypothetical catchments examined here the occurrence of extreme floods is more likely in summer, and less extreme floods may occur throughout the year (Figure 12). Qualitatively, this reproduces the narrowing down in the timing of annual maximum floods with increasing return period illustrated by some of the Austrian example catchments, particularly the Teufelmühle and the Wienerbruck catchments (Figures 1a and 1c). This type of analysis facilitates the assessment of floods that occur in an unusual season. For example, a recent flood at the Salzach in Salzburg, Austria, that occurred in March 2002 [Godina et al., 2003] corresponds to a return period of 100 years if only March floods are considered, while the same flood peak if it had occurred in July had been associated with a return period of only 2 years. This is very similar to the hypothetical example in Figure 12a corresponding to a flood peak of about 2 mm/h.

[55] The rainfall-runoff model used in the study is rather simple, as the main motivation is to present the generalized derived flood frequency framework, and validate it using results from the Monte Carlo approach. For applications to real catchments a more complex runoff model will likely be needed that is able to represent nonlinearities in both runoff generation and runoff routing. It is straightforward to extend the quasi-analytical framework to nonlinear models, however, equation (25) needs to be solved iteratively as in the derived flood frequency model of Blöschl and Sivapalan [1997], for example. It is also likely that due to the space-time intermittency of rainfall, there will be some interdependence between the contributing area of the catchment and the distributions of the duration of flood producing storms and this may also depend on the scale. Similarly, the rainfall model would have to be calibrated to each climate region of interest. These issues are left for further research.

[56] Even though the results presented here only dealt with hypothetical catchments, the results give us hope that with the proper rainfall and runoff models, we can almost fully capture the effects of seasonality in derived flood frequency models, and through these obtain valuable insights into the process controls of flood frequency. This will be extremely valuable for flood frequency regionalizations, since it enables us to regionalize individual components of the flood frequency curves (i.e., rainfall and catchment characteristics) which may be physically more meaningful than to regionalize the complete annual flood frequency curve. The results have also important implications for climate/land use change analyses. It is likely that climate change will not only impact on the flood magnitude but also on the seasonal distribution of flood occurrence. In fact, it is likely that the more obvious first indication of climate and land use change effects will be through a change in the seasonal patterns of flood occurrence. The flood frequency curve is an integrated response but we do need to isolate the controls to understand climate/land use changes and other nonstationarities.

[57] It is important to re-emphasize the purpose of this work. There have been a number of studies that have examined flood frequencies (or precipitation frequencies) separately for different seasons which were then combined into annual curves [e.g., Buishand and Demaré, 1990; Rasmussen and Rosbjerg, 1991]. These studies have generally found that separating flood frequency curves into seasons does not necessarily provide a better estimation of the annual flood frequency curve because of the larger number of parameters that need to be estimated in the case of a seasonal model. It should be noted that a better estimation of the annual flood frequency curve at gauged locations is not the main focus of this paper. The value of the proposed approach lies in the potential of isolating process controls and using known controls in a consistent way. This is in the spirit of examining and using the causative mechanisms of floods based on surface and upper weather maps of House and Hirschboeck [1997] and the flood process types of Merz and Blöschl [2003]. The method will enable hydrologists to interpret differences in observed flood frequency behavior between different climatic regions and can help to decipher climatic and catchment controls on flood frequencies, to classify variability between climatic regions, and generally to extrapolate information relating to floods from gauged to ungauged catchments.

Appendix A:: Within-Storm Temporal Patterns

[58] We used explicit within-storm temporal patterns for two purposes: to estimate the χ ratio in the quasi-analytical approach, and to generate rainfall time series in the Monte Carlo approach.

[59] To define the temporal patterns of rainfall intensity for each storm we used the idea of mass curves [Huff, 1967; Chow et al., 1988]. A normalized mass curve is defined by

display math

where t* = t/tr (0 ≤ t* ≤ 1), i is the mean rainfall intensity for the storm, ι(t′) is the instantaneous rainfall intensity (within-storm pattern) at time t′ internal to the storm (0 ≤ t′ ≤ tr), and tr is storm duration.

[60] In order to generate realistic within-storm intensity patterns, we use a stochastic model capable of producing normalized mass curves satisfying the statistical characteristics described above. The model has a multiplicative structure similar to that used in random cascades. At the first level of disaggregation H(t*) is given as follows:

display math

where w1 is a random number with the property that 0 ≤ w1 ≤ 1. The corresponding temporal pattern consists of two rectangles of height 2w1 and 2(1 − w1). At the next, second, level of disaggregation, the mass curve is obtained using two additional independent random variables w2 and w3 as follows:

display math

[61] This corresponds to a temporal pattern with four rectangles. The disaggregation process is continued in this way to higher levels of disaggregation until the desired fine timescale (1 hour in this case) is reached. The independent random numbers w1, w2, … wk, .. (0 ≤ wk ≤ 1 for any k) are drawn from the beta distribution, fW (w), which is given by

display math

[62] When θ1 = θ2, fW (w) is symmetrical around w = 0.5, and this results in normalized mass curves the median of which is a straight line, namely the 1:1 line. The actual magnitude of θ1 = θ2 then controls the extent of variability around the median curve. High values of θ1 = θ2 give rise to smaller variability since then much of the weight of fW (w) is concentrated around w = 0.5. Low values of θ1 = θ2 cause fW (w) to be concentrated, bimodally, around w = 1 and w = 0, which causes larger variability of mass curves above and below the 1:1 line. For our purposes we chose values of θ1 = θ2 = 1 since the stochastically generated mass curves matched the 10%, 50% and 90% percentile mass curves estimated from data.

[63] Finally, the stochastically generated normalized mass curves are converted to within-storm temporal patterns of rainfall intensity using

display math

where ι(t′) is the storm rainfall intensity over the period (t′ − Δ, t′), Δ is the time step (1 hour used in the paper), and 0 ≤ t′ ≤ tr.

Acknowledgments

[64] This research was supported by the Austrian Science Foundation (FWF), project P14478-TEC, and the Austrian Academy of Sciences, project HÖ 18. The first author acknowledges the hospitality of Vienna University of Technology, where much of this work was completed while on his study leave. CWR reference ED 1158 MS.

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