- Email: [email protected]
A short time-step point rainfall stochastic model
~ Pergamon
Wal. Sci. Tech. Vol. 37, No. II, pp. 37-4S, 1998.
IAWQ 1998 Publishedby Elsevier Science LId. Printed in Oreal Britain. All rights reserved 0273-1223/98 $19'00 + 0'00 @
PH: S0273-1223(98)OO314-X
A SHORT TIME-STEP POINT RAINFALL STOCHASTIC MODEL V. Thauvin*, E. Gaume* and C. Roux** • ENPc/CERGRENE, 6 et 8 avenue Blaise Pascal, Champs Sur Marne. 77455 Marne La Vallee Cedex2. France •• SEl'UDE. 11 bd Pershing, 75017 Paris. France
ABSTRACT This paper deals with the development of a point rainfall stochastic model. which generates synthetic sequences of S min rainfall rates. This time step is necessary to derive correctly the short hydrological response of urban catchments. The model simulates dry and rainy sequences in alternance. Particular attention has been paid to intense periods which are of utmost importance as far as flooding is concerned. This leads us to describe a rain event as a juxtaposition of inter-showers and showers. The intensity of an inter-shower is assumed to remain constant. Showers are defined as a succession of « peaks" with a single maximum and a trapezoidal shape. The model has been calibrated with raingauge data from the Seine Saint Denis area, situated near Paris. in France. This paper presents the definition of the model variables, the calibration of their probability distributions and the initial validation results. The conclusion insists on the fact that for the model to be completed. hydrological validation is necessary. ~ 1998 Published by Elsevier Science Ltd. All rights reserved
KEYWORDS Model;rainfall;simulation; stochastic; urban hydrology. INTRODUCTION Since the beginningof the eighties, point rainfall stochastic models have been used to provide hydrological models with long term rainfall series in order to allow for long term discharge simulations. Simulation results can then be devoted to predetermination of flooding and to design issues. Nevertheless. most of the models have been developed to be applied on rural catchments, and they run with a time step of one hour (see for example Tourasse (1981) and Cemesson (1993) who conducted research in this way). Rainfall stochastic models may also be utilised to design urban sewer systems. in the place of usual techniques like the use of a limited number of measured or synthetic design rainfalls. This utilisation requires rainfall models with a time step shorter than one hour (typically 5 min), since the hydrological response of urban catchment to rainfall input is short. Preliminary work about point rainfall stochastic models with short time steps (10 min) can be found in Marien and Vandewiele (1986). For such a model to be useful for design issues, it is necessaryto go deeply into the description of the form of rainfall events. Followingthis research. the topic of our work is to study the feasibility of building a model capable of reproducing the temporal structure of rainfall with a time step compatible with urban hydrologic applications. e.g. 5 min. Particular attentionhas been paid to the analysisof dependencies between the different variables of the model.
37
38
V. lHAUVIN et al.
Two remarks can be made before goingfurther. I. This workconstitutes the first phase of the development of a new tool intended for urban sewer network design. Indeed such a tool will be of great interest if a large number of its parameters can be regionalized: either they can be considered as constant values, or they depend on variables easily available, for example, daily rainfall depth, or regional parameters such as location, monthly rainfall depth, elevation and so on. In theseconditions the rainfall model can be usedeven in placeswithout raingauge observations. 2. The model reproduces point rainfall, in other words the spatial variability is not taken into account. Thus its results can be used only for applications on smallurbancatchments (a few squarekilometres). This paper presents firstly, the identification of the descriptive variables of a 5 min rainfall series, secondly, the calibration of the model parameters, and thirdly, the first validation made on the model results. The conclusion ends with the exposition of the principle of a hydrological validation whichappearsnecessary to validate the assumptions made whilerepresenting the formof rain events. CHARACfERISTICS OF A POINT RAINFALL SERIES The time rainfall series is considered as groups of positive values with a defined structure, and each group being the resultof a random process. There are two broadcategories of models: the aggregation models (see Rodriguez-Iturbe et al., 1987) and the composition models (see Croley et al., 1978). In the first case. the hypothesis is that each storm event is composed of successive cells. Each cell is associated with a period of rainfall of random duration and of constant, or non-constant. random intensity. Total rainfall intensity during a time step is formed by adding the contributions from all cells. In the second case, the rainfall series is considered as a juxtaposition of entities. which represent the main features of the hyetograph, that is to say dry periods and rainy periods.One periodis described by its duration, its rainfall depth and its shape.These characteristics are treatedas random variables. Intensity
«
it II
Inter-shower
"
Shower Rainevent
..
.,
Inter-shower
Dry weather
Rain event
period Figure I. Representation of a rainfall series of intensities.
The model presented in this paper is a composition model. The rainfall series is supposed to be a juxtaposition of embedded structures (Figure 1). 1. Dry periodsand rain events occur in aItemance. They are distinguished by a threshold value of 0.2 mm'h, applied after a moving average, calculated on a period of one hour. Moreover a rain event is retained if its cumulative rainfall exceeds 2 mm, Below this value. rain events do not produce significant runoff. and consequently they can not form discharges in the sewerdrainage network. Thus about 30% of the total year rainfall appearsas dry periodsin the model. 2. A rain event is composed of showers and inter-showers. The showers are the most intense parts of a rain event. They are defined by a threshold value of 3.2 mmlh, applied after a moving average. calculatedon a
39
Short time-steppoint rainfall stochasticmodel
period of 15 min. Showers and inter-showers are assumed to be produced by two different random processes. In these conditionsa better adjustment of the probability functions is obtained. 3. A shower consists of a succession of peaks. A peak is a rainy period between two significant minima. A minimum is reached when it is a unique exception in an ascending or a descending part of the series. The less trivial cases of equality betweentwo values arealso considered. A peak presents a single maximum. One of the crucial features of the hyetograph is its form. The shape of an inter-shower is assumed to be rectangular. It is then describedby its duration and its mean intensity. The shape of a shower depends on the number of peaks it contains. Each peak is represented by a trapezoidal shape. and the mean intensity is consideredequal on each side of the maximum intensity. Given the above definitions, the entities that compose a 5 min rainfallseries are expressed by the following variables: D Na Dis lis Np D, Is Dp Ip Imax Lp Sp
the durationof the dry weatherperiodbetweentwo rain events; the numberof showersper event; the durationof the inter-showers; the mean intensityof the inter-showers; the numberof peaksper shower; the durationof the showers; the mean intensity of the showers; the durationof the peaks; the mean intensityof the peaks; the maximumintensity of the peaks; the locationof Imax; the shape of the peaks.
The variables were identified by means of tipping bucket raingauge data, which consisted in cumulative rainfall over I min and with a precision of 0.1 or 0.2 mm. These came from the dense network of the Seine Saint Denis area, situated in the North East of Paris, in France. Seven stations with 16or 17 years of records were available. Roux and Desbordes (1996) have shown that the 7 series are equivalent to a 50 year point rainfall record. The raw data were treated to obtain series of 5 min time step intensities. After which, a regional analysis was carried out in order to determine the probability distributions of the descriptive variables of a hyetograph. The methodconsists in computingthe first moments of each local sample (mean, variance). and afterwards in making an arithmetic mean to establish the regional moments. These were then used to derive the regional statistical distributions by means of the method of the moments. Some of these distributions were found to depend on the season.andloron the value of another variable. Table I. Theoretical probability distributions used in the model
Gamma(a,~. y)
Domainof random variable x : real, n : integer OSx<+-
Geometric(p) Beta(p. q)
k>O OSxSI
Nameand parameters
Probability density function or discrete probability f(x) =(x - "(t' . e'''' 'Y>i\, I (I ~ I r(a» a>O.lhO P(k) - pO-p)'"' ft .
(x)
=x~' .O-x)'"
. I'(p + q) I r(p) I r(q) p>O.g >0
40
V.THAUVlN et aI.
PROBABILITY MODELS FOR THE RAINFALL CHARACTERISTICS Theoretical probability distributions In follow up to previous research a small number of theoretical probability distributions have been needed to represent the descriptive variables of a rainfall series. These are the Gamma distribution, the Geometrical distribution and the Beta distribution (see Table I for their expression). The quality of the adjustment is verified graphically, and also by computing the theoretical mean and variance. Theoretical distribution is retained according to its capability to model correctly all the quantiles of the observed distribution. Dependencies and derived yariables The dependencies between the successive values d a variable and between the main descriptive variables are studied. In order to point out non-linear links, the lack of dependence is tested by verifying the equality of the distributions of one variable according to the value of another variable. The Wilcoxon rank sum test has been used to compare the distributions. It is a robust test, although it is not very powerful. But it was chosen because of the large positive dissymmetry of the observed distributions. The main results show that firstly, no temporal dependence could be detected between successive values of the model variables, and secondly, new conditional variables have to be defined.
Inter-showers. The duration and the mean intensity of the inter-showers depend on their location inside the storm. The new variables defined are (DiJI and (liJI if the inter-shower forms a storm event without any shower (Na=O), (Dish and (Iish when it is situated before the first shower, (Dish and (Iish if it is located between two showers and (Dis)4 and (I is)4 if it occurs after the last shower. Furthermore, dependencies have been found between (Dis)2 and the mean intensity of the folIowing shower, as well as between (Disl4 and the mean intensity of the preceding shower. On the contrary, no relation was found between the mean intensity of the inter-showers and the mean intensity of the showers belonging to the same storm event.
Showers. The duration and the mean intensity of a shower are defined in relation to the number of peaks per shower and the number of showers per storm event. Thus (D,h, (DJ2 and (Dsh corresponds to the duration of three types of showers which have either one peak.stwo peaks, or more. (IJI is the mean intensity of a single shower (N.=!) and (15)2 is the mean intensity of a shower on the other cases (N.>!). The number of peaks per shower is conditioned by the location of the shower inside the storm. Two variables are necessary to describe it: (Npl2 if the shower is located between two others and (Npll in all other cases.
Peaks. Due to their definition, the characteristics of the peaks are linked to the characteristics of the showers they belong to. This point is described in detail in the folIowing section. Seasons. The dependence on the seasons is also considered. It is tested if the monthly distributions are homogeneous. Few variables appear to be seasonal: N a, (D iJ2' (Di')4' (1,)1 and (15)2' and consequently the mean intensities of the peaks which belong to showers of types I and 2. Two seasons are distinguished: winter from October to April and summer from May to September. Model parameter estimation The parameter estimation is carried out for each variable with respect to dependencies on the season and on other variables. Conditional distributions are derived by means of conditional means and variances.
Gamma distribution. The Gamma distribution with 3 parameters is convenient to describe 10 variables of the model (Table 2). It should be noted that the dry duration between two events is modelled by a single distribution whereas many previous works needed three distributions to describe it It is the opinion of the authors that a combination of several distributions makes the adjustment less accurate. Moreover a single
Short time-step point rainfall stochastic model
41
distribution is convenient to reproduce quite satisfactory different quantile values (Figure 2). Th is is both due to our definition of the dry periods, which does not include the very short dry periods inside a storm event, and to the Gamma distribution propert ies, that allow for correctly reproducing the high frequencie s when the observed distribution presents a spreaded tail. The dependence between the expectation of the duration of the inter-showers «Dish and (DiJ~ and the mean intensity of the follow ing or preceding shower is taken into account by fitting a non linear regression with 3 parameters. This relation is seasonal. Thus a total of 14 parameters is necessary to fully describe the duration of the inter-s howers.
0
~ ~
n. ", I 0 '
I oj o
o
- 2" - - 3
•
([)-60)/13900
Figure 2. Distribution of dry weather period duration. Observations and Gamma distribution, 0-=0.45.
Table 2. Variables described by the Gamma distribution with 3 parameters. Units are min for the durations and mm/h for the intensities Variables Duration of dry weather periods between two rain events : 0
Comments D~60
(Dis) I ~ 60, (Dish ~ 0, (Dish ~ IS, (Dish ~ 0 (a) seasonal (b) conditional to the mean intensity of the following shower (c) condit ional to the mean intensity of the preceding shower (d) an additional parameter models the percentage of intershowers with a null duration (0.) 1~ 0, (Dsh ~ 10, (Dsh ~ 0 Duration of showers : (e) the duration of a shower containing more than two peaks (Os)" ro,», (Dsh!e) is derived by multiply ing (D s)1 by the number of peaks Np (I.), > 5, (Ish> 5 Mean intensity of showers : (a) seasonal (ls),I'l(O and (I,hl. )(O (I) an additional parameter models the percentage of showers with mean intensity less than 5 mmfh
Geometrical distribution. The Geometrical distribution is used to model 3 variables : the numbe r of showers inside an event , Na' and the number of peaks per shower: (N p) I and (N p)2' The proportion of events without
42
V. THAUVIN et 0/ .
showers was found to be strongly dependent on the season (with 51% for winter and 24.5% for summer), whereas the monthly distribution of the showers per event (N a ~ I) did not show any significant difference.
Beta distribution. Table 3 summarises the variables which are represented by a Beta distribution. This is well adapted to reproduce bounded variables. They are in this case expressed as a proportion of the upper limit, the value of the lower limit being previously removed. This distribution is also adequate to describe the characteristic variables of a peak (Figure 3). The duration and the rainfall depth of a peak is expressed as a proportion of shower duration in the first case and of shower rainfall depth in the second case. It is assumed that the average proportion of the shower duration (respectively the shower rainfall depth) is equivalent for each of the peaks.
o
", 1
oj
o
00
02
04
06 et duration
Flfst pea k.duration over &h
08
Figure 3. Distribution of peak duration. Observations and Beta(4, 4) distribution.
Table 3. Descriptive variables represented by a Beta distribution with 2 parameters and normed variables. Units are min for the durations and mmfh f~r the intensities. Variables Mean intensity of inter-showers: (Ii.), (a), (lish.
n,», (li.)4
Duration of peaks: (Dp)j, with je l to N, Mean intensity of peaks: (Ip)j, with je l to Np
Normed variables and comments ((lis); - 0.2) / 3, i = I to 4 (a) conditional to the duration (Di.), (Dp)j / parameters p, q
», =p.(N p -I)
((lp)j . (D p»)) / (Is . Os) parameters p, q = p.Il), - (DpW (Dp)j
Maximum intensity of peaks : (Imax)j. with j= I to N p
Location of maximum inten sity : L,
L, is a proportion of the total duration of the peak
The calibration of the descriptive variables of a peak was carried out with sufficient accuracy for showers containing up to three peaks. For the showers made up of more peaks the same probability functions as those of three peak showers were used. FIRST VALIDATION AND DISCUSSION The first validation consists in verifying that the variables simulated by the rainfall model present statistical distributions similar to those of the observed variables . The model is composed of three kinds of variables: first order variables, which are generated by a single theoretical distribution, second order variables. where
43
Short time-step point rainfall stochastic model
the distribution is conditi onal to another variable. and finally third order variables. which depend on two other variables. This first validation is of particular importance for the second order and third order variables. Examples of results are given for each kind of variables : D, the duration of dry weather periods between two rain events (Table 4 and Figure 4). Ip' the mean intensity of the peaks, which is derived conditionally to the mean intensity of the shower it belong s to (Table 5 and Figure 5), and lmax. the maximum intensity of the peaks, which is defined relatively to both the mean intensity of the shower and the mean intensity of the peak (Table 6). As to what concern s first order variables. random simulation is carried out. As regards second and third order variables, the process used in the rainfall model is applied. A simulation of 50 years is carried out.
o
3 4 (D-60Y 13900
1
6
Figure 4. Duration of dry weather periods . 20 year simulation and Gamma distribut ion, a=0.45 .
Table 4. Some characteristics of the observed and simulated series related to dry weather period durations D (min )
Mean (min)
Observed Simulated
6306 6250
Annual number of Annual number of Annual events including number of events including showers (winter) showers (summer) events 23.2 25.5 81.1 23.4 25.7 81.8
Table 5. Observed and simulated frequencies of the mean intensity of the first peale Case A: showers containing one peak. Case B: showers containing more than one peak Mean intensity (rom/h) >10 > 20 > 30 >40 > 50
Observed frequencies
Simulated frequencies
(%)
(%)
A
B
A
B
33.7 9.01 3.14 1.24 0.533
18.5 3.45 1.72 0.69 0
33.5 9.01 3.07 1.15 0.489
30.7 7.80 2.15 0.539 0.268
Figure 5. Mean intensity of first peaks . Observed distribution and simulated distribution (20 year simulation with Beta(4 , 4) distribution).
Table 6. Observed and simulated frequencies of the maximal intensity of the first peak for all the showers Maximal intensity (mmlh) >20 >40 >60 >80
> 100
Observed frequencies
(%) 16.6 3.80 1.48 0.638 0.275
Simulated frequencies (%) 16.3 3.58 1.50 0.633 0.290
The verifications made on the first order variables confirm the quality of the probability functions' calibration by the method of the moments. Moreover, for most of second and third order variables the adequation between the simulations and the observations was quite satisfactory. However the modelling of some peak variables was of poorer quality . Indeed, as regards the mean intensity, significant differences were found between simulated and observed values depending on the position of the peale Thus the results for the second peak of a shower indicate that the Beta distribution overestimates the observed values, as much as the intensity is low. For example it can be seen from table 5 that the frequency value of the 1020 mmlh class is of 18.5% for the observations and of 30.7% for the simulations. An explanation of this fact can be the following: when a shower contains more than one peak, the manner in which the mean intensities of a peak are simulated implies that at least one of the values is equal to or greater than the mean intensity of the shower. But the observations show the opposite tendency. This leads the authors to believe that the dependencies concerning the mean intensity of showers were misrepresented. It was assumed that it depends on the number of showers per event whereas there probably exists a more important dependence on the number of peaks . CONCLUSION A point rainfall stochastic model with a 5 min time step was presented. It generates a synthetic rainfall series as an alternance of dry and wet spells . The wet spells are fully described, with particular care taken in representing the shape of the most intense periods . As a first step, the analysis, which was carried out using many long series of observed rainfall data, took into account all the links between the different variables.
Shorttime-step pointrainfall stochastic model
45
This resulted in a complex model, which included 48 parameters, some of which were seasonal. For that reason this model has to be at present considered as a prototype. When used in the Seine Saint Denis area, it is capable of simulating patterns of rain events likely to provoke seriously harmful effects on the sewer system. But in order for the model to be useful, its parameters have to be regionalized. The next two steps before the regionalization would be firstly, to reduce the number of parameters by means of a sensitivity analysis, secondly, to study the links between the variables of the model and regional variables, such as monthly rainfall depth, location, elevation and so on. The results of the first validation indicate that the probability distributions of second and third order variables are on the whole correctly reproduced. However, it is not possible when using this kind of validation to point out the consequences of the hypothesis made on the rain event form while building the model. For example, "significant" minima have been defined, and thus secondary minima were not taken into account. Moreover the shape of a shower has been simplified. These assumptions could have led to eliminate rainfall features which are important as far as the runoff production is concerned. Work in progress consists in the hydrological validation. It will verify that the assumptions made, in particular about the form of showers, have no consequence when the model is used to design sewer networks. Both synthetic series of rainfall and the observed series of rainfall used for calibration will serve to reconstitute long term simulations of flows at the outlet of a small urban catchment with a rainfall-runoff model. The statistics of peak discharges and of overflow volumes will be compared, and the equivalency in both cases will be verified. ACKNOWLEDGEMENTS This research was funded by the Ministere de I'Environnement and the Agence de I'Eau SeineNormandie.The authors thank the Direction de l'Eau et de l'Assainissement de Seine Saint Denis for supplying data.
REFERENCES Cerncsson, F. (1993). Modele simple de prlditermination des crue« de frlqrumces courante a rare sur petits bassins versants mlditerranlens. PhDthesis,University of Montpellier II. Croley,T. E., Eli. R. N. and Cryer. 1. D. (1978). Ralstom Creek hourly precipitation model. Water Resources Research. 14(3). 485-490. Marien. J. L. and Vandewiele, G. L. (1986). A point rainfall generator with internal stormstructure. Water Resources Research, 22(4).475-482. Rodriguel-Iturbc, I.. COli, D. R. and Isham, V. (1987). Somemodels for rainfall basedon stochastic pointprocesses. Proc. R. Soc. Lond. Series A(41O), 269·288 ROUll, C. and Desbordes, M. (1996). Rainfall-frequency curves with a recent urban dense rainfall measurement network. Atmospheric Research, 41. 163-176. Tourassc, P. (1981). Analyses spatiales et umporelles de pricipitations et utilisation opirationnelle dans /In systeme de prlvision des crues. Application aux rigionsdvenoles. PhDthesis, University of Grenoble.
Wal. Sci. Tech. Vol. 37, No. II, pp. 37-4S, 1998.
IAWQ 1998 Publishedby Elsevier Science LId. Printed in Oreal Britain. All rights reserved 0273-1223/98 $19'00 + 0'00 @
PH: S0273-1223(98)OO314-X
A SHORT TIME-STEP POINT RAINFALL STOCHASTIC MODEL V. Thauvin*, E. Gaume* and C. Roux** • ENPc/CERGRENE, 6 et 8 avenue Blaise Pascal, Champs Sur Marne. 77455 Marne La Vallee Cedex2. France •• SEl'UDE. 11 bd Pershing, 75017 Paris. France
ABSTRACT This paper deals with the development of a point rainfall stochastic model. which generates synthetic sequences of S min rainfall rates. This time step is necessary to derive correctly the short hydrological response of urban catchments. The model simulates dry and rainy sequences in alternance. Particular attention has been paid to intense periods which are of utmost importance as far as flooding is concerned. This leads us to describe a rain event as a juxtaposition of inter-showers and showers. The intensity of an inter-shower is assumed to remain constant. Showers are defined as a succession of « peaks" with a single maximum and a trapezoidal shape. The model has been calibrated with raingauge data from the Seine Saint Denis area, situated near Paris. in France. This paper presents the definition of the model variables, the calibration of their probability distributions and the initial validation results. The conclusion insists on the fact that for the model to be completed. hydrological validation is necessary. ~ 1998 Published by Elsevier Science Ltd. All rights reserved
KEYWORDS Model;rainfall;simulation; stochastic; urban hydrology. INTRODUCTION Since the beginningof the eighties, point rainfall stochastic models have been used to provide hydrological models with long term rainfall series in order to allow for long term discharge simulations. Simulation results can then be devoted to predetermination of flooding and to design issues. Nevertheless. most of the models have been developed to be applied on rural catchments, and they run with a time step of one hour (see for example Tourasse (1981) and Cemesson (1993) who conducted research in this way). Rainfall stochastic models may also be utilised to design urban sewer systems. in the place of usual techniques like the use of a limited number of measured or synthetic design rainfalls. This utilisation requires rainfall models with a time step shorter than one hour (typically 5 min), since the hydrological response of urban catchment to rainfall input is short. Preliminary work about point rainfall stochastic models with short time steps (10 min) can be found in Marien and Vandewiele (1986). For such a model to be useful for design issues, it is necessaryto go deeply into the description of the form of rainfall events. Followingthis research. the topic of our work is to study the feasibility of building a model capable of reproducing the temporal structure of rainfall with a time step compatible with urban hydrologic applications. e.g. 5 min. Particular attentionhas been paid to the analysisof dependencies between the different variables of the model.
37
38
V. lHAUVIN et al.
Two remarks can be made before goingfurther. I. This workconstitutes the first phase of the development of a new tool intended for urban sewer network design. Indeed such a tool will be of great interest if a large number of its parameters can be regionalized: either they can be considered as constant values, or they depend on variables easily available, for example, daily rainfall depth, or regional parameters such as location, monthly rainfall depth, elevation and so on. In theseconditions the rainfall model can be usedeven in placeswithout raingauge observations. 2. The model reproduces point rainfall, in other words the spatial variability is not taken into account. Thus its results can be used only for applications on smallurbancatchments (a few squarekilometres). This paper presents firstly, the identification of the descriptive variables of a 5 min rainfall series, secondly, the calibration of the model parameters, and thirdly, the first validation made on the model results. The conclusion ends with the exposition of the principle of a hydrological validation whichappearsnecessary to validate the assumptions made whilerepresenting the formof rain events. CHARACfERISTICS OF A POINT RAINFALL SERIES The time rainfall series is considered as groups of positive values with a defined structure, and each group being the resultof a random process. There are two broadcategories of models: the aggregation models (see Rodriguez-Iturbe et al., 1987) and the composition models (see Croley et al., 1978). In the first case. the hypothesis is that each storm event is composed of successive cells. Each cell is associated with a period of rainfall of random duration and of constant, or non-constant. random intensity. Total rainfall intensity during a time step is formed by adding the contributions from all cells. In the second case, the rainfall series is considered as a juxtaposition of entities. which represent the main features of the hyetograph, that is to say dry periods and rainy periods.One periodis described by its duration, its rainfall depth and its shape.These characteristics are treatedas random variables. Intensity
«
it II
Inter-shower
"
Shower Rainevent
..
.,
Inter-shower
Dry weather
Rain event
period Figure I. Representation of a rainfall series of intensities.
The model presented in this paper is a composition model. The rainfall series is supposed to be a juxtaposition of embedded structures (Figure 1). 1. Dry periodsand rain events occur in aItemance. They are distinguished by a threshold value of 0.2 mm'h, applied after a moving average, calculated on a period of one hour. Moreover a rain event is retained if its cumulative rainfall exceeds 2 mm, Below this value. rain events do not produce significant runoff. and consequently they can not form discharges in the sewerdrainage network. Thus about 30% of the total year rainfall appearsas dry periodsin the model. 2. A rain event is composed of showers and inter-showers. The showers are the most intense parts of a rain event. They are defined by a threshold value of 3.2 mmlh, applied after a moving average. calculatedon a
39
Short time-steppoint rainfall stochasticmodel
period of 15 min. Showers and inter-showers are assumed to be produced by two different random processes. In these conditionsa better adjustment of the probability functions is obtained. 3. A shower consists of a succession of peaks. A peak is a rainy period between two significant minima. A minimum is reached when it is a unique exception in an ascending or a descending part of the series. The less trivial cases of equality betweentwo values arealso considered. A peak presents a single maximum. One of the crucial features of the hyetograph is its form. The shape of an inter-shower is assumed to be rectangular. It is then describedby its duration and its mean intensity. The shape of a shower depends on the number of peaks it contains. Each peak is represented by a trapezoidal shape. and the mean intensity is consideredequal on each side of the maximum intensity. Given the above definitions, the entities that compose a 5 min rainfallseries are expressed by the following variables: D Na Dis lis Np D, Is Dp Ip Imax Lp Sp
the durationof the dry weatherperiodbetweentwo rain events; the numberof showersper event; the durationof the inter-showers; the mean intensityof the inter-showers; the numberof peaksper shower; the durationof the showers; the mean intensity of the showers; the durationof the peaks; the mean intensityof the peaks; the maximumintensity of the peaks; the locationof Imax; the shape of the peaks.
The variables were identified by means of tipping bucket raingauge data, which consisted in cumulative rainfall over I min and with a precision of 0.1 or 0.2 mm. These came from the dense network of the Seine Saint Denis area, situated in the North East of Paris, in France. Seven stations with 16or 17 years of records were available. Roux and Desbordes (1996) have shown that the 7 series are equivalent to a 50 year point rainfall record. The raw data were treated to obtain series of 5 min time step intensities. After which, a regional analysis was carried out in order to determine the probability distributions of the descriptive variables of a hyetograph. The methodconsists in computingthe first moments of each local sample (mean, variance). and afterwards in making an arithmetic mean to establish the regional moments. These were then used to derive the regional statistical distributions by means of the method of the moments. Some of these distributions were found to depend on the season.andloron the value of another variable. Table I. Theoretical probability distributions used in the model
Gamma(a,~. y)
Domainof random variable x : real, n : integer OSx<+-
Geometric(p) Beta(p. q)
k>O OSxSI
Nameand parameters
Probability density function or discrete probability f(x) =(x - "(t' . e'''' 'Y>i\, I (I ~ I r(a» a>O.lhO P(k) - pO-p)'"' ft .
(x)
=x~' .O-x)'"
. I'(p + q) I r(p) I r(q) p>O.g >0
40
V.THAUVlN et aI.
PROBABILITY MODELS FOR THE RAINFALL CHARACTERISTICS Theoretical probability distributions In follow up to previous research a small number of theoretical probability distributions have been needed to represent the descriptive variables of a rainfall series. These are the Gamma distribution, the Geometrical distribution and the Beta distribution (see Table I for their expression). The quality of the adjustment is verified graphically, and also by computing the theoretical mean and variance. Theoretical distribution is retained according to its capability to model correctly all the quantiles of the observed distribution. Dependencies and derived yariables The dependencies between the successive values d a variable and between the main descriptive variables are studied. In order to point out non-linear links, the lack of dependence is tested by verifying the equality of the distributions of one variable according to the value of another variable. The Wilcoxon rank sum test has been used to compare the distributions. It is a robust test, although it is not very powerful. But it was chosen because of the large positive dissymmetry of the observed distributions. The main results show that firstly, no temporal dependence could be detected between successive values of the model variables, and secondly, new conditional variables have to be defined.
Inter-showers. The duration and the mean intensity of the inter-showers depend on their location inside the storm. The new variables defined are (DiJI and (liJI if the inter-shower forms a storm event without any shower (Na=O), (Dish and (Iish when it is situated before the first shower, (Dish and (Iish if it is located between two showers and (Dis)4 and (I is)4 if it occurs after the last shower. Furthermore, dependencies have been found between (Dis)2 and the mean intensity of the folIowing shower, as well as between (Disl4 and the mean intensity of the preceding shower. On the contrary, no relation was found between the mean intensity of the inter-showers and the mean intensity of the showers belonging to the same storm event.
Showers. The duration and the mean intensity of a shower are defined in relation to the number of peaks per shower and the number of showers per storm event. Thus (D,h, (DJ2 and (Dsh corresponds to the duration of three types of showers which have either one peak.stwo peaks, or more. (IJI is the mean intensity of a single shower (N.=!) and (15)2 is the mean intensity of a shower on the other cases (N.>!). The number of peaks per shower is conditioned by the location of the shower inside the storm. Two variables are necessary to describe it: (Npl2 if the shower is located between two others and (Npll in all other cases.
Peaks. Due to their definition, the characteristics of the peaks are linked to the characteristics of the showers they belong to. This point is described in detail in the folIowing section. Seasons. The dependence on the seasons is also considered. It is tested if the monthly distributions are homogeneous. Few variables appear to be seasonal: N a, (D iJ2' (Di')4' (1,)1 and (15)2' and consequently the mean intensities of the peaks which belong to showers of types I and 2. Two seasons are distinguished: winter from October to April and summer from May to September. Model parameter estimation The parameter estimation is carried out for each variable with respect to dependencies on the season and on other variables. Conditional distributions are derived by means of conditional means and variances.
Gamma distribution. The Gamma distribution with 3 parameters is convenient to describe 10 variables of the model (Table 2). It should be noted that the dry duration between two events is modelled by a single distribution whereas many previous works needed three distributions to describe it It is the opinion of the authors that a combination of several distributions makes the adjustment less accurate. Moreover a single
Short time-step point rainfall stochastic model
41
distribution is convenient to reproduce quite satisfactory different quantile values (Figure 2). Th is is both due to our definition of the dry periods, which does not include the very short dry periods inside a storm event, and to the Gamma distribution propert ies, that allow for correctly reproducing the high frequencie s when the observed distribution presents a spreaded tail. The dependence between the expectation of the duration of the inter-showers «Dish and (DiJ~ and the mean intensity of the follow ing or preceding shower is taken into account by fitting a non linear regression with 3 parameters. This relation is seasonal. Thus a total of 14 parameters is necessary to fully describe the duration of the inter-s howers.
0
~ ~
n. ", I 0 '
I oj o
o
- 2" - - 3
•
([)-60)/13900
Figure 2. Distribution of dry weather period duration. Observations and Gamma distribution, 0-=0.45.
Table 2. Variables described by the Gamma distribution with 3 parameters. Units are min for the durations and mm/h for the intensities Variables Duration of dry weather periods between two rain events : 0
Comments D~60
(Dis) I ~ 60, (Dish ~ 0, (Dish ~ IS, (Dish ~ 0 (a) seasonal (b) conditional to the mean intensity of the following shower (c) condit ional to the mean intensity of the preceding shower (d) an additional parameter models the percentage of intershowers with a null duration (0.) 1~ 0, (Dsh ~ 10, (Dsh ~ 0 Duration of showers : (e) the duration of a shower containing more than two peaks (Os)" ro,», (Dsh!e) is derived by multiply ing (D s)1 by the number of peaks Np (I.), > 5, (Ish> 5 Mean intensity of showers : (a) seasonal (ls),I'l(O and (I,hl. )(O (I) an additional parameter models the percentage of showers with mean intensity less than 5 mmfh
Geometrical distribution. The Geometrical distribution is used to model 3 variables : the numbe r of showers inside an event , Na' and the number of peaks per shower: (N p) I and (N p)2' The proportion of events without
42
V. THAUVIN et 0/ .
showers was found to be strongly dependent on the season (with 51% for winter and 24.5% for summer), whereas the monthly distribution of the showers per event (N a ~ I) did not show any significant difference.
Beta distribution. Table 3 summarises the variables which are represented by a Beta distribution. This is well adapted to reproduce bounded variables. They are in this case expressed as a proportion of the upper limit, the value of the lower limit being previously removed. This distribution is also adequate to describe the characteristic variables of a peak (Figure 3). The duration and the rainfall depth of a peak is expressed as a proportion of shower duration in the first case and of shower rainfall depth in the second case. It is assumed that the average proportion of the shower duration (respectively the shower rainfall depth) is equivalent for each of the peaks.
o
", 1
oj
o
00
02
04
06 et duration
Flfst pea k.duration over &h
08
Figure 3. Distribution of peak duration. Observations and Beta(4, 4) distribution.
Table 3. Descriptive variables represented by a Beta distribution with 2 parameters and normed variables. Units are min for the durations and mmfh f~r the intensities. Variables Mean intensity of inter-showers: (Ii.), (a), (lish.
n,», (li.)4
Duration of peaks: (Dp)j, with je l to N, Mean intensity of peaks: (Ip)j, with je l to Np
Normed variables and comments ((lis); - 0.2) / 3, i = I to 4 (a) conditional to the duration (Di.), (Dp)j / parameters p, q
», =p.(N p -I)
((lp)j . (D p»)) / (Is . Os) parameters p, q = p.Il), - (DpW (Dp)j
Maximum intensity of peaks : (Imax)j. with j= I to N p
Location of maximum inten sity : L,
L, is a proportion of the total duration of the peak
The calibration of the descriptive variables of a peak was carried out with sufficient accuracy for showers containing up to three peaks. For the showers made up of more peaks the same probability functions as those of three peak showers were used. FIRST VALIDATION AND DISCUSSION The first validation consists in verifying that the variables simulated by the rainfall model present statistical distributions similar to those of the observed variables . The model is composed of three kinds of variables: first order variables, which are generated by a single theoretical distribution, second order variables. where
43
Short time-step point rainfall stochastic model
the distribution is conditi onal to another variable. and finally third order variables. which depend on two other variables. This first validation is of particular importance for the second order and third order variables. Examples of results are given for each kind of variables : D, the duration of dry weather periods between two rain events (Table 4 and Figure 4). Ip' the mean intensity of the peaks, which is derived conditionally to the mean intensity of the shower it belong s to (Table 5 and Figure 5), and lmax. the maximum intensity of the peaks, which is defined relatively to both the mean intensity of the shower and the mean intensity of the peak (Table 6). As to what concern s first order variables. random simulation is carried out. As regards second and third order variables, the process used in the rainfall model is applied. A simulation of 50 years is carried out.
o
3 4 (D-60Y 13900
1
6
Figure 4. Duration of dry weather periods . 20 year simulation and Gamma distribut ion, a=0.45 .
Table 4. Some characteristics of the observed and simulated series related to dry weather period durations D (min )
Mean (min)
Observed Simulated
6306 6250
Annual number of Annual number of Annual events including number of events including showers (winter) showers (summer) events 23.2 25.5 81.1 23.4 25.7 81.8
Table 5. Observed and simulated frequencies of the mean intensity of the first peale Case A: showers containing one peak. Case B: showers containing more than one peak Mean intensity (rom/h) >10 > 20 > 30 >40 > 50
Observed frequencies
Simulated frequencies
(%)
(%)
A
B
A
B
33.7 9.01 3.14 1.24 0.533
18.5 3.45 1.72 0.69 0
33.5 9.01 3.07 1.15 0.489
30.7 7.80 2.15 0.539 0.268
Figure 5. Mean intensity of first peaks . Observed distribution and simulated distribution (20 year simulation with Beta(4 , 4) distribution).
Table 6. Observed and simulated frequencies of the maximal intensity of the first peak for all the showers Maximal intensity (mmlh) >20 >40 >60 >80
> 100
Observed frequencies
(%) 16.6 3.80 1.48 0.638 0.275
Simulated frequencies (%) 16.3 3.58 1.50 0.633 0.290
The verifications made on the first order variables confirm the quality of the probability functions' calibration by the method of the moments. Moreover, for most of second and third order variables the adequation between the simulations and the observations was quite satisfactory. However the modelling of some peak variables was of poorer quality . Indeed, as regards the mean intensity, significant differences were found between simulated and observed values depending on the position of the peale Thus the results for the second peak of a shower indicate that the Beta distribution overestimates the observed values, as much as the intensity is low. For example it can be seen from table 5 that the frequency value of the 1020 mmlh class is of 18.5% for the observations and of 30.7% for the simulations. An explanation of this fact can be the following: when a shower contains more than one peak, the manner in which the mean intensities of a peak are simulated implies that at least one of the values is equal to or greater than the mean intensity of the shower. But the observations show the opposite tendency. This leads the authors to believe that the dependencies concerning the mean intensity of showers were misrepresented. It was assumed that it depends on the number of showers per event whereas there probably exists a more important dependence on the number of peaks . CONCLUSION A point rainfall stochastic model with a 5 min time step was presented. It generates a synthetic rainfall series as an alternance of dry and wet spells . The wet spells are fully described, with particular care taken in representing the shape of the most intense periods . As a first step, the analysis, which was carried out using many long series of observed rainfall data, took into account all the links between the different variables.
Shorttime-step pointrainfall stochastic model
45
This resulted in a complex model, which included 48 parameters, some of which were seasonal. For that reason this model has to be at present considered as a prototype. When used in the Seine Saint Denis area, it is capable of simulating patterns of rain events likely to provoke seriously harmful effects on the sewer system. But in order for the model to be useful, its parameters have to be regionalized. The next two steps before the regionalization would be firstly, to reduce the number of parameters by means of a sensitivity analysis, secondly, to study the links between the variables of the model and regional variables, such as monthly rainfall depth, location, elevation and so on. The results of the first validation indicate that the probability distributions of second and third order variables are on the whole correctly reproduced. However, it is not possible when using this kind of validation to point out the consequences of the hypothesis made on the rain event form while building the model. For example, "significant" minima have been defined, and thus secondary minima were not taken into account. Moreover the shape of a shower has been simplified. These assumptions could have led to eliminate rainfall features which are important as far as the runoff production is concerned. Work in progress consists in the hydrological validation. It will verify that the assumptions made, in particular about the form of showers, have no consequence when the model is used to design sewer networks. Both synthetic series of rainfall and the observed series of rainfall used for calibration will serve to reconstitute long term simulations of flows at the outlet of a small urban catchment with a rainfall-runoff model. The statistics of peak discharges and of overflow volumes will be compared, and the equivalency in both cases will be verified. ACKNOWLEDGEMENTS This research was funded by the Ministere de I'Environnement and the Agence de I'Eau SeineNormandie.The authors thank the Direction de l'Eau et de l'Assainissement de Seine Saint Denis for supplying data.
REFERENCES Cerncsson, F. (1993). Modele simple de prlditermination des crue« de frlqrumces courante a rare sur petits bassins versants mlditerranlens. PhDthesis,University of Montpellier II. Croley,T. E., Eli. R. N. and Cryer. 1. D. (1978). Ralstom Creek hourly precipitation model. Water Resources Research. 14(3). 485-490. Marien. J. L. and Vandewiele, G. L. (1986). A point rainfall generator with internal stormstructure. Water Resources Research, 22(4).475-482. Rodriguel-Iturbc, I.. COli, D. R. and Isham, V. (1987). Somemodels for rainfall basedon stochastic pointprocesses. Proc. R. Soc. Lond. Series A(41O), 269·288 ROUll, C. and Desbordes, M. (1996). Rainfall-frequency curves with a recent urban dense rainfall measurement network. Atmospheric Research, 41. 163-176. Tourassc, P. (1981). Analyses spatiales et umporelles de pricipitations et utilisation opirationnelle dans /In systeme de prlvision des crues. Application aux rigionsdvenoles. PhDthesis, University of Grenoble.