A SIMPLE SCALING CHARATERISTICS OF RAINFALL IN TIME AND SPACE TO DERIVE INTENSITY DURATION FREQUENCY RELATIONSHIPS

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1 Annual Journal of Hyraulic Engineering, JSCE, Vol.51, 27, February A SIMPLE SCALING CHARATERISTICS OF RAINFALL IN TIME AND SPACE TO DERIVE INTENSITY DURATION FREQUENCY RELATIONSHIPS Le Minh NHAT 1, Yasuto TACHIKAWA 2, Takahiro SAYAMA 3 an Kaoru TAKARA 4 1 Stuent Member, Grauate stuent, Dept. of Urban an Environmental Eng., Kyoto University (Kyoto , Japan) nhat@floo.pri.kyoto-u.ac.jp 2 Member of JSCE, Dr. Eng., Associate Professor, DPRI, Kyoto University (Gokasho, Uji , Japan) 3 Member of JSCE, M. Eng., Assistant Professor, DPRI, Kyoto University (Gokasho, Uji , Japan) 4 Fellow of JSCE, Dr. Eng., Professor, DPRI, Kyoto University (Gokasho, Uji , Japan) The properties of time an space scale invariance of rainfall are investigate an applie to Intensity-Duration-Freuency (IDF) relationships. The hypothesis of the simple scaling is examine in time an space in Yoo River basin an the simple scaling properties in time are confirme, which is use to erive IDF relationships for short-uration rainfall of several hours from the statistical characteristics of aily ata only. The erive IDF matches well with the one erive from historical observe ata. Also, the simple scaling properties in space are examine. It is foun that two ranges less than 1 km 2 an more than 1 km 2 show ifferent scale properties. Key Wors: rainfall intensity, scaling invariance, IDF curves, esign rainfall. 1. INTRODUCTION The Intensity Duration Freuency (IDF) relationship of heavy storms is one of the most important hyrologic tools utilize by engineers for esigning floo alleviation an rainage structures in urban an urban areas. Local IDF euations are often estimate on the basis of recors of intensities abstracte from rainfall epths of ifferent urations, observe at a given recoring rainfall gauging station. In some regions, there may exist a number of recoring rainfall gauging stations operating for a time perio sufficiently long to yiel a reliable estimation of the IDF relationships; in many other regions, especially in eveloping countries, however, these stations are either non-existent or their sample sizes are too small. Because aily precipitation ata is the most accessible an abunant source of rainfall information, it seems natural, at least for the regions where ata at higher time resolution are scarce, to evelop an apply methos to erive the IDF characteristics of short-uration events from aily rainfall statistics. Over the last two ecaes, concepts of scale invariance have come to the fore in both moelling an ata analysis in hyrological precipitation research. Gupta an Waymire 1) stuie rainfall spatial variability by introucing the concepts of simple an multiple scaling to characterise the probabilistic structure of the precipitation processes. Kuzuha et al. 2) showe the scaling framework an regional floo freuency analysis. Burlano an Rosso 3) showe that both the simple scaling an multiscaling lognormal moels can be use to erive Depth Duration Freuency (DDF) curves of point precipitation. Menabe et al. 4) evelope a simple scaling methoology to use aily rainfall statistics to infer the IDF curves for rainfall uration less than 1 ay. The scaling hypothesis was verifie by fitting the moel to two ifferent sets of ata (from Australia an South Afric. Yu et al. 5) is an example of methoology in which the theories of scaling properties an employe to infer the IDF characteristics of short-uration rainfall from aily ata. Until now, time scaling characteristics have been stuie by many researchers, while space scaling properties that will link to erivation of Area-Intensity-Duration-Freuency (AIDF), are not well stuie. Thus, in this paper, the properties of time an space scale invariance of rainfall are examine in the Yoo River catchment. Then the IDF relationships for short-uration rainfall from aily ata are erive accoring to Menabe 3) an compare with the obtaine from the traitional metho. The paper is organize as follows: The next

2 section presents the theoretical backgroun as relate to the time an space scale invariance properties of short-uration rainfall. In section 3, these properties are verifie for ata observe at rainfall gauging stations locate in the Yoo River catchment in Japan. The IDF relationship, erive from aily rainfall ata, then compare to the traitional estimate curve in section 4 an conclusions are given in the last section. 2. SIMPLE SCALING CHARATERISTICS OF RAINFALL IN TIME AND SPACE In this section, a general theoretical framework for the simple scaling is introuce. The scaling or scale-invariant moels enable us to transform ata from one temporal or spatial moel to another one, an thus, help to overcome the ifficulty of inaeuate ata. A natural process fulfills the simple scaling property if the unerlying probability istribution of some physical measurements at one scale is ientical to the istribution at another scale, multiplie by a factor that is a power function of the ratio of the two scales. The basic theoretical evelopment of scaling has been investigate by many authors incluing Gupta an Waymire 1) an Kuzuha et al. 2) Let X(t) an X( λ t) enote measurements at two istinct time or spatial scales t an λ t, respectively. Definition of scaling of the probability of the X(t) is ist H X ( t) = λ X ( λt) (1) ist The euality = refers to ientical probability istributions in both sie of the euations, λ enotes a scale factor an H is a scaling exponent. Gupta an Waymire 1) introuce the notions of strict an wie sense simple scaling. The strict sense simple scaling in E.(1) implies that X ( t) an H ( λ X ( λt)) have the same probability istribution. The wie sense simple scaling is expresse as they have the same moments, i.e. H [ X( t) ] λ E[ X( λt ] E ) = (2) The scaling exponent, H can be estimate from the slope of linear regression relationship between the log transforme values of moment, log E [ X (λ t) ] an scale parameters log λ for various orer of moment. This is efinition of a wie sense simple scaling 1). A wie sense simple scaling with t=1 an λ t = λ, is given by H [ X( 1) ] λ E[ X( λ ] E ) = (3) If the scaling exponent H is not constant an changes probabilistically, E.(3) are escribe as ( ) [ (1) EX ] = λ K EX [ ( λ) ] (4) where K() is a function of the moment orer. The proceure to test the suitability of scale invariant moel to escribe the rainfall process is briefly escribe in Fig.1. The moments E [ X (λt) ] are plotte on the logarithmic chart versus the scale λ for ifferent moments orer. The slope K() is plotte on the linear chart versus the moment orer. If the resulting graph is a straight line, the fiel is simple scaling, while in other cases, the multi-scaling approach has to be consiere 1). Log [moment] K () Simple scaling Fig.1 Simple an multiscaling in term of statistical moments. First step, moments of ifferent orers are plotte as function of scale in a log-log plot. From the slope, values of the function K() are obtaine. If K() is linear, the process is simple scaling. If K() is non-linear, the process is multiscaling. (1) Time scaling of rainfall intensity Let the ranom variable I() the maximum annual value of local rainfall intensity over a uration. It is efine as: t+ / 2 1 I( ) = max X ( ξ ) ξ (5) t 1 year t / 2 where X (ξ ) is a time continuous stochastic process representing rainfall intensity an is uration. It is suppose that I() represents the Annual Maximum (AMRI) of uration, efine by the maximum value of moving average of with of the continuous rainfall process. Here, some concepts are introuce about scaling of the probability istribution of ranom functions. A generic ranom function I() is enote by simple scaling properties if it obeys the following: ist H D I( ) = I( D) (6) D is a aggregate time uration, i.e.: 2, 3,..., 24 hours. D Defining the scaling ratio is λ = ist H I ( ) = λ I( λ ) (7) Log [scale] K () Mutil-scaling

3 The E.(7) is rewritten in terms of the moments of orer about the origin, enote by E [ I( ) ]. The resulting expression is: ( ) H E I = λ EI( λ ) (8) [ ] [ ] If one assumes the wie sense simple scaling exists, the istribution of IDF for short-uration of rainfall intensity can be erive from aily rainfall. (2) Space scaling of rainfall intensity Let consier a continuous precipitation process I(, which represents the maximum annual value of average rainfall intensity over a uration, an an area a. It is efine as I(, 1 max t 1 year a t+ / 2 t / 2 a = X ( ξ, ω) ωξ (9) where X ( ξ, ω) is a time-space continuous stochastic process representing rainfall intensity. Since the probability of extreme events is usually examine, the maximum annual rainfall intensity can be efine as the maximum value of a moving average (with area span) in a given year. The concepts are introuce about scaling of the probability istribution of ranom functions. A generic ranom function I(,,with fixe uration, is enote by the simple scaling properties of space if it obeys the following relationship: ist H A a Ia (, ) = IA (, ) a Defining the space scaling ratio is ist λ a = A a (1) Ha Ia (, ) = λa I (, λ (11) This also implies that the moments of any orer are scale-invariance. The resulting expression is (, ) H a E Ia = λ EI (, λ (12) [ ] a [ a ] where enotes the orer of the moment. If one assumes the wie sense simple scaling exists, the AIDF curves can be erive by eveloping a general framework base on the estimation of a common scaling exponent for any freuency level. This approach is use to erive the istribution of the AIDF where ata for the spatial scale of interest oes not exist. 3. SCALE INVARIANCE PROPERTIES OF RAINFALL IN TIME AND SPACE IN YODO RIVER CATCHMENT, JAPAN The scaling properties of rainfall ata was investigate by computing the moment for each uration an area an then by examining the log-log plots of the moments against their uration an area. Fig.2 an Fig.3 illustrate the relationships between moments versus uration (time) an area (space) an a plot of the scaling exponents versus the orer of moments for the Yoo River catchment. In orer to examine the time scale invariance of rainfall in the Yoo River catchment, the four stations Hikone, Hirakata, Ieno an Kamo were selecte from Yoo River catchment. The recor lengths are 21 years an range from 1982 to 22. The analysis was performe on annual maximum rainfall series for storm urations from 1 hour to 24 hours, with λ =1, Fig.2 shows the relationships between the log-transforme values of moment of various orers against values uration at the Hikone station, the slopes of linear regression efine the scaling exponents H. Fig.2b) shows the scaling exponent ecreases with the orer of moment an a linear relationship exists between scaling exponents an orers of moment, which implies that the property of wie sense simple scaling of rainfall intensity exists in this station. In Table 1, the results of the scaling exponent factor H of the four stations in the Yoo River catchment are shown with the high coefficients of etermination for each station ranging from.98 to 1. The result inicates a strong valiity of the simple scaling property of the extreme rainfall in time series. Sample moment of orer 1.E+9 1.E+8 1.E+7 1.E+6 1.E+5 HIKONE Station 1.E+4 6=5 1.E+3 5=4 1.E+2 4=3 3=2 1.E+1 2=1 1=.5 1.E Duration (hour) K() y = -.658x -.13 R 2 =.9999 Hikone Station b) Fig.2 The time scale invariance at the Hikone station, the Yoo catchment, Japan Relationship between moment of orer an uration (hour) b) Relationship between K() an the sample moment orer.

4 M om ent E[I(6,A)] 1.E+8 1.E+7 1.E+6 1.E+5 1.E+4 1.E+3 1.E+2 1.E+1 1.E Area (km 2 ) K() y = x R 2 =.99 A=2.25 to 1 Km 2 y = -.474x R 2 =.99 A=1 to b) Fig.3 The space scale invariance at the Yoo River catchment for 6 hours. Relationship between moment of orer an area A (km 2 ). b) Relationship between K() an the sample moment orer. Table 1. The scaling exponent factor of the four stations in the Yoo River catchment Name of the station No of years recor Scaling exponent H R 2 Hikone Hirakata Ieno Kamo To examine the space scaling invariance, at first hourly ata from 1982 to 22 with 1.5 x 1.5 km spatial resolution, which is generate by AMeDAS rain gauge ata an covers the Yoo River basin are arrange. To obtain spatial mean rainfall intensity the Otorii station is put at the center of the area, an the area is expane in the form of circle with increase raius by.5 km. The ata use for examine spatial scaling factor are annual maximum rainfall series for the area from 2.25 km 2 to km 2, with λ a =1, an fixe uration. In Fig.3 a ifference of the egree of steepness in the slopes for the area (2.25 km 2 to 1 km 2 ) an larger area (1 km 2 to 5 km 2 ) area is foun, which inicates that two ifferent scaling space regimes exist for rainfall space scaling. A steeper slope is foun in the smaller area compare to the lager area. The plots inicate that the relationships between moments an areas are linear having two ifferent slopes with a breaking point at 1 km 2. This property suggests an existence of the two ifferent regimes with a transition in storm ynamics from high variability convective storms with less than 1 km 2 to frontal storms in an area larger than 1 km DERIVATION OF IDF FOR SHORT DURATION All forms of the generalize IDF relationships assume that rainfall epth or intensity is inversely relate to the uration of a storm raise to a power, or scale factor 6). There are several commonly use functions foun in the literature of hyrology applications 7),8),9),1). Koutsoyiannis et al. 11) have moifie the IDF relationship for a given return perio as particular cases, using the following general empirical formula w i = (13) ( +θ) where i enotes the rainfall intensity for uration an w, θ an represent non-negative coefficients. In fact, these arguments justify the formulation of the following general moel for the IDF relationships: a( T ) i = (14) b( ) In Euation (14), b() = ( + θ) with θ> an <<1, whereas a(t) is completely efine by the probability istribution function of the maximum rainfall intensity. The form of Euation (14) is consistent with most IDF empirical euations estimate for many locations 12) : For example Nhat et al. 13) establishe the IDF curves for precipitation in the monsoon area of Vietnam. The ranom variable rainfall intensity I() for uration, has a cumulative probability istribution CDF, which is given by 1 Pr( I( ) i) = F ( i) = 1 (15) T ( i) Accoring to the scaling theory by Menabe et al. 4), the scaling property in a strict sense can be written explicitly using the CDF:

5 H F() i = Fλ [ ] λ i (16) For many parametric forms, left han sie of Euation (16) may be expresse in terms of stanar variant, as in i μ F ( i) = F (17) σ where F (.) is a function inepenent of. Uner this form, it can be euce from Euation (16) that H μ = λ μ λ (18) H σ = λ σ λ (19) Substituting Euations (17), (18) an (19) into Euation (15) an investing with respect to i, one obtains: H H 1 μλ ( ) ( ) (1 1/ ) λ + σλ λ F T it, = (2) H By eualing Euation (2) to the general moel for IDF relationship, given by Euation (13), it is easy to verify that = H (21) θ = (22) b ( ) = (23) at = + F T (24) H H 1 ( ) μλ ( ) ( ) (1 1/ ) λ σλ λ 1 μ + σf (1 1/ T ) i, T = (25) where ( H μ = μλ ) λ an ( H σ = σλ ) λ are constants. It is worthwhile to note that the simple scaling hypothesis leas to the euality between the scale factor an the exponent in the expression relating rainfall intensity an uration. Application of the simple scaling moel for rainfall intensities at the Hikone station, the scale factor can be estimate H =.658. The IDF relationship for a short uration rainfall can be euce from aily ata by Euation (25) with the estimates of μ D an σ with D=24h. From D 24-hour ata collecte at the Hikone recoring station, the sample of 21 years of 24 hours annual maximum rainfall intensity yiels, which can be estimate μ D =24 = an σ D=24 = Back with these estimates an the Gumbel inverse function, the euce IDF relationship for the location of Hikone may be written as Euation (25) with μ = λ μ 24 = an σ = λ σ = then ln( ln(1 1/ T )) i = (26).65 The rainfall Intensity Duration Freuency curves for the Hikone station can be reconstructe by scaling methos with Euation (26). Figure 4 shows well matching of the IDF relationships calculate by Euation (26) an plots obtaine by the historical ata as the return perios from 5 to 1 years. The scale factor H along with the parameters σ an μ, may be interprete as regional climatic characteristics Years return perio Scaling metho Years return perio Scaling metho Years return perio Scaling metho Years return perio Scaling metho Fig.4 The Freuency Curves at the Hikone station, the Yoo Catchments, Japan by the scaling metho. b) c)

6 Rainfall intensity(mm/hr) 7 65 Area=11.25km Year return perio Rainfall intensity(mm/hr) 7 65 b) Area=992.25km Year return perio Fig. 5 The Area Freuency Curves at the Yoo River catchment, Japan by the scaling metho. Area=11.25Km 2 an b)area= km 2. The AIDF curves can be erive by the simple scaling metho. For example, with a fixe area eual to km 2 an a scaling factor H=-.51; μ = λ μ 24 = 25.8; σ = λ σ 24 = 8.22 are foun, the AIDF curves can be erive by: ln( ln(1 1/ T )) i = (27).514 For an area of km 2, the IDF curve can be constructe with a scaling factor H = -.46: ln( ln(1 1/ T )) i = (28).46 The Figure 5 shows a result of the AIDF base on the scaling metho in the Yoo River catchment by the euations (27) an (28). 5. CONCLUSION The major finings of the present stuy can be summarize as follows: The properties of the time an space scale invariance of rainfall uantiles are examine in the Yoo River basin: For time scaling, rainfall properties follows a simple scaling; For space scaling, a simple scaling process with the two ifferent regimes of 2.25 km 2 to 1 km 2 an 1 km 2 to 5 km 2 are foun. Then, accoring to Menabe et al. 4) the rainfall IDF curves for a short uration (hourly) were erive from 24-hour ata. The simple scaling properties are verifie by local ata; the IDF relationships are euce from aily rainfall, which shows goo results in comparison with the IDF curves obtaine from at-site short-uration rainfall ata. Results of this stuy are of significant practical importance because statistical rainfall inferences can be mae with the use of a simple scaling in time an space. Furthermore, aily ata are more wiely available from stanar rain gauge measurements, but ata for short urations are often not available for the reuire site. Further works will focus on a combination between the time scaling an the space scaling an theoretical escription of the AIDF applicable for a less gauge basin. ACKNOWLEDGMENT: This stuy was supporte by Grant-in-Ai for Scientific Research (C) (PI: Assoc. Prof. Y. Tachikawa, Kyoto Univ.). REFERENCES 1) Gupta, V. K. an Waymire, E.: Multiscaling properties of spatial rainfall an river flow istributions. Journal of Geophysical Research, 95(D3): , ) Kuzuha, Y., Komatsu, Y., Tomosugi, K., Kishii, T.: Regional Floo Freuency Analysis, Scaling an PUB, Journal Japan Soc. Hyrol. an Water Resources Vol.18,No.4, pp , 25. 3) Burlano, P. an Rosso, R.: Scaling an multiscaling moels of epth-uration-freuency curves for storm precipitation. Journal of Hyrology, 187, pp , ) Menabe, M., See, A., Pegram, G.: A simple scaling moel for extreme rainfall. Water Resources Research, Vol. 35, No.1, pp , ) Yu, P. S., Yang, T.C. an Lin, C. S.: Regional rainfall intensity formulas base on scaling property of rainfall Journal of Hyrology, 295 (1-4), pp , 24. 6) Chow, V.T., Maiment, D.R. & Mays, L.W: Applie Hyrology, McGraw-Hill, ) Chen, C. L.: Rainfall intensity uration freuency formulas, Journal of Hyraulic Engineering, ASCE, 19(12), pp , ) Hershfie, D. M.: Estimating the Probable Maximum Precipitation, Journal of the Hyraulic Division, Proceeing of the ASCE, HY5, pp , ) Van Nguyen, V. T., Nguyen, T.-D., Ashkar, F.: Regional freuency analysis of extreme rainfalls, Water Sci. Technol. 45(2), pp.75 81, 22. 1) Bell, F. C.: Generalize rainfall uration freuency relationships. Journal of Hyraulic Div., ASCE, 95(1), pp , ) Koutsoyiannis, D., Manetas, A.: A mathematical framework for stuying rainfall intensity uration freuency relationships, Journal of Hyrology, 26, pp , ) Kothyari, U. C. an Grae, R.J.: Rainfall intensity uration freuency formula for Inia, J. Hyr. Eng., ASCE, 118(2), pp , ) Nhat, L. M., Tachikawa, Y., Takara, K.: Establishment of Intensity-Duration-Freuency curves for precipitation in the monsoon area of Vietnam, Annuals of Disas. Prev. Res. Inst., Kyoto Univ., No. 49B, 26. (Receive September 3, 26)