Storm rainfall. Lecture content. 1 Analysis of storm rainfall 2 Predictive model of storm rainfall for a given

Transcription

1 Storm rainfall Lecture content 1 Analysis of storm rainfall 2 Predictive model of storm rainfall for a given max rainfall depth 1 rainfall duration and return period à Depth-Duration-Frequency curves 2 3 Storm rainfall distribution in time à design hyetograph 4 Areal variation of storm rainfall à areal reduction factor Skript: - Ch. IV, Ch. VII, 1 to 4 (all sections) Areal Reduction Factor [%] 30 1 hour 4 Area [km 2 ] 3 1

2 Importance of storm rainfall analysis and modelling Storm rainfall is the trigger of many phenomena leading to important water resources problems and natural hazards - floods - urban drainage - soil erosion - landslides Mathematical models of storm rainfall are the design input for a number of water infrastructures and the prediction of natural hazard occurrences - natural hazards à risk analysis à protection measures and risk mitigation 2

3 Analysis of storm rainfall 3

4 Objective of the analysis to extract from rainfall data (time series), the extreme values extreme values for different durations h 9.2 mm 12.5 mm 21.3 mm 36.4 mm to build a mathematical model that computes the rainfall depth, H*, for a given storm duration, T*, and a given return period, R à Depth-Duration-Frequency curve RAINFALL DEPTH 2 R = 10 years STORM DURATION 4

5 Extraction of extreme values from raw rainfall data to time series of rainfall intensity rainfall intensity, X(t) T Z t t+t 2 ( ) = X τ t T 2 ( )dτ { } for different T H = max Z ( t), L = 1 year time, t to annual maxima of rainfall depth, H, for a given duration T H i,j = H 1990, T=20 durations, j year, i h mm 12.5 mm 21.3 mm 36.4 mm 5

6 Estimation of DDFs 6

7 Empirical Depth-Duration-Frequency curves (1) Sample data table of the annual maxima of storm rainfall for different durations ê Ordered data table of the annual maxima of storm rainfall for different durations 7

8 Empirical Depth-Duration-Frequency curves (1) Compute the empirical non-exceedance frequency, F, using a convenient plotting position, e.g. Gringorten F( j) = j 0.44 N and the corresponding return period, R, as R( j) = 1 1 F j ( ) j F(j) R(j) 8

9 Empirical Depth-Duration-Frequency curves (2) The annual maxima for a given R exhibit a power law behaviour Interpolation through a power law 100" 90" 80" H T ( R) = a( R)T n( R) F(j=19) à R 34 years F(j=18) à R 12 years à empirical Depth Duration Frequency (DDF) curve Rainfall'Depth'[mm]' 70" 60" 50" 40" 30" 20" 10" 0" Interpola5ng"power"law" Observa5ons,"2nd"lowest"frequency" Observa5ons,"lowest"frequency" Interpola5ng"power"law" 0" 500" 1000" 1500" 2000" Dura2on'[minutes]' The interpolating power law allows to estimate the rainfall depth for any given duration and for the empirical frequency for which the DDF is estimated For higher n.exc.. frequencies (=higher return periods) it is necessary to carry out a PROBABILISTIC ANALYSIS à DDF estimation by means of the QUANTILE REGRESSION METHOD 9

10 DDF through quantile regression method - summary Step Purpose / Actions Outcome 1 Statistical analysis of annual maxima of storm rainfall To compute basic statistics relative frequency function cumulative frequency function Statistical characterisation of the sample data 2 Select an extreme value probability distribution, F X (x), to describe the sample data across all the durations 3 Compute the rainfall quantiles from the selected distribution for the selected durations and return periods 4 Estimate the DDF from the quantiles computed in 3 To fit F X (x) to the data statistical tests probability paper plot parameter estimation To extend the basis of values on which to estimate the DDF To calibrate a(r) and n(r) by means of regression of the values log(h R,T ) computed in 3 against the values of log(t) Fitted F X (x) for each duration H T H R,T computed from P X (x) for different R and T H R,T Calibrated DDF H T F R ( R) = a( R)T n( R) 10

11 DDF through quantile regression method Selection of the probability distribution (step 2) - Extreme Value Type I (EV I, Gumbel distribution) F X ( x) = exp exp x u α u, α parameters - the selected distribution holds for all the durations DDF equation - the DDF follows a power law H T ( R) = a( R)T n R ( ) à log H T R - the parameters a and n depend on the return period R ( ) = loga( R) + n R ( )logt log H T log T 11

12 DDF through quantile regression method Step 1 Estimation of the statistics of the sample data ( à sample moments) - mean - variance N ˆm T = 1 N ŝ 2 T = 1 N 1 i=1 H T,i N i=1 ( H T,i ˆm T ) 2 - standard deviation ŝ T = ŝ T 2 Estimation of the parameters of the EV I distribution - method of moments α T = α T u T = u T ( ŝ T ) ( ) ŝ T, ˆm T à α T = 6 π ŝt u T = ˆm T α T - the estimation is carried out for each duration T 12

13 ! DDF through quantile regression method Step 3 Compute the rainfall quantiles for each duration and for selected return periods F HT ( ) = exp exp h u T h T è by defining the Gumbel reduced variate α h T ( F ) = u α ln ln 1 T T F é F = R 1 R R y R = ln ln R 1 è h T R ( ) = u T α T ln ln R R 1 observed extremes one can write h T ( R) = u T α T y R 13

14 DDF through quantile regression method Step 4 Estimate the DDF from the quantiles computed by means of the EV I distribution R = 50 years h 1 ( R) = u 1 α 1 ln ln h 12 ( R) = u 12 α 12 ln ln H T ( R) = a( R)T n( R) ê log H T ( R) = loga( R) + n( R)logT ê for T=1 è log (T)=0 è log H T ( R) = loga( R) n corresponds to the slope of the regression line 14

15 DDF through quantile regression method Numerical example Station Davos ( ) 1 Statistical moments estimation 2 EV I parameter estimation (method of moments) 1h 3h 6h 12h 24h Statistical moments mean [mm] st. dev. [mm] ˆm T = 1 N ŝ T = N i=1 1 N 1 H T,i N i=1 ( H T,i ˆm T ) 2 EV I parameters α u α T = 6 π ŝt u T = ˆm T α T 15

16 DDF through quantile regression method Numerical example Station Davos ( ) 3Computation of EV I rainfall quantiles h T ( R) = u T α T ln ln R R 1 Rainfall depth from EV I [mm] R = 2.33 years R = 50 years 1h 3h 6h 12h 24h e.g. h 1 (50) à ( ) = u 1 α 1 ln ln h = ln ln = 29.6 mm 16

17 DDF through quantile regression method Numerical example Station Davos ( ) 4Estimation of the DDF for R=50 years H T ( R) = a( R)T n R ( ) log H T R ( ) = loga( R) + n R ( )logt è è a(50) = mm h-n n(50) = h 1 ( 50) = 29.6 mm ln( 29.6) = 3.39 T = 1: ln H T ( 50) = ln( 28.47) + n( R)ln( 1) =

18 DDFs vs IDFs (Intensity Duration-Frequency curves) The DDF concept and model can be also applied to rainfall intensities ê IDFs, Intensity-Duration-Frequency curves I T [mm/h] H T ( R) = a( R)T n R ( ) è I R T é I = H T ( ) = a R ( )T n R ( ) 1 I 1 I 2 I 3 Return Period, R t 1 t 2 t 3 t T t 18

19 DDF Hydrological Atlas of Switzerland method Template to estimate the rainfall depth from 1 h and 24 h DDF curves Concept: Read parameters A, B, C and D on maps - T=1h, R=2.33 years à C - T=1h, R=100 years à A - T=24h, R=2.33 years à D - T=24h, R=100 years à B EV I 24 h Compute DDF for desired R and T using the equations H 1,R = 1.14 C 0.14 A + 1h A C 1 ln ln R H 24,R = 1.14 D 0.14 B + B D 1 ln ln R For durations between 1 and 24 hrs interpolate linearly N.B. The diagram depends on the valid distribution in the region 19

20 DDF Hydrological Atlas of Switzerland method Regions of validity of EV I and EV II distributions 20

21 DDF Hydrological Atlas of Switzerland method Regionalisation of rainfall depth for T=1 h and R=2.33 years map for estimation of C 21

22 DDF Hydrological Atlas of Switzerland method Regionalisation of rainfall depth for T=1 h and R=100 years map for estimation of A 22

23 DDF Hydrological Atlas of Switzerland method Regionalisation of rainfall depth for T=24 h and R=2.33 years map for estimation of D 23

24 DDF Hydrological Atlas of Switzerland method Regionalisation of rainfall depth for T=24 h and R=100 years map for estimation of B 24

25 Design hyetograph 25

26 From DDFs to design storm hyetograph DDFs provide a rainfall depth, H, for a given duration, T, and given return period R H T (R) They do not provide the distribution of the storm event through time à no hyetograph Actual rainfall events exhibit temporal variability A realistic rainfall input for any design procedure (e.g. storm drainage, flood estimation, etc.) requires to transform H T (R) into a hyetograph that mimics actual rainfall events ê Design hyetograph à synthetic hyetograph Most of the synthetic hyetographs are empirical 26

27 US Soil Conservation Service (SCS) synthetic hyetograph The SCS hyetograph defines four types of storms based on a 24 hour storm and associates to each of them a different rainfall distribution The concept is based on the definition of the storm advancement ratio, i.e. the proportion of rainfall amount progressively falling within the 24 h event Fraction of 24 h rainfall The concept can be easily extended to any storm duration (24 h = 100% storm duration) [Chow et al., 1988] time [hours] [Chow et al., 1988] 27!

28 Huff s synthetic hyetograph The concept is based on time distribution patterns developed for four probability groups most severe (1 st quartile) à least severe (4 th quartile) Storms within each quartile are classified according to their probability of occurrence in 9 classes from 10% to 90% 10% probability à equaled or exceeded in 90% of the observed storms Cumulative percent of rainfall Cumulative percent of storm time [Chow et al., 1998] Percent total storm rainfall 50% probability à equaled or exceeded in about half of the observed storms 90% probability à equaled or exceeded in 10% of the observed storms Cumulative percent of storm time 28

29 Triangular synthetic hyetograph The concept is based on the assumption of a trinagular shape for the hyetograph The shape of the triangle is dictated by the storm advancement coefficient r = t a /T d, being T d the storm duration and t a the time to peak intensity The peak intensity is given as i d = 2P/T d where P is the rainfall depth estimated from the DDF curve The recession time is: t b = T d - t a = (1-r) T d Typical values of r: [Chow et al., 1998] 29

30 Synthetic hyetograph based on the Instantaneous Intensity Method Based on the storm advancement coefficient concept r = t a /T d Two curves are defined to describe the intensity i a =f(t a ) for the time before the peak i b =f(t b ) for the time after the peak The total amount of rainfall estimated from the DDF, P, is rt d ( 1 rt d ) P = i T d = f ( t a )dt a + f ( t b )dt b where i 0 is the average rainfall intensity for T d 0 i a = f ( t a ) i b = f ( t b ) Symmetry of f(t a ) and f(t b ) is assumed à dp = f ( t a ) = f ( t b ) Problem with IDF expression dt d for t à t a I T à solution: solve in discrete form for a fixed Δt I T ( R) = a( R)T n( R) 1 30

31 Alternating Block synthetic hyetograph The synthetic hyetograph is developed from the IDF curve I T [mm/h] The total amount of precipitation P = i T d T d = n Δt Construction: occurs in the intensity is read from the IDF curve for each duration, i.e. Δt à I 1, 2Δt à I 2, 3Δt à I 3, I 1 I 2 I3 Return Period, R the rainfall depth for the first Δt is given by P 1 = I 1 Δt Δt 2Δt 3Δt T d t the subsequent rainfall depths for each Δt are obtained taking the difference between successive rainfall depth values, i.e. P 2 = I 2 2Δt P 1 P 3 = I 3 3Δt (P 1 +P 2 ) The blocks P 1, P 2, P 3, are reordered with the max intensity at the center of the hyetograph and the remaining blocks alternating to the right and left I T ( R) = a( R)T n( R) 1 31

32 Areal variation of storm rainfall 32

33 Spatial variability of storm rainfall Storm rainfall has been recognised since long time to be variable in space Intensity declines from the storm centre towards the periphery of the storm Modern radar based observations confirm this Neglecting the intensity reduction in design problems can lead to overestimation of the rainfall input ê There is need to identify a technique to extend the design storm rainfall from the point (station) to a given area ê Areal Reduction Factor (ARF) PA = r ( A,Td, R ) P0 PA areal storm rainfall P0 point storm rainfall ARF = r = A area, Td storm duration, R return period 33

34 Spatial variability of storm rainfall The difference between point and areal storm rainfall increases for increasing area for low values of precipitation for increasing convective character decreases for increasing storm duration Most of the literature methods consider ARF independent from R à r = r(a, T d ) Most of the ARF formulations are of empirical nature 34

35 ARF general equation Point DDF Areal DDF H T = a T n H T, A = a T n where ( ) a = a 1+ a 1 A b 1 + a 2A b 2 n = n + c 1 A d 1 and a 1, b 1, a 2, b 2, c 1, d 1 are empirical parameters to be estimated from data The resulting ARF is: r = H T, A = ( 1+ a 1 A b 1 H + a 2 Ab 2 )T c 1A d 1 T 35

36 ARF template of US Weather Bureau (1958) r = H T, A H T = ( 1 e 1.1T e 1.1T 0.25 ) 0.01A Area [mi 2 ] Percent of point rainfall for given area h 1 h 3 h 6 h Area [km 2 ] 36

37 ARF template of Swiss Hydrological Atlas (1) 37

38 ARF template of Swiss Hydrological Atlas (2) 38

39 ARF template of Swiss Hydrological Atlas (3) 39

40 Storm rainfall example of application of knowledge Engineering Problem: Design of drainage systems (e.g. roof, parking place,...) Solution Compute the rainfall volume for an accepted level of risk (return period) Method Depth-Duration-Frequency curves 2 40

41 Storm rainfall example of application of knowledge Engineering Problem: Assessment of the risk of collapse of a hillslope Solution For a known critical rainfall amount i.e. landslide triggering rainfall depth computed from a landslide model compute the return period of the mobilizing rainfall Method 2 Depth-Duration-Frequency curves 41