1. A Scale-Invariant Hyetograph Model for Stormwater Drainage Design
Ke-Sheng Cheng and En-Ching Hsu
Department of Bioenvironmental Systems Engineering
National Taiwan University
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

2. The Role of A Hyetograph in Hydrologic Design
Rainfall frequency analysis
Design storm hyetograph
Rainfall-runoff modeling
Total rainfall depth
Time distribution of total rainfall
Runoff hydrograph
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

3. Characteristics of Storm Hyetographs
Although the shapes of storm hyetographs vary significantly, many studies have shown that dimensionless hyetographs are storm-type specific (Huff, 1967; Eagleson, 1970).
In general, convective and frontal-type storms tend to have their peak rainfall rates near the beginning of the rainfall processes, while cyclonic events have the peak rainfall somewhere in the central third of the storm duration.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

4. Representation of a Storm Hyetograph
Rainfall depth process Dimensionless hyetograph
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

5. Design Storm Hyetograph Models
Duration-specific hyetograph models
Keifer and Chu, 1957;
Pilgrim and Cordery, 1975;
Yen and Chow, 1980;
SCS, 1986.
Koutsoyiannis and Foufoula-Georgiou (1993) presented evidence that dimensionless hyetographs are scale invariant.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

6. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Objective of the Study
The goal of this study is to propose a hyetograph modeling approach that have the following properties:
Representative of the dominant storm type (storm-type-specific);
Allowing translation between storms of different durations (scale-invariant);
Characterizing the random nature of rainfall processes;
Having the maximum likelihood of occurrence.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

7. Selecting Storm Events for Hyetograph Design
If rainfall data of both types were simultaneously utilized in order to develop design storm hyetographs, quite likely an average hyetograph results which characterizes the temporal rainfall variation of neither storm type.
Selecting the real storm events that gave rise to the annual maximum rainfalls, the so-called annual maximum events, to develop design hyetographs.
Annual maximum events tend to occur in certain periods of the year and tend to emerge from the same storm type.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

8. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Annual maximum rainfall data in Taiwan strongly indicate that a single annual maximum event often is responsible for the annual maximum rainfall depths of different design durations. In some situations, single annual maximum event even produced annual maximum rainfalls for many nearby raingauge stations.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

9. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Annual Maximum Events
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

10. Selecting Storm Events for Hyetograph Design
Using only the annual maximum events has two advantages:
to focus on events of the same dominant storm type,
to develop the design storm hyetographs using largely the same annual maximum events that are employed in constructing the IDF curves.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

11. Simple Scaling Model for Storm Events – Instantaneous Rainfall
Let represent the instantaneous rainfall intensity at time t of a storm with duration D.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

12. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Simple Scaling Model for Storm Events – Incremental & Cumulative Rainfall
Incremental rainfall
Cumulative rainfall
Total rainfall
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

13. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Simple Scaling Model for Storm Events – Incremental & Cumulative Rainfall
, h(t,D), and h(D,D) all have the simple scaling property with scaling exponent H+1, i.e.,
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

14. IDF Curves and the Scaling Property
The event-average rainfall intensity of a design storm with duration D and recurrence interval T can be represented by
From the scaling property of total rainfall
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

15. IDF Curves and Random Variables
is a random variable and represents the total depth of a storm with duration D.
is the (1-p)th quantile (p =1/T) of the random variable, i.e.,
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

16. Random Variable Interpretation of IDF Curves
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

17. IDF Curves and the Scaling Property
Horner’s Equation:
D >> b , particularly for long-duration events.
Neglecting b
C = - H
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

18. Theoretical Basis for Using Dimensionless Hyetographs
in view of the simple scaling characteristics, the normalized rainfall rates of storms of different event durations are identically distributed.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

19. Theoretical Basis for Using Dimensionless Hyetographs
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

20. Gauss-Markov Model of Dimensionless Hyetographs
Assume that the process {Y(i): i = 1, 2, …, n} is a Gauss-Markov process.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

21. Gauss-Markov Model of Dimensionless Hyetographs
By the Markov property,
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

22. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Modeling Objectives
An ideal hyetograph should not only access the random nature of the rainfall process but also the extreme characteristics of the peak rainfall.
Our objective is to find the hyetograph {yi , i = 1,2,…, n} that
Maximize lnL, and
y*: peak rainfall rate, t*: time-to-peak
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

23. Lagrange Multiplier Technique
The objectives can be achieved by introducing two Lagrange multipliers  and m , and minimizing the following expression:
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

24. Laboratory for Remote Sensing Hydrology and Spatial Modeling
SIGM Model System
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

25. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Model Applications
Two raingauge stations in Northern Taiwan.
Annual maximum events that produced annual maximum rainfall depths of 6-, 12-, 18-, 24-, 48-, and 72-hr design durations were collected.
All event durations were divided into twenty-four equal periods i (i=1,2,…,24, D = event duration, =D/24).
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

26. Parameters for Distributions of Normalized Rainfalls.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

27. Evidence of Nonstationarity
In general,
Autocovariance function of a stationary process:
For a non-stationary process, the autocovariance function is NOT independent of t.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

28. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

29. Calculation of Autocorrelation Coefficients of a Nonstationary Process
The lag-k correlation coefficients = correl.(Y(i), Y(i-k)) of the normalized rainfalls were estimated by
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

30. Normality Check for Normalized Rainfalls by Kolmogorov-Smirnov Test
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

31. Significance Test for Lag-1 and Lag-2 Autocorrelation Coefficients
If = 0, then
has a t-distribution with (N-2) degree of freedom.
= autocorrel (Y(i), Y(i-k))
At significance level , the null hypothesis
is rejected if
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

32. Significance Test for Lag-1 and Lag-2 Autocorrelation Coefficients
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

33. Significance Test for Lag-1 and Lag-2 Autocorrelation Coefficients
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

34. SIGM Hyetograph - Hosoliau
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

35. SIGM Hyetograph-Wutuh
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

36. Other Hyetograph Models
Average Rank Model (Pilgrim and Cordery, 1975)
Triangular Hyetographs (Yen and Chow, 1980)
Alternating Block Approach (IDF-Based)
Peak-Aligned Approach (Yeh and Han,1990)
Clustering Approach (TPC)
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

37. Hyetographs of TPC’s Clustering Approach
Three major clusters account for 94% of the total events.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

38. Hyetographs of TPC’s Clustering Approach
Average hyetograph of the three major clusters (94% of total events).
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

39. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Model Evaluation
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

40. Laboratory for Remote Sensing Hydrology and Spatial Modeling
SIGM Model Validation
Over thirty years of annual maximum events were utilized.
Results were also compared against other hyetograph models.
Validation parameters
Peak rainfall rates
Peak flow rates
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

41. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Event Validation:
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

42. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Event Validation:
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

43. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Event Validation:
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

44. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Event Validation:
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

45. Comparison of Peak Rainfall Rates
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

46. Comparison of Peak Flows
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

47. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Conclusions
The SIGM hyetograph is the most suitable hyetograph model for the study area.
Overall, it yields lowest RMSE of peak rainfall and peak flow estimates and its performance is more consistent across all gauges than other models.
Although development of the alternating block and average rank models are computationally easier than the SIGM model; application of these two models may encounter difficulties.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University

48. Laboratory for Remote Sensing Hydrology and Spatial Modeling
Conclusions
The average rank model is duration-specific and requires rainfall data of real storms; however, gathering enough storms of the same duration may not always be possible.
Although it dose not require rainfall data of real storms, the alternating block model is dependent on both duration and return period, and many hyetographs may need to be developed for various design storms.
The SIGM hyetograph is storm-type-specific, scale-invariant and a unique hyetograph can be easily applied to design storms of various durations.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University