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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
The Role of A Hyetograph
in Hydrologic Design
Rainfall frequency
analysis
Design storm
hyetograph
Total rainfall depth
Rainfall-runoff
modeling
Runoff hydrograph
Time distribution
of total rainfall
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
Characteristics of Storm
Hyetographs


Although the shapes of storm hyetographs
vary significantly, many studies have shown
that dimensionless hyetographs are stormtype 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
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
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
Objective of the Study

1.
2.
3.
4.
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
Selecting Storm Events for
Hyetograph Design
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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

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
Annual Maximum Events
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
Selecting Storm Events for
Hyetograph Design

1.
2.
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
Simple Scaling Model for Storm
Events – Instantaneous Rainfall
Let  (t , D) represent the instantaneous rainfall
intensity at time t of a storm with duration D.
d
{ (t , D)}{ H  (t , D)},
 0
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
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
Simple Scaling Model for Storm
Events – Incremental &
Cumulative Rainfall

X  (i, D), 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
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
IDF Curves and Random
Variables


h( D, D) is a random variable and represents
the total depth of a storm with duration D.
hT ( D, 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
Random Variable Interpretation
of IDF Curves
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
IDF Curves and the Scaling
Property

Horner’s Equation:
aT m
iT ( D) 
( D  b)c

D >> b , particularly for long-duration events.
Neglecting b
iT ( D)  ciT (D)

C=-H
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
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
Theoretical Basis for Using
Dimensionless Hyetographs
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
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
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
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
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
SIGM Model System
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
Model Applications
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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
Parameters for Distributions of
Normalized Rainfalls.
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
Evidence of Nonstationarity

In general,
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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
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
Calculation of Autocorrelation
Coefficients of a Nonstationary
Process

The lag-k correlation coefficients  k (i)= 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
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
Significance Test for Lag-1 and
Lag-2 Autocorrelation Coefficients


If  k (i) = 0, then t  rk (i )
N 2
1  rk2 (i )
has a t-distribution with (N-2) degree of freedom.
 k (i) = autocorrel (Y(i), Y(i-k))
At significance level , the null hypothesis
H 0 : k (i)  0 is rejected if t  t1 / 2, N 2 .
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
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
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
SIGM Hyetograph - Hosoliau
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
SIGM Hyetograph-Wutuh
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
Other Hyetograph Models
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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
Hyetographs of TPC’s
Clustering Approach
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
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
Model Evaluation
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
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
Event Validation: 54-08-18
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
Event Validation: 63-09-15
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
Event Validation: 69-08-27
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
Event Validation: 77-09-16
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
Comparison of Peak Rainfall Rates
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
Comparison of Peak Flows
Laboratory for Remote Sensing Hydrology and Spatial Modeling
Department of Bioenvironmental Systems Engineering, National Taiwan University
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
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, scaleinvariant 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