Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University A Scale-Invariant

Laboratory for Remote Sensing Hydrology and Spatial ModelingDepartment of Bioenvironmental Systems Engineering, National Taiwan University

A Scale-Invariant Hyetograph Model for Stormwater Drainage Design

Ke-Sheng Cheng and En-Ching Hsu

Department of Bioenvironmental Systems Engineering

National Taiwan University


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


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.


Representation of a Storm Hyetograph

Rainfall depth process Dimensionless hyetograph


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) pr

esented evidence that dimensionless hyetographs are scale invariant.


Objective of the Study

The goal of this study is to propose a hyetograph modeling approach that have the following properties:

1. Representative of the dominant storm type (storm-type-specific);

2. Allowing translation between storms of different durations (scale-invariant);

3. Characterizing the random nature of rainfall processes;

4. Having the maximum likelihood of occurrence.


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.


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.


Annual Maximum Events


Selecting Storm Events for Hyetograph Design

Using only the annual maximum events has two advantages:

1. to focus on events of the same dominant storm type,

2. to develop the design storm hyetographs using largely the same annual maximum events that are employed in constructing the IDF curves.


Simple Scaling Model for Storm Events – Instantaneous Rainfall

Let represent the instantaneous rainfall intensity at time t of a storm with duration D.

),( Dt

0)},,({)},({ DtDt Hd


Simple Scaling Model for Storm Events – Incremental & Cumulative Rainfall Incremental rainfall

Cumulative rainfall

Total rainfall


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.,

),( DiX


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


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.,

),( DDh

),( DDhT


Random Variable Interpretation of IDF Curves


IDF Curves and the Scaling Property

Horner’s Equation:

D >> b , particularly for long-duration events. Neglecting b

C = - H

c

m

T bD

aTDi

)()(

)()( DiDi Tc

T


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.


Theoretical Basis for Using Dimensionless Hyetographs


Gauss-Markov Model of Dimensionless Hyetographs

Assume that the process {Y(i): i = 1, 2, …, n} is a Gauss-Markov process.


Gauss-Markov Model of Dimensionless Hyetographs

By the Markov property,


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


Lagrange Multiplier Technique

The objectives can be achieved by introducing two Lagrange multipliers and m , and minimizing the following expression:


SIGM Model System


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).


Parameters for Distributions of Normalized Rainfalls.


Evidence of Nonstationarity In general,

Autocovariance function of a stationary process:

For a non-stationary process, the autocovariance function is NOT independent of t.



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

)(ik


Normality Check for Normalized Rainfalls by Kolmogorov-Smirnov Test


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 .

)(ik)(1

2)(

2 ir

Nirt

kk

)(ik

0)(:0 iH k 2,2/1 Ntt






SIGM Hyetograph - Hosoliau


SIGM Hyetograph-Wutuh


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)


Hyetographs of TPC’s Clustering Approach


Hyetographs of TPC’s Clustering Approach

Average hyetograph of the three major clusters (94% of total events).


Model Evaluation


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


Event Validation: 54-08-18








Comparison of Peak Rainfall Rates


Comparison of Peak Flows


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.


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.

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Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University A Scale-Invariant