1. Stochastic Hydrology Simple scaling in temporal variation of rainfalls
Professor Ke-sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University

2. The random sample often is an annual maximum series of total rainfall depth with respect to a pre-specified duration tr.
The annual maximum series is obtained by continuously moving a window of size tr along the time axis and selecting the maximum total values within the window in each year.
The pre-specified durations, also known as the design durations, are artificially designated durations which are used to determine the corresponding annual maximum depths.
By contrast, event durations are actual raining periods of time of real storm events.
7/12/2019
Dept. of Bioenvironmental Systems Engineering, National Taiwan University

3. Event duration vs design duration
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

4. Event duration vs design duration
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

5. Event duration vs design duration
The 24-hr annual max rainfall is contributed by two separated storms.
7/12/2019
Dept. of Bioenvironmental Systems Engineering, National Taiwan University

6. It has been found (Cheng et al
It has been found (Cheng et al., 2001) that annual maximum events tend to occur in certain periods of the year (such as a few months or a season) 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, a single annual maximum event even produced annual maximum rainfalls for many nearby raingauge stations.
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

7. Actual occurrence dates of annual maximum events
* Design durations
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

8. The usage of annual maximum series for rainfall frequency analysis is more of an intelligent and convenient engineering practice and the annual maximum data do not provide much information about the characteristics of the duration and total depth of real storm events.
7/12/2019
Dept. of Bioenvironmental Systems Engineering, National Taiwan University

9. DDF curves under scaling property
Development of scaling , or scale-invariant, models enable us to transform data from one temporal or spatial model to another and help to overcome the problem of inadequacy of the available data.
The simple scaling property of a natural process implies that the statistical distribution of some physical measurement at one scale is identical to the distribution at another scale multiplied by a factor that is a power function of the ratio of the two scales.
7/12/2019
Dept. of Bioenvironmental Systems Engineering, National Taiwan University

10. The simple scaling property
Let X(t) and X(t) be random variables of physical measurements made at two different time or spatial scales t and t, respectively. If the following relation holds
for all ratios  > 0, we say that X(t) has the simple scaling property. The denotes identity in statistical distributions and n is the scaling exponent.
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

11. Scaling (Scale-Invariant)
Strict Sense Simple Scaling
Wide Sense Simple Scaling (Second- Order Simple Scaling)
Multiple Scaling
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

12. Let represent the instantaneous rainfall intensity at time t of a storm with duration D.
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

13. Simple Scaling Model for Storm Events – Incremental & Cumulative Rainfall
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

14. Simple Scaling Model for Storm Events – Incremental & Cumulative Rainfall
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

15. 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
7/12/2019
Dept. of Bioenvironmental Systems Engineering, National Taiwan University

16. IDF Curves and the Scaling Property
Horner’s Equation:
D >> b , particularly for long-duration events.
Neglecting b
C = - H
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Dept. of Bioenvironmental Systems Engineering, National Taiwan University

17. Theoretical Basis for Usage of Dimensionless Hyetographs
in view of the simple scaling characteristics, the normalized rainfall rates of storms of different event durations are identically distributed.
7/12/2019
Dept. of Bioenvironmental Systems Engineering, National Taiwan University

18. Theoretical Basis for Usage of Dimensionless Hyetographs
The simple scaling property of rainfalls leads to a scale-invariant dimensionless hyetograph.
7/12/2019
Dept. of Bioenvironmental Systems Engineering, National Taiwan University