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

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

Event duration vs design duration 7/12/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University

Event duration vs design duration 7/12/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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

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

Actual occurrence dates of annual maximum events * Design durations 7/12/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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

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

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

Scaling (Scale-Invariant) Strict Sense Simple Scaling Wide Sense Simple Scaling (Second- Order Simple Scaling) Multiple Scaling 7/12/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University

Let represent the instantaneous rainfall intensity at time t of a storm with duration D. 7/12/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University

Simple Scaling Model for Storm Events – Incremental & Cumulative Rainfall 7/12/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University

Simple Scaling Model for Storm Events – Incremental & Cumulative Rainfall 7/12/2019 Dept. 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 7/12/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University

IDF Curves and the Scaling Property Horner’s Equation: D >> b , particularly for long-duration events. Neglecting b C = - H 7/12/2019 Dept. of Bioenvironmental Systems Engineering, National Taiwan University

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

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