Hydrological Frequency Analysis Extracting Data for Frequency Analysis

Hydrological Frequency Analysis Extracting Data for Frequency Analysis Professor Ke-sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

Data Series for Hydrological Frequency Analysis • Complete duration series • Partial duration series (Peak-over-threshold, POT, series) • Annual exceedance series • Extreme value series • annual maximum series • annual minimum series • Block max/min series 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 2

Annual maximum series (Block maximum series) Threshold of annual exceedance series 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 3

• The concept of durational events • • • Hydrological events such as storms and floods are durational events. Event durations vary from one event to another. Durational events do not occur on regular time intervals. There may be different types of events. There are no clear criteria for partitioning of individual events. Hydrologists often use the minimum interarrival time (MIT) (or, minimum inter-event time) for event partitioning. • The above data series do not consider the event duration and event occurrences. 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 4

Flood Flow Data • Xia-Yun daily flow • Easy to determine the annual maximum value. 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 5

Extreme Rainfalls – Annual Maximum Series • Hourly rainfall data • Annual maximum rainfalls of various design durations are desired. • A moving window is used to select the annual maximum rainfall. 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 6

24 -hr Annual Maximum Rainfall 1969, Jia-Yi Station (164. 3 mm) Meiyu Rainfalls 7/29/2019 Laboratory for Remote Sensing Hydrology and Spatial Modeling Dept. of Bioenvironmental Systems Engineering, NTU 7

Extreme Rainfalls – POT Series • 3 -day POT rainfall extremes How can we select POT data of 3 -day rainfall extremes? • Size of the rolling window • Minimum interval between peaks R 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 8

Volpi, et al. , 2019. Save hydrological observations! Return period estimation without data decimation. 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 9

Modeling Occurrences of Extremes • The number of POT extreme values in a period of time can be modeled by a Poisson random process. • The interarrival time of a purely random process has an exponential distribution. 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 10

Poisson Distribution • The Poisson distribution provides a realistic model for many random phenomena for which the number of event occurrences within a given scope (time, length, area, volume) is of interest. For example, the number of fatal traffic accidents per day in Taipei, the number of meteorites that collide with a satellite during a single orbit, the number of defects per unit of some material, the number of flaws per unit length of some wire, etc. 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 11

• Probability mass function of the Poisson distribution 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 12

Exponential distribution (negative exponential distribution) • Probability density function of the exponential distribution 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 13

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Block Maximum Data Series 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 15

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A Further Look Into the Extremal Type Theorem • Block maximum data series • Maximum values of blocks of the same size. • Data in the block are IID. • The speed of convergence of Mn to the limiting GEV distribution depends on the underlying distribution F - for example, convergence is slow for maxima of n Gaussian variables (Coles and Davison, 2008). 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 18

The Concept of Durational Events • Based on the Extremal Type Theorem, the limiting distribution of the annual maximum rainfalls is a GEV distribution. • Is it justified? • The annual maximum rainfall in a year is the result of • • Storm occurrences (number of storms in a year) Storm duration Total amount and temporal variation Dominant storm types • Partitioning of durational events from historical data series • Interarrival time vs interevent time 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 19

General concept of hydrological frequency analysis • Hydrological frequency analysis is the work of determining the magnitude of hydrological variables that corresponds to a given exceedance probability. Frequency analysis can be conducted for many hydrological variables including floods, rainfalls, and droughts. • The work can be better understood by treating the interested variable as a random variable. 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 20

• Let X represent the hydrological (random) variable under investigation. A value xc is chosen such that an event is said to occur if X assumes a value exceeding xc. Every time when a random experiment (or a trial) is conducted the event may or may not occur. • We are interested in the number of Bernoulli trials in which the first success occur. This can be described by the geometric distribution. 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 21

Geometric distribution • Geometric distribution represents the probability of obtaining the first success in x independent and identical Bernoulli trials. 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 22

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Average number of trials to achieve the first success. Recurrence interval vs return period 10/18/2021 Dept. of Bioenvironmental Systems Engineering, National Taiwan University 24

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