**Professor Ke-Sheng Cheng (Email: rslab@ntu.edu.tw)**

RSLAB_BSE_NTU

No. 1, Section 4, Roosevelt Road

Bioenvironmental Syst. Eng., National Taiwan University

- To introduce fundamental concept of random variables, random vector and random function, and their applications in hydrology.
- To demonstrate the stochastic nature of many hydrological variables and processes and how hydrological parameters can be estimated by statistical methods.
- To discuss the uncertainties involved in parameter estimation and introduce the techniques of stochastic simulation for quantification of uncertainties.
- To demonstrate how hydrological processes can be characterized using stochastic models.

**Datasets**(1) Extreme rainfall data

Event-max rainfall data in Taiwan

24-hour annual-max rainfall data in Taiwan

(2) Streamflow data

**40-year (1964 - 2003) flow data at the Xia-Yun station (Xia_Yuan_DailyFlow.csv)**(3) Reservoir inflow data

(4) Hourly rainfalls of storm events (

**hourly_rainfall_depths.xlsx**) (Hourly typhoon rainfall data at Bamboo Lake,**BambooLake_Typhoon_Rainfall.xslx**)(5) Daily rainfall data

**1. Stochastic simulation of univariate random variables**Random Number Generation in R

Pseudo Random Number Generator (PRNG)

Probability Integral Transformation

Acceptance/Rejection Method

Frequency-factor-based Method

**2. Hydrological frequency analysis - 1****PPT - Fundamentals of Hydrological Frequency Analysis**General concept

General equation for frequency analysis

Data series for frequency analysis

Parameter estimation

Techniques for goodness-of-fit test

Selection of best-fit distribution

IDF curve fitting

### 3. Hydrological frequency analysis - 2

Goodness-of-fit test using moment ratios diagram

L-moments and L-moment ratios diagram (LMRD)

Establishing acceptance region for L-moment ratios

- Normal distribution
- Gumbel distribution
- Pearson Type III distribution

**References**- Wu, Y.C., Liou, J.J., Cheng, K.S., 2012. Establishing acceptance regions for
*L*-moments based goodness-of-fit tests for the Pearson type III distribution.*Stochastic Environmental Research and Risk Assessment*, 26: 873-885, DOI 10.1007/s00477-011-0519-z. - Liou, J.J., Wu, Y.C., Cheng, K.S., 2008. Establishing acceptance regions for L-moments-based goodness-of-fit test by stochastic simulation.
*Journal of Hydrology*, Vol. 355, No.1-4, 49-62. (doi:10.1016/j.jhydrol.2008.02.023).

**Working problems WP-3 [Solutions of Problems 1 and 2]**R code for LMRD-GOF plotting (LMRD-GOF_Plotting.R)

**Comparison of power of different GOF test methods (GOF_Power_Comparison.xlsx)**### 3A. Event-based rainfall frequency analysis (PPT)

Probability distribution of number of storm occurrences

- Typhoons [Poisson]
- Meiyu events [Over dispersed]
- Convective storm events
- Winter frontal events

**4. Hydrological frequency analysis - 3 (Simple scaling)****PPT - Simple Scaling in Rainfalls**Annual maximum events

Simple scaling and multiple scaling

IDF Curves and the Scaling Property

Theoretical Basis for Usage of Dimensionless Hyetographs

Simple scaling DDF (skipped)Multiple scaling DDF (skipped)

**5. Hydrological frequency analysis - 4 (Regional Frequency Analysis, RFA)**Fundamental concept of regional frequency analysis

The index-flood approach

General procedures of regional frequency analysis

Situations for application of RFA

Regional frequency analysis with presence of extraordinary rainfalls**6. Design storm hyetograph****PPT**-**SSGM design storm hyetograph**Alternating block hyetograph

Average rank hyetograph

Simple scaling Gauss-Markov hyetograph (SSGM)

### 7. Stochastic simulation of bivariate distributions

### 8. Random process and stochastic convergence

Random (stochastic) process

Characterizing a random process

Stationary random process

Equality of random processesStochastic convergence

- Sure convergence (convergence everywhere)
- Almost-sure convergence (convergence with probability 1)
- Mean-square convergence
- Convergence in probability
- Convergence in distribution

Ergodic theorem

Examples of stochastic processes

- iid random process
- Random walk process
- Gaussian process
- Autoregressive (AR) random process
- The 1-D Brownian motion

### 9. Introduction to time series model - the autoregressive (AR) model

Autogressiove model - general form

Characteristics of AR(1) and AR(2) models

Time series modeling in R (

**A good reference book: Time Series Analysis and Its Applications With R Examples by RH Shumway and DS Stoffer. Springer**)AR(1), Gauss-Markov process, and bivariate normal distribution

Stream flow series modeling**Working problems 5**(**Xia_Yun_DailyFlow.csv**) [**WP-5 Reference Code**]Flow persistence and the Hurst phenomenon

Flow duration curve

### 10. Hydrological time series

*PPT*Stream flow series

10-day-period (TDP) rainfall series

Standardized Precipitation Index (SPI) for drought monitoring, early warning, and forecasting

### 11. Gamma random field simulation

*PPT - Gamma Random Field Simulation*Characterizing a random field

Sequential Gaussian Random Field Simulation (SGS)

Gamma random field simulation

Potential applications

**Stochastic simulation of multispectral remote sensing images**### 12. Multisite stream flow simulation

### 13. Rainfall frequency analysis with consideration of spatial correlation - Estimating the return period of a multisite-extreme event

### 14. Stochastic storm rainfall simulation model

### 15. Assessing the impact of climate change on rainfall extremes

### 16. Statistical downscaling of GCM outputs

### 17. Hydrological forecasting and model performance evaluation

*PPT - Evaluation of hydrological model performance considering uncertainties*Sources of uncertainty and uncertainty in model performance

Persistence in flood flow series

Criteria for model performance evaluation (MPE)

Coefficient of efficiency (CE), coefficient of persistence (CP), and bench coefficient

Theorectial asymptotic relationship between CE and CP

Misuse of CE and CP for MPE of realtime flood forecasting

Theoretical CE-CP relationships of the AR(1) and AR(2) models

Demonstration of MPE using model-based bootstrap samples

### 18. Bootstrap resampling

**RSLAB - NTU**

**Prof. Ke-Sheng Cheng **

RSLAB_BSE_NTU

No. 1, Section 4, Roosevelt Road

Bioenvironmental Syst. Eng., National Taiwan University