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

RSLAB_BSE_NTU

No. 1, Section 4, Roosevelt Road

Bioenvironmental Syst. Eng., National Taiwan University

**Objectives**

- 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 and annual-max rainfall data of two rainfall stations in Taiwan.

**(2) Reservoir inflow data****(3) Streamflow data****(4) Hourly rainfalls of storm events****(5) Daily rainfall data****1. Univariate Simulation***Pseudo Random Number Generator (PRNG)**Probability Integral Transformation**Acceptance/Rejection Method**Frequency-factor-based Method**Random Number Generation in R***Reference****2. H****ydrological Frequency Analysis (I)**General interpretation of hydrological frequency analysis

Data series for frequency analysis

Parameters estimation

Techniques for goodness-of-fit tests

- Probability plotting
- Chi-square GOF test
- Kolmogorov-Smirnov GOF test

Selection of the best-fit distribution

- Information criteria

**3. Hydrological Frequency Analysis (II)***L-Moment and L-Moment Ratios Diagram (LMRD)*- Moments and moment-ratios diagram
- LMRD of various distributions

*Confidence Interval (Region) vs Acceptance Interval (Region)*- Definitions of confidence interval and acceptance interval
- Stochastic simulation of sample L-moment ratios

*Establishing acceptance regions of LMRD*- Normal and Gumbel distributions
- Pearson Type III
- R code for LMRD GOF tests (
**Sample data**,**R-code**)

**References:****4. Hydrological Frequency Analysis (III)***Event duration vs design duration**Simple scaling**Simple scaling modeling of storm rainfalls**The simple scaling property of IDF curves**Simple-scaling DDF curves**Multiple-scaling DDF curves*### 5. Hydrological Frequency Analysis (IV)

*Regional frequency analysis*Fundamental concept of regional frequency analysis

The index-flood approach

The frequency-factor approach

Demonstrating advantage of RFA using simulated data

**6. Stochastic Simulation of Bivariate Distributions***Bivariate normal distribution*- Using conditional density
- Using Principal Component Transformation

*Bivariate gamma distribution**Bivariate exponential distribution***7.****Random Processes***Characterization of a random process**Equality of random processes**Stochastic convergence**Ergodic theorem**Examples of random processes*### 8.

**Modeling Storm Rainfall Process***Hyetigraphs**Simple scaling model for storm events**Annual maximum events**Gauss-Markov model of dimensionless hyetographs**Translating hyetographs between storms of different durations*### 9. Random Field Simulation

*Sequential Gaussian random field simulation**Gamma random field simulation**Applications***10. Stochastic Storm Rainfall Simulation Model****PPT-1, PPT-2***stochastic weather generator**Elements of the storm rainfall process**GCM models and the effect of climate changes on rainfalls**Stochastic modeling of the storm rainfall process*### 11. Model performance evaluation - Realtime flood forecasting

*Sources of uncertainties**Uncertainties in model outputs**Uncertainties in model performance evaluation**Persistence in flood flow series**Commonly used criteria for model performance evaluation**Demonstration of uncertainties in modeling**Examples of flood forecasting model performance evaluation**Naive forecasting vs ANN forecasting*

*Asymptotic relationship between CE and CP**Sample-dependent relationship**Model-dependent relationship*

*A CE-CP coupled MPE criterion**Bootstrap resampling*

**RSLAB - NTU**

**Prof. Ke-Sheng Cheng **

RSLAB_BSE_NTU

No. 1, Section 4, Roosevelt Road

Bioenvironmental Syst. Eng., National Taiwan University