1. Hydrologic Statistics
04/04/2006
Hydrologic Statistics
Reading: Chapter 11 in Applied Hydrology
Some slides by Venkatesh Merwade

2. Classification based on randomness.
Hydrologic Models
Classification based on randomness.
Deterministic (eg. Rainfall runoff analysis)
Analysis of hydrological processes using deterministic approaches
Hydrological parameters are based on physical relations of the various components of the hydrologic cycle.
Do not consider randomness; a given input produces the same output.
Stochastic (eg. flood frequency analysis)
Probabilistic description and modeling of hydrologic phenomena
Statistical analysis of hydrologic data.

3. Probability A measure of how likely an event will occur
A number expressing the ratio of favorable outcome to the all possible outcomes
Probability is usually represented as P(.)
P (getting a club from a deck of playing cards) = 13/52 = 0.25 = 25 %
P (getting a 3 after rolling a dice) = 1/6

4. Random Variable
Random variable: a quantity used to represent probabilistic uncertainty
Incremental precipitation
Instantaneous streamflow
Wind velocity
Random variable (X) is described by a probability distribution
Probability distribution is a set of probabilities associated with the values in a random variable’s sample space

6. Sampling terminology
Sample: a finite set of observations x1, x2,….., xn of the random variable
A sample comes from a hypothetical infinite population possessing constant statistical properties
Sample space: set of possible samples that can be drawn from a population
Event: subset of a sample space
Example
Population: streamflow
Sample space: instantaneous streamflow, annual maximum streamflow, daily average streamflow
Sample: 100 observations of annual max. streamflow
Event: daily average streamflow > 100 cfs

7. Types of sampling
Random sampling: the likelihood of selection of each member of the population is equal
Pick any streamflow value from a population
Stratified sampling: Population is divided into groups, and then a random sampling is used
Pick a streamflow value from annual maximum series.
Uniform sampling: Data are selected such that the points are uniformly far apart in time or space
Pick steamflow values measured on Monday midnight
Convenience sampling: Data are collected according to the convenience of experimenter.
Pick streamflow during summer

8. Summary statistics Also called descriptive statistics
If x1, x2, …xn is a sample then
Mean,
m for continuous data
Variance,
s2 for continuous data
s for continuous data
Standard deviation,
Coeff. of variation,
Also included in summary statistics are median, skewness, correlation coefficient,

10. Graphical display Time Series plots Histograms/Frequency distribution
Cumulative distribution functions
Flow duration curve

11. Colorado River near Austin
Time series plot
Plot of variable versus time (bar/line/points)
Example. Annual maximum flow series
Colorado River near Austin

12. Histogram
Plots of bars whose height is the number ni, or fraction (ni/N), of data falling into one of several intervals of equal width
Interval = 25,000 cfs
Interval = 50,000 cfs
Interval = 10,000 cfs
Dividing the number of occurrences with the total number of points will give Probability Mass Function

14. Using Excel to plot histograms
1) Make sure Analysis Tookpak is added in Tools.
This will add data analysis command in Tools
2) Fill one column with the data, and another with the intervals (eg. for 50 cfs interval, fill 0,50,100,…)
3) Go to ToolsData AnalysisHistogram
4) Organize the plot in a presentable form (change fonts, scale, color, etc.)

15. Probability density function
Continuous form of probability mass function is probability density function
pdf is the first derivative of a cumulative distribution function

17. Cumulative distribution function
Cumulate the pdf to produce a cdf
Cdf describes the probability that a random variable is less than or equal to specified value of x
P (Q ≤ 50000) = 0.8
P (Q ≤ 25000) = 0.4

22. Flow duration curve
A cumulative frequency curve that shows the percentage of time that specified discharges are equaled or exceeded.
Steps
Arrange flows in chronological order
Find the number of records (N)
Sort the data from highest to lowest
Rank the data (m=1 for the highest value and m=N for the lowest value)
Compute exceedance probability for each value using the following formula
Plot p on x axis and Q (sorted) on y axis

23. Flow duration curve in Excel
Median flow

24. Statistical analysis Regression analysis Mass curve analysis
Flood frequency analysis
Many more which are beyond the scope of this class!

25. Linear Regression
A technique to determine the relationship between two random variables.
Relationship between discharge and velocity in a stream
Relationship between discharge and water quality constituents
A regression model is given by :
yi = ith observation of the response (dependent variable)
xi = ith observation of the explanatory (independent) variable
b0 = intercept
b1 = slope
ei = random error or residual for the ith observation
n = sample size

26. Least square regression
We have x1, x2, …, xn and y1,y2, …, yn observations of independent and dependent variables, respectively.
Define a linear model for yi,
Fit the model (find b0 and b1) such at the sum of the squares of the vertical deviations is minimum
Minimize
Regression applet:

27. Linear Regression in Excel
Steps:
Prepare a scatter plot
Fit a trend line
Data are for Brazos River near Highbank, TX
Alternatively, one can use ToolsData AnalysisRegression

28. Coefficient of determination (R2)
It is the proportion of observed y variation that can be explained by the simple linear regression model
Total sum of squares, Ybar is the mean of yi
Error sum of squares
The higher the value of R2, the more successful is the model in explaining y variation.
If R2 is small, search for an alternative model (non linear or multiple regression model) that can more effectively explain y variation