Hydrologic Statistics

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Hydrologic Statistics
04/04/2006

• Reading Chapter 11 in Applied Hydrology
• Some slides by Venkatesh Merwade

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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.

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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

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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 variables sample space

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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 gt 100 cfs

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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

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Summary statistics

• Also called descriptive statistics
• If x1, x2, xn is a sample then

m for continuous data
Mean,
s2 for continuous data
Variance,
s for continuous data
Standard deviation,
Coeff. of variation,
Also included in summary statistics are median,
skewness, correlation coefficient,
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Graphical display

• Time Series plots
• Histograms/Frequency distribution
• Cumulative distribution functions
• Flow duration curve

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Time series plot

• Plot of variable versus time (bar/line/points)
• Example. Annual maximum flow series

Colorado River near Austin
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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

Dividing the number of occurrences with the total
number of points will give Probability Mass
Function
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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.)
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Probability density function

• Continuous form of probability mass function is
  probability density function

pdf is the first derivative of a cumulative
distribution function
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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
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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 (m1 for the highest value and mN
  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

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Flow duration curve in Excel
Median flow
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Statistical analysis

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

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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
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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 http//www.math.csusb.edu/facul
ty/stanton/m262/regress/regress.html
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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

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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