Hydrologic Statistics

Reading: Chapter 11 in Applied Hydrology

Some slides by Venkatesh Merwade

04/04/2006

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Hydrologic Models

• 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.

Classification based on randomness.

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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 variable’s 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 ExampleExample

Population: streamflowPopulation: streamflow Sample space: instantaneous streamflow, annual Sample space: instantaneous streamflow, annual

maximum streamflow, daily average streamflow maximum streamflow, daily average streamflow Sample: 100 observations of annual max. streamflowSample: 100 observations of annual max. streamflow Event: daily average streamflow > 100 cfsEvent: daily average streamflow > 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

n

iixn

X1

1

2

1

2

1

1

n

ii Xx

nS

2SS

X

SCV

Mean,

Variance,

Standard deviation,

Coeff. of variation,

for continuous data

for continuous data

for continuous data

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

0

100

200

300

400

500

600

1905 1908 1918 1927 1938 1948 1958 1968 1978 1988 1998

Year

An

nu

al M

ax F

low

(10

3 c

fs)

Colorado River near Austin

0

100

200

300

400

500

600

1900 1900 1900 1900 1900 1900 1900

Year

An

nu

al M

ax F

low

(10

3 c

fs)

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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

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200 250 300 350 400 450 500

Annual max flow (103 cfs)

No

. of

occ

ure

nce

s Interval = 50,000 cfs

0

10

20

30

40

50

60

Annual max flow (103 cfs)

No

. of

occ

ure

nce

s

Interval = 25,000 cfs

0

5

10

15

20

25

30

0 50 100 150 200 250 300 350 400 450 500

Annual max flow (103 cfs)

No

. of

occ

ure

nce

s

Interval = 10,000 cfs

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 ToolsData AnalysisHistogram

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

0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200 250 300 350 400 450 500

Annual max flow (103 cfs)

No

. of

occ

ure

nce

s

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 100 200 300 400 500 600

Annual max flow (103 cfs)

Pro

bab

ility

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

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600

Annual max flow (103 cfs)

Pro

bab

ility

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.

StepsSteps Arrange flows in chronological order Arrange flows in chronological order Find the number of records (N)Find the number of records (N) Sort the data from highest to lowest Sort the data from highest to lowest Rank the data (m=1 for the highest value and m=N for the lowest value)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 Compute exceedance probability for each value using the following

formulaformula

Plot p on x axis and Q (sorted) on y axisPlot p on x axis and Q (sorted) on y axis

1100

N

mp

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Flow duration curve in Excel

0

100

200

300

400

500

600

0 20 40 60 80 100

% of time Q will be exceeded

Q (

1000

cfs

) 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 :A regression model is given by :

yi = ith observation of the response (dependent variable)

xi = ith observation of the explanatory (independent) variable

0 = intercept

1 = slope

i = random error or residual for the ith observation

n = sample size

nixy iii ,...,2,110

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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

nixy ii ,...,2,1ˆ 10

nixyyy iiii ,...,2,1)(ˆ 210

2

Regression applet: http://www.math.csusb.edu/faculty/stanton/m262/regress/regress.html

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Linear Regression in Excel

• Steps: – Prepare a scatter plot– Fit a trend line

TDS = 0.5946(sp. Cond) - 15.709R2 = 0.9903

0

300

600

900

1200

1500

1800

0 500 1000 1500 2000 2500 3000

Specific Conductance ( S/cm)

TD

S (

mg

/L)

Alternatively, one can use ToolsAlternatively, one can use ToolsData Data AnalysisAnalysisRegressionRegression

Data are for Brazos River near Highbank, TX

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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

SST

SSER 12

2)( yySST i Total sum of squares, Ybar is the mean of yi

2)ˆ( ii yySSE Error sum of squares

The higher the value of RThe higher the value of R22, the more successful is the model in explaining y , the more successful is the model in explaining y variation.variation.

If RIf R22 is small, search for an alternative model (non linear or multiple is small, search for an alternative model (non linear or multiple regression model) that can more effectively explain y variationregression model) that can more effectively explain y variation