lec 9-impacts of cc on hydrological variables distributions

8/13/2019 Lec 9-Impacts of Cc on Hydrological Variables Distributions

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LECTURE OUTLINE

COMPARING TIME SERIES OFHYDROLOGICAL VARIABLES

BASIC STATISTICS BOXPLOTS

THE QUANTILE-QUANTLETRANSFORMATION


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DISCREPANCIES BETWEEN OBSERVED ANDSIMULATED VARIABLES DISTRIBUTIONS


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DISCREPANCIES BETWEEN OBSERVED ANDSIMULATED VARIABLES DISTRIBUTIONS

0.4

0.6

0.8

1

C D F

Month 5

Observations-CurrentRaw simulations-currentQQ-Corrected-current

4

6

8

10

P D F

Month 5

Observations-CurrentRaw simulations-currentQQ-Corrected-current

0 10 20 30 40 50 60 70 80 900

0.2

Precipitation (mm)

0 5 10 15 20 25 30

0

2

Precipitation (mm)

0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

Precipitation (mm)

C D F

Raw simulations-FutureQQ-Corrected-Future

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Precipitation (mm)

C D F

Raw simulations-FutureQQ-Corrected-Future


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COMPARING HYDROLOGICALVARIABLES BEFORE AND AFTER CC

Options: Visual observation Basic statistics like the mean, the variance and the

skewness Use boxplots Estimate the CDFs and PDFs

The two last options are preferred as theyprovide much more complete information onthe variables


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INFERENCE OF THE FULLDISTRIBUTION


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Frequency Histogram36

27

Probability that Q is 10,000 to 15, 000 = 17.3%

Prob that Q < 20,000 = 1.3 + 17.3 + 36 = 54.6%

17.3

1.3

8

1.3

9


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Probability Distributions

CDF is the most useful form for analysis

ii

F ( x1)=

P (

x

x1)=

f ( x)

x 1

dxP( x

1 x x

2) = F ( x

2) F ( x

1)


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PROBLEMS IN INFERINGTHE FULL DISTRIBUTION Can be done only if the sample size is infinite When the sample size is finite, one usulayy has

to assume a parent distribution

- ubut the size of the sample limits the precision ofthe estimation.

Only the full distribution allows thr calculation ofextreme high or low events The assumed parent distribution will affect the

estimation of the extreme events


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BASIC STATISTICS


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MEAN AND VARIANCE FROM THEDATA

x = 1n

x ii

n

Mean of Data

s x2 =

1

n 1( xi x)

2 Varianc

Std Dev. S x = (S x 2 )1/2


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SKEWNESS

Used to evaluate high or low data points - floodor drought data

Skewness 33 third central moment

C s =

n

(n 1)( n 2)

( x i x)3

s x3 skewness coeff.

Coeff of Var =


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Mean, Median, Mode Positive Skew moves mean to right

Negative Skew moves mean to left

Normal Distn has mean = median = mode

Median has highest prob. of occurrence


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Skewed PDF - Long Right Tail


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BOXPLOTS


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BoxplotsBoxplots also called box and whisker plots Present the 5 number summary in graphical

form


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BoxplotBoxplot


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BoxplotBoxplot

Q1 and Q2 are thequartiles (25% of the data

Q2


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BoxplotBoxplot


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Shapes from boxplots Shapes of distributions from boxplots

Whiskers indicate the long tail of a distribution Skewed or symmetric.


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Long whisker to the right

Long tailto theri ht


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Both whiskersabout the same


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Long whisker to the left

Long tail tothe left


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APPLICATION TO THE SN

0 50 100 150 200 250 300 3500

0.2

0.4

0.6

0.8

1Month5:CDF

CURRENT

FUTURE

Month5: PDF

0 50 100 150 200 250 300 3500

0.002

0.004

0.006

0.008

0.01

CURRENTFUTURE

0

200

400

CURRENT FUTURE

Month5:Boxplots


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APPLICATION TO YOURPROJECT


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EXEMPLE INPUT DATA


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THE QUANTILE-QUANTILETRANSFORMATION


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THE QUANTILE-QUANTILE METHOD

The purpose of the quantile quatile method isto correct the distribution of the GCM output

Builds a correspondence table between the

magnitude Can be easily implemented in Excel


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EXEMPLE: OBSERVED AND SIMULATED PCPAT LAT13.43 LON 0.52

0

50

100

150

200

KNMI

-50

-5.00E-050.00E+005.00E-051.00E-041.50E-042.00E-042.50E-043.00E-043.50E-044.00E-04

07/05/1990 19/09/1991 31/01/1993 15/06/1994 28/10/1995 11/03/1997 24/07/1998 06/12/1999

observed


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QUANTILE-QUANTILE Select the same number of data points from each

precipitation series, ideally from the same period In the exemple we will take the 3650 first values of

each series

Paste them in Excel and sort them in ascendingorder. The two columns give you the correspondence

between the GCM data and observations


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A N T I L E

The excel Vlookup function can do the interpolation for youFor serious application consider a more precise interpolationmethod (ex. Matlab interp2)

Q

U A N T I L E

- Q U

=VLOOKUP(F2,$A$2:$B$4

017,2,TRUE)


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Assignment 3 Apply the Delta to precipitation on your

watershed Apply the Quantile-Quantile transformation to

Compare the distributions on the twotransformations the the observed distributions

Comment on the ability of the twotransformations to correctly reproduceextreme events

lec 9-impacts of cc on hydrological variables distributions

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