Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Probability Plotting - Stochastic Hydrology - Lecture Notes, Study notes of Mathematical Statistics

The main points i the stochastic hydrology are listed below:Probability Plotting, Graphical Construction, Normal Distribution Table, Arithmetic Scale Plot, Transformation Plot, Plotting Position, Empirical Methods, California Method, Formulae for Exceedence Probability

Typology: Study notes

2012/2013

Uploaded on 04/20/2013

sathyanarayana
sathyanarayana 🇮🇳

4.4

(21)

140 documents

1 / 73

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Graphical construction:
Graphical construction is done by transforming the
arithmetic scale to probability scale so that a
straight line is obtained when cumulative
distribution function is plotted.
The transformation technique is explained with the
normal distribution.
Consider the coordinates from the standardized
normal distribution table.
3%
Probability Plotting
Docsity.com
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c
pf1d
pf1e
pf1f
pf20
pf21
pf22
pf23
pf24
pf25
pf26
pf27
pf28
pf29
pf2a
pf2b
pf2c
pf2d
pf2e
pf2f
pf30
pf31
pf32
pf33
pf34
pf35
pf36
pf37
pf38
pf39
pf3a
pf3b
pf3c
pf3d
pf3e
pf3f
pf40
pf41
pf42
pf43
pf44
pf45
pf46
pf47
pf48
pf49

Partial preview of the text

Download Probability Plotting - Stochastic Hydrology - Lecture Notes and more Study notes Mathematical Statistics in PDF only on Docsity!

Graphical construction:

  • Graphical construction is done by transforming the arithmetic scale to probability scale so that a straight line is obtained when cumulative distribution function is plotted.
  • The transformation technique is explained with the normal distribution.
  • Consider the coordinates from the standardized normal distribution table. 3

Probability Plotting

4 z 0 2 4 6 8 0 0 0.008 0.016 0.0239 0. 0.1 0.0398 0.0478 0.0557 0.0636 0. 0.2 0.0793 0.0871 0.0948 0.1026 0. 0.3 0.1179 0.1255 0.1331 0.1406 0. 0.4 0.1554 0.1628 0.17 0.1772 0. 0.5 0.1915 0.1985 0.2054 0.2123 0. 0.6 0.2257 0.2324 0.2389 0.2454 0. 0.7 0.258 0.2642 0.2704 0.2764 0. 0.8 0.2881 0.2939 0.2995 0.3051 0. 0.9 0.3159 0.3212 0.3264 0.3315 0. 1 0.3413 0.3461 0.3508 0.3554 0.

Normal Distribution Tables

z (Partial tables shown here)

Arithmetic scale plot: 6

Probability Plotting

-­‐ -­‐ -­‐ 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Z F(Z)

0.01 0.05 0.1 0.2 (^) 0.4 0.5 0.6 0.7 0.8 0.9 (^) 0.95 0. Transformation plot: 7

Probability Plotting

F(Z) (probability scale) F(Z) (arithmetic scale)

  • The purpose is to check if a data set fits the particular distribution.
  • The plot can be used for interpolation, extrapolation and comparison purposes.
  • The plot can be used for estimating magnitudes with specified return periods.
  • Extrapolation must be attempted only when a reasonable fit is assured for the distribution. 9

Probability Plotting

PLOTTING POSITION

  • Arrange the given series of data in descending order.
  • Assign a order number to each of the data (termed as rank of the data).
  • Let n is the total no. of values to be plotted and m is the rank of a value, the exceedence probability (p) of the m th largest value is obtained by various plotting position formulae.
  • The return period (T) of the event is calculated by T = 1/p
  • Plot magnitude of event verses the exceedence probability (p) or 1 – p or the return period T. 12

Plotting Position

Formulae for exceedence probability: California Method: Limitations

  • Produces a probability of 100% for m = n 13

Plotting Position

( (^) m )

m

p X x

n

Hazen s formula: Chegodayev s formula: 15

Plotting Position

( )

m

m

p X x

n

( )

m

m

p X x

n

Weibull s formula:

  • Most commonly used method.
  • If n values are distributed uniformly between 0 and 100 percent probability, then there must be n+ intervals, n–1 between the data points and 2 at the ends.
  • Indicates a return period T one year longer than the period of record for the largest value. 16

Plotting Position

( )

m

m

p X x

n

  • Cunnane (1978) studied the various available plotting position methods based on unbiasedness and minimum variance criteria.
  • If large number of equally sized samples are plotted, the average of the plotted points for each value of m lie on the theoretical distribution line.
  • Minimum variance plotting minimizes the variance of the plotted points about the theoretical line. 18

Plotting Position

Ref: Cunnane, C., Unbiased plotting positions – a review, J,Hydrol., Vol. 37, pp.205-222, 1978

  • For normally distributed data, Blom s plotting position formula (b = 3/8) is commonly used.
  • For Extreme Value Type I distribution, the Gringorten formula (b = 0.44) is used.
  • All the relationships give similar values near the center of the distribution but may vary near the tails considerably.
  • Predicting extreme events depend on the tails of the distribution. 19

Plotting Position

  1. The observations should be equally spaced on the frequency scale
  2. The plotting position should be analytically simple and easy to use. 21

Plotting Position

Ref: Gumbel, E. J., 1958. Statistics of Extreme Values, Columbia University Press, New York

The Weibull plotting position formula meets all the 5 of the above criteria. 1.All the observations can be plotted since the plotting positions range from 1/(n+1) (which is greater than 0) to n/(n+1) (which is less than 1). 2.The relationship lies between (m – 1)/n and m/n for all values of m and n. 22

Plotting Position

( )

m

m

p X x

n

Ref: Statistical methods in Hydrology by C.T.Haan, Iowa state university press