1. HYDROLOGY Lecture 12 Probability
Assoc.Prof. dr.tarkan erdik

2. What it is – Descriptive statistics
Unit Hydrograph
Descriptive statistics include the numbers, tables, charts, and graphs used to describe, organize, summarize, and present raw data.
central tendency (location) of data. i.e. where data tend to fall as measured by the mean, median, and mode.
dispersion (variability) of data. i.e. how spread out data are as measured by the variance and its square root, the standard deviation.
skew (symmetry) of data i.e. how concentrated data are at the low or high end of the scale as measured by the skew index.
kurtosis (peakedness) of data. i.e. how concentrated data are around a single value as measured by the kurtosis index.

6. Coefficient of variation: Coefficient of variation is a nondimensional parameter defined as the standart deviation to its mean

7. FREQUENCY ANALYSIS- Frequency Analysis of Continuous Variables

8. Dicle during 1956-75 are given below:
Example: Please calculate probability values of The flood data (m3/s) of the river
Dicle during are given below: i
Unit Hydrograph
Year
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965 X
2324
6300
2340
2080
2262
1250
3014
7910
4350
2630
Year
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975 X
8820
4516
4866
6450
2250
3450
5300
963
2571
The probability of being equal to or remaining smaller than xm is calculated by the Weibull formula, Eq

9. m 1 2 3 4 5 6 7 8 9 10 xm
963
1250
2080
2250
2262
2340
2424
2571
2630
3014
F(xm)
0.048
0.095
0.143
0.190
0.238
0.286
0.333
0.381
0.429
0.475 i
Unit Hydrograph m 11 12 13 14 15 16 17 18 19 20 xm
3450
4350
4516
4866
5300
5772
6300
6450
7910
8820
F(xm)
0.524
0.571
0.619
0.667
0.714
0.762
0.810
0.857
0.905
0.952
The probability that the flood discharge exceeds m3/s can be read from the table as 1-F(8820)=1-0.95=0.05 .

10. Dataset (original) Year Peak discharge (m3/s) 1996 2500 1997 4500 1998
3000
1999
2980
2000
2100
2001
1250
2002
3014
2003
8500
2004
4350
2005
6200
2006
7500
2007
4550
2008
4866
2009
6450
2010
6052
2011
2012
3580
2013
5335
2014
650
2015
2978 i
Please calculate the probability that the flood discharge
exceeds m3/s
Unit Hydrograph

11. Excedence probability is 1-0.76=0.24
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0.76
6052 m3/s
Excedence probability is =0.24

12. Probability Distributions
It has been observed that certain functions F(x) and f(x) can successfully express the distributions of many random variables.

13. Several continuous distributions play useful roles in engineering as in numerous other disciplines. The more important ones are: Uniform, Normal, Exponential, Gamma, Beta, Weibull, Lognormal distributions.

14. Uniform Distribution
The simplest type of continuous distribution is the uniform. As implied by the name, the pdf is constant over a given interval (for example, from a to b, where a < b). f(x)=constant, F(x)=cx; c is constant.

15. Normal Distributions
The normal distribution arose originally in the study of experimental errors.
Such errors pertain to unavoidable differences between observations when an experiment is repeated under similar conditions.

16. An alternative term is noise, which is used in telecommunication engineering and elsewhere when referring to the difference between the true state of nature and the signal received. The uncertainties which are manifest in the errors may arise from different causes that are not easily identifiable.
THE NORMAL DISTRIBUTION IS AN IDEAL CANDIDATE TO REPRESENT SUCH ERRORS WHEN THEY ARE OF AN ADDITIVE NATURE.

17. A large number of random variables encountered in practical applications fit to the Normal (Gaussian) distribution with the following probability density function:
This distribution is shown briefly as N(, 2).
It has two parameters: X and X .
Normal distribution is symmetrical (Cs=0), with a kurtosis coefficient equal to 3 (K=3).
The mode and median are equal to the mean because of the symmetry.
The sample values of x and sX can be taken as the estimates of the parameters X and X

18. (a) Probability density function, f X (x), and (b) probability distribution function, FX (x), of X for m=0 and σ=1

19. Since the normal distribution is symmetrical, this table is prepared for the positive values of Z only.
The probabilities of Z exceeding a certain positive value z, F1(z)=A are given.
For positive z we can compute the probability of nonexceedance as F(z)=1-F1(z), and for negative z we have F(z)=F1(z) because of symmetry around the mean 0.
Probability distribution function of standard normal distribution

20. The probability density function is bell-shaped around the mean X .
The mode and median are equal to the mean because of the symmetry.
The probabilities of the normal variable to remain in the intervals around the mean of width one, two and three standard deviations are equal to 0.683, and (nearly 1), respectively.

21. The ordinate axis of this probability paper is scaled such that the cumulative distribution function of the normal distribution appears as a straight line
The probability paper of the normal distribution

22. How can we tell if a variable is normally distributed?
1. The first check is by sketching the cumulative frequency distribution of the data on the normal probability paper. We can resume normal distribution if the plot is nearly a straight line.
2. The closeness of the skewness coefficient Cs of the sample to 0 (its absolute value below 0.10 or 0.05) and that of the kurtosis coefficient K to 3 (between 2.5 and 3.5) are further checks.
3. If the data pass these tests, the assumption of the normal distribution can be tested by statistical tests to be discussed in Chapter 6.

23. The normal distribution is certainly not valid in many cases because the variable is skewed. Most hydrologic variables (such as the discharge in a stream, the precipitation depth at a location) are NOT symmetrically distributed. For such variables, distributions other than normal must be used.

24. Because Y = ln(X) has a normal distribution, we can use the tables of the normal distribution to determine a probability or solve an inverse problem.
In the computations, the values for the Y variable taken from Table 4.1 are transformed into the X values by :
x = ey

25. The parameters Y and Y can be estimated in two ways:
They can be computed from the logarithms of the observations of the X variable, OR
They can be estimated from the equation below using the computed values of X and X. This approach preserves the parameters of the original variable.

26. EXTREME VALUE DISTRIBUTION
In civil engineering sometimes it is required to estimate the distribution of the extreme (i.e. maximum or minimum) values of the relevant random variable.

27. Bridge Design (Flood discharge)
Example:
Bridge Design (Flood discharge)
Photos:

28. EXAMPLE Spilways (Flood discharge) http://alviassociates.com
EXAMPLE
Spilways (Flood discharge)

29. Coastal structures (Wave loads)
Example
Coastal structures (Wave loads)

30. In these type of engineering problems Gumbel Distribution can be employed. FY(y) = exp -exp [- (y-)]  Herein the parameters of Y are:

31. The probability paper of the Gumbel distribution can be prepared by scaling the ordinate axis as double logarithmic, by taking the logarithm of above equation twice: (y-) = -ln [-ln FY(y)]
The Gumbel probability paper resulting from this linearized cdf function shows that the variables y and
ln[-ln(Fy(y)] are linearly related.

32. EXAMPLE
In California the magnitudes by Richter scale of the annual most severe earthquakes are given below in the increasing order. F(xm) values are computed by Eq. (3.7) as F(xm)=m/(N+1). Please compute the probability of occurrence of an earthquake with a magnitude higher than 7.0: m 1 2 3 4 5 6 7 8 9 10 xm
4.9
5.3
5.4
5.5
5.6
105 F(xm)
3125
6250
9375
12500
15625
18750
21875
25000
28125
31250 m 11 12 13 14 15 16 17 18 19 20 xm
5.8
5.9
6.0
105 F(xm)
34375
37500
40625
43750
46875
50000
53125
56250
59375
62500 m 21 22 23 24 25 26 27 28 29 30 31 xm
6.2
6.3
6.4
6.5
7.1
105 F(xm)
65625
68750
71875
75000
78125
81250
84375
87500
90625
93750
96875

33. Plotting the computed F(x) values on the Gumbel probability paper
Plotting of the earthquake magnitudes on the Gumbel probability paper

34. The parameters of the earthquake magnitudes are computed as:
X  x = 6.0 X  sX = 0.58
The parameters of the Gumbel distribution are:
 = / X =  = X -
= 5.74
Let us compute the probability of occurrence of an earthquake with a magnitude higher than 7.0:
P(X>7) = 1-F(7) = 1-exp -exp [-2.21 ( )] = 0.06

35. Return period of the earthquake
The return period of the earthquake with a magnitude of 8 can be computed as:
F(8) = exp -exp [-2.21 ( )] = 0.993
T = 1/[1-F(8)] = 143 years
T = 1/[1-F(8)] = 143 years
From Gumbel probability paper, the approximate value of the probability of the occurrence of an earthquake with a magnitude higher than 7 is:
1-0.93=0.07,
and the return period of an earthquake of magnitude 8 as approximately 100 years.
CONCLUSION: For small samples it is recommended to correct the parameter estimates of the Gumbel distribution

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