Prediction of Design Wind Speeds: Gumbel and Gringorten Methods, Slides of Environmental Law and Policy

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• Historical :
  1928. Fisher and Tippett. Three asymptotic extreme value distributions

1954. Gumbel method of fitting extremes. Still widely used for windspeeds.
1955. Jenkinson. Generalized extreme value distribution
1956. Simiu. First comprehensive analysis of U.S. historical extreme wind speeds. Sampling errors.
1957. Gomes and Vickery. Separation of storm types
1958. Davison and Smith. Excesses over threshold method.
1959. Peterka and Shahid. Re-analysis of U.S. data - ‘superstations’

• Generalized Extreme Value distribution (G.E.V.) :

Type I, II : U is unlimited as c.d.f. reduces (reduced variate increases)

Type III: U has an upper limit

0

2

4

6

8

-3 -2 -1 0 1 2 3 4

Reduced variate : -ln[-ln(FU(U)]

(U-u)/a

Type I k = 0 Type III k = +0. Type II k = -0.

(In this way of plotting, Type I appears as a straight line)

• Return Period (mean recurrence interval) :

Unit : depends on population from which extreme value is selected

A 50-year return-period wind speed has an probability of exceedence of 0.02 in any one year

Return Period, R = Probability of exceedence

1 1 F (U)

1  U



e.g. for annual maximum wind speeds, R is in years

it should not be interpreted as occurring regularly every 50 years

or average rate of exceedence of 1 in 50 years

• Gumbel method - for fitting Type I E.V.D. to recorded extremes
  • procedure
• Assign probability of non-exceedence
• Extract largest wind speed in each year
• Rank series from smallest to largest m=1,2…..to N

 (^) N 1 

m

p

• Form reduced variate : y = - loge (-loge p)
• Plot U versus y, and draw straight line of best fit, using least squares method (linear regression) for example

• Gringorten method
  • Gringorten formula is ‘unbiased’ :
  • same as Gumbel but uses different formula for p
  • Gumbel formula is ‘biased’ at top and bottom ends

 (^) N 1 - 0.88

m-0.

p

• Otherwise the method is the same as the Gumbel method

 (^) N 0. 12 

m-0.

• Gringorten method -example
  • Baton Rouge Annual maximum gust speeds 1970-

BATON ROUGE ANNUAL MAXIMA 1970- y = 6.24x + 49.

0

20

40

60

80

-2 -1 0 1 2 3 4 reduced variate (Gringorten) -ln(-ln(p))

Gust wind speed (mph)

• Gringorten method -example

• Baton Rouge Annual maximum gust speeds 1970-

Mode = 49. Predicted values Slope = 6. Return Period UR(mph) 10 63. 20 67. 50 73. 100 78. 200 82. 500 88. 1000 92.

 

  

 

      ) R U u a log log (1-^1 e e

• Separation by storm type
• Probability of annual max. wind being less than Uext due to any storm type =

Probability of annual max. wind from storm type 1 being less than Uext

 Probability of annual max. wind from storm type 2 being less than Uext

 etc…. (assuming statistical independence)

• In terms of return period,



 

  

    

 

  1 2

1 1 1 1 1 1 Rc R R

R 1 is the return period for a given wind speed from type 1 storms etc.

• Wind direction effects If wind speed data is available as a function of direction, it is very useful to analyse it this way, as structural responses are usually quite sensitive to wind direction

Probability of annual max. wind speed (response) from any direction being less than Uext =

Probability of annual max. wind speed (response)from direction 1 being less than Uext Probability of annual max. wind speed (response)from direction 2 being less than Uext  etc…. (assuming statistical independence of directions)

• In terms of return periods,

Ri is the return period for a given wind speed from direction sector i

  

 

 

 

   

 

 

N Ra (^) i 1 Ri

1 1 1 1 

• Compositing data (‘superstations’)

Example of a superstation (Peterka and Shahid ASCE 1978) :

3931 FORT POLK, LA 1958 -

3937 LAKE CHARLES, LA 1970 - 1990

12884 BOOTHVILLE, LA 1972 - 1981

12916 NEW ORLEANS, LA 1950 - 1990

12958 NEW ORLEANS, LA 1958 - 1990

13934 ENGLAND, LA 1956 - 1990

13970 BATON ROUGE, LA 1971 - 1990

93906 NEW ORLEANS, LA 1948 - 1957

193 station-years of combined data

• Excesses (peaks) over threshold approach

Uses all values from independent storms above a minimum defined threshold

Example : all thunderstorm winds above 20 m/s at a station

• Procedure :
• several threshold levels of wind speed are set :u 0 , u 1 , u 2 , etc. (e.g. 20, 21, 22 …m/s)
• the exceedences of the lowest level by the maximum wind speed in each storm are identified and the average number of crossings per year, , are calculated
• the differences (U-u 0 ) between each storm wind and the threshold level u 0 are calculated and averaged (only positive excesses are counted)
  • previous step is repeated for each level, u 1 , u 2 etc, in turn
• mean excess for each threshold level is plotted against the level
• straight line is fitted

• Excesses (peaks) over threshold approach
• Average number of excesses above lowest threshold, uo per annum = 

= u 0 + [1-(R)-k]/k

• Upper limit to UR as R  for positive k
  • UR= u 0 +( /k)

≈ u 0 + value of x exceeded with a probability, (1/ R)

• Average number of excesses above uo in R years = R
• R-year return period wind speed, UR = u 0 + value of x with average rate of exceedence of 1 in R years

• Excesses (peaks) over threshold approach
• Example of plot of mean excess versus threshold level :

Negative slope indicates positive k (extreme wind speed has upper limit )

MOREE Downburst Gusts

1

2

3

4

y = -0.139x + 4.

0 5 10 15 Threshold (m/s)

Average excess (m/s)