GUIDE WAVE ANALYSIS AND FORECASTING - WMO

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136 6. Rayleigh distribution Probability paper All lines have slope O.5. Moments estimators f( x) x ⎛ x ⎞ = exp ⎜ – ⎟ α ⎝ α ⎠ α, x > = Fx ( ) = x< ⎛ x ⎞ = – exp ⎜ – ⎟ ⎝ α ⎠ x > = α = ⎛ − ⎞ ⎝ ⎠ π 2 2 2 2 0 0 otherwise 0 0 2 1 2 2 0 mean 2 2 π variance 2α1 4 x = log −log 1−p y = log( h) α˜ = Maximum likelihood estimators ˜α = [ ( ) ] x π / 2 ∑ x 2n i 2 The two-parameter Rayleigh is produced by replacing x by x – θ and is defined on [θ, ∞). Estimation is more complex. The Rayleigh distribution is a special case of the Weibull distribution (β = 2, α = √2α). 7. Generalized extreme value (GEV) distribution Fx ( ) = 0 x< α + < 0 β and θ θ ⎡ x – = exp ⎢ ⎛ α ⎞ ⎢ – ⎜1– θ ⎟ ⎝ β ⎠ ⎣⎢ = x > + > = + [ – ( + ) ] ≠ = + ( = = . ...) = = ⎛ 1 α 0 1 1 0 0 5772 0 2 ⎞ 2 ⎜ ⎟ ( 1+ 2 )+ ⎝ ⎠ β θ θ α β and mean Γ θ θ θ α γ β γ Euler’sconstant θ β variance [ Γ θ Γ ( 1+ θ) ] θ θ ≠ 0 For methods of estimation, see NERC (1975). 2 GUIDE TO WAVE ANALYSIS AND FORECASTING 1 θ ⎤ ⎥ ⎥ ⎦⎥ 2 = β = 0 6 π θ
8. Fisher-Tippett Type I (FT-I) (or Gumbel) distribution Probability paper Moments estimators f( x) x x exp – Fx x – – exp – – ( ) exp – exp – – 1 ⎡ α ⎛ α ⎞⎤ = ⎢ ⎜ ⎟⎥ β ⎣ β ⎝ β ⎠⎦ β > 0 ⎡ ⎛ α ⎞⎤ = ⎢ ⎜ ⎟⎥ ⎣ ⎝ β ⎠⎦ = α+ γ β ( γ = = 0. 5772...) 2 2 = β 6 π mean Euler’sconstant variance x = – log −log( p) y = h α = x – γ β s β = 6 π Maximum likelihood estimators ⎡1 ⎛ x ⎞⎤ i α = – βlog ⎢ ∑exp ⎜ – ⎟⎥ ⎣n ⎝ β ⎠⎦ ⎛ xi ⎞ ∑ xi exp ⎜ – ⎟ 1 ⎝ β ⎠ β = ∑xi– n ⎛ xi ⎞ ∑ exp ⎜ – ⎟ ⎝ β ⎠ These equations have to be solved numerically. A detailed description and comparison of the various methods of estimation for the FT-I distribution are given in Carter and Challenor (1983). The FT-I is a special case of the GEV (θ = 0). 9. Fisher-Tippett Type III (FT-III) distribution f( x) [ ] – – x – x = exp – ⎛ ⎞ ⎡ β ⎛ ⎞ ⎝ ⎠ ⎢ ⎝ ⎠ ⎣⎢ = θ 1 1 θ α α α 0 β β ⎤ ⎥ x < θ ⎦⎥ otherwise Fx ( ) ⎡ β – x = exp – ⎛θ ⎞ ⎤ ⎢ ⎝ ⎠ ⎥ ⎣⎢ α ⎦⎥ x < θ = 1 x > θ mean ⎛ 1 ⎞ = θ– αΓ⎜ + 1⎟ ⎝ β ⎠ ⎡ 2 2 ⎛ 2 ⎞ ⎧ ⎛ 1 ⎞⎫ ⎤ variance= α ⎢Γ ⎜ + 1⎟ – ⎨Γ ⎜ + 1⎟ ⎬ ⎥ ⎢ ⎝ β ⎠ ⎩ ⎝ β ⎠ ⎣ ⎭ ⎥ ⎦ ANNEX III 137
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WORLD METEOROLOGICAL ORGANIZATION G
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© 1998, World Meteorological Organ
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IV Chapter 7 - WAVES IN SHALLOW WAT
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ACKNOWLEDGEMENTS The revision of th
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VIII has been included particularly
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X considerable attention. Annex III
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2 η Crest Zero level a H = 2a a Tr
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4 Figure 1.5 — Paths of the water
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6 When waves propagate into shallow
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8 1.3.2 Wave groups and group veloc
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10 wavelengths in a given sea state
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12 for instance, a frequency of 0.1
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14 in which E(f) is the variance de
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16 2.1.1 Wind and pressure analyses
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18 GUIDE TO WAVE ANALYSIS AND FOREC
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20 Figure 2.2(a) (right) — Usual
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22 As a quick approximation of ocea
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24 GUIDE TO WAVE ANALYSIS AND FOREC
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26 Gr G Gr ∇p C Cnf ∇p C Cnf
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28 POINT C — The effect of warm a
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30 a general sense, and can be appl
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32 Free atmosphere Ekman layer Cons
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3.1 Introduction This chapter gives
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ange of directions. Also, waves at
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small enough that swell can survive
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Figure 3.7 — Structure of spectra
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4.1 Introduction CHAPTER 4 WAVE FOR
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necessary to forecast waves for a p
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TABLE 4.4 Additional wave informati
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TABLE 4.6 Ranges of swell periods a
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this situation, the angular spreadi
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the energy flux is c gH 2 . This is
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α0, degrees Solution: 80° 70° 60
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5.1 Introduction National Meteorolo
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only once, since it is usual to sto
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Frequency 0.050 0.067 0.083 0.100 0
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The models may differ in several re
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The CH class may include many semi-
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6.1 Introductory remarks Since the
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in some applications, the zero up/d
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OPERATIONAL WAVE MODELS 71 Figure 6
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give large differences in the compa
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Wind (m/s) Waves (m) Buoy/GSOWM Wav
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TABLE 6.2 Numerical wave models ope
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TABLE 6.2 (cont.) Country Name of m
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7.1 Introduction The evolution of w
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water boundary into shallow water.
Page 95 and 96: eflected waves, although the latter
Page 97 and 98: Energy (cm 2 /Hz) Energy (cm 2 /Hz)
Page 99 and 100: 8.1 Introduction Wave data are ofte
Page 101 and 102: matter and timing the intervals bet
Page 103 and 104: a Directional Waverider (Barstow et
Page 105 and 106: 80°N 60°N 40°N 20°N 0° -20°S
Page 107 and 108: a combination of sea-surface temper
Page 109 and 110: 8.7.1 Digital analysis of wave reco
Page 111 and 112: 9.1 Introduction Chapter 8 describe
Page 113 and 114: Presentation of data and wave clima
Page 115 and 116: the monsoon season of June to Septe
Page 117 and 118: Probability of non-exceedance case
Page 119 and 120: individual wave height derived usin
Page 121 and 122: emote, data-sparse areas. However,
Page 123 and 124: TABLE 9.2 (cont.) Country Source of
Page 125 and 126: For a storm hindcast, select the mo
Page 127 and 128: TABLE 9.3 (cont.) Country Model use
Page 129 and 130: ANNEX I ABBREVIATIONS AND KEY TO SY
Page 131 and 132: Abbreviation/ Definition Symbol MOS
Page 133 and 134: CODE FORM: D . . . . D or IIiii* SE
Page 135 and 136: sensors, spectral peak wave period
Page 137 and 138: cs1cs1 The ratio of the spectral de
Page 139 and 140: M jM j n mn m n smn sm n bn bn b P
Page 141 and 142: 1744 Im Indicator for method of cal
Page 143 and 144: The following distributions are bri
Page 145: 4. Weibull distribution Probability
Page 149 and 150: ANNEX IV THE PNJ (PIERSON-NEUMANN-J
Page 151 and 152: ANNEX IV 141 Duration graph. Distor
Page 153 and 154: REFERENCES Abbott, M. B., H. H. Pet
Page 155 and 156: REFERENCES 145 Carter, D. J. T., P.
Page 157 and 158: REFERENCES 147 Gumbel, E. J., 1958:
Page 159 and 160: REFERENCES 149 Maat, N., C. Kraan a
Page 161 and 162: Slutz, R. J., S. J. Lubker, J. D. H
Page 163 and 164: SELECTED BIBLIOGRAPHY CERC, 1984: C
Page 165 and 166: 156 Energy . . . . . . . . . . . .
Page 167 and 168: 158 steepness . . . . . . . . . . .
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GUIDE WAVE ANALYSIS AND FORECASTING - WMO

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