GUIDE WAVE ANALYSIS AND FORECASTING - WMO

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134 2. Log-normal distribution The two-parameter log-normal distribution is given by: Probability paper Maximum likelihood estimators ˆα = 1 n ∑ log xi ˆβ = The three-parameter log-normal is produced by replacing x by x – θ and is defined on [θ, ∞). Estimation is more complex. 3. Gamma distribution Probability paper cannot be used for estimation with the gamma distribution. Moments estimators f( x) 2 1 ⎡ (log x – α) ⎤ = exp ⎢– 2 ⎥ xβ 2π ⎣ 2β ⎦ x > 0 = 0 ⎡ ⎤ ⎢α ⎥ ⎣ ⎦ 2 exp 2 exp – 1 β otherwise mean 2 = exp + 2 variance= exp 2α β β – x = Φ ( p) y = h ⎡ x ⎢Φ( x) = ∫– ∞ ⎣⎢ ⎛ x ⎞ ⎤ exp ⎜ ⎟ dx⎥ ⎝ ⎠ ⎦⎥ – 1 log( ) 2 1 2π 2 f( x) x x = α exp(– ) β Γ ( α) β α, β, x > 0 = 0 otherwise mean = αβ 2 variance = αβ x α˜ = 2 s 2 ˜ s β = x 1 n ∑(log x i – ˆα ) 2 Maximum likelihood estimators 2 α – 1 1 ∑ log x = log + n x = ˆ ( ˆ i β Ψ α) αβ ˆ ˆ GUIDE TO WAVE ANALYSIS AND FORECASTING [ ] [ ] ( [ ] ) where Ψ(x) = Γ′(x)/Γ(x). These equations have to be solved numerically. The three-parameter gamma is produced by replacing x by x – θ and is defined on [θ, ∞). Estimation is more complex.
4. Weibull distribution Probability paper x = log −log 1−p y = log( h) Maximum likelihood estimators These equations have to be solved numerically. The three-parameter Weibull is produced by replacing x by x – θ and is defined on [θ, ∞). Estimation is more complex. 5. Exponential distribution Probability paper f( x) – x x = exp – , , x ⎛ β 1 ⎞ ⎡ β 1 ⎛ ⎞ ⎤ β ⎝ ⎠ ⎢ ⎝ ⎠ ⎥ α α ⎣⎢ α ⎦⎥ α β > 0 = 0 otherwise Fx ( ) ⎡ β x – ⎛ ⎞ ⎤ =1– exp ⎢ ⎝ ⎠ ⎥ ⎣⎢ α ⎦⎥ x > 0 = 0 x < 0 mean ⎛ 1 ⎞ = αΓ ⎜ + 1⎟ ⎝ β ⎠ 2 ⎡ 2 ⎛ 2 ⎞ ⎧ ⎛ 1 ⎞⎫ ⎤ variance = α ⎢Γ ⎜ + 1⎟ – ⎨Γ⎜+ 1⎟ ⎬ ⎥ ⎢ ⎝ β ⎠ ⎩ ⎝ β ⎠ ⎣ ⎭ ⎥ ⎦ ˆ log( ) log( ) – – β β = β ∑ ⎛ xi xi ∑ xi ⎞ ⎜ ⎟ ⎝ ∑ x n ⎠ f( x) 1 x = exp ⎛ – ⎞ α ⎝ α⎠ , x > 0 = 0 Fx ( ) = 0 x< 0 x = 1– exp ⎛ – ⎞ ⎝ ⎠ x > 0 = 2 = α otherwise α mean α variance α x = – log( 1– p) y = h Moments and maximum likelihood estimators α˜= αˆ= x 1 β β αˆ = ∑ ⎛ xi ⎞ ⎜ ⎟ ⎝ n ⎠ [ ( ) ] i 1 ANNEX III 135 (Any line must go through the origin.) The two-parameter exponential is produced by replacing x by x – θ and is defined on [θ, ∞). Estimation is more complex. The exponential distribution is a special case of the Gamma (α = 1) and Weibull (β = 1) distributions.
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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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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: The following distributions are bri
Page 147 and 148: 8. Fisher-Tippett Type I (FT-I) (or
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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