LTWCPDF Long Term Wave Climate PDF of significant wave height and wave period CALL: f = ltwcpdf(T,Hs,Pt,Ph,def) f = density T = peak period, Tp or mean zerodowncrossing period, Tz (sec) Hs = significant wave height (meter) Pt = [A1 A2 A3 B1 B2 B3 B4] parameters for the conditional distribution of T given Hs (default [1.59 0.42 2 0.005 0.09 0.13 1.34]) Ph = [C1 C2 C3 Hc M1 S1], parameters for the marginal distribution for Hs (default [2.82 1.547 0 3.27 0.836 0.376]) def = defines the parametrization used (default 0) The probability distribution is given as the product: f(T,Hs)=f(T|Hs)*f(Hs) where f(T|Hs) = wlognpdf(T,my(Hs),S(Hz)) with mean and variance my(Hs) = A1 + A2*log(Hs+A3) if mod(def,2)==0 A1 + A2*Hs^A3 if mod(def,2)==1 S(Hs) = B1+B2*exp(-B3*Hs.^B4) For def <=1: f(Hs) = wlognpdf(Hs,M1,S1) for Hs <= Hc = wweibpdf(Hs-C3,C1,C2) for Hs > Hc For def >1: f(Hs) = wlognpdf(Hs,M1,S1) for Hs <= Hc = wggampdf(Hs,C1,C2,C3) for Hs > Hc (Suggested by Ochi) The default values for T and Hs are suitable for the Northern North Sea for peak period, Tp and significant waveheight, Hs. With a suitable change of parameters for Pt this model fit mean wave period, Tz, also reasonably well. The size of f is the common size of T and Hs NOTE: - by specifying nan's in the vectors Pt or Ph default values will be used. - if length(Pt) or length(Ph) is shorter than the parameters needed then the default values are used for the parameters not specified. - For tables of fitted parameter values to specific sites see the end of this file (type ltwcpdf.m) Example: % Set C1 = 2.73, Hc = 3.95 and the rest to their default values Ph = [2.73, NaN,NaN, 3.95]; x = linspace(0, 15); [T,H] = meshgrid(x); f = ltwcpdf(T,H,[],Ph); contour(x,x,f) See also wweibpdf, wlognpdf, wggampdf
Generalized Gamma probability density function | |
Lognormal cumulative distribution function | |
Lognormal probability density function | |
Weibull probability density function | |
Check if all input arguments are either scalar or of common size. | |
Display message and abort function. |
001 function p = ltwcpdf(t,h,Pt1,Ph1, def,condon) 002 %LTWCPDF Long Term Wave Climate PDF of significant wave height and wave period 003 % 004 % CALL: f = ltwcpdf(T,Hs,Pt,Ph,def) 005 % 006 % f = density 007 % T = peak period, Tp or mean zerodowncrossing period, Tz (sec) 008 % Hs = significant wave height (meter) 009 % Pt = [A1 A2 A3 B1 B2 B3 B4] parameters for the conditional 010 % distribution of T given Hs 011 % (default [1.59 0.42 2 0.005 0.09 0.13 1.34]) 012 % Ph = [C1 C2 C3 Hc M1 S1], parameters for the marginal 013 % distribution for Hs (default [2.82 1.547 0 3.27 0.836 0.376]) 014 % def = defines the parametrization used (default 0) 015 % 016 % The probability distribution is given as the product: 017 % 018 % f(T,Hs)=f(T|Hs)*f(Hs) 019 % where 020 % 021 % f(T|Hs) = wlognpdf(T,my(Hs),S(Hz)) with mean and variance 022 % my(Hs) = A1 + A2*log(Hs+A3) if mod(def,2)==0 023 % A1 + A2*Hs^A3 if mod(def,2)==1 024 % S(Hs) = B1+B2*exp(-B3*Hs.^B4) 025 % 026 % For def <=1: 027 % f(Hs) = wlognpdf(Hs,M1,S1) for Hs <= Hc 028 % = wweibpdf(Hs-C3,C1,C2) for Hs > Hc 029 % For def >1: 030 % f(Hs) = wlognpdf(Hs,M1,S1) for Hs <= Hc 031 % = wggampdf(Hs,C1,C2,C3) for Hs > Hc (Suggested by Ochi) 032 % 033 % The default values for T and Hs are suitable for the Northern North 034 % Sea for peak period, Tp and significant waveheight, Hs. 035 % With a suitable change of parameters for Pt this model fit mean wave 036 % period, Tz, also reasonably well. 037 % 038 % The size of f is the common size of T and Hs 039 % 040 % NOTE: - by specifying nan's in the vectors Pt or Ph default values 041 % will be used. 042 % - if length(Pt) or length(Ph) is shorter than the parameters 043 % needed then the default values are used for the parameters 044 % not specified. 045 % - For tables of fitted parameter values to specific sites see 046 % the end of this file (type ltwcpdf.m) 047 % 048 % Example: % Set C1 = 2.73, Hc = 3.95 and the rest to their default values 049 % Ph = [2.73, NaN,NaN, 3.95]; x = linspace(0, 15); 050 % [T,H] = meshgrid(x); 051 % f = ltwcpdf(T,H,[],Ph); 052 % contour(x,x,f) 053 % 054 % See also wweibpdf, wlognpdf, wggampdf 055 056 % References: 057 % Haver, S (1980) 058 % 'Analysis of uncertainties related to the stochastic modelling of 059 % Ocean waves' 060 % Ph.D. thesis, Norwegian Institute of Technology, NTH, Trondheim, Norway 061 % 062 % Haver,S and Nyhus, K. A. (1986) 063 % 'A wave climate description for long term response calculation.' 064 % In Proc. of OMAE'86, tokyo, Japan, pp. 27-34 065 % 066 % Sagli, Gro (2000) 067 % "Model uncertainty and simplified estimates of long term extremes of 068 % hull girder loads in ships" 069 % Ph.D. thesis, Norwegian University of Science and Technology, NTNU, 070 % Trondheim, Norway, pp 45--47 071 % 072 % Michel K. Ochi (1998), 073 % "OCEAN WAVES, The stochastic approach", 074 % OCEAN TECHNOLOGY series 6, Cambridge, pp 127. 075 % (Generalized Gamma distribution for wave height) 076 % 077 % Bitner-Gregersen, E.M. and Hagen, ุ. (2000) 078 % "Aspects of joint distribution for metocean Phenoma at the Norwegian 079 % Continental shelf." 080 % In Proc. of OMAE'2000, 081 082 % tested on: matlab 5.2 083 % history: 084 % revised pab 04.11.2000 085 % -updated calls to new wstats functions 086 % by Per A. Brodtkorb 13.05.2000 087 088 089 090 091 092 error(nargchk(2,6,nargin)) 093 094 if nargin < 5|isempty(def), def=0; end 095 if nargin < 6|isempty(condon), condon=0; end 096 097 Ph=[2.82 1.547 0 3.27 0.836 0.376]; % default values 098 if nargin < 4|isempty(Ph), 099 100 else 101 nh=length(Ph1); 102 ind=find(~isnan(Ph1(1:min(nh,6)))); 103 if any(ind) % replace default values with those from input data 104 Ph(ind)=Ph1(ind); 105 end 106 end 107 108 Pt=[1.59 0.42 2 0.005 0.09 0.13 1.34]; % default values 109 if nargin < 3|isempty(Pt), 110 else 111 nt=length(Pt1); 112 ind=find(~isnan(Pt1(1:min(nt,7)))); 113 if any(ind) % replace default values with those from input data 114 Pt(ind) = Pt1(ind); 115 end 116 end 117 118 [errorcode,t,h] = comnsize(t,h); 119 if errorcode > 0 120 error('Requires non-scalar arguments to match in size.'); 121 end 122 123 124 % Model parameters for the marginal distribution for Hz: 125 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 126 % scale, shape and location parameter for weibull distribution 127 % or shape,shape and scale parameter for the generalized Gamma distribution 128 C1=Ph(1);C2=Ph(2);C3=Ph(3); 129 % Split value, mean and standard deviation for Lognormal distribution 130 Hc=Ph(4);myHz=Ph(5);sHz=Ph(6); 131 132 % Model parameters for the conditional distribution of Tp or Tz given Hz 133 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 134 A1=Pt(1);A2=Pt(2);A3=Pt(3); 135 B1=Pt(4);B2=Pt(5);B3=Pt(6);B4=Pt(7); 136 % Mean and Variance of the lognormal distribution 137 if mod(abs(def),2)==0 138 myTp = A1+A2*log(h+A3); 139 else 140 myTp = A1+A2*h.^A3; 141 end 142 sTp =(B1+B2*exp(-B3*h.^B4)); 143 144 p=zeros(size(h)); 145 k0=find(t>0 & h>0); 146 if any(k0) 147 if condon==5, 148 p(k0) = wlogncdf(t(k0),myTp(k0),sTp(k0)); 149 elseif condon ==10 150 p(k0) = 1; 151 else 152 p(k0) = wlognpdf(t(k0),myTp(k0),sTp(k0)); 153 end 154 end 155 156 157 158 if condon==0 | condon==5 | condon ==10 159 k=find(0<h & h<=Hc ); 160 if any(k) 161 p(k)=p(k).*wlognpdf(h(k),myHz,sHz); 162 end 163 k1=find(h>Hc); 164 if any(k1) 165 if def<=1, 166 % NB! weibpdf must be modified to correspond to 167 % pdf=x^(b-1)/a^b*exp(-(x/a)^b) 168 p(k1)=p(k1).*wweibpdf(h(k1)-C3,C1,C2); 169 else 170 p(k1)=p(k1).*wggampdf(h(k1),C1,C2,C3); 171 end 172 end 173 end 174 175 switch condon, 176 case 0, % regular pdf is returned 177 case 1, %pdf conditioned on Hz ie. p(Tp|Hz) 178 case 3, % secret option used by ltwcstat: returns Tp*p(Tp|Hz) 179 p = t.*p 180 case 4, % secret option used by ltwcstat: returns Tp.^2*p(Tp|Hz) 181 p = t.^2.*p 182 case 5, % p(Hz)*P(Tp|Hz) is returned special case used by ltwccdf 183 case 10 % p(Hz) is returned 184 185 otherwise error('unknown option') 186 end 187 188 189 190 191 % The following is taken from Sagli (2000): 192 % Parameters for the long term description of the wave climate (Tp|Hs): 193 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 194 % 195 % Area Pt = [ A1 A2 A3 B1 B2 B3 B4] def 196 % จจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจ 197 % Northern North Sea: [1.59, 0.42, 2.00, 0.005, 0.09, 0.13, 1.34] 0 198 % Aasgard : [1.72, 0.34, 0.46, 0.005, 0.10, 0.29, 1.00] 1 199 % Sleipner : [0.23, 1.69, 0.15, 0.005, 0.12, 0.40, 1.00] 1 200 % Troms I : [1.35, 0.61, 0.34, 0.005, 0.12, 0.36, 1.00] 1 201 % Statfjord/Brent : [0.40, 1.59, 0.15, 0.005, 0.12, 0.34, 1.00] 1 202 % Ekofisk : [1.00, 0.90, 0.25, 0.005, 0.10, 0.44, 1.00] 1 203 % Ekofisk Zone II : [0.03, 1.81, 0.15, 0.005, 0.16, 0.58, 1.00] 1 204 % Ekofisk Zone III : [0.03, 1.81, 0.15, 0.005, 0.16, 0.58, 1.00] 1 205 % Ekofisk Zone IV : [1.41, 0.38, 0.54, 0.010, 0.08, 0.41, 1.00] 1 206 % Ekofisk Zone VI : [0.00, 1.72, 0.15, 0.010, 0.14, 1.00, 1.00] 1 207 % 208 % 209 % Parameters for the long term description of the wave climate (Hs): 210 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 211 % 212 % Area Ph = [ C1 C2 C3 Hc M1 S1 ] def 213 % จจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจ 214 % Northern North Sea: [2.82, 1.55, 0.00, 3.27, 0.84, 0.61^2 ] 0 215 % Aasgard : [2.73, 1.45, 0.00, 3.95, 0.84, 0.59^2 ] 1 216 % Sleipner : [2.38, 1.44, 0.00, 3.00, 0.69, 0.60^2 ] 1 217 % Troms I : [2.26, 1.36, 0.00, 4.25, 0.72, 0.56^2 ] 1 218 % Statfjord/Brent : [2.82, 1.55, 0.00, 3.27, 0.84, 0.61^2 ] 1 219 % Ekofisk : [2.08, 1.37, 0.00, 2.65, 0.52, 0.67^2 ] 1 220 % Ekofisk Zone II : [2.12, 1.37, 0.00, 2.70, 0.54, 0.67^2 ] 1 221 % Ekofisk Zone III : [2.05, 1.55, 0.00, 2.15, 0.50, 0.65^2 ] 1 222 % Ekofisk Zone IV : [1.56, 1.46, 0.00, 2.00, 0.25, 0.62^2 ] 1 223 % Ekofisk Zone VI : [1.18, 1.42, 0.00, 1.25, 0.07, 0.70^2 ] 1 224 % 225 226 227 % The following is taken from Bitner-Gregersen and Hagen (2000): 228 % Description of sites: 229 % V๘ring plateau: (position 67, 27'N 5, 58'E water depth 1460m) NORWAVE 230 % buoy every 3rd hour for a period of 2.3 years 231 % 232 233 % Parameters for the long term description of the wave climate (Tp|Hs): 234 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 235 % 236 % Area Pt = [ A1 A2 A3 B1 B2 B3 B4] def 237 % จจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจ 238 % V๘ring M : [1.718, 0.194, 0.642, -288.28, 288.52,0.279e-5 , 1] 1 239 % V๘ring M/H : [1.670, 0.220, 0.617, - 93.13, 93.37,0.330e-4 , 1] 1 240 % V๘ring H : [0.474, 1.438, 0.186, 0.047, 0.46, 0.476 , 1] 1 241 % Haltenbanken : [1.921, 0.171, 0.665, 0.064, 0.327, 0.227 , 1] 1 242 243 % Parameters for the long term description of the wave climate (Tz|Hs) 244 % (mean zero-down crossing period given Hz) : 245 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 246 % 247 % Area Pt = [ A1 A2 A3 B1 B2 B3 B4] def 248 % จจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจ 249 % Statfjord&Gullfaks: [1.790, 0.110, 0.759, 0.106, 0.361,0.969 , 1] 1 250 % Statfjord Platform: [1.771, 0.080, 0.902, 0.001, 0.239,0.169 , 1] 1 251 % Ekofisk : [0.611, 0.902, 0.289, 0.071, 0.211,0.606 , 1] 1 252 253 254 % Parameters for the long term description of the wave climate (Hs): 255 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 256 % 257 % Area Ph = [ C1 C2 C3 Hc ] def 258 % จจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจ 259 % V๘ring M : [2.371, 1.384, 0.755, 0 ] 1 260 % V๘ring M/H : [2.443, 1.405, 0.820, 0 ] 1 261 % V๘ring H : [2.304, 1.383, 0.899, 0 ] 1 262 % Haltenbanken : [2.154, 1.273, 0.763, 0 ] 1 263 % Statfjord&Gullfaks: [2.264, 1.398, 0.969, 0 ] 1 264 % Statfjord Platform: [2.502, 1.527, 0.657, 0 ] 1 265 % Ekofisk : [1.597, 1.184, 0.852, 0 ] 1 266 267 268 % Fitted parameters by splitting the data into wind Sea and swell: 269 270 % Parameters for the long term description of the wave climate (Tp|Hs): 271 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 272 % 273 % Area Pt = [ A1 A2 A3 B1 B2 B3 B4] def 274 % จจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจ 275 % V๘ring M : [-0.707, 2.421, 0.126, 0.076, 0.063, 0.162, 1] 1 (Wind Sea) 276 % V๘ring M : [ 1.501, 0.882, 0,135, 0.126, 0.000, 0.000, 1] 1 (Swell) 277 % Haltenbanken : [-4383.9, 4385.6, 0.948e-4, 0.045, 0.369,0.780,1] 1 (Wind Sea) 278 % Haltenbanken : [ 2.257, 0.088, 0.746, 0.010, 0.229, 0.167, 1] 1 (Swell) 279 280 % Parameters for the long term description of the wave climate (Hs): 281 %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 282 % 283 % Area Ph = [ C1 C2 C3 Hc ] def 284 % จจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจจ 285 % V๘ring M : [2.170, 1.382, 0.203, 0 ] 1 (Wind sea) 286 % V๘ring M : [1.606, 1.252, 0.229, 0 ] 1 (Swell) 287 % Haltenbanken : [1.473, 1.032, 0.241, 0 ] 1 (Wind sea) 288 % Haltenbanken : [1.661, 1.117, 0.240, 0 ] 1 (Swell) 289 290 291 292 293
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