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ltwcpdf

PURPOSE ^

Long Term Wave Climate PDF of significant wave height and wave period

SYNOPSIS ^

p = ltwcpdf(t,h,Pt1,Ph1, def,condon)

DESCRIPTION ^

 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

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

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

Mathematical Statistics
Centre for Mathematical Sciences
Lund University with Lund Institute of Technology

Comments or corrections to the WAFO group


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