#!/usr/bin/env python """ Models module ------------- Dispersion relation ------------------- k2w - Translates from wave number to frequency w2k - Translates from frequency to wave number Model spectra ------------- Bretschneider - Bretschneider spectral density. Jonswap - JONSWAP spectral density McCormick - McCormick spectral density. OchiHubble - OchiHubble bimodal spectral density model. Tmaspec - JONSWAP spectral density for finite water depth Torsethaugen - Torsethaugen double peaked (swell + wind) spectrum model Wallop - Wallop spectral density. demospec - Loads a precreated spectrum of chosen type jonswap_peakfact - Jonswap peakedness factor Gamma given Hm0 and Tp jonswap_seastate - jonswap seastate from windspeed and fetch Directional spreading functions ------------------------------- Spreading - Directional spreading function. """ from __future__ import absolute_import, division import warnings from scipy.interpolate import interp1d import scipy.optimize as optimize import scipy.integrate as integrate import scipy.special as sp from scipy.fftpack import fft import numpy as np from numpy import (inf, atleast_1d, newaxis, any, minimum, maximum, array, asarray, exp, log, sqrt, where, pi, arange, linspace, sin, cos, abs, sinh, isfinite, mod, expm1, tanh, cosh, finfo, ones, ones_like, isnan, zeros_like, flatnonzero, sinc, hstack, vstack, real, flipud, clip) from ..wave_theory.dispersion_relation import w2k, k2w # @UnusedImport from .core import SpecData1D, SpecData2D __all__ = ['Bretschneider', 'Jonswap', 'Torsethaugen', 'Wallop', 'McCormick', 'OchiHubble', 'Tmaspec', 'jonswap_peakfact', 'jonswap_seastate', 'spreading', 'w2k', 'k2w', 'phi1'] _EPS = finfo(float).eps def sech(x): return 1.0 / cosh(x) def _gengamspec(wn, N=5, M=4): ''' Return Generalized gamma spectrum in dimensionless form Parameters ---------- wn : arraylike normalized frequencies, w/wp. N : scalar defining the decay of the high frequency part. M : scalar defining the spectral width around the peak. Returns ------- S : arraylike spectral values, same size as wn. The generalized gamma spectrum in non- dimensional form is defined as: S = G0.*wn.**(-N).*exp(-B*wn.**(-M)) for wn > 0 = 0 otherwise where B = N/M C = (N-1)/M G0 = B**C*M/gamma(C), Normalizing factor related to Bretschneider form Note that N = 5, M = 4 corresponds to a normalized Bretschneider spectrum. Examples -------- >>> import wafo.spectrum.models as wsm >>> import numpy as np >>> wn = np.linspace(0,4,5) >>> wsm._gengamspec(wn, N=6, M=2) array([ 0. , 1.16765216, 0.17309961, 0.02305179, 0.00474686]) See also -------- Bretschneider Jonswap, Torsethaugen References ---------- Torsethaugen, K. (2004) "Simplified Double Peak Spectral Model for Ocean Waves" In Proc. 14th ISOPE ''' w = atleast_1d(wn) S = zeros_like(w) k = flatnonzero(w > 0.0) if k.size > 0: B = N / M C = (N - 1.0) / M # A = Normalizing factor related to Bretschneider form # A = B**C*M/gamma(C) # S[k] = A*wn[k]**(-N)*exp(-B*wn[k]**(-M)) logwn = log(w.take(k)) logA = (C * log(B) + log(M) - sp.gammaln(C)) S.put(k, exp(logA - N * logwn - B * exp(-M * logwn))) return S class ModelSpectrum(object): def __init__(self, Hm0=7.0, Tp=11.0, **kwds): self.Hm0 = Hm0 self.Tp = Tp self.type = 'ModelSpectrum' def tospecdata(self, w=None, wc=None, nw=257): ''' Return SpecData1D object from ModelSpectrum Parameter --------- w : arraylike vector of angular frequencies used in discretization of spectrum wc : scalar cut off frequency (default 33/Tp) nw : int number of frequencies Returns ------- S : SpecData1D object member attributes of model spectrum are copied to S.workspace ''' if w is None: if wc is None: wc = 33. / self.Tp w = linspace(0, wc, nw) S = SpecData1D(self.__call__(w), w) try: h = self.h S.h = h except: pass S.labels.title = self.type + ' ' + S.labels.title S.workspace = self.__dict__.copy() return S def chk_seastate(self): ''' Check if seastate is valid ''' if self.Hm0 < 0: raise ValueError('Hm0 can not be negative!') if self.Tp <= 0: raise ValueError('Tp must be positve!') if self.Hm0 == 0.0: warnings.warn('Hm0 is zero!') self._chk_extra_param() def _chk_extra_param(self): pass class Bretschneider(ModelSpectrum): ''' Bretschneider spectral density model Member variables ---------------- Hm0 : significant wave height (default 7 (m)) Tp : peak period (default 11 (sec)) N : scalar defining decay of high frequency part. (default 5) M : scalar defining spectral width around the peak. (default 4) Parameters ---------- w : array-like angular frequencies [rad/s] The Bretschneider spectrum is defined as S(w) = A * G0 * wn**(-N)*exp(-N/(M*wn**M)) where G0 = Normalizing factor related to Bretschneider form A = (Hm0/4)**2 / wp (Normalization factor) wn = w/wp wp = 2*pi/Tp, angular peak frequency This spectrum is a suitable model for fully developed sea, i.e. a sea state where the wind has been blowing long enough over a sufficiently open stretch of water, so that the high-frequency waves have reached an equilibrium. In the part of the spectrum where the frequency is greater than the peak frequency (w>wp), the energy distribution is proportional to w**-5. The spectrum is identical with ITTC (International Towing Tank Conference), ISSC (International Ship and Offshore Structures Congress) and Pierson-Moskowitz, wave spectrum given Hm0 and Tm01. It is also identical with JONSWAP when the peakedness factor, gamma, is one. For this spectrum, the following relations exist between the mean period Tm01 = 2*pi*m0/m1, the peak period Tp and the mean zero-upcrossing period Tz: Tm01 = 1.086*Tz, Tp = 1.408*Tz and Tp=1.2965*Tm01 Examples -------- >>> import wafo.spectrum.models as wsm >>> S = wsm.Bretschneider(Hm0=6.5,Tp=10) >>> S((0,1,2,3)) array([ 0. , 1.69350993, 0.06352698, 0.00844783]) See also -------- Jonswap, Torsethaugen ''' def __init__(self, Hm0=7.0, Tp=11.0, N=5, M=4, chk_seastate=True, **kwds): self.type = 'Bretschneider' self.Hm0 = Hm0 self.Tp = Tp self.N = N self.M = M if chk_seastate: self.chk_seastate() def __call__(self, wi): ''' Return Bretschnieder spectrum ''' w = atleast_1d(wi) if self.Hm0 > 0: wp = 2 * pi / self.Tp wn = w / wp S = (self.Hm0 / 4.0) ** 2 / wp * _gengamspec(wn, self.N, self.M) else: S = zeros_like(w) return S def jonswap_peakfact(Hm0, Tp): ''' Jonswap peakedness factor, gamma, given Hm0 and Tp Parameters ---------- Hm0 : significant wave height [m]. Tp : peak period [s] Returns ------- gamma : Peakedness parameter of the JONSWAP spectrum Details ------- A standard value for GAMMA is 3.3. However, a more correct approach is to relate GAMMA to Hm0 and Tp: D = 0.036-0.0056*Tp/sqrt(Hm0) gamma = exp(3.484*(1-0.1975*D*Tp**4/(Hm0**2))) This parameterization is based on qualitative considerations of deep water wave data from the North Sea, see Torsethaugen et. al. (1984) Here GAMMA is limited to 1..7. NOTE: The size of GAMMA is the common shape of Hm0 and Tp. Examples -------- >>> import wafo.spectrum.models as wsm >>> import pylab as plb >>> Tp,Hs = plb.meshgrid(range(4,8),range(2,6)) >>> gam = wsm.jonswap_peakfact(Hs,Tp) >>> Hm0 = plb.linspace(1,20) >>> Tp = Hm0 >>> [T,H] = plb.meshgrid(Tp,Hm0) >>> gam = wsm.jonswap_peakfact(H,T) >>> v = plb.arange(0,8) >>> h = plb.contourf(Tp,Hm0,gam,v);h=plb.colorbar() >>> Hm0 = plb.arange(1,11) >>> Tp = plb.linspace(2,16) >>> T,H = plb.meshgrid(Tp,Hm0) >>> gam = wsm.jonswap_peakfact(H,T) >>> h = plb.plot(Tp,gam.T) >>> h = plb.xlabel('Tp [s]') >>> h = plb.ylabel('Peakedness parameter') >>> plb.close('all') See also -------- jonswap ''' Hm0, Tp = atleast_1d(Hm0, Tp) x = Tp / sqrt(Hm0) gam = ones_like(x) k1 = flatnonzero(x <= 5.14285714285714) if k1.size > 0: # limiting gamma to [1 7] xk = x.take(k1) D = 0.036 - 0.0056 * xk # approx 5.061*Hm0**2/Tp**4*(1-0.287*log(gam)) # gamma gam.put(k1, minimum(exp(3.484 * (1.0 - 0.1975 * D * xk ** 4.0)), 7.0)) return gam def jonswap_seastate(u10, fetch=150000., method='lewis', g=9.81, output='dict'): ''' Return Jonswap seastate from windspeed and fetch Parameters ---------- U10 : real scalar windspeed at 10 m above mean water surface [m/s] fetch : real scalar fetch [m] method : 'hasselman73' seastate according to Hasselman et. al. 1973 'hasselman76' seastate according to Hasselman et. al. 1976 'lewis' seastate according to Lewis and Allos 1990 g : real scalar accelaration of gravity [m/s**2] output : 'dict' or 'list' Returns ------- seastate: dict where Hm0 : significant wave height [m] Tp : peak period [s] gamma : jonswap peak enhancement factor. sigmaA, sigmaB : jonswap spectral width parameters. Ag : jonswap alpha, normalization factor. Example -------- >>> import wafo.spectrum.models as wsm >>> fetch = 10000; u10 = 10 >>> ss = wsm.jonswap_seastate(u10, fetch, output='dict') >>> for key in sorted(ss.keys()): key, ss[key] ('Ag', 0.016257903375341734) ('Hm0', 0.51083679198275533) ('Tp', 2.7727680999585265) ('gamma', 2.4824142635861119) ('sigmaA', 0.07531733139517202) ('sigmaB', 0.09191208451225134) >>> S = wsm.Jonswap(**ss) >>> S.Hm0 0.51083679198275533 # Alternatively >>> ss1 = wsm.jonswap_seastate(u10, fetch, output='list') >>> S1 = wsm.Jonswap(*ss1) >>> S1.Hm0 0.51083679198275533 See also -------- Jonswap References ---------- Lewis, A. W. and Allos, R.N. (1990) JONSWAP's parameters: sorting out the inconscistencies. Ocean Engng, Vol 17, No 4, pp 409-415 Hasselmann et al. (1973) Measurements of Wind-Wave Growth and Swell Decay during the Joint North Sea Project (JONSWAP). Ergansungsheft, Reihe A(8), Nr. 12, Deutschen Hydrografischen Zeitschrift. Hasselmann et al. (1976) A parametric wave prediction model. J. phys. oceanogr. Vol 6, pp 200-228 ''' # The following formulas are from Lewis and Allos 1990: zeta = g * fetch / (u10 ** 2) # dimensionless fetch, Table 1 # zeta = min(zeta, 2.414655013429281e+004) if method.startswith('h'): if method[-1] == '3': # Hasselman et.al (1973) A = 0.076 * zeta ** (-0.22) # dimensionless peakfrequency, Table 1 ny = 3.5 * zeta ** (-0.33) # dimensionless surface variance, Table 1 epsilon1 = 9.91e-8 * zeta ** 1.1 else: # Hasselman et.al (1976) A = 0.0662 * zeta ** (-0.2) ny = 2.84 * zeta ** (-0.3) # dimensionless peakfrequency, Table 1 # dimensionless surface variance, Eq.4 epsilon1 = 1.6e-7 * zeta sa = 0.07 sb = 0.09 gam = 3.3 else: A = 0.074 * zeta ** (-0.22) # Eq. 10 ny = 3.57 * zeta ** (-0.33) # dimensionless peakfrequency, Eq. 11 # dimensionless surface variance, Eq.12 epsilon1 = 3.512e-4 * A * ny ** (-4.) * zeta ** (-0.1) sa = 0.05468 * ny ** (-0.32) # Eq. 13 sb = 0.078314 * ny ** (-0.16) # Eq. 14 gam = maximum(17.54 * zeta ** (-0.28384), 1) # Eq. 15 Tp = u10 / (ny * g) # Table 1 Hm0 = 4 * sqrt(epsilon1) * u10 ** 2. / g # Table 1 if output[0] == 'l': return Hm0, Tp, gam, sa, sb, A else: return dict(Hm0=Hm0, Tp=Tp, gamma=gam, sigmaA=sa, sigmaB=sb, Ag=A) class Jonswap(ModelSpectrum): ''' Jonswap spectral density model Member variables ---------------- Hm0 : significant wave height (default 7 (m)) Tp : peak period (default 11 (sec)) gamma : peakedness factor determines the concentraton of the spectrum on the peak frequency. Usually in the range 1 <= gamma <= 7. default depending on Hm0, Tp, see jonswap_peakedness) sigmaA : spectral width parameter for w1: N : scalar defining decay of high frequency part. (default 5) M : scalar defining spectral width around the peak. (default 4) method : String defining method used to estimate Ag when gamma>1 'integration': Ag = 1/gaussq(Gf*ggamspec(wn,N,M),0,wnc) (default) 'parametric' : Ag = (1+f1(N,M)*log(gamma)**f2(N,M))/gamma 'custom' : Ag = Ag wnc : wc/wp normalized cut off frequency used when calculating Ag by integration (default 6) Parameters ---------- w : array-like angular frequencies [rad/s] Description ----------- The JONSWAP spectrum is defined as S(w) = A * Gf * G0 * wn**(-N)*exp(-N/(M*wn**M)) where G0 = Normalizing factor related to Bretschneider form A = Ag * (Hm0/4)**2 / wp (Normalization factor) Gf = j**exp(-.5*((wn-1)/s)**2) (Peak enhancement factor) wn = w/wp wp = angular peak frequency s = sigmaA for wn <= 1 sigmaB for 1 < wn j = gamma, (j=1, => Bretschneider spectrum) The JONSWAP spectrum is assumed to be especially suitable for the North Sea, and does not represent a fully developed sea. It is a reasonable model for wind generated sea when the seastate is in the so called JONSWAP range, i.e., 3.6*sqrt(Hm0) < Tp < 5*sqrt(Hm0) The relation between the peak period and mean zero-upcrossing period may be approximated by Tz = Tp/(1.30301-0.01698*gamma+0.12102/gamma) Examples --------- >>> import pylab as plb >>> import wafo.spectrum.models as wsm >>> S = wsm.Jonswap(Hm0=7, Tp=11,gamma=1) >>> w = plb.linspace(0,5) >>> h = plb.plot(w,S(w)) >>> S2 = wsm.Bretschneider(Hm0=7, Tp=11) >>> all(abs(S(w)-S2(w))<1.e-7) True >>> plb.close('all') See also -------- Bretschneider Tmaspec Torsethaugen References ----------- Torsethaugen et al. (1984) Characteristica for extreme Sea States on the Norwegian continental shelf. Report No. STF60 A84123. Norwegian Hydrodyn. Lab., Trondheim Hasselmann et al. (1973) Measurements of Wind-Wave Growth and Swell Decay during the Joint North Sea Project (JONSWAP). Ergansungsheft, Reihe A(8), Nr. 12, Deutschen Hydrografischen Zeitschrift. ''' def __init__(self, Hm0=7.0, Tp=11.0, gamma=None, sigmaA=0.07, sigmaB=0.09, Ag=None, N=5, M=4, method='integration', wnc=6.0, chk_seastate=True): self.type = 'Jonswap' self.Hm0 = Hm0 self.Tp = Tp self.N = N self.M = M self.sigmaA = sigmaA self.sigmaB = sigmaB self.gamma = gamma self.Ag = Ag self.method = method self.wnc = wnc if self.gamma is None or not isfinite(self.gamma) or self.gamma < 1: self.gamma = jonswap_peakfact(Hm0, Tp) self._preCalculateAg() if chk_seastate: self.chk_seastate() def _chk_extra_param(self): Tp = self.Tp Hm0 = self.Hm0 gam = self.gamma outsideJonswapRange = Tp > 5 * sqrt(Hm0) or Tp < 3.6 * sqrt(Hm0) if outsideJonswapRange: txt0 = ''' Hm0=%g,Tp=%g is outside the JONSWAP range. The validity of the spectral density is questionable. ''' % (Hm0, Tp) warnings.warn(txt0) if gam < 1 or 7 < gam: txt = ''' The peakedness factor, gamma, is possibly too large. The validity of the spectral density is questionable. ''' warnings.warn(txt) def _localspec(self, wn): Gf = self.peak_e_factor(wn) return Gf * _gengamspec(wn, self.N, self.M) def _preCalculateAg(self): ''' PRECALCULATEAG Precalculate normalization. ''' if self.gamma == 1: self.Ag = 1.0 self.method = 'parametric' elif self.Ag is not None: self.method = 'custom' if self.Ag <= 0: raise ValueError('Ag must be larger than 0!') elif self.method[0] == 'i': # normalizing by integration self.method = 'integration' if self.wnc < 1.0: raise ValueError('Normalized cutoff frequency, wnc, ' + 'must be larger than one!') area1, unused_err1 = integrate.quad(self._localspec, 0, 1) area2, unused_err2 = integrate.quad(self._localspec, 1, self.wnc) area = area1 + area2 self.Ag = 1.0 / area elif self.method[1] == 'p': self.method = 'parametric' # Original normalization # NOTE: that Hm0**2/16 generally is not equal to intS(w)dw # with this definition of Ag if sa or sb are changed from the # default values N = self.N M = self.M gammai = self.gamma parametersOK = (3 <= N and N <= 50) or ( 2 <= M and M <= 9.5) and (1 <= gammai and gammai <= 20) if parametersOK: f1NM = 4.1 * \ (N - 2 * M ** 0.28 + 5.3) ** (-1.45 * M ** 0.1 + 0.96) f2NM = (2.2 * M ** (-3.3) + 0.57) * \ N ** (-0.58 * M ** 0.37 + 0.53) - 1.04 * M ** (-1.9) + 0.94 self.Ag = (1 + f1NM * log(gammai) ** f2NM) / gammai # elseif N == 5 && M == 4, # options.Ag = (1+1.0*log(gammai).**1.16)/gammai # options.Ag = (1-0.287*log(gammai)) # options.normalizeMethod = 'Three' # elseif N == 4 && M == 4, # options.Ag = (1+1.1*log(gammai).**1.19)/gammai else: raise ValueError('Not knowing the normalization because N, ' + 'M or peakedness parameter is out of bounds!') if self.sigmaA != 0.07 or self.sigmaB != 0.09: warnings.warn('Use integration to calculate Ag when ' + 'sigmaA!=0.07 or sigmaB!=0.09') def peak_e_factor(self, wn): ''' PEAKENHANCEMENTFACTOR ''' w = maximum(atleast_1d(wn), 0.0) sab = where(w > 1, self.sigmaB, self.sigmaA) wnm12 = 0.5 * ((w - 1.0) / sab) ** 2.0 Gf = self.gamma ** (exp(-wnm12)) return Gf def __call__(self, wi): ''' JONSWAP spectral density ''' w = atleast_1d(wi) if (self.Hm0 > 0.0): N = self.N M = self.M wp = 2 * pi / self.Tp wn = w / wp Ag = self.Ag Hm0 = self.Hm0 Gf = self.peak_e_factor(wn) S = ((Hm0 / 4.0) ** 2 / wp * Ag) * Gf * _gengamspec(wn, N, M) else: S = zeros_like(w) return S def phi1(wi, h, g=9.81): ''' Factor transforming spectra to finite water depth spectra. Input ----- w : arraylike angular frequency [rad/s] h : scalar water depth [m] g : scalar acceleration of gravity [m/s**2] Returns ------- tr : arraylike transformation factors Example: ------- Transform a JONSWAP spectrum to a spectrum for waterdepth = 30 m >>> import wafo.spectrum.models as wsm >>> S = wsm.Jonswap() >>> w = np.arange(3.0) >>> S(w)*wsm.phi1(w,30.0) array([ 0. , 1.0358056 , 0.03796281]) Reference --------- Buows, E., Gunther, H., Rosenthal, W. and Vincent, C.L. (1985) 'Similarity of the wind wave spectrum in finite depth water: 1 spectral form.' J. Geophys. Res., Vol 90, No. C1, pp 975-986 ''' w = atleast_1d(wi) if h == inf: # % special case infinite water depth return ones_like(w) k1 = w2k(w, 0, inf, g=g)[0] dw1 = 2.0 * w / g # % dw/dk|h=inf k2 = w2k(w, 0, h, g=g)[0] k2h = k2 * h den = where(k1 == 0, 1, (tanh(k2h) + k2h / cosh(k2h) ** 2.0)) dw2 = where(k1 == 0, 0, dw1 / den) # dw/dk|h=h0 return where(k1 == 0, 0, (k1 / k2) ** 3.0 * dw2 / dw1) class Tmaspec(Jonswap): ''' JONSWAP spectrum for finite water depth Member variables ---------------- h = water depth (default 42 [m]) g : acceleration of gravity [m/s**2] Hm0 = significant wave height (default 7 [m]) Tp = peak period (default 11 (sec)) gamma = peakedness factor determines the concentraton of the spectrum on the peak frequency. Usually in the range 1 <= gamma <= 7. default depending on Hm0, Tp, see getjonswappeakedness) sigmaA = spectral width parameter for w1: N = scalar defining decay of high frequency part. (default 5) M = scalar defining spectral width around the peak. (default 4) method = String defining method used to estimate Ag when gamma>1 'integrate' : Ag = 1/gaussq(Gf.*ggamspec(wn,N,M),0,wnc) (default) 'parametric': Ag = (1+f1(N,M)*log(gamma)^f2(N,M))/gamma 'custom' : Ag = Ag wnc = wc/wp normalized cut off frequency used when calculating Ag by integration (default 6) Parameters ---------- w : array-like angular frequencies [rad/s] Description ------------ The evaluated spectrum is S(w) = Sj(w)*phi(w,h) where Sj = jonswap spectrum phi = modification due to water depth The concept is based on a similarity law, and its validity is verified through analysis of 3 data sets from: TEXEL, MARSEN projects (North Sea) and ARSLOE project (Duck, North Carolina, USA). The data include observations at water depths ranging from 6 m to 42 m. Example -------- >>> import wafo.spectrum.models as wsm >>> import pylab as plb >>> w = plb.linspace(0,2.5) >>> S = wsm.Tmaspec(h=10,gamma=1) # Bretschneider spectrum Hm0=7, Tp=11 >>> o=plb.plot(w,S(w)) >>> o=plb.plot(w,S(w,h=21)) >>> o=plb.plot(w,S(w,h=42)) >>> plb.show() >>> plb.close('all') See also --------- Bretschneider, Jonswap, phi1, Torsethaugen References ---------- Buows, E., Gunther, H., Rosenthal, W., and Vincent, C.L. (1985) 'Similarity of the wind wave spectrum in finite depth water: 1 spectral form.' J. Geophys. Res., Vol 90, No. C1, pp 975-986 Hasselman et al. (1973) Measurements of Wind-Wave Growth and Swell Decay during the Joint North Sea Project (JONSWAP). Ergansungsheft, Reihe A(8), Nr. 12, deutschen Hydrografischen Zeitschrift. ''' def __init__(self, Hm0=7.0, Tp=11.0, gamma=None, sigmaA=0.07, sigmaB=0.09, Ag=None, N=5, M=4, method='integration', wnc=6.0, chk_seastate=True, h=42, g=9.81): self.g = g self.h = h super(Tmaspec, self).__init__(Hm0, Tp, gamma, sigmaA, sigmaB, Ag, N, M, method, wnc, chk_seastate) self.type = 'TMA' def phi(self, w, h=None, g=None): if h is None: h = self.h if g is None: g = self.g return phi1(w, h, g) def __call__(self, w, h=None, g=None): jonswap = super(Tmaspec, self).__call__(w) return jonswap * self.phi(w, h, g) class Torsethaugen(ModelSpectrum): ''' Torsethaugen double peaked (swell + wind) spectrum model Member variables ---------------- Hm0 : significant wave height (default 7 (m)) Tp : peak period (default 11 (sec)) wnc : wc/wp normalized cut off frequency used when calculating Ag by integration (default 6) method : String defining method used to estimate normalization factors, Ag, in the the modified JONSWAP spectra when gamma>1 'integrate' : Ag = 1/quad(Gf.*gengamspec(wn,N,M),0,wnc) 'parametric': Ag = (1+f1(N,M)*log(gamma)**f2(N,M))/gamma Parameters ---------- w : array-like angular frequencies [rad/s] Description ----------- The double peaked (swell + wind) Torsethaugen spectrum is modelled as S(w) = Ss(w) + Sw(w) where Ss and Sw are modified JONSWAP spectrums for swell and wind peak, respectively. The energy is divided between the two peaks according to empirical parameters, which peak that is primary depends on parameters. The empirical parameters are found for classes of Hm0 and Tp, originating from a dataset consisting of 20 000 spectra divided into 146 different classes of Hm0 and Tp. (Data measured at the Statfjord field in the North Sea in a period from 1980 to 1989.) The range of the measured Hm0 and Tp for the dataset are from 0.5 to 11 meters and from 3.5 to 19 sec, respectively. Preliminary comparisons with spectra from other areas indicate that some of the empirical parameters are dependent on geographical location. Thus the model must be used with care for other areas than the North Sea and sea states outside the area where measured data are available. Example ------- >>> import wafo.spectrum.models as wsm >>> import pylab as plb >>> w = plb.linspace(0,4) >>> S = wsm.Torsethaugen(Hm0=6, Tp=8) >>> h=plb.plot(w,S(w),w,S.wind(w),w,S.swell(w)) See also -------- Bretschneider Jonswap References ---------- Torsethaugen, K. (2004) "Simplified Double Peak Spectral Model for Ocean Waves" In Proc. 14th ISOPE Torsethaugen, K. (1996) Model for a doubly peaked wave spectrum Report No. STF22 A96204. SINTEF Civil and Environm. Engineering, Trondheim Torsethaugen, K. (1994) 'Model for a doubly peaked spectrum. Lifetime and fatigue strength estimation implications.' International Workshop on Floating Structures in Coastal zone, Hiroshima, November 1994. Torsethaugen, K. (1993) 'A two peak wave spectral model.' In proceedings OMAE, Glasgow ''' def __init__(self, Hm0=7, Tp=11, method='integration', wnc=6, gravity=9.81, chk_seastate=True, **kwds): self.type = 'Torsethaugen' self.Hm0 = Hm0 self.Tp = Tp self.method = method self.wnc = wnc self.gravity = gravity self.wind = None self.swell = None if chk_seastate: self.chk_seastate() self._init_spec() def __call__(self, w): ''' TORSETHAUGEN spectral density ''' return self.wind(w) + self.swell(w) def _chk_extra_param(self): Hm0 = self.Hm0 Tp = self.Tp if Hm0 > 11 or Hm0 > max((Tp / 3.6) ** 2, (Tp - 2) * 12 / 11): txt0 = '''Hm0 is outside the valid range. The validity of the spectral density is questionable''' warnings.warn(txt0) if Tp > 20 or Tp < 3: txt1 = '''Tp is outside the valid range. The validity of the spectral density is questionable''' warnings.warn(txt1) def _init_spec(self): ''' Initialize swell and wind part of Torsethaugen spectrum ''' monitor = 0 Hm0 = self.Hm0 Tp = self.Tp gravity1 = self.gravity # m/s**2 min = minimum # @ReservedAssignment max = maximum # @ReservedAssignment # The parameter values below are found comparing the # model to average measured spectra for the Statfjord Field # in the Northern North Sea. Af = 6.6 # m**(-1/3)*sec AL = 2 # sec/sqrt(m) Au = 25 # sec KG = 35 KG0 = 3.5 KG1 = 1 # m r = 0.857 # 6/7 K0 = 0.5 # 1/sqrt(m) K00 = 3.2 M0 = 4 B1 = 2 # sec B2 = 0.7 B3 = 3.0 # m S0 = 0.08 # m**2*s S1 = 3 # m # Preliminary comparisons with spectra from other areas indicate that # the parameters on the line below can be dependent on geographical # location A10 = 0.7 A1 = 0.5 A20 = 0.6 A2 = 0.3 A3 = 6 Tf = Af * (Hm0) ** (1.0 / 3.0) Tl = AL * sqrt(Hm0) # lower limit Tu = Au # upper limit # Non-dimensional scales # New call pab April 2005 El = min(max((Tf - Tp) / (Tf - Tl), 0), 1) # wind sea Eu = min(max((Tp - Tf) / (Tu - Tf), 0), 1) # Swell if Tp < Tf: # Wind dominated seas # Primary peak (wind dominated) Nw = K0 * sqrt(Hm0) + K00 # high frequency exponent Mw = M0 # spectral width exponent Rpw = min((1 - A10) * exp(-(El / A1) ** 2) + A10, 1) Hpw = Rpw * Hm0 # significant waveheight wind Tpw = Tp # primary peak period # peak enhancement factor gammaw = KG * (1 + KG0 * exp(-Hm0 / KG1)) * \ (2 * pi / gravity1 * Rpw * Hm0 / (Tp ** 2)) ** r gammaw = max(gammaw, 1) # Secondary peak (swell) Ns = Nw # high frequency exponent Ms = Mw # spectral width exponent Rps = sqrt(1.0 - Rpw ** 2.0) Hps = Rps * Hm0 # significant waveheight swell Tps = Tf + B1 gammas = 1.0 if monitor: if Rps > 0.1: print(' Spectrum for Wind dominated sea') else: print(' Spectrum for pure wind sea') else: # swell dominated seas # Primary peak (swell) Ns = K0 * sqrt(Hm0) + K00 # high frequency exponent Ms = M0 # spectral width exponent Rps = min((1 - A20) * exp(-(Eu / A2) ** 2) + A20, 1) Hps = Rps * Hm0 # significant waveheight swell Tps = Tp # primary peak period # peak enhancement factor gammas = KG * (1 + KG0 * exp(-Hm0 / KG1)) * \ (2 * pi / gravity1 * Hm0 / (Tf ** 2)) ** r * (1 + A3 * Eu) gammas = max(gammas, 1) # Secondary peak (wind) Nw = Ns # high frequency exponent Mw = M0 * (1 - B2 * exp(-Hm0 / B3)) # spectral width exponent Rpw = sqrt(1 - Rps ** 2) Hpw = Rpw * Hm0 # significant waveheight wind C = (Nw - 1) / Mw B = Nw / Mw G0w = B ** C * Mw / sp.gamma(C) # normalizing factor # G0w = exp(C*log(B)+log(Mw)-gammaln(C)) # G0w = Mw/((B)**(-C)*gamma(C)) if Hpw > 0: Tpw = (16 * S0 * (1 - exp(-Hm0 / S1)) * (0.4) ** Nw / (G0w * Hpw ** 2)) ** (-1.0 / (Nw - 1.0)) else: Tpw = inf # Tpw = max(Tpw,2.5) gammaw = 1 if monitor: if Rpw > 0.1: print(' Spectrum for swell dominated sea') else: print(' Spectrum for pure swell sea') if monitor: if (3.6 * sqrt(Hm0) <= Tp & Tp <= 5 * sqrt(Hm0)): print(' Jonswap range') print('Hm0 = %g' % Hm0) print('Ns, Ms = %g, %g Nw, Mw = %g, %g' % (Ns, Ms, Nw, Mw)) print('gammas = %g gammaw = ' % (gammas, gammaw)) print('Rps = %g Rpw = %g' % (Rps, Rpw)) print('Hps = %g Hpw = %g' % (Hps, Hpw)) print('Tps = %g Tpw = %g' % (Tps, Tpw)) # G0s=Ms/((Ns/Ms)**(-(Ns-1)/Ms)*gamma((Ns-1)/Ms )) #normalizing factor # Wind part self.wind = Jonswap(Hm0=Hpw, Tp=Tpw, gamma=gammaw, N=Nw, M=Mw, method=self.method, chk_seastate=False) # Swell part self.swell = Jonswap(Hm0=Hps, Tp=Tps, gamma=gammas, N=Ns, M=Ms, method=self.method, chk_seastate=False) class McCormick(Bretschneider): ''' McCormick spectral density model Member variables ---------------- Hm0 = significant wave height (default 7 (m)) Tp = peak period (default 11 (sec)) Tz = zero-down crossing period (default 0.8143*Tp) M = scalar defining spectral width around the peak. (default depending on Tp and Tz) Parameters ---------- w : array-like angular frequencies [rad/s] Description ----------- The McCormick spectrum parameterization is a modification of the Bretschneider spectrum and defined as S(w) = (M+1)*(Hm0/4)^2/wp*(wp./w)^(M+1)*exp(-(M+1)/M*(wp/w)^M) where Tp/Tz=(1+1/M)^(1/M)/gamma(1+1/M) Example: -------- >>> import wafo.spectrum.models as wsm >>> S = wsm.McCormick(Hm0=6.5,Tp=10) >>> S(range(4)) array([ 0. , 1.87865908, 0.15050447, 0.02994663]) See also -------- Bretschneider Jonswap, Torsethaugen References: ----------- M.E. McCormick (1999) "Application of the Generic Spectral Formula to Fetch-Limited Seas" Marine Technology Society, Vol 33, No. 3, pp 27-32 ''' def __init__(self, Hm0=7, Tp=11, Tz=None, M=None, chk_seastate=True): self.type = 'McCormick' self.Hm0 = Hm0 self.Tp = Tp if Tz is None: Tz = 0.8143 * Tp self.Tz = Tz if chk_seastate: self.chk_seastate() if M is None and self.Hm0 > 0: self._TpdTz = Tp / Tz M = 1.0 / optimize.fminbound(self._localoptfun, 0.01, 5) self.M = M self.N = M + 1.0 def _localoptfun(self, x): # LOCALOPTFUN Local function to optimize. y = 1.0 + x return (y ** (x) / sp.gamma(y) - self._TpdTz) ** 2.0 class OchiHubble(ModelSpectrum): ''' OchiHubble bimodal spectral density model. Member variables ---------------- Hm0 : significant wave height (default 7 (m)) par : integer defining the parametrization (default 0) 0 : The most probable spectrum 1,2,...10 : gives 95% Confidence spectra The OchiHubble bimodal spectrum is modelled as S(w) = Ss(w) + Sw(w) where Ss and Sw are modified Bretschneider spectra for swell and wind peak, respectively. The OH spectrum is a six parameter spectrum, all functions of Hm0. The values of these parameters are determined from a analysis of data obtained in the North Atlantic. The source of the data is the same as that for the development of the Pierson-Moskowitz spectrum, but analysis is carried out on over 800 spectra including those in partially developed seas and those having a bimodal shape. From a statistical analysis of the data, a family of wave spectra consisting of 11 members is generated for a desired sea severity (Hm0) with the coefficient of 0.95. A significant advantage of using a family of spectra for design of marine systems is that one of the family members yields the largest response such as motions or wave induced forces for a specified sea severity, while another yields the smallest response with confidence coefficient of 0.95. Examples -------- >>> import wafo.spectrum.models as wsm >>> S = wsm.OchiHubble(par=2) >>> S(range(4)) array([ 0. , 0.90155636, 0.04185445, 0.00583207]) See also -------- Bretschneider, Jonswap, Torsethaugen References: ---------- Ochi, M.K. and Hubble, E.N. (1976) 'On six-parameter wave spectra.' In Proc. 15th Conf. Coastal Engng., Vol.1, pp301-328 ''' def __init__(self, Hm0=7, par=1, chk_seastate=True): self.type = 'Ochi Hubble' self.Hm0 = Hm0 self.Tp = 1 self.par = par self.wind = None self.swell = None if chk_seastate: self.chk_seastate() self._init_spec() def __call__(self, w): return self.wind(w) + self.swell(w) def _init_spec(self): hp = array([[0.84, 0.54], [0.84, 0.54], [0.84, 0.54], [0.84, 0.54], [0.84, 0.54], [0.95, 0.31], [0.65, 0.76], [0.90, 0.44], [0.77, 0.64], [0.73, 0.68], [0.92, 0.39]]) wa = array([[0.7, 1.15], [0.93, 1.5], [0.41, 0.88], [0.74, 1.3], [0.62, 1.03], [0.70, 1.50], [0.61, 0.94], [0.81, 1.60], [0.54, 0.61], [0.70, 0.99], [0.70, 1.37]]) wb = array([[0.046, 0.039], [0.056, 0.046], [0.016, 0.026], [0.052, 0.039], [0.039, 0.030], [0.046, 0.046], [0.039, 0.036], [0.052, 0.033], [0.039, 0.000], [0.046, 0.039], [0.046, 0.039]]) Lpar = array([[3.00, 1.54, -0.062], [3.00, 2.77, -0.112], [2.55, 1.82, -0.089], [2.65, 3.90, -0.085], [2.60, 0.53, -0.069], [1.35, 2.48, -0.102], [4.95, 2.48, -0.102], [1.80, 2.95, -0.105], [4.50, 1.95, -0.082], [6.40, 1.78, -0.069], [0.70, 1.78, -0.069]]) Hm0 = self.Hm0 Lpari = Lpar[self.par] Li = hstack((Lpari[0], Lpari[1] * exp(Lpari[2] * Hm0))) Hm0i = hp[self.par] * Hm0 Tpi = 2 * pi * exp(wb[self.par] * Hm0) / wa[self.par] Ni = 4 * Li + 1 Mi = [4, 4] self.swell = Bretschneider(Hm0=Hm0i[0], Tp=Tpi[0], N=Ni[0], M=Mi[0]) self.wind = Bretschneider(Hm0=Hm0i[1], Tp=Tpi[1], N=Ni[1], M=Mi[1]) def _chk_extra_param(self): if self.par < 0 or 10 < self.par: raise ValueError('Par must be an integer from 0 to 10!') class Wallop(Bretschneider): '''Wallop spectral density model. Member variables ---------------- Hm0 = significant wave height (default 7 (m)) Tp = peak period (default 11 (sec)) N = shape factor, i.e. slope for the high frequency % part (default depending on Hm0 and Tp, see below) Parameters ---------- w : array-like angular frequencies [rad/s] Description ----------- The WALLOP spectrum parameterization is a modification of the Bretschneider spectrum and defined as S(w) = A * G0 * wn**(-N)*exp(-N/(4*wn**4)) where G0 = Normalizing factor related to Bretschneider form A = (Hm0/4)^2 / wp (Normalization factor) wn = w/wp wp = 2*pi/Tp, angular peak frequency N = abs((log(2*pi^2)+2*log(Hm0/4)-2*log(Lp))/log(2)) Lp = wave length corresponding to the peak frequency, wp. If N=5 it becomes the same as the JONSWAP spectrum with peak enhancement factor gamma=1 or the Bretschneider (Pierson-Moskowitz) spectrum. Example: -------- >>> import wafo.spectrum.models as wsm >>> S = wsm.Wallop(Hm0=6.5, Tp=10) >>> S(range(4)) array([ 0.00000000e+00, 9.36921871e-01, 2.76991078e-03, 7.72996150e-05]) See also -------- Bretschneider Jonswap, Torsethaugen References: ----------- Huang, N.E., Long, S.R., Tung, C.C, Yuen, Y. and Bilven, L.F. (1981) "A unified two parameter wave spectral model for a generous sea state" J. Fluid Mechanics, Vol.112, pp 203-224 ''' def __init__(self, Hm0=7, Tp=11, N=None, chk_seastate=True): self.type = 'Wallop' self.Hm0 = Hm0 self.Tp = Tp self.M = 4 if N is None: wp = 2. * pi / Tp kp = w2k(wp, 0, inf)[0] # wavenumber at peak frequency Lp = 2. * pi / kp # wave length at the peak frequency N = abs((log(2. * pi ** 2.) + 2 * log(Hm0 / 4) - 2.0 * log(Lp)) / log(2)) self.N = N if chk_seastate: self.chk_seastate() class Spreading(object): ''' Directional spreading function. Parameters ---------- theta, w : arrays angles and angular frequencies given in radians and rad/s, respectively. Lenghts are Nt and Nw. wc : real scalar cut over frequency Returns ------- D : 2D array Directonal spreading function. size Nt X Nw. The principal direction of D is always along the x-axis. phi0 : real scalar Parameter defining the actual principal direction of D. Member variables ---------------- type : string (default 'cos-2s') type of spreading function, see options below 'cos-2s' : N(S)*[cos((theta-theta0)/2)]**(2*S) (0 < S) 'Box-car' : N(A)*I( -A < theta-theta0 < A) (0 < A < pi) 'von-Mises' : N(K)*exp(K*cos(theta-theta0)) (0 < K) 'Poisson' : N(X)/(1-2*X*cos(theta-theta0)+X**2) (0 < X < 1) 'sech-2' : N(B)*sech(B*(theta-theta0))**2 (0 < B) 'wrapped-normal': [1 + 2*sum exp(-(n*D1)^2/2)*cos(n*(theta-theta0))]/(2*pi) (0 < D1) (N(.) = normalization factor) (the first letter is enough for unique identification) theta0 : callable, matrix or a scalar defines average direction given in radians at every angular frequency. (length 1 or length == length(wn)) (default 0) method : string or integer Defines function used for direcional spreading parameter: 0, None : S(wn) = s_a, frequency independent 1, 'mitsuyasu': S(wn) frequency dependent (default) where S(wn) = s_a *(wn)**m_a, for wn_lo <= wn < wn_c = s_b *(wn)**m_b, for wn_c <= wn < wn_up = 0 otherwise 2, 'donelan' : B(wn) frequency dependent 3, 'banner' : B(wn) frequency dependent where B(wn) = S(wn) for wn_lo <= wn < wn_up = s_b*wn_up**m_b, for wn_up <= wn and method = 2 = sc*F(wn) for wn_up <= wn and method = 3 where F(wn) = 10^(-0.4+0.8393*exp(-0.567*log(wn^2))) and sc is scalefactor to make the spreading funtion continous. wn_lo, wn_c, wn_up: real scalars (default 0, 1, inf) limits used in the function defining the directional spreading parameter, S() or B() defined above. wn_c is the normalized cutover frequency s_a, s_b : real scalars maximum spread parameters (default [15 15]) m_a, m_b : real scalars shape parameters (default [5 -2.5]) SPREADING return a Directional spreading function. Here the S- or B-parameter, of the COS-2S and SECH-2 spreading function, respectively, is used as a measure of spread. All the parameters of the other distributions are related to this parameter through the first Fourier coefficient, R1, of the directional distribution as follows: R1 = S/(S+1) or S = R1/(1-R1). where Box-car spreading : R1 = sin(A)/A Von Mises spreading: R1 = besseli(1,K)/besseli(0,K), Poisson spreading : R1 = X sech-2 spreading : R1 = pi/(2*B*sinh(pi/(2*B)) Wrapped Normal : R1 = exp(-D1^2/2) A value of S = 15 corresponds to 'box' : A=0.62, 'sech-2' : B=0.89 'von-mises' : K=8.3, 'poisson': X=0.94 'wrapped-normal': D=0.36 The COS2S is the most frequently used spreading in engineering practice. Apart from the current meter/pressure cell data in WADIC all instruments seem to support the 'cos2s' distribution for heavier sea states, (Krogstad and Barstow, 1999). For medium sea states a spreading function between COS2S and POISSON seem appropriate, while POISSON seems appropriate for swell. For the COS2S Mitsuyasu et al. parameterized SPa = SPb = 11.5*(U10/Cp) where Cp = g/wp is the deep water phase speed at wp and U10 the wind speed at reference height 10m. Hasselman et al. (1980) parameterized mb = -2.33-1.45*(U10/Cp-1.17). Mitsuyasu et al. (1975) showed that SP for wind waves varies from 5 to 30 being a function of dimensionless wind speed. However, Goda and Suzuki (1975) proposed SP = 10 for wind waves, SP = 25 for swell with short decay distance and SP = 75 for long decay distance. Compared to experiments Krogstad et al. (1998) found that m_a = 5 +/- _EPS and that -1< m_b < -3.5. Values given in the litterature: [s_a s_b m_a m_b wn_lo wn_c wn_up] (Mitsuyasu: s_a == s_b) (cos-2s) [15 15 5 -2.5 0 1 3 ] (Hasselman: s_a ~= s_b) (cos-2s) [6.97 9.77 4.06 -2.3 0 1.05 3 ] (Banner : s_a ~= s_b) (sech2) [2.61 2.28 1.3 -1.3 0.56 0.95 1.6] Examples -------- >>> import wafo.spectrum.models as wsm >>> import pylab as plb >>> D = wsm.Spreading('cos2s',s_a=10.0) # Make directionale spectrum >>> S = wsm.Jonswap().tospecdata() >>> SD = D.tospecdata2d(S) >>> h = SD.plot() >>> w = plb.linspace(0,3,257) >>> theta = plb.linspace(-pi,pi,129) >>> t = plb.contour(D(theta,w)[0].squeeze()) # Make frequency dependent direction spreading >>> theta0 = lambda w: w*plb.pi/6.0 >>> D2 = wsm.Spreading('cos2s',theta0=theta0) >>> t = plb.contour(D2(theta,w)[0]) # Plot all spreading functions >>> alltypes = ('cos2s','box','mises','poisson','sech2','wrap_norm') >>> for ix in range(len(alltypes)): ... D3 = wsm.Spreading(alltypes[ix]) ... t = plb.figure(ix) ... t = plb.contour(D3(theta,w)[0]) ... t = plb.title(alltypes[ix]) >>> plb.close('all') See also -------- mkdspec, plotspec, spec2spec References --------- Krogstad, H.E. and Barstow, S.F. (1999) "Directional Distributions in Ocean Wave Spectra" Proceedings of the 9th ISOPE Conference, Vol III, pp. 79-86 Goda, Y. (1999) "Numerical simulation of ocean waves for statistical analysis" Marine Tech. Soc. Journal, Vol. 33, No. 3, pp 5--14 Banner, M.L. (1990) "Equilibrium spectra of wind waves." J. Phys. Ocean, Vol 20, pp 966--984 Donelan M.A., Hamilton J, Hui W.H. (1985) "Directional spectra of wind generated waves." Phil. Trans. Royal Soc. London, Vol A315, pp 387--407 Hasselmann D, Dunckel M, Ewing JA (1980) "Directional spectra observed during JONSWAP." J. Phys. Ocean, Vol.10, pp 1264--1280 Mitsuyasu, H, et al. (1975) "Observation of the directional spectrum of ocean waves using a coverleaf buoy." J. Physical Oceanography, Vol.5, No.4, pp 750--760 Some of this might be included in help header: cos-2s: NB! The generally strong frequency dependence in directional spread makes it questionable to run load tests of ships and structures with a directional spread independent of frequency (Krogstad and Barstow, 1999). ''' # Parameterization of B # def = 2 Donelan et al freq. parametrization for 'sech2' # def = 3 Banner freq. parametrization for 'sech2' # (spa ~= spb) (sech-2) [2.61 2.28 1.3 -1.3 0.56 0.95 1.6] # def __init__(self, type='cos-2s', theta0=0, # @ReservedAssignment method='mitsuyasu', s_a=15., s_b=15., m_a=5., m_b=-2.5, wn_lo=0.0, wn_c=1., wn_up=inf): self.type = type self.theta0 = theta0 self.method = method self.s_a = s_a self.s_b = s_b self.m_a = m_a self.m_b = m_b self.wn_lo = wn_lo self.wn_c = wn_c self.wn_up = wn_up methods = dict(n=None, m='mitsuyasu', d='donelan', b='banner') methodslist = (None, 'mitsuyasu', 'donelan', 'banner') if isinstance(self.method, str): if not self.method[0] in methods: raise ValueError('Unknown method') self.method = methods[self.method[0]] elif self.method is None: pass else: if method < 0 or 3 < method: method = 2 self.method = methodslist[method] self._spreadfun = dict(c=self.cos2s, b=self.box, m=self.mises, v=self.mises, p=self.poisson, s=self.sech2, w=self.wrap_norm) self._fourierdispatch = dict(b=self.fourier2a, m=self.fourier2k, v=self.fourier2k, p=self.fourier2x, s=self.fourier2b, w=self.fourier2d) def __call__(self, theta, w=1, wc=1): spreadfun = self._spreadfun[self.type[0]] return spreadfun(theta, w, wc) def chk_input(self, theta, w=1, wc=1): # [s_par,TH,phi0,Nt] = ''' CHK_INPUT CALL [s_par,TH,phi0,Nt] = inputchk(theta,w,wc) ''' wn = atleast_1d(w / wc) theta = theta.ravel() Nt = len(theta) # Make sure theta is from -pi to pi phi0 = 0.0 theta = mod(theta + pi, 2 * pi) - pi if hasattr(self.theta0, '__call__'): th0 = self.theta0(wn.flatten()) else: th0 = atleast_1d(self.theta0).flatten() Nt0 = th0.size Nw = wn.size isFreqDepDir = (Nt0 == Nw) if isFreqDepDir: # frequency dependent spreading and/or # frequency dependent direction # make sure -pi<=TH= 1): raise ValueError('POISSON spreading: X value must be less than 1') return X def fourier2a(self, r1): ''' Returns the solution of R1 = sin(A)/A. ''' A0 = flipud(linspace(0, pi + 0.1, 1025)) funA = interp1d(sinc(A0 / pi), A0) A0 = funA(r1.ravel()) A = asarray(A0) # Newton-Raphson da = ones_like(r1) max_count = 100 ix = flatnonzero(A) for unused_iy in range(max_count): Ai = A[ix] da[ix] = (sin(Ai) - Ai * r1[ix]) / (cos(Ai) - r1[ix]) Ai = Ai - da[ix] # Make sure that the current guess is larger than zero. A[ix] = Ai + 0.5 * (da[ix] - Ai) * (Ai <= 0.0) ix = flatnonzero( (abs(da) > sqrt(_EPS) * abs(A)) * (abs(da) > sqrt(_EPS))) if ix.size == 0: if any(A > pi): raise ValueError( 'BOX-CAR spreading: The A value must be less than pi') return A.clip(min=1e-16, max=pi) warnings.warn('Newton raphson method did not converge.') return A.clip(min=1e-16) # Avoid division by zero def fourier2k(self, r1): ''' Returns the solution of R1 = besseli(1,K)/besseli(0,K), ''' def fun0(x): return sp.ive(1, x) / sp.ive(0, x) K0 = hstack((linspace(0, 10, 513), linspace(10.00001, 100))) funK = interp1d(fun0(K0), K0) K0 = funK(r1.ravel()) k1 = flatnonzero(isnan(K0)) if (k1.size > 0): K0[k1] = 0.0 K0[k1] = K0.max() ix0 = flatnonzero(r1 != 0.0) K = zeros_like(r1) for ix in ix0: K[ix] = optimize.fsolve(lambda x: fun0(x) - r1[ix], K0[ix]) return K def fourier2b(self, r1): ''' Returns the solution of R1 = pi/(2*B*sinh(pi/(2*B)). ''' B0 = hstack((linspace(_EPS, 5, 513), linspace(5.0001, 100))) funB = interp1d(self._r1ofsech2(B0), B0) B0 = funB(r1.ravel()) k1 = flatnonzero(isnan(B0)) if (k1.size > 0): B0[k1] = 0.0 B0[k1] = max(B0) ix0 = flatnonzero(r1 != 0.0) B = zeros_like(r1) def fun(x): return 0.5 * pi / (sinh(.5 * pi / x)) - x * r1[ix] for ix in ix0: B[ix] = abs(optimize.fsolve(fun, B0[ix])) return B def fourier2d(self, r1): ''' Returns the solution of R1 = exp(-D**2/2). ''' r = clip(r1, 0., 1.0) return where(r <= 0, inf, sqrt(-2.0 * log(r))) def spread_par_s(self, wn): ''' Return spread parameter, S, of COS2S function Parameters ---------- wn : array_like normalized frequencies. Returns ------- S : ndarray spread parameter of COS2S functions ''' if self.method is None: # no frequency dependent spreading, # but possible frequency dependent direction s = atleast_1d(self.s_a) else: wn_lo = self.wn_lo wn_up = self.wn_up wn_c = self.wn_c spa = self.s_a spb = self.s_b ma = self.m_a mb = self.m_b # Mitsuyasu et. al and Hasselman et. al parametrization of # frequency dependent spreading s = where(wn <= wn_c, spa * wn ** ma, spb * wn ** mb) s[wn <= wn_lo] = 0.0 k = flatnonzero(wn_up < wn) if k.size > 0: if self.method[0] == 'd': # Donelan et. al. parametrization for B in SECH-2 s[k] = spb * (wn_up) ** mb # Convert to S-paramater in COS-2S distribution r1 = self.r1ofsech2(s) s = r1 / (1. - r1) elif self.method[0] == 'b': # Banner parametrization for B in SECH-2 s3m = spb * (wn_up) ** mb s3p = self._donelan(wn_up) # Scale so that parametrization will be continous scale = s3m / s3p s[k] = scale * self.donelan(wn[k]) r1 = self.r1ofsech2(s) # Convert to S-paramater in COS-2S distribution s = r1 / (1. - r1) else: s[k] = 0.0 if any(s < 0): raise ValueError('The COS2S spread parameter, S(w), ' + 'value must be larger than 0') return s def _donelan(self, wn): ''' High frequency decay of B of sech2 paramater ''' return 10.0 ** (-0.4 + 0.8393 * exp(-0.567 * log(wn ** 2))) def _r1ofsech2(self, B): ''' R1OFSECH2 Computes R1 = pi./(2*B.*sinh(pi./(2*B))) ''' realmax = finfo(float).max tiny = 1. / realmax x = clip(2. * B, tiny, realmax) xk = pi / x return where(x < 100., xk / sinh(xk), -2. * xk / (exp(xk) * expm1(-2. * xk))) def tospecdata2d(self, specdata=None, theta=None, wc=0.52, nt=51): ''' MKDSPEC Make a directional spectrum frequency spectrum times spreading function CALL: Snew=mkdspec(S,D,plotflag) Snew = directional spectrum (spectrum struct) S = frequency spectrum (spectrum struct) (default jonswap) D = spreading function (special struct) (default spreading([],'cos2s')) plotflag = 1: plot the spectrum, else: do not plot (default 0) Creates a directional spectrum through multiplication of a frequency spectrum and a spreading function: S(w,theta)=S(w)*D(w,theta) The spreading structure must contain the following fields: .S (size [np 1] or [np nf]) and .theta (length np) optional fields: .w (length nf), .note (memo) .phi (rotation-azymuth) NB! S.w and D.w (if any) must be identical. Example ------- >>> import wafo.spectrum.models as wsm >>> S = wsm.Jonswap().tospecdata() >>> D = wsm.Spreading('cos2s') >>> SD = D.tospecdata2d(S) >>> h = SD.plot() See also spreading, rotspec, jonswap, torsethaugen ''' if specdata is None: specdata = Jonswap().tospecdata() if theta is None: pi = np.pi theta = np.linspace(-pi, pi, nt) else: L = abs(theta[-1] - theta[0]) if abs(L - pi) > _EPS: raise ValueError('theta must cover all angles -pi -> pi') nt = len(theta) if nt < 40: warnings.warn('Number of angles is less than 40. ' + 'Spreading too sparsely sampled!') w = specdata.args S = specdata.data D, phi0 = self(theta, w=w, wc=wc) if D.ndim != 2: # frequency dependent spreading D = D[:, None] SD = D * S[None, :] Snew = SpecData2D(SD, (w, theta), type='dir', freqtype=specdata.freqtype) Snew.tr = specdata.tr Snew.h = specdata.h Snew.phi = phi0 Snew.norm = specdata.norm # Snew.note = specdata.note + ', spreading: %s' % self.type return Snew def _test_some_spectra(): S = Jonswap() w = arange(3.0) S(w) * phi1(w, 30.0) S1 = S.tospecdata(w) S1.plot() import pylab as plb w = plb.linspace(0, 2.5) S = Tmaspec(h=10, gamma=1) # Bretschneider spectrum Hm0=7, Tp=11 plb.plot(w, S(w)) plb.plot(w, S(w, h=21)) plb.plot(w, S(w, h=42)) plb.show() plb.close('all') w, th = plb.ogrid[0:4, 0:6] k, k2 = w2k(w, th) plb.plot(w, k, w, k2) plb.show() plb.close('all') w = plb.linspace(0, 2, 100) S = Torsethaugen(Hm0=6, Tp=8) plb.plot(w, S(w), w, S.wind(w), w, S.swell(w)) S1 = Jonswap(Hm0=7, Tp=11, gamma=1) w = plb.linspace(0, 2, 100) plb.plot(w, S1(w)) plb.show() plb.close('all') Hm0 = plb.arange(1, 11) Tp = plb.linspace(2, 16) T, H = plb.meshgrid(Tp, Hm0) gam = jonswap_peakfact(H, T) plb.plot(Tp, gam.T) plb.xlabel('Tp [s]') plb.ylabel('Peakedness parameter') Hm0 = plb.linspace(1, 20) Tp = Hm0 [T, H] = plb.meshgrid(Tp, Hm0) gam = jonswap_peakfact(H, T) v = plb.arange(0, 8) plb.contourf(Tp, Hm0, gam, v) plb.colorbar() plb.show() plb.close('all') def _test_spreading(): import pylab as plb pi = plb.pi w = plb.linspace(0, 3, 257) theta = plb.linspace(-pi, pi, 129) D2 = Spreading('cos2s', theta0=lambda w: w * plb.pi / 6.0) d1 = D2(theta, w)[0] _t = plb.contour(d1.squeeze()) pi = plb.pi D = Spreading('wrap_norm', s_a=10.0) w = plb.linspace(0, 3, 257) theta = plb.linspace(-pi, pi, 129) d1 = D(theta, w) plb.contour(d1[0]) plb.show() def test_docstrings(): import doctest print('Testing docstrings in %s' % __file__) doctest.testmod(optionflags=doctest.NORMALIZE_WHITESPACE) def main(): if False: # True: # _test_some_spectra() else: test_docstrings() if __name__ == '__main__': main()