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analytical_solution_viscoacoustic_Carcione_version1.f90
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analytical_solution_viscoacoustic_Carcione_version1.f90
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program analytical_solution
!! DK DK to compare to our finite-difference codes from SEISMIC_CPML or SOUNDVIEW,
!! DK DK we divide the source by 4 * PI * cp^2 to get the right amplitude (our convention being to use a source of amplitude 1,
!! DK DK while the convention used by Carcione in his 1988 paper is to use a source of amplitude 4 * PI * cp^2
! this program implements the analytical solution for a viscoacoustic medium
! from Carcione et al., Wave propagation simulation in a linear viscoacoustic medium,
! Geophysical Journal, vol. 93, p. 393-407 (1988)
!! DK DK Dimitri Komatitsch, CNRS Marseille, France, April 2017
!! DK DK adapted from a program written for the viscoelastic case by Jose' M. Carcione.
implicit none
! compute the non-viscoacoustic case as a reference if needed, i.e. turn attenuation off
logical, parameter :: TURN_ATTENUATION_OFF = .false. ! .true.
!! DK DK Dimitri Komatitsch, CNRS Marseille, France, October 2015:
!! DK DK by default I turned off the fix for attenuation causality (using the unrelaxed velocities
!! DK DK as reference instead of the relaxed ones) because it is not useful any more,
!! DK DK this modification was not consistent with the calculations of the tau values
!! DK DK made by Carcione et al. 1988 and by Carcione 1993.
!! Comment from Quentin Brissaud, March 2018:
!! This flag will tell the code that the input velocities are the relaxed one (omega -> zero frequency)
!! instead of the unrelaxed ones (by default omega -> + infinity)
logical, parameter :: FIX_ATTENUATION_CAUSALITY = .true.
integer, parameter :: iratio = 64
integer, parameter :: nfreq = 524288
integer, parameter :: nt = iratio * nfreq
double precision, parameter :: freqmax = 1000.d0 ! 225.d0
!! DK DK to print the velocity if we want to display the curve of how velocity varies with frequency
!! DK DK for instance to compute the unrelaxed velocity in the Zener model
! double precision, parameter :: freqmax = 20000.d0
double precision, parameter :: freqseuil = 0.00005d0
double precision, parameter :: pi = 3.141592653589793d0
! for the solution in time domain
integer it
real wsave(4*nt+15)
complex c(nt)
!! DK DK for my slow inverse Discrete Fourier Transform using a double loop
complex :: input(nt), i_imaginary_constant
integer :: j,m
! density of the medium
double precision, parameter :: rho = 2000.d0
! definition position recepteur Carcione
double precision x1,x2
! Definition source Dimitri
double precision, parameter :: f0 = 35.d0
double precision, parameter :: t0 = 1.2d0 / f0
! Definition source Carcione
! double precision f0,t0,eta,epsil
! parameter(f0 = 50.d0)
! parameter(t0 = 0.06d0)
! parameter(epsil = 1.d0)
! parameter(eta = 0.5d0)
! number of Zener standard linear solids in parallel
! integer, parameter :: L_mech = 5
integer, parameter :: L_mech = 3
! DK DK I implemented a very simple and slow inverse Discrete Fourier Transform
! DK DK at some point, for verification, using a double loop. I keep it just in case.
! DK DK For large number of points it is extremely slow because of the double loop.
! DK DK Thus there is no reason to turn this flag on.
logical, parameter :: USE_SLOW_FOURIER_TRANSFORM = .false.
! attenuation constants from Carcione 1988 GJI vol 95 p 604
double precision, dimension(L_mech) :: tau_epsilon_nu1, tau_sigma_nu1
! this value comes from page 397 of Carcione et al., Wave propagation simulation in a linear viscoacoustic medium,
! Geophysical Journal, vol. 93, p. 393-407 (1988)
double precision, parameter :: vp = 2000.d0
double precision, parameter :: M_relaxed = rho*vp**2
integer :: ifreq,i_mech,iposition
double precision :: deltafreq,freq,omega,omega0,deltat,time,a,sum_of_coefficients
double complex :: comparg,sum_to_compute
! Fourier transform of the Ricker wavelet source
double complex fomega(0:nfreq)
! real and imaginary parts
double precision ra(0:nfreq),rb(0:nfreq)
! spectral amplitude
double precision ampli(0:nfreq)
! analytical solution for the single scalar component (pressure)
double complex phi1(-nfreq:nfreq)
! external functions
double complex, external :: u1
! modules elastiques
double complex :: MC, V1
! ********** end of variable declarations ************
!! DK DK July 2018: values computed to fit Q = 65 for the example I designed for the "SOUNDVIEW" finite-difference code
tau_epsilon_nu1 = (/ 2.408158185805540d-002, 4.699608990946073d-003, 9.567997872679109d-004/)
tau_sigma_nu1 = (/ 2.256014638685252d-002, 4.508471279793884d-003, 8.937876403997143d-004/)
! position of the receiver
do iposition = 1,3
if (iposition == 1) then
x1 = +200.
x2 = +200.
else if (iposition == 2) then
x1 = +500.
x2 = +500.
else
!!!!!!!! x1 = +800.
!!!!!!!! x2 = +800.
!! DK DK modified to fall exactly on a grid point
x1 = +801.
x2 = +801.
endif
print *,'Pressure source located at the origin (0,0)'
print *,'Receiver located in (x,z) = ',x1,x2
if (TURN_ATTENUATION_OFF) then
print *,'BEWARE: computing the acoustic reference solution (i.e., without attenuation) instead of the viscoacoustic solution'
else
print *,'Computing the viscoacoustic solution'
endif
! step in frequency
deltafreq = freqmax / dble(nfreq)
! define parameters for the Ricker source
omega0 = 2.d0 * pi * f0
a = pi**2 * f0**2
deltat = 1.d0 / (freqmax*dble(iratio))
print *,'deltat = ',deltat
! define the spectrum of the source
do ifreq=0,nfreq
freq = deltafreq * dble(ifreq)
omega = 2.d0 * pi * freq
! typo in equation (B10) of Carcione et al., Wave propagation simulation in a linear viscoacoustic medium,
! Geophysical Journal, vol. 93, p. 393-407 (1988), the exponential is of -i omega t0,
! fixed here by adding the minus sign
comparg = dcmplx(0.d0,-omega*t0)
! definir le spectre du Ricker de Carcione avec cos()
! equation (B10) of Carcione et al., Wave propagation simulation in a linear viscoacoustic medium,
! Geophysical Journal, vol. 93, p. 393-407 (1988)
! fomega(ifreq) = pi * dsqrt(pi/eta) * (1.d0/omega0) * cdexp(comparg) * ( dexp(- (pi*pi/eta) * (epsil/2 - omega/omega0)**2) &
! + dexp(- (pi*pi/eta) * (epsil/2 + omega/omega0)**2) )
! definir le spectre d'un Ricker classique (centre en t0)
fomega(ifreq) = dsqrt(pi) * cdexp(comparg) * omega**2 * dexp(-omega**2/(4.d0*a)) / (2.d0 * dsqrt(a**3))
!! DK DK to compare to our finite-difference codes from SEISMIC_CPML or SOUNDVIEW,
!! DK DK we divide the source by 4 * PI * cp^2 to get the right amplitude (our convention being to use a source of amplitude 1,
!! DK DK while the convention used by Carcione in his 1988 paper is to use a source of amplitude 4 * PI * cp^2
fomega(ifreq) = fomega(ifreq) / (4.d0 * PI * vp**2)
ra(ifreq) = dreal(fomega(ifreq))
rb(ifreq) = dimag(fomega(ifreq))
! prendre le module de l'amplitude spectrale
ampli(ifreq) = dsqrt(ra(ifreq)**2 + rb(ifreq)**2)
enddo
! sauvegarde du spectre d'amplitude de la source en Hz au format Gnuplot
open(unit=10,file='spectrum_of_the_source_used.gnu',status='unknown')
do ifreq = 0,nfreq
freq = deltafreq * dble(ifreq)
write(10,*) sngl(freq),sngl(ampli(ifreq))
enddo
close(10)
! ************** calcul solution analytique ****************
! d'apres Carcione GJI vol 95 p 611 (1988)
do ifreq=0,nfreq
freq = deltafreq * dble(ifreq)
omega = 2.d0 * pi * freq
! critere ad-hoc pour eviter singularite en zero
if (freq < freqseuil) omega = 2.d0 * pi * freqseuil
! equation (16) of Carcione et al., Wave propagation simulation in a linear viscoacoustic medium,
! Geophysical Journal, vol. 93, p. 393-407 (1988)
sum_to_compute = dcmplx(0.d0,0.d0)
do i_mech = 1,L_mech
sum_to_compute = sum_to_compute + dcmplx(1.d0,omega*tau_epsilon_nu1(i_mech)) / dcmplx(1.d0,omega*tau_sigma_nu1(i_mech))
enddo
!! DK DK Quentin Brissaud in March 2018 added the 1/L factor here (it is missing in Carcione's older papers)
MC = M_relaxed * (1.d0 + (1./L_mech)*(-L_mech + sum_to_compute))
! use more standard infinite frequency (unrelaxed) reference,
! in which waves slow down when attenuation is turned on,
! or use far less standard zero frequency (relaxed) reference,
! in which waves speed up when attenuation is turned on
if (FIX_ATTENUATION_CAUSALITY) then
sum_of_coefficients = 0.d0
do i_mech = 1,L_mech
sum_of_coefficients = sum_of_coefficients + tau_epsilon_nu1(i_mech) / tau_sigma_nu1(i_mech)
enddo
!! DK DK Quentin Brissaud in March 2018 added the 1/L factor here (it is missing in Carcione's older papers)
MC = MC / (1.d0 + (1./L_mech)*(-L_mech + sum_of_coefficients))
endif
! equation (18) of Carcione et al., Wave propagation simulation in a linear viscoacoustic medium,
! Geophysical Journal, vol. 93, p. 393-407 (1988)
V1 = cdsqrt(MC / rho)
! compute the non-viscoacoustic case as a reference if needed, i.e. turn attenuation off
if (TURN_ATTENUATION_OFF) V1 = cdsqrt(dcmplx(M_relaxed,0.d0) / rho)
! calcul de la solution analytique en frequence
phi1(ifreq) = u1(omega,V1,x1,x2) * fomega(ifreq)
enddo
! take the conjugate value for negative frequencies
do ifreq=-nfreq,-1
phi1(ifreq) = dconjg(phi1(-ifreq))
enddo
! save the result in the frequency domain
! open(unit=11,file='cmplx_phi',status='unknown')
! do ifreq=-nfreq,nfreq
! freq = deltafreq * dble(ifreq)
! write(11,*) sngl(freq),sngl(dreal(phi1(ifreq))),sngl(dimag(phi1(ifreq)))
! enddo
! close(11)
! ***************************************************************************
! Calculation of the time domain solution (using routine "cfftb" from Netlib)
! ***************************************************************************
! ****************
! Compute pressure
! ****************
! initialize FFT arrays
call cffti(nt,wsave)
! clear array of Fourier coefficients
do it = 1,nt
c(it) = cmplx(0.,0.)
enddo
! use the Fourier values for pressure
c(1) = cmplx(phi1(0))
do ifreq=1,nfreq-2
c(ifreq+1) = cmplx(phi1(ifreq))
c(nt+1-ifreq) = conjg(cmplx(phi1(ifreq)))
enddo
! perform the inverse FFT for pressure
if (.not. USE_SLOW_FOURIER_TRANSFORM) then
call cfftb(nt,c,wsave)
else
! DK DK I implemented a very simple and slow inverse Discrete Fourier Transform here
! DK DK at some point, for verification, using a double loop. I keep it just in case.
! DK DK For large number of points it is extremely slow because of the double loop.
input(:) = c(:)
! imaginary constant "i"
i_imaginary_constant = (0.,1.)
do it = 1,nt
if (mod(it,1000) == 0) print *,'FFT inverse it = ',it,' out of ',nt
j = it
c(j) = cmplx(0.,0.)
do m = 1,nt
c(j) = c(j) + input(m) * exp(2.d0 * PI * i_imaginary_constant * dble((m-1) * (j-1)) / nt)
enddo
enddo
endif
! in the inverse Discrete Fourier transform one needs to divide by N, the number of samples (number of time steps here)
c(:) = c(:) / nt
! value of a time step
deltat = 1.d0 / (freqmax*dble(iratio))
! to get the amplitude right, we need to divide by the time step
c(:) = c(:) / deltat
! save time result inverse FFT for pressure
if (iposition == 1) then
if (TURN_ATTENUATION_OFF) then
open(unit=11,file='pressure_time_analytical_solution_acoustic_200.dat',status='unknown')
else
open(unit=11,file='pressure_time_analytical_solution_viscoacoustic_200.dat',status='unknown')
endif
else if (iposition == 2) then
if (TURN_ATTENUATION_OFF) then
open(unit=11,file='pressure_time_analytical_solution_acoustic_500.dat',status='unknown')
else
open(unit=11,file='pressure_time_analytical_solution_viscoacoustic_500.dat',status='unknown')
endif
else
if (TURN_ATTENUATION_OFF) then
open(unit=11,file='pressure_time_analytical_solution_acoustic_800.dat',status='unknown')
else
open(unit=11,file='pressure_time_analytical_solution_viscoacoustic_800.dat',status='unknown')
endif
endif
do it=1,nt
! DK DK Dec 2011: subtract t0 to be consistent with the SPECFEM2D code
time = dble(it-1)*deltat - t0
! the seismograms are very long due to the very large number of FFT points used,
! thus keeping the useful part of the signal only (the first six seconds of the seismogram)
if (time >= 0.d0 .and. time <= 6.d0) write(11,*) sngl(time),real(c(it))
enddo
close(11)
print *,'Maximum positive amplitude of the time-domain solution = ',maxval(real(c(:)))
print *
enddo ! of loop on the three positions of the receiver
end
! -----------
double complex function u1(omega,v1,x1,x2)
implicit none
double precision omega
double complex v1
double complex G1
external G1
double precision x1,x2,r
! source-receiver distance
r = dsqrt(x1**2 + x2**2)
! equation (B8a) of Carcione et al., Wave propagation simulation in a linear viscoacoustic medium,
! Geophysical Journal, vol. 93, p. 393-407 (1988)
u1 = G1(r,omega,v1)
end
! -----------
double complex function G1(r,omega,v1)
implicit none
double precision r,omega
double complex v1
double complex hankel0
external hankel0
double precision pi
parameter (pi = 3.141592653589793d0)
! equation (B8a) of Carcione et al., Wave propagation simulation in a linear viscoacoustic medium,
! Geophysical Journal, vol. 93, p. 393-407 (1988)
G1 = hankel0(omega*r/v1) * dcmplx(0.d0,-pi)
end
! -----------
double complex function hankel0(z)
implicit none
double complex z
! on utilise la routine NAG appelee S17DLE (simple precision)
integer ifail,nz
complex result
ifail = -1
call S17DLE(2,0.0,cmplx(z),1,'U',result,nz,ifail)
if (ifail /= 0) stop 'S17DLE failed in hankel0'
if (nz > 0) print *,nz,' termes mis a zero par underflow'
hankel0 = dcmplx(result)
end
! ***************** routine de FFT pour signal en temps ****************
! FFT routine taken from Netlib
subroutine CFFTB (N,C,WSAVE)
DIMENSION C(1) ,WSAVE(1)
if (N == 1) return
IW1 = N+N+1
IW2 = IW1+N+N
CALL CFFTB1 (N,C,WSAVE,WSAVE(IW1),WSAVE(IW2))
END
subroutine CFFTB1 (N,C,CH,WA,IFAC)
DIMENSION CH(1) ,C(1) ,WA(1) ,IFAC(1)
NF = IFAC(2)
NA = 0
L1 = 1
IW = 1
DO 116 K1=1,NF
IP = IFAC(K1+2)
L2 = IP*L1
IDO = N/L2
IDOT = IDO+IDO
IDL1 = IDOT*L1
if (IP /= 4) goto 103
IX2 = IW+IDOT
IX3 = IX2+IDOT
if (NA /= 0) goto 101
CALL PASSB4 (IDOT,L1,C,CH,WA(IW),WA(IX2),WA(IX3))
goto 102
101 CALL PASSB4 (IDOT,L1,CH,C,WA(IW),WA(IX2),WA(IX3))
102 NA = 1-NA
goto 115
103 if (IP /= 2) goto 106
if (NA /= 0) goto 104
CALL PASSB2 (IDOT,L1,C,CH,WA(IW))
goto 105
104 CALL PASSB2 (IDOT,L1,CH,C,WA(IW))
105 NA = 1-NA
goto 115
106 if (IP /= 3) goto 109
IX2 = IW+IDOT
if (NA /= 0) goto 107
CALL PASSB3 (IDOT,L1,C,CH,WA(IW),WA(IX2))
goto 108
107 CALL PASSB3 (IDOT,L1,CH,C,WA(IW),WA(IX2))
108 NA = 1-NA
goto 115
109 if (IP /= 5) goto 112
IX2 = IW+IDOT
IX3 = IX2+IDOT
IX4 = IX3+IDOT
if (NA /= 0) goto 110
CALL PASSB5 (IDOT,L1,C,CH,WA(IW),WA(IX2),WA(IX3),WA(IX4))
goto 111
110 CALL PASSB5 (IDOT,L1,CH,C,WA(IW),WA(IX2),WA(IX3),WA(IX4))
111 NA = 1-NA
goto 115
112 if (NA /= 0) goto 113
CALL PASSB (NAC,IDOT,IP,L1,IDL1,C,C,C,CH,CH,WA(IW))
goto 114
113 CALL PASSB (NAC,IDOT,IP,L1,IDL1,CH,CH,CH,C,C,WA(IW))
114 if (NAC /= 0) NA = 1-NA
115 L1 = L2
IW = IW+(IP-1)*IDOT
116 continue
if (NA == 0) return
N2 = N+N
DO 117 I=1,N2
C(I) = CH(I)
117 continue
END
subroutine PASSB (NAC,IDO,IP,L1,IDL1,CC,C1,C2,CH,CH2,WA)
DIMENSION CH(IDO,L1,IP) ,CC(IDO,IP,L1), &
C1(IDO,L1,IP) ,WA(1) ,C2(IDL1,IP), &
CH2(IDL1,IP)
IDOT = IDO/2
NT = IP*IDL1
IPP2 = IP+2
IPPH = (IP+1)/2
IDP = IP*IDO
if (IDO < L1) goto 106
DO 103 J=2,IPPH
JC = IPP2-J
DO 102 K=1,L1
DO 101 I=1,IDO
CH(I,K,J) = CC(I,J,K)+CC(I,JC,K)
CH(I,K,JC) = CC(I,J,K)-CC(I,JC,K)
101 continue
102 continue
103 continue
DO 105 K=1,L1
DO 104 I=1,IDO
CH(I,K,1) = CC(I,1,K)
104 continue
105 continue
goto 112
106 DO 109 J=2,IPPH
JC = IPP2-J
DO 108 I=1,IDO
DO 107 K=1,L1
CH(I,K,J) = CC(I,J,K)+CC(I,JC,K)
CH(I,K,JC) = CC(I,J,K)-CC(I,JC,K)
107 continue
108 continue
109 continue
DO 111 I=1,IDO
DO 110 K=1,L1
CH(I,K,1) = CC(I,1,K)
110 continue
111 continue
112 IDL = 2-IDO
INC = 0
DO 116 L=2,IPPH
LC = IPP2-L
IDL = IDL+IDO
DO 113 IK=1,IDL1
C2(IK,L) = CH2(IK,1)+WA(IDL-1)*CH2(IK,2)
C2(IK,LC) = WA(IDL)*CH2(IK,IP)
113 continue
IDLJ = IDL
INC = INC+IDO
DO 115 J=3,IPPH
JC = IPP2-J
IDLJ = IDLJ+INC
if (IDLJ > IDP) IDLJ = IDLJ-IDP
WAR = WA(IDLJ-1)
WAI = WA(IDLJ)
DO 114 IK=1,IDL1
C2(IK,L) = C2(IK,L)+WAR*CH2(IK,J)
C2(IK,LC) = C2(IK,LC)+WAI*CH2(IK,JC)
114 continue
115 continue
116 continue
DO 118 J=2,IPPH
DO 117 IK=1,IDL1
CH2(IK,1) = CH2(IK,1)+CH2(IK,J)
117 continue
118 continue
DO 120 J=2,IPPH
JC = IPP2-J
DO 119 IK=2,IDL1,2
CH2(IK-1,J) = C2(IK-1,J)-C2(IK,JC)
CH2(IK-1,JC) = C2(IK-1,J)+C2(IK,JC)
CH2(IK,J) = C2(IK,J)+C2(IK-1,JC)
CH2(IK,JC) = C2(IK,J)-C2(IK-1,JC)
119 continue
120 continue
NAC = 1
if (IDO == 2) return
NAC = 0
DO 121 IK=1,IDL1
C2(IK,1) = CH2(IK,1)
121 continue
DO 123 J=2,IP
DO 122 K=1,L1
C1(1,K,J) = CH(1,K,J)
C1(2,K,J) = CH(2,K,J)
122 continue
123 continue
if (IDOT > L1) goto 127
IDIJ = 0
DO 126 J=2,IP
IDIJ = IDIJ+2
DO 125 I=4,IDO,2
IDIJ = IDIJ+2
DO 124 K=1,L1
C1(I-1,K,J) = WA(IDIJ-1)*CH(I-1,K,J)-WA(IDIJ)*CH(I,K,J)
C1(I,K,J) = WA(IDIJ-1)*CH(I,K,J)+WA(IDIJ)*CH(I-1,K,J)
124 continue
125 continue
126 continue
return
127 IDJ = 2-IDO
DO 130 J=2,IP
IDJ = IDJ+IDO
DO 129 K=1,L1
IDIJ = IDJ
DO 128 I=4,IDO,2
IDIJ = IDIJ+2
C1(I-1,K,J) = WA(IDIJ-1)*CH(I-1,K,J)-WA(IDIJ)*CH(I,K,J)
C1(I,K,J) = WA(IDIJ-1)*CH(I,K,J)+WA(IDIJ)*CH(I-1,K,J)
128 continue
129 continue
130 continue
END
subroutine PASSB2 (IDO,L1,CC,CH,WA1)
DIMENSION CC(IDO,2,L1) ,CH(IDO,L1,2), &
WA1(1)
if (IDO > 2) goto 102
DO 101 K=1,L1
CH(1,K,1) = CC(1,1,K)+CC(1,2,K)
CH(1,K,2) = CC(1,1,K)-CC(1,2,K)
CH(2,K,1) = CC(2,1,K)+CC(2,2,K)
CH(2,K,2) = CC(2,1,K)-CC(2,2,K)
101 continue
return
102 DO 104 K=1,L1
DO 103 I=2,IDO,2
CH(I-1,K,1) = CC(I-1,1,K)+CC(I-1,2,K)
TR2 = CC(I-1,1,K)-CC(I-1,2,K)
CH(I,K,1) = CC(I,1,K)+CC(I,2,K)
TI2 = CC(I,1,K)-CC(I,2,K)
CH(I,K,2) = WA1(I-1)*TI2+WA1(I)*TR2
CH(I-1,K,2) = WA1(I-1)*TR2-WA1(I)*TI2
103 continue
104 continue
END
subroutine PASSB3 (IDO,L1,CC,CH,WA1,WA2)
DIMENSION CC(IDO,3,L1) ,CH(IDO,L1,3), &
WA1(1) ,WA2(1)
DATA TAUR,TAUI /-.5,.866025403784439/
if (IDO /= 2) goto 102
DO 101 K=1,L1
TR2 = CC(1,2,K)+CC(1,3,K)
CR2 = CC(1,1,K)+TAUR*TR2
CH(1,K,1) = CC(1,1,K)+TR2
TI2 = CC(2,2,K)+CC(2,3,K)
CI2 = CC(2,1,K)+TAUR*TI2
CH(2,K,1) = CC(2,1,K)+TI2
CR3 = TAUI*(CC(1,2,K)-CC(1,3,K))
CI3 = TAUI*(CC(2,2,K)-CC(2,3,K))
CH(1,K,2) = CR2-CI3
CH(1,K,3) = CR2+CI3
CH(2,K,2) = CI2+CR3
CH(2,K,3) = CI2-CR3
101 continue
return
102 DO 104 K=1,L1
DO 103 I=2,IDO,2
TR2 = CC(I-1,2,K)+CC(I-1,3,K)
CR2 = CC(I-1,1,K)+TAUR*TR2
CH(I-1,K,1) = CC(I-1,1,K)+TR2
TI2 = CC(I,2,K)+CC(I,3,K)
CI2 = CC(I,1,K)+TAUR*TI2
CH(I,K,1) = CC(I,1,K)+TI2
CR3 = TAUI*(CC(I-1,2,K)-CC(I-1,3,K))
CI3 = TAUI*(CC(I,2,K)-CC(I,3,K))
DR2 = CR2-CI3
DR3 = CR2+CI3
DI2 = CI2+CR3
DI3 = CI2-CR3
CH(I,K,2) = WA1(I-1)*DI2+WA1(I)*DR2
CH(I-1,K,2) = WA1(I-1)*DR2-WA1(I)*DI2
CH(I,K,3) = WA2(I-1)*DI3+WA2(I)*DR3
CH(I-1,K,3) = WA2(I-1)*DR3-WA2(I)*DI3
103 continue
104 continue
END
subroutine PASSB4 (IDO,L1,CC,CH,WA1,WA2,WA3)
DIMENSION CC(IDO,4,L1) ,CH(IDO,L1,4), &
WA1(1) ,WA2(1) ,WA3(1)
if (IDO /= 2) goto 102
DO 101 K=1,L1
TI1 = CC(2,1,K)-CC(2,3,K)
TI2 = CC(2,1,K)+CC(2,3,K)
TR4 = CC(2,4,K)-CC(2,2,K)
TI3 = CC(2,2,K)+CC(2,4,K)
TR1 = CC(1,1,K)-CC(1,3,K)
TR2 = CC(1,1,K)+CC(1,3,K)
TI4 = CC(1,2,K)-CC(1,4,K)
TR3 = CC(1,2,K)+CC(1,4,K)
CH(1,K,1) = TR2+TR3
CH(1,K,3) = TR2-TR3
CH(2,K,1) = TI2+TI3
CH(2,K,3) = TI2-TI3
CH(1,K,2) = TR1+TR4
CH(1,K,4) = TR1-TR4
CH(2,K,2) = TI1+TI4
CH(2,K,4) = TI1-TI4
101 continue
return
102 DO 104 K=1,L1
DO 103 I=2,IDO,2
TI1 = CC(I,1,K)-CC(I,3,K)
TI2 = CC(I,1,K)+CC(I,3,K)
TI3 = CC(I,2,K)+CC(I,4,K)
TR4 = CC(I,4,K)-CC(I,2,K)
TR1 = CC(I-1,1,K)-CC(I-1,3,K)
TR2 = CC(I-1,1,K)+CC(I-1,3,K)
TI4 = CC(I-1,2,K)-CC(I-1,4,K)
TR3 = CC(I-1,2,K)+CC(I-1,4,K)
CH(I-1,K,1) = TR2+TR3
CR3 = TR2-TR3
CH(I,K,1) = TI2+TI3
CI3 = TI2-TI3
CR2 = TR1+TR4
CR4 = TR1-TR4
CI2 = TI1+TI4
CI4 = TI1-TI4
CH(I-1,K,2) = WA1(I-1)*CR2-WA1(I)*CI2
CH(I,K,2) = WA1(I-1)*CI2+WA1(I)*CR2
CH(I-1,K,3) = WA2(I-1)*CR3-WA2(I)*CI3
CH(I,K,3) = WA2(I-1)*CI3+WA2(I)*CR3
CH(I-1,K,4) = WA3(I-1)*CR4-WA3(I)*CI4
CH(I,K,4) = WA3(I-1)*CI4+WA3(I)*CR4
103 continue
104 continue
END
subroutine PASSB5 (IDO,L1,CC,CH,WA1,WA2,WA3,WA4)
DIMENSION CC(IDO,5,L1) ,CH(IDO,L1,5), &
WA1(1) ,WA2(1) ,WA3(1) ,WA4(1)
DATA TR11,TI11,TR12,TI12 /.309016994374947,.951056516295154, &
-.809016994374947,.587785252292473/
if (IDO /= 2) goto 102
DO 101 K=1,L1
TI5 = CC(2,2,K)-CC(2,5,K)
TI2 = CC(2,2,K)+CC(2,5,K)
TI4 = CC(2,3,K)-CC(2,4,K)
TI3 = CC(2,3,K)+CC(2,4,K)
TR5 = CC(1,2,K)-CC(1,5,K)
TR2 = CC(1,2,K)+CC(1,5,K)
TR4 = CC(1,3,K)-CC(1,4,K)
TR3 = CC(1,3,K)+CC(1,4,K)
CH(1,K,1) = CC(1,1,K)+TR2+TR3
CH(2,K,1) = CC(2,1,K)+TI2+TI3
CR2 = CC(1,1,K)+TR11*TR2+TR12*TR3
CI2 = CC(2,1,K)+TR11*TI2+TR12*TI3
CR3 = CC(1,1,K)+TR12*TR2+TR11*TR3
CI3 = CC(2,1,K)+TR12*TI2+TR11*TI3
CR5 = TI11*TR5+TI12*TR4
CI5 = TI11*TI5+TI12*TI4
CR4 = TI12*TR5-TI11*TR4
CI4 = TI12*TI5-TI11*TI4
CH(1,K,2) = CR2-CI5
CH(1,K,5) = CR2+CI5
CH(2,K,2) = CI2+CR5
CH(2,K,3) = CI3+CR4
CH(1,K,3) = CR3-CI4
CH(1,K,4) = CR3+CI4
CH(2,K,4) = CI3-CR4
CH(2,K,5) = CI2-CR5
101 continue
return
102 DO 104 K=1,L1
DO 103 I=2,IDO,2
TI5 = CC(I,2,K)-CC(I,5,K)
TI2 = CC(I,2,K)+CC(I,5,K)
TI4 = CC(I,3,K)-CC(I,4,K)
TI3 = CC(I,3,K)+CC(I,4,K)
TR5 = CC(I-1,2,K)-CC(I-1,5,K)
TR2 = CC(I-1,2,K)+CC(I-1,5,K)
TR4 = CC(I-1,3,K)-CC(I-1,4,K)
TR3 = CC(I-1,3,K)+CC(I-1,4,K)
CH(I-1,K,1) = CC(I-1,1,K)+TR2+TR3
CH(I,K,1) = CC(I,1,K)+TI2+TI3
CR2 = CC(I-1,1,K)+TR11*TR2+TR12*TR3
CI2 = CC(I,1,K)+TR11*TI2+TR12*TI3
CR3 = CC(I-1,1,K)+TR12*TR2+TR11*TR3
CI3 = CC(I,1,K)+TR12*TI2+TR11*TI3
CR5 = TI11*TR5+TI12*TR4
CI5 = TI11*TI5+TI12*TI4
CR4 = TI12*TR5-TI11*TR4
CI4 = TI12*TI5-TI11*TI4
DR3 = CR3-CI4
DR4 = CR3+CI4
DI3 = CI3+CR4
DI4 = CI3-CR4
DR5 = CR2+CI5
DR2 = CR2-CI5
DI5 = CI2-CR5
DI2 = CI2+CR5
CH(I-1,K,2) = WA1(I-1)*DR2-WA1(I)*DI2
CH(I,K,2) = WA1(I-1)*DI2+WA1(I)*DR2
CH(I-1,K,3) = WA2(I-1)*DR3-WA2(I)*DI3
CH(I,K,3) = WA2(I-1)*DI3+WA2(I)*DR3
CH(I-1,K,4) = WA3(I-1)*DR4-WA3(I)*DI4
CH(I,K,4) = WA3(I-1)*DI4+WA3(I)*DR4
CH(I-1,K,5) = WA4(I-1)*DR5-WA4(I)*DI5
CH(I,K,5) = WA4(I-1)*DI5+WA4(I)*DR5
103 continue
104 continue
END
subroutine CFFTI (N,WSAVE)
DIMENSION WSAVE(1)
if (N == 1) return
IW1 = N+N+1
IW2 = IW1+N+N
CALL CFFTI1 (N,WSAVE(IW1),WSAVE(IW2))
END
subroutine CFFTI1 (N,WA,IFAC)
DIMENSION WA(1) ,IFAC(1) ,NTRYH(4)
DATA NTRYH(1),NTRYH(2),NTRYH(3),NTRYH(4)/3,4,2,5/
NL = N
NF = 0
J = 0
101 J = J+1
if (J-4) 102,102,103
102 NTRY = NTRYH(J)
goto 104
103 NTRY = NTRY+2
104 NQ = NL/NTRY
NR = NL-NTRY*NQ
if (NR) 101,105,101
105 NF = NF+1
IFAC(NF+2) = NTRY
NL = NQ
if (NTRY /= 2) goto 107
if (NF == 1) goto 107
DO 106 I=2,NF
IB = NF-I+2
IFAC(IB+2) = IFAC(IB+1)
106 continue
IFAC(3) = 2
107 if (NL /= 1) goto 104
IFAC(1) = N
IFAC(2) = NF
TPI = 6.28318530717959
ARGH = TPI/FLOAT(N)
I = 2
L1 = 1
DO 110 K1=1,NF
IP = IFAC(K1+2)
LD = 0
L2 = L1*IP
IDO = N/L2
IDOT = IDO+IDO+2
IPM = IP-1
DO 109 J=1,IPM
I1 = I
WA(I-1) = 1.
WA(I) = 0.
LD = LD+L1
FI = 0.
ARGLD = FLOAT(LD)*ARGH
DO 108 II=4,IDOT,2
I = I+2
FI = FI+1.
ARG = FI*ARGLD
WA(I-1) = COS(ARG)
WA(I) = SIN(ARG)
108 continue
if (IP <= 5) goto 109
WA(I1-1) = WA(I-1)
WA(I1) = WA(I)
109 continue
L1 = L2
110 continue
END
subroutine CFFTF (N,C,WSAVE)
DIMENSION C(1) ,WSAVE(1)
if (N == 1) return
IW1 = N+N+1
IW2 = IW1+N+N
CALL CFFTF1 (N,C,WSAVE,WSAVE(IW1),WSAVE(IW2))
END
subroutine CFFTF1 (N,C,CH,WA,IFAC)
DIMENSION CH(1) ,C(1) ,WA(1) ,IFAC(1)
NF = IFAC(2)
NA = 0
L1 = 1
IW = 1
DO 116 K1=1,NF
IP = IFAC(K1+2)
L2 = IP*L1
IDO = N/L2
IDOT = IDO+IDO
IDL1 = IDOT*L1
if (IP /= 4) goto 103
IX2 = IW+IDOT
IX3 = IX2+IDOT
if (NA /= 0) goto 101
CALL PASSF4 (IDOT,L1,C,CH,WA(IW),WA(IX2),WA(IX3))
goto 102
101 CALL PASSF4 (IDOT,L1,CH,C,WA(IW),WA(IX2),WA(IX3))
102 NA = 1-NA
goto 115
103 if (IP /= 2) goto 106
if (NA /= 0) goto 104
CALL PASSF2 (IDOT,L1,C,CH,WA(IW))
goto 105
104 CALL PASSF2 (IDOT,L1,CH,C,WA(IW))
105 NA = 1-NA
goto 115
106 if (IP /= 3) goto 109
IX2 = IW+IDOT
if (NA /= 0) goto 107
CALL PASSF3 (IDOT,L1,C,CH,WA(IW),WA(IX2))
goto 108
107 CALL PASSF3 (IDOT,L1,CH,C,WA(IW),WA(IX2))
108 NA = 1-NA
goto 115
109 if (IP /= 5) goto 112
IX2 = IW+IDOT
IX3 = IX2+IDOT
IX4 = IX3+IDOT
if (NA /= 0) goto 110
CALL PASSF5 (IDOT,L1,C,CH,WA(IW),WA(IX2),WA(IX3),WA(IX4))
goto 111
110 CALL PASSF5 (IDOT,L1,CH,C,WA(IW),WA(IX2),WA(IX3),WA(IX4))
111 NA = 1-NA
goto 115
112 if (NA /= 0) goto 113
CALL PASSF (NAC,IDOT,IP,L1,IDL1,C,C,C,CH,CH,WA(IW))
goto 114
113 CALL PASSF (NAC,IDOT,IP,L1,IDL1,CH,CH,CH,C,C,WA(IW))
114 if (NAC /= 0) NA = 1-NA
115 L1 = L2
IW = IW+(IP-1)*IDOT
116 continue
if (NA == 0) return
N2 = N+N
DO 117 I=1,N2
C(I) = CH(I)
117 continue
END
subroutine PASSF (NAC,IDO,IP,L1,IDL1,CC,C1,C2,CH,CH2,WA)
DIMENSION CH(IDO,L1,IP) ,CC(IDO,IP,L1), &
C1(IDO,L1,IP) ,WA(1) ,C2(IDL1,IP), &
CH2(IDL1,IP)
IDOT = IDO/2
NT = IP*IDL1
IPP2 = IP+2
IPPH = (IP+1)/2
IDP = IP*IDO
if (IDO < L1) goto 106
DO 103 J=2,IPPH
JC = IPP2-J
DO 102 K=1,L1
DO 101 I=1,IDO
CH(I,K,J) = CC(I,J,K)+CC(I,JC,K)
CH(I,K,JC) = CC(I,J,K)-CC(I,JC,K)
101 continue
102 continue
103 continue
DO 105 K=1,L1
DO 104 I=1,IDO
CH(I,K,1) = CC(I,1,K)
104 continue
105 continue
goto 112
106 DO 109 J=2,IPPH
JC = IPP2-J
DO 108 I=1,IDO
DO 107 K=1,L1
CH(I,K,J) = CC(I,J,K)+CC(I,JC,K)
CH(I,K,JC) = CC(I,J,K)-CC(I,JC,K)
107 continue
108 continue
109 continue
DO 111 I=1,IDO
DO 110 K=1,L1
CH(I,K,1) = CC(I,1,K)
110 continue
111 continue
112 IDL = 2-IDO
INC = 0
DO 116 L=2,IPPH
LC = IPP2-L
IDL = IDL+IDO
DO 113 IK=1,IDL1
C2(IK,L) = CH2(IK,1)+WA(IDL-1)*CH2(IK,2)
C2(IK,LC) = -WA(IDL)*CH2(IK,IP)
113 continue
IDLJ = IDL
INC = INC+IDO
DO 115 J=3,IPPH
JC = IPP2-J
IDLJ = IDLJ+INC
if (IDLJ > IDP) IDLJ = IDLJ-IDP
WAR = WA(IDLJ-1)
WAI = WA(IDLJ)
DO 114 IK=1,IDL1
C2(IK,L) = C2(IK,L)+WAR*CH2(IK,J)
C2(IK,LC) = C2(IK,LC)-WAI*CH2(IK,JC)
114 continue
115 continue
116 continue
DO 118 J=2,IPPH
DO 117 IK=1,IDL1
CH2(IK,1) = CH2(IK,1)+CH2(IK,J)
117 continue
118 continue
DO 120 J=2,IPPH
JC = IPP2-J
DO 119 IK=2,IDL1,2
CH2(IK-1,J) = C2(IK-1,J)-C2(IK,JC)
CH2(IK-1,JC) = C2(IK-1,J)+C2(IK,JC)
CH2(IK,J) = C2(IK,J)+C2(IK-1,JC)
CH2(IK,JC) = C2(IK,J)-C2(IK-1,JC)
119 continue
120 continue
NAC = 1
if (IDO == 2) return
NAC = 0
DO 121 IK=1,IDL1