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haskattn.f
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c----------------------------------------------------------------------
c
c Compute the Green's functions for a 1D medium using TH (Thomson-Haskell).
c
c This version allows for attenuating medium by using complex wavespeeds.
c
c It also normalizes the stress using the shear modulus of the half space
c in order to balance the magnitudes of the entries in the 'a' propagaion
c matrix.
c
c This routine is the computative "engine" of the T-H method.
c It impliments the concepts in Haskell (1953; H53) and Haskell
c (1962; H62) to compute the Green's functions in layered media
c do to an external plane wave source of unit amplitude
c
c Arguments:
c df The frequency
c isrc source number used to determine source type (P or S)
c IERR error flag
c
c This version for 1 frequency = df
c
c This version allows the addition of a fluid layer (vs = 0)
c
c Author: S. Roecker 4/09
c
subroutine haskattn(nl,nzpts,srctyp,lflip, df,freq0,ain,
; z1d,vp1d,vs1d,rh1d,qa1d,qb1d, depth,
; ugrn1f,wgrn1f, ierr)
implicit none
character(2), intent(in) :: srctyp
real*8, intent(in) :: z1d(nl),vp1d(nl),vs1d(nl),rh1d(nl),
; qa1d(nl), qb1d(nl), depth(nzpts),ain,df,
; freq0
integer*4, intent(in) :: nl,nzpts
logical*4, intent(in) :: lflip
complex*16, intent(out) :: ugrn1f(nzpts),wgrn1f(nzpts)
integer*4, intent(out) :: ierr
complex*16, allocatable :: r1(:), r2(:), aag(:), aag1(:),
; xh(:), xh1(:),
; s01(:),c01(:), s02(:), c02(:)
real*8, allocatable :: h1d(:), h(:), tt(:)
complex*16 alph, beta
real*8 vsvpnl
real*8 coa
character(1) ctype
complex*16 x,y,ag,ag1,tm,
& pd,qd,pm,qm,cp,cq
complex*16 c11,c12,c21,c22,
& c31,c32,c41,c42
complex*16 b11,b12,b21,b22,
& b31,b32,b41,b42
complex*16 d11,d12,d21,d22,
& d31,d32,d41,d42
complex*16 a11,a12,a13,a14,a21,a22,a23,a24,
& a31,a32,a33,a34,a41,a42,a43,a44
complex*16 e11, e13, e22, e24, e33, e31, e42, e44,
; z01, z02, eye, up, wp, cova, covb
real*8 cv, pic, hc, dept, dloc, ztop, pi180, pi
parameter(pi180 = 0.017453292519943295d0,
; pi = 3.1415926535897932384626434d0)
parameter(eye = dcmplx(0.d0, 1.d0))
integer*4 i, m, iz
integer*4 ibov, ibovo, ifin, idir, izb, ize
complex*16 vcmplx
!
!...... check angle of incidence
if (ain.ge.90.d0 .or. ain.lt.0.d0) then
write(*,*) 'haskattn: Angle of incidence invalid:',ain
ugrn1f(1:nzpts) = dcmplx(0.d0,0.d0)
wgrn1f(1:nzpts) = dcmplx(0.d0,0.d0)
ierr = 1
return
endif
c----incident wave type
ierr = 0
ctype = 'P'
if (srctyp.eq.'PS' .or. srctyp.eq.'ps' .or.
; srctyp.eq.'pS' .or. srctyp.eq.'Ps') ctype = 'S'
c----horizontal wavespeed
c NB: it seems to me that if we do the dispersion correction we use the apparent
c wavespeed from the original vp1d/vs1d and ain(isrc), so to be consistent we should
c use vp1d and vs1d here. Otherwise we should either redefine ain, or define cv elsewhere.
if (ctype.eq.'P') then
cv = vp1d(nl)/dsin(ain*pi180)
else
cv = vs1d(nl)/dsin(ain*pi180)
endif
pic = 2.d0*pi*df/cv
c pic = df/cv
c write(*,*) ' cv, pic = ', cv, pic
!
!...... may need to flip coordinate system in z
allocate(h(nl-1))
h(1:nl-1) = 0.d0
allocate(h1d(nl))
h1d(1:nl) = 0.d0
ztop = z1d(1)
do i=1,nl
if (lflip) then
h1d(i) = ztop - z1d(i)
else
h1d(i) = z1d(i)
endif
enddo
c----thickness and complex wavespeeds array
do i = 1, nl-1
h(i) = h1d(i+1) - h1d(i)
enddo
!
!...... set space
allocate(r1(nl+1))
allocate(r2(nl+1))
allocate(aag(nl+1))
allocate(aag1(nl+1))
allocate(xh(nl+1))
allocate(xh1(nl+1))
allocate(s01(nl+1))
allocate(c01(nl+1))
allocate(s02(nl+1))
allocate(c02(nl+1))
allocate(tt(nl+1))
c----Loop over the model, setting up layer specific values
do m = 1,nl
alph = vcmplx(df,freq0,vp1d(m),qa1d(m)) !cmplx(vp1d(m), vp1di(m))
beta = vcmplx(df,freq0,vs1d(m),qb1d(m)) !cmplx(vs1d(m), vs1di(m))
c--- c/alpha and c/beta
cova = dcmplx(cv,0.d0)/alph
c---set up a bogus value for covb in case of vs = 0
if (cdabs(beta).ne.0.d0) then
covb = dcmplx(cv,0.d0)/beta
else
covb = dcmplx(1.d0,0.d0)
endif
c----r1 and r2 are r(alpha) and r(beta)
c----aag is gamma, aag1 is gamma-1
x = cova*cova - dcmplx(1.d0,0.d0)
r1(m) = cdsqrt(x)
x = covb*covb - dcmplx(1.d0,0.d0)
r2(m) = cdsqrt(x)
aag(m) = dcmplx(2.d0,0.d0)/(covb*covb)
aag1(m) = aag(m) - dcmplx(1.d0,0.d0)
if(m.lt.nl) then
tt(m) = (rh1d(m)/rh1d(nl))*(cv/vs1d(nl))*(cv/vs1d(nl))
xh(m) = aag(m)*aag(m)
xh1(m) = aag1(m)*aag1(m)
hc = pic*h(m)
c----z01, z02 are Pm and Qm in H53
z01 = dcmplx(hc,0.d0)*r1(m)
z02 = dcmplx(hc,0.d0)*r2(m)
x = cdexp(eye*z01)
c01(m) = dcmplx(0.5d0,0.d0)*(x + dcmplx(1.d0,0.d0)/x)
s01(m) = dcmplx(0.5d0,0.d0)*(x - dcmplx(1.d0,0.d0)/x)
x = cdexp(eye*z02)
c02(m) = dcmplx(0.5d0,0.d0)*(x + dcmplx(1.d0,0.d0)/x)
s02(m) = dcmplx(0.5d0,0.d0)*(x - dcmplx(1.d0,0.d0)/x)
c write(*,*) m, c01(m), s01(m), e2(m), c02(m), s02(m)
c read(*,*) pause
endif
enddo
c---Now propagate to the surface to compute the surface displacements
c---The following should be equivalent to the inverse E matrix for
c the half space (see Eqn 2.16 of H53).
vsvpnl = vs1d(nl)/vp1d(nl)
vsvpnl = vsvpnl*vsvpnl
coa = cv/vp1d(nl)
e11 = -dcmplx(2.d0*vsvpnl,0.d0)
e13 = dcmplx(vsvpnl,0.d0)
e22 = dcmplx(coa*coa,0.d0)*aag1(nl)/r1(nl)
e24 = dcmplx(vsvpnl,0.d0)/r1(nl)
e31 = aag1(nl)/(aag(nl)*r2(nl))
e33 = -dcmplx(1.d0,0.d0)/(dcmplx(2.d0,0.d0)*r2(nl))
e42 = dcmplx(1.d0,0.d0)
e44 = dcmplx(0.5d0,0.d0)
c write(*,*) ' nl, x, e11, e33 = ', nl, x, e11, e33
c----The 4 x 2 b matrix will hold the multiplication
c of all the a matrices. This initialization allows
c the first two columns of the a matrix to be saved
c the first time around.
b11 = dcmplx(1.d0,0.d0)
b12 = dcmplx(0.d0,0.d0)
b21 = dcmplx(0.d0,0.d0)
b22 = dcmplx(1.d0,0.d0)
b31 = dcmplx(0.d0,0.d0)
b32 = dcmplx(0.d0,0.d0)
b41 = dcmplx(0.d0,0.d0)
b42 = dcmplx(0.d0,0.d0)
c----Loop over all layers
do 7 m = 1,nl-1
pd = s01(m)/r1(m)
qd = s02(m)/r2(m)
pm = s01(m)*r1(m)
qm = s02(m)*r2(m)
c
c c1, cp = cos(Pm)
c c2, cq = cos(Qm)
c s1 = sin(Pm)
c s2 = sin(Qm)
c The following lines are a clever way of
c upgrading the sin and cosine for the next
c frequency. Essentally we are evaluating:
c cos(f + df) = cos(f)cos(df) - sin(f)sin(df)
c sin(f + df) = sin(f)cos(df) + cos(f)sin(df)
c
cp = c01(m)
cq = c02(m)
tm = dcmplx(tt(m),0.d0)
ag = aag(m)
ag1 = aag1(m)
x = xh(m)
y = xh1(m)
c
c These "a" terms those shown on page 21 of H53
c
if (vs1d(m).ne.0.d0) then
a11 = ag*cp - ag1*cq
a12 = ag1*pd + ag*qm
a13 = (cq - cp)/tm
a14 = (pd + qm)/tm
a21 = -ag*pm - ag1*qd
a22 = ag*cq - ag1*cp
a23 = (pm + qd)/tm
a24 = a13
a31 = tm*ag*ag1*(cp - cq)
a32 = tm*(y*pd + x*qm)
a33 = a22
a34 = a12
a41 = tm*(x*pm + y*qd)
a42 = a31
a43 = a21
a44 = a11
else
c---special forms for fluid layer from eqn 6.3 of H53. Note there
c seems to be a sign error on a12 in his paper, however.
a11 = dcmplx(0.d0,0.d0)
a12 = -pd
a13 = -cp/tm
a14 = dcmplx(0.d0,0.d0)
a21 = dcmplx(0.d0,0.d0)
a22 = cp
a23 = pm/tm
a24 = dcmplx(0.d0,0.d0)
a31 = dcmplx(0.d0,0.d0)
a32 = tm*pd
a33 = cp
a34 = dcmplx(0.d0,0.d0)
a41 = dcmplx(0.d0,0.d0)
a42 = dcmplx(0.d0,0.d0)
a43 = dcmplx(0.d0,0.d0)
a44 = dcmplx(0.d0,0.d0)
endif
c
c D is the accumulation of the multiplication of the A
c matrices as we descend through the layers. Note that
c because we start at the surface, the stresses are 0 and
c so the vector we multiply is (u, w, 0, 0). This means
c that for the entire problem we need save only the first
c two columns of the multiplication.
c
c
d11 = a11*b11 + a12*b21 + a13*b31 + a14*b41
d12 = a11*b12 + a12*b22 + a13*b32 + a14*b42
d21 = a21*b11 + a22*b21 + a23*b31 + a24*b41
d22 = a21*b12 + a22*b22 + a23*b32 + a24*b42
d31 = a31*b11 + a32*b21 + a33*b31 + a34*b41
d32 = a31*b12 + a32*b22 + a33*b32 + a34*b42
d41 = a41*b11 + a42*b21 + a43*b31 + a44*b41
d42 = a41*b12 + a42*b22 + a43*b32 + a44*b42
b11 = d11
b12 = d12
b21 = d21
b22 = d22
b31 = d31
b32 = d32
b41 = d41
b42 = d42
7 continue
c
c Now multiply A by inverse E of the last layer.
c C is then the same as J in H53.
c11 = e11*b11 + e13*b31
c12 = e11*b12 + e13*b32
c21 = e22*b21 + e24*b41
c22 = e22*b22 + e24*b42
c31 = e31*b11 + e33*b31
c32 = e31*b12 + e33*b32
c41 = e42*b21 + e44*b41
c42 = e42*b22 + e44*b42
c ag id the denominator
ag = (c11 - c21)*(c32-c42) - (c12 - c22)*(c31 - c41)
c---t for P waves
if (ctype.eq.'P') then
tm = dcmplx(2.d0*cv,0.d0)/(dcmplx(vp1d(nl),0.d0)*ag)
c---t for S waves
else
tm = dcmplx(cv,0.d0)/(dcmplx(vs1d(nl),0.d0)*ag)
endif
c
c wr, wi are the real and imaginary components of vertical
c velocity
c ur, ui are the real and imaginary components of horizontal
c velocity
c See equations 5 and 6 of H62. Note that D = |D| e(i theta), so
c 1/D = e(-i theta)/|D| = (x - iy)/|D|^2 = D*/(DD*)
c
c
c rs1 and rs2 are the real and imaginary parts of the
c velocity ratio u/w for P and w/u for S. Note that
c u/w = uw*/(ww*) = (ur*wr + ui*wi + i(ui*wr - ur*wi))/ww*
c w/u = wu*/(uu*) = (wr*ur + wi*ui + i(wi*ur - wr*ui))/uu*
c
c---P terms: The "W" terms are reversed in sign because a postive
c "L" component will will be in the (-Z, +R) direction as defined by
c H53. Note that this does NOT change the sense of Z - it is still
c positive down.
c
c The idea is that the impulse is (sind, cosd) which would be a pulse in
c the +X, +Z direction, but we want the response to (+x, -z) or (sind, -cosd)
c so we revers the sign on the Z component.
c
if (ctype.eq.'P') then
c--this from equations 10 and 11 of H62, modified as discussed at the top of page 4752
wp = tm*(c31 - c41)
up = tm*(c42 - c32)
else
c---S terms: these are the Haskell conventions on W and U. A source in the +H
c direction would be (+Z, +R) in the Haskell sense, so we retain the signs
c on W and U. Equations 16 and 17 in H62.
wp = tm*(c21 - c11)
up = tm*(c12 - c22)
endif
c---these are the surface displacements
c write(*,*) ' wp, up = ', wp, up
c read(*,*) pause
c---now propagate backwards, looping over the grid points
ibovo = 1
ifin = 0
ugrn1f(1:nzpts) = dcmplx(0.d0,0.d0)
wgrn1f(1:nzpts) = dcmplx(0.d0,0.d0)
if (lflip) then
izb = nzpts
ize = 1
idir =-1
else
izb = 1
ize = nzpts
idir = 1
endif
!
!...... need to be careful, distributed models may not start at top
if (dabs(depth(izb)- z1d(1)).lt.1.11d-7) then
ugrn1f(izb) = up
wgrn1f(izb) = wp
if (lflip) izb = izb + idir
endif
c---copy over response at the surface
do iz =izb,ize,idir ! 2, nz
!dept = dz*(iz-1) + zorig
if (lflip) then
dept = ztop - depth(iz)
else
dept = depth(iz)
endif
c----Loop over the model, setting up layer specific values
c----dep(1) should the top of the model (e.g., zero) so start at dep(2)
do i = 2,nl
if (dept.lt.h1d(i)) go to 4
enddo
i = nl
4 ibov = i - 1
dloc = dept - h1d(i-1)
if (dloc.lt.0.d0) then
WRITE(*,*) 'haskattn: Error in haskgrn dloc is < 0! ',dloc
ierr = 1
return
endif
c write(*,*) ' iz, dept, dloc, ibov = ', iz, dept, dloc, ibov
c----see if we have entered a new layer. If so, finish off the previous one
if (ibov.ne.ibovo) then
ibovo = ibov
if (ibov.gt.1) then
ifin = ifin + 1
hc = pic*h(ifin)
c----z01, z02 are Pm and Qm in H53
z01 = dcmplx(hc,0.d0)*r1(ifin)
z02 = dcmplx(hc,0.d0)*r2(ifin)
x = cdexp(eye*z01)
c01(ifin) = dcmplx(0.5d0,0.d0)*(x + dcmplx(1.d0,0.d0)/x)
s01(ifin) = dcmplx(0.5d0,0.d0)*(x - dcmplx(1.d0,0.d0)/x)
x = cdexp(eye*z02)
c02(ifin) = dcmplx(0.5d0,0.d0)*(x + dcmplx(1.d0,0.d0)/x)
s02(ifin) = dcmplx(0.5d0,0.d0)*(x - dcmplx(1.d0,0.d0)/x)
end if
end if
c----current layer terms
hc = pic*dloc
c----z01, z02 are Pm and Qm in H53
z01 = dcmplx(hc,0.d0)*r1(ibov)
z02 = dcmplx(hc,0.d0)*r2(ibov)
x = cdexp(eye*z01)
c01(ibov) = dcmplx(0.5d0,0.d0)*(x + dcmplx(1.d0,0.d0)/x)
s01(ibov) = dcmplx(0.5d0,0.d0)*(x - dcmplx(1.d0,0.d0)/x)
x = cdexp(eye*z02)
c02(ibov) = dcmplx(0.5d0,0.d0)*(x + dcmplx(1.d0,0.d0)/x)
s02(ibov) = dcmplx(0.5d0,0.d0)*(x - dcmplx(1.d0,0.d0)/x)
c---The following should be equivalent to the inverse E matrix for
c the half space (see Eqn 2.16 of H53).
if (ibov.eq.nl) then
vsvpnl = vs1d(nl)/vp1d(nl)
vsvpnl = vsvpnl*vsvpnl
coa = cv/vp1d(nl)
e11 = -dcmplx(2.d0*vsvpnl,0.d0)
e13 = dcmplx(vsvpnl,0.d0)
e22 = dcmplx(coa*coa,0.d0)*aag1(nl)/r1(nl)
e24 = dcmplx(vsvpnl,0.d0)/r1(nl)
e31 = aag1(nl)/(aag(nl)*r2(nl))
e33 = -dcmplx(1.d0,0.d0)/(dcmplx(2.d0,0.d0)*r2(nl))
e42 = dcmplx(1.d0,0.d0)
e44 = dcmplx(0.5d0,0.d0)
endif
c----The 4 x 2 b matrix will hold the multiplication
c of all the a matrices. This initialization allows
c the first two columns of the a matrix to be saved
c the first time around.
c---this part you only need do when you enter a new layer. Save
c results up to ifin then then do one ore iteration for the current layer.
b11 = dcmplx(1.d0,0.d0)
b12 = dcmplx(0.d0,0.d0)
b21 = dcmplx(0.d0,0.d0)
b22 = dcmplx(1.d0,0.d0)
b31 = dcmplx(0.d0,0.d0)
b32 = dcmplx(0.d0,0.d0)
b41 = dcmplx(0.d0,0.d0)
b42 = dcmplx(0.d0,0.d0)
c----Loop over all layers
do m = 1,ibov
pd = s01(m)/r1(m)
qd = s02(m)/r2(m)
pm = s01(m)*r1(m)
qm = s02(m)*r2(m)
c
c c1, cp = cos(Pm)
c c2, cq = cos(Qm)
c s1 = sin(Pm)
c s2 = sin(Qm)
c The following lines are a clever way of
c upgrading the sin and cosine for the next
c frequency. Essentally we are evaluating:
c cos(f + df) = cos(f)cos(df) - sin(f)sin(df)
c sin(f + df) = sin(f)cos(df) + cos(f)sin(df)
c
cp = c01(m)
cq = c02(m)
tm = dcmplx(tt(m),0.d0)
ag = aag(m)
ag1 = aag1(m)
x = xh(m)
y = xh1(m)
c
c These "a" terms those shown on page 21 of H53
c
if (vs1d(m).ne.0.d0) then
a11 = ag*cp - ag1*cq
a12 = ag1*pd + ag*qm
a13 = (cq - cp)/tm
a14 = (pd + qm)/tm
a21 = -ag*pm - ag1*qd
a22 = ag*cq - ag1*cp
a23 = (pm + qd)/tm
a24 = a13
a31 = tm*ag*ag1*(cp - cq)
a32 = tm*(y*pd + x*qm)
a33 = a22
a34 = a12
a41 = tm*(x*pm + y*qd)
a42 = a31
a43 = a21
a44 = a11
else
c---special forms for fluid layer from eqn 6.3 of H53. Note there
c seems to be a sign error on a12 in his paper, however.
a11 = dcmplx(0.d0,0.d0)
a12 = -pd
a13 = -cp/tm
a14 = dcmplx(0.d0,0.d0)
a21 = dcmplx(0.d0,0.d0)
a22 = cp
a23 = pm/tm
a24 = dcmplx(0.d0,0.d0)
a31 = dcmplx(0.d0,0.d0)
a32 = tm*pd
a33 = cp
a34 = dcmplx(0.d0,0.d0)
a41 = dcmplx(0.d0,0.d0)
a42 = dcmplx(0.d0,0.d0)
a43 = dcmplx(0.d0,0.d0)
a44 = dcmplx(0.d0,0.d0)
endif
c D is the accumulation of the multiplication of the A
c matrices as we descend through the layers. Note that
c because we start at the surface, the stresses are 0 and
c so the vector we multiply is (u, w, 0, 0). This means
c that for the entire problem we need save only the first
c two columns of the multiplication.
c
c
d11 = a11*b11 + a12*b21 + a13*b31 + a14*b41
d12 = a11*b12 + a12*b22 + a13*b32 + a14*b42
d21 = a21*b11 + a22*b21 + a23*b31 + a24*b41
d22 = +a21*b12 + a22*b22 + a23*b32 + a24*b42
d31 = a31*b11 + a32*b21 + a33*b31 + a34*b41
d32 = a31*b12 + a32*b22 + a33*b32 + a34*b42
d41 = a41*b11 + a42*b21 + a43*b31 + a44*b41
d42 = a41*b12 + a42*b22 + a43*b32 + a44*b42
b11 = d11
b12 = d12
b21 = d21
b22 = d22
b31 = d31
b32 = d32
b41 = d41
b42 = d42
enddo
!ugrn1f(iz) = b11*ugrn1f(1) + b12*wgrn1f(1)
!wgrn1f(iz) = b21*ugrn1f(1) + b22*wgrn1f(1)
ugrn1f(iz) = b11*up + b12*wp
wgrn1f(iz) = b21*up + b22*wp
enddo !end loop over nodes
!
!...... free space
deallocate(r1)
deallocate(r2)
deallocate(aag)
deallocate(aag1)
deallocate(xh)
deallocate(xh1)
deallocate(s01)
deallocate(c01)
deallocate(s02)
deallocate(c02)
deallocate(tt)
return
end
COMPLEX*16 FUNCTION VCMPLX(FREQ,FREQ0,V,Q)
!
! Calculates the complex velocity given a reference frequency,
! velocity, and the reciprocal of the quality factor. Based on
! SWRs dispersion.f and Aki and Richards pg 175 formula 5.94.
!
! INPUT MEANING
! ----- -------
! FREQ current frequency (Hz)
! FREQ0 reference frequency (Hz)
! V velocity Vp or Vs (m/s)
! Q dispersion factor (Qp or Qs)
!
! OUTPUT MEANING
! ------ -------
! VCMPLX complex velocity
!
!.... variable declarations
REAL*8, INTENT(IN) :: FREQ, FREQ0, V, Q
REAL*8 FAC1, FACT, VRNEW, VINEW
REAL*8 PI
PARAMETER(PI = 3.1415926535897932384626434D0)
!
!----------------------------------------------------------------------!
!
!.... checks
IF (FREQ0.EQ.0.D0 .OR. DABS(Q - 9999.D0).LT.1.D-5) THEN
VCMPLX = DCMPLX(V,0.D0)
RETURN
ENDIF
IF (Q.EQ.0.D0) THEN
VCMPLX = DCMPLX(V,0.D0)
WRITE(*,*) 'vcmplx: 0 quality factor!, returning real velocity'
RETURN
ENDIF
IF (FREQ.GT.FREQ0)
;WRITE(*,*) 'vcmplx: Warning freq > freq0, velocity increase!'
FAC1 = DLOG(FREQ/FREQ0)
FACT = 1.D0 + 1.D0/(PI*Q)*FAC1
IF (FACT.LT.0.1D0) FACT = 0.1D0 !a 90 percent reduction in vel
VRNEW = V*FACT !Re{ Aki and Richards 5.94}
VINEW =-V/(2.D0*Q) !Im{ Aki And Richards 5.94}
VCMPLX = DCMPLX(VRNEW,VINEW)
RETURN
END