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rams0.5.f
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rams0.5.f
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program rams
c
c ******************************************************************
c ***** Range-dependent Acoustic Model, Version 0.5s, 13-Sep-00 ****
c ******************************************************************
c
c This code was developed by Michael D. Collins at the Naval
c Research Laboratory in Washington, DC. It solves range-dependent
c seismo-ocean acoustics problems with the split-step Pade
c algorithm [M. D. Collins, J. Acoust. Soc. Am. 93, 1736-1742
c (1993)].
c
c Version 0.5s contains a correction to a bug in the dimension of
c quantities passed to subroutines fndrt and guerre that Laurie
c Fialkowski noticed.
c
c Version 0.4s contains a correction to a minor bug in subroutine
c guerre that Dave King noticed (amp1 and amp2 were declared
c twice) and a few other minor improvements.
c
c Version 0.3s contains a new root-finding subroutine.
c
c Version 0.2s contains a correction of a bug in the call to solve,
c which Peter Mignerey helped to locate. The bug only caused
c problems with certain compilers.
c
c Version 0.1s contains two improvements:
c
c (1) An improved self starter. Stability is improved by using the
c factor (1-i*X)**2 instead of (1+X)**2 to smooth the delta function.
c The factor (1+X)**2 is nearly singular for some problems involving
c deep water and/or weak attenuation. Numerical problems associated
c with this singularity were detected by Eddie Scheer of Woods Hole
c Oceanographic Institute.
c
c (2) Elimination of underflow problems. A very small number is
c added to the solution in subroutine solve to prevent underflow,
c which can adversely affect run time on some computers. This
c improvement was suggested by Ed McDonald of the SACLANT Undersea
c Research Centre.
c
complex ci,lamb,mub,u,v,r1,r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,s5,s6,
> s7,t1,t2,t3,t4,t5,t6,t7,pd1,pd2,g0
real k0,lamw
c
c mr=bathymetry points, mz=depth grid, mp=pade terms.
c
parameter (mr=100,mz=10000,mp=10)
dimension rb(mr),zb(mr),cw(mz),cp(mz),cs(mz),rhob(mz),attnp(mz),
> attns(mz),lamw(mz),lamb(mz),mub(mz),u(mz),v(mz),tlg(mz),
> r1(mz,mp),r2(mz,mp),r3(mz,mp),r4(mz,mp),r5(mz,mp),r6(mz,mp),
> r7(mz,mp),s1(mz,mp),s2(mz,mp),s3(mz,mp),s4(mz,mp),s5(mz,mp),
> s6(mz,mp),s7(mz,mp),t1(6,mp),t2(6,mp),t3(6,mp),t4(6,mp),
> t5(6,mp),t6(6,mp),t7(6,mp),pd1(mp),pd2(mp)
c
open(unit=1,status='old',file='rams.in')
open(unit=2,status='unknown',file='tl.line')
open(unit=3,status='unknown',file='tl.grid',form='unformatted')
c
call setup(mr,mz,nz,mp,np,mdr,ndr,ndz,iz,nzplt,lz,ib,ir,dir,dr,
> dz,pi,eta,eps,omega,rmax,c0,k0,ci,r,rp,rb,zb,cw,cp,cs,rhob,
> attnp,attns,lamw,lamb,mub,u,v,r1,r2,r3,r4,r5,r6,r7,s1,s2,s3,
> s4,s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,pd1,pd2,irot,theta,g0,tlg)
c
c March the seismo-acoustic field out in range.
c
1 call updat(mr,mz,nz,mp,np,iz,zs,ib,dr,dz,eta,omega,rmax,c0,k0,ci,
> r,rp,rb,zb,cw,cp,cs,rhob,attnp,attns,lamw,lamb,mub,u,r1,r2,r3,
> r4,r5,r6,r7,s1,s2,s3,s4,s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,pd1,
> pd2)
call solve(mz,nz,mp,np,iz,u,v,r1,r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,
> s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,g0)
r=r+dr
call outpt(mz,mdr,ndr,ndz,iz,nzplt,lz,ir,dir,eps,r,u,tlg)
if(r.lt.rmax)go to 1
c
close(1)
close(2)
close(3)
c
stop
end
c
c Initialize the parameters, seismo-acoustic field, and matrices.
c
subroutine setup(mr,mz,nz,mp,np,mdr,ndr,ndz,iz,nzplt,lz,ib,ir,
> dir,dr,dz,pi,eta,eps,omega,rmax,c0,k0,ci,r,rp,rb,zb,cw,cp,
> cs,rhob,attnp,attns,lamw,lamb,mub,u,v,r1,r2,r3,r4,r5,r6,r7,
> s1,s2,s3,s4,s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,pd1,pd2,irot,theta,
> g0,tlg)
complex ci,g0,u(mz),v(mz),lamb(mz),mub(mz),r1(mz,mp),r2(mz,mp),
> r3(mz,mp),r4(mz,mp),r5(mz,mp),r6(mz,mp),r7(mz,mp),s1(mz,mp),
> s2(mz,mp),s3(mz,mp),s4(mz,mp),s5(mz,mp),s6(mz,mp),s7(mz,mp),
> t1(6,mp),t2(6,mp),t3(6,mp),t4(6,mp),t5(6,mp),t6(6,mp),
> t7(6,mp),pd1(mp),pd2(mp),nu
real k0,rb(mr),zb(mr),cw(mz),cp(mz),cs(mz),rhob(mz),attnp(mz),
> attns(mz),lamw(mz),tlg(mz)
c
read(1,*)
read(1,*)freq,zs,zr
read(1,*)rmax,dr,ndr
read(1,*)zmax,dz,ndz,zmplt
read(1,*)c0,np,irot,theta
c
i=1
1 read(1,*)rb(i),zb(i)
if(rb(i).lt.0.0)go to 2
i=i+1
go to 1
2 zb(i)=zb(i-1)
rb(i)=rmax+2.0*dr
c
pi=4.0*atan(1.0)
ci=cmplx(0.0,1.0)
nu=ci
eta=1.0/(40.0*pi*alog10(exp(1.0)))
eps=1.0e-20
ib=1
mdr=0
r=dr
omega=2.0*pi*freq
ri=1.0+zr/dz
ir=ifix(ri)
dir=ri-float(ir)
k0=omega/c0
nz=zmax/dz-0.5
nzplt=zmplt/dz-0.5
z=zb(1)
iz=z/dz
c
do 3 i=1,2*nz+4
u(i)=0.0
v(i)=0.0
3 continue
lz=0
do 4 i=1+ndz,nzplt,ndz
lz=lz+1
4 continue
write(3)lz
c
c The initial profiles and starting field.
c
call profl(mz,nz,ci,dz,eta,omega,rmax,rp,cw,cp,cs,rhob,attnp,
> attns,lamw,lamb,mub)
call selfs(mz,nz,ci,mp,np,iz,zs,dr,dz,pi,c0,k0,omega,rhob,lamw,
> lamb,mub,u,v,r1,r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,s5,s6,s7,t1,t2,
> t3,t4,t5,t6,t7,pd1,pd2,nu)
call outpt(mz,mdr,ndr,ndz,iz,nzplt,lz,ir,dir,eps,r,u,tlg)
c
c The propagation matrices.
c
call epade(mp,np,1,1,k0,c0,dr,pd1,pd2,irot,theta,g0,nu)
call matrc(mz,nz,mp,np,iz,iz,dz,k0,omega,rhob,lamw,lamb,mub,r1,
> r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,
> pd1,pd2)
c
return
end
c
c Set up the profiles.
c
subroutine profl(mz,nz,ci,dz,eta,omega,rmax,rp,cw,cp,cs,rhob,
> attnp,attns,lamw,lamb,mub)
complex ci,mub(mz),lamb(mz)
real cw(mz),cp(mz),cs(mz),rhob(mz),attnp(mz),attns(mz),lamw(mz)
c
call zread(mz,nz,dz,cw)
call zread(mz,nz,dz,cp)
call zread(mz,nz,dz,cs)
call zread(mz,nz,dz,rhob)
call zread(mz,nz,dz,attnp)
call zread(mz,nz,dz,attns)
rp=2.0*rmax
read(1,*,end=1)rp
c
1 do 2 i=1,nz+2
lamw(i)=cw(i)**2
lamb(i)=rhob(i)*((cp(i)/(1.0+ci*eta*attnp(i)))**2-
> 2.0*(cs(i)/(1.0+ci*eta*attns(i)))**2)
mub(i)=rhob(i)*(cs(i)/(1.0+ci*eta*attns(i)))**2
2 continue
c
return
end
c
c Profile reader and interpolator.
c
subroutine zread(mz,nz,dz,prof)
real prof(mz)
c
do 1 i=1,nz+2
prof(i)=-1.0
1 continue
read(1,*)zi,profi
prof(1)=profi
i=1.5+zi/dz
prof(i)=profi
iold=i
2 read(1,*)zi,profi
if(zi.lt.0.0)go to 3
i=1.5+zi/dz
if(i.eq.iold)i=i+1
prof(i)=profi
iold=i
go to 2
3 prof(nz+2)=prof(i)
i=1
j=1
4 i=i+1
if(prof(i).lt.0.0)go to 4
if(i-j.eq.1)go to 6
do 5 k=j+1,i-1
prof(k)=prof(j)+float(k-j)*(prof(i)-prof(j))/float(i-j)
5 continue
6 j=i
if(j.lt.nz+2)go to 4
c
return
end
c
c Output transmission loss.
c
subroutine outpt(mz,mdr,ndr,ndz,iz,nzplt,lz,ir,dir,eps,r,u,tlg)
complex ur,u(mz)
real tlg(mz)
c
ur=(1.0-dir)*u(2*ir-1)+dir*u(2*ir+1)
tl=-20.0*alog10(cabs(ur)+eps)+10.0*alog10(r+eps)
write(2,*)r,tl
write(*,*)r,tl
c
mdr=mdr+1
if(mdr.eq.ndr)then
mdr=0
c
j=0
iflag=1
do 1 i=1+ndz,nzplt,ndz
ur=u(2*i-1)
j=j+1
tlg(j)=-20.0*alog10(cabs(ur)+eps)+10.0*alog10(r+eps)
c
c Mark the ocean bottom.
c
c if((i.gt.iz).and.(iflag.eq.1))then
c tlg(j)=0.0
c iflag=0
c end if
c
1 continue
write(3)(tlg(j),j=1,lz)
end if
c
return
end
c
c Matrix updates and energy conservation.
c
subroutine updat(mr,mz,nz,mp,np,iz,zs,ib,dr,dz,eta,omega,rmax,c0,
> k0,ci,r,rp,rb,zb,cw,cp,cs,rhob,attnp,attns,lamw,lamb,mub,u,r1,
> r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,
> pd1,pd2)
complex ci,lamb(mz),mub(mz),u(mz),r1(mz,mp),r2(mz,mp),r3(mz,mp),
> r4(mz,mp),r5(mz,mp),r6(mz,mp),r7(mz,mp),s1(mz,mp),s2(mz,mp),
> s3(mz,mp),s4(mz,mp),s5(mz,mp),s6(mz,mp),s7(mz,mp),t1(6,mp),
> t2(6,mp),t3(6,mp),t4(6,mp),t5(6,mp),t6(6,mp),t7(6,mp),pd1(mp),
> pd2(mp)
real k0,rb(mr),zb(mr),cw(mz),cp(mz),cs(mz),rhob(mz),attnp(mz),
> attns(mz),lamw(mz)
c
c Varying bathymetry.
c
jz=iz
z=zb(ib)+(r+0.5*dr-rb(ib))*(zb(ib+1)-zb(ib))/(rb(ib+1)-rb(ib))
iz=z/dz
if(iz.ne.jz)then
call matrc(mz,nz,mp,np,iz,jz,dz,k0,omega,rhob,lamw,lamb,mub,r1,
> r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,
> pd1,pd2)
c
c An approximate energy-flux correction.
c
if(iz.lt.jz)then
fact=sqrt(cw(iz)**3)/sqrt(rhob(iz)*cp(iz)**3)
do 1 i=iz+1,jz
u(2*i+1)=u(2*i+1)*fact
1 continue
c
else
fact=sqrt(rhob(iz)*cp(iz)**3)/sqrt(cw(iz)**3)
do 2 i=jz+1,iz
u(2*i+1)=u(2*i+1)*fact
2 continue
end if
c
end if
c
if(r.ge.rb(ib+1))ib=ib+1
c
c Varying profiles.
c
if(r.ge.rp)then
call profl(mz,nz,ci,dz,eta,omega,rmax,rp,cw,cp,cs,rhob,attnp,
> attns,lamw,lamb,mub)
call matrc(mz,nz,mp,np,iz,iz,dz,k0,omega,rhob,lamw,lamb,mub,r1,
> r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,
> pd1,pd2)
end if
c
return
end
c
c The self starter.
c
subroutine selfs(mz,nz,ci,mp,np,iz,zs,dr,dz,pi,c0,k0,omega,rhob,
> lamw,lamb,mub,u,v,r1,r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,s5,s6,s7,
> t1,t2,t3,t4,t5,t6,t7,pd1,pd2,nu)
complex ci,g0,u(mz),v(mz),lamb(mz),mub(mz),r1(mz,mp),r2(mz,mp),
> r3(mz,mp),r4(mz,mp),r5(mz,mp),r6(mz,mp),r7(mz,mp),s1(mz,mp),
> s2(mz,mp),s3(mz,mp),s4(mz,mp),s5(mz,mp),s6(mz,mp),s7(mz,mp),
> t1(6,mp),t2(6,mp),t3(6,mp),t4(6,mp),t5(6,mp),t6(6,mp),
> t7(6,mp),pd1(mp),pd2(mp),nu
real k0,rhob(mz),lamw(mz)
c
c Conditions for the delta function.
c
si=1.0+zs/dz
is=ifix(si)
dis=si-float(is)
u(2*is-1)=(1.0-dis)*sqrt(2.0*pi/k0)/dz
u(2*is+1)=dis*sqrt(2.0*pi/k0)/dz
c
c Divide the delta function by (1-nu*X)**2 to get a smooth rhs.
c
pd1(1)=0.0
pd2(1)=-nu
g0=(1.0,0.0)
call matrc(mz,nz,mp,1,iz,iz,dz,k0,omega,rhob,lamw,lamb,mub,r1,
> r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,
> pd1,pd2)
call solve(mz,nz,mp,1,iz,u,v,r1,r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,
> s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,g0)
call solve(mz,nz,mp,1,iz,u,v,r1,r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,
> s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,g0)
c
c Apply the operator (1-nu*X)**2*(1+X)**(-1/4)*exp(ci*k0*r*sqrt(1+X)).
c
call epade(mp,np,1,0,k0,c0,dr,pd1,pd2,0,0.0,g0,nu)
call matrc(mz,nz,mp,np,iz,iz,dz,k0,omega,rhob,lamw,lamb,mub,r1,
> r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,
> pd1,pd2)
call solve(mz,nz,mp,np,iz,u,v,r1,r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,
> s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,g0)
c
return
end
c
c The heptadiagonal matrices.
c
subroutine matrc(mz,nz,mp,np,iz,jz,dz,k0,omega,rhob,lamw,lamb,
> mub,r1,r2,r3,r4,r5,r6,r7,s1,s2,s3,s4,s5,s6,s7,t1,t2,t3,t4,t5,
> t6,t7,pd1,pd2)
complex l1,l2,l3,l4,l5,l6,l7,m1,m2,m3,m4,m5,m6,m7,a11,a12,a21,
> a22,a23,a31,a32,a33,a34,a35,lamb(mz),mub(mz),r1(mz,mp),
> r2(mz,mp),r3(mz,mp),r4(mz,mp),r5(mz,mp),r6(mz,mp),r7(mz,mp),
> s1(mz,mp),s2(mz,mp),s3(mz,mp),s4(mz,mp),s5(mz,mp),s6(mz,mp),
> s7(mz,mp),t1(6,mp),t2(6,mp),t3(6,mp),t4(6,mp),t5(6,mp),
> t6(6,mp),t7(6,mp),pd1(mp),pd2(mp)
real k0,rhob(mz),lamw(mz)
c
c New matrices when iz.eq.jz.
c
if(iz.eq.jz)then
ia=2
ib=nz-2
c
c Updated matrices when iz.ne.jz.
c
else
ia=min(iz,jz)
ib=max(iz,jz)
end if
c
c Eq. (4) of jasa 86, 1459-1464 in fluid layer.
c
do 2 i=ia-1,iz
l1=0.0
l2=(lamw(i)+lamw(i+1))/12.0
l3=0.0
l4=(lamw(i)+6.0*lamw(i+1)+lamw(i+2))/12.0
l5=0.0
l6=(lamw(i+1)+lamw(i+2))/12.0
l7=0.0
m1=0.0
m2=lamw(i+1)/dz**2+omega**2/6.0+
> 0.5*(lamw(i)-lamw(i+1))/dz**2+
> 0.5*(lamw(i)-lamw(i+1))/dz**2
m3=0.0
m4=-2.0*lamw(i+1)/dz**2+2.0*omega**2/3.0+
> 0.5*(2.0*lamw(i+1)-lamw(i)-lamw(i+2))/dz**2+
> 0.5*(lamw(i)-2.0*lamw(i+1)+lamw(i+2))/dz**2
m5=0.0
m6=lamw(i+1)/dz**2+omega**2/6.0+
> 0.5*(lamw(i+2)-lamw(i+1))/dz**2+
> 0.5*(lamw(i+2)-lamw(i+1))/dz**2
m7=0.0
c
do 1 j=1,np
r1(2*i-1,j)=k0**2*l1+pd2(j)*(m1-k0**2*l1)
r2(2*i-1,j)=k0**2*l2+pd2(j)*(m2-k0**2*l2)
r3(2*i-1,j)=k0**2*l3+pd2(j)*(m3-k0**2*l3)
r4(2*i-1,j)=k0**2*l4+pd2(j)*(m4-k0**2*l4)
r5(2*i-1,j)=k0**2*l5+pd2(j)*(m5-k0**2*l5)
r6(2*i-1,j)=k0**2*l6+pd2(j)*(m6-k0**2*l6)
r7(2*i-1,j)=k0**2*l7+pd2(j)*(m7-k0**2*l7)
c
s1(2*i-1,j)=k0**2*l1+pd1(j)*(m1-k0**2*l1)
s2(2*i-1,j)=k0**2*l2+pd1(j)*(m2-k0**2*l2)
s3(2*i-1,j)=k0**2*l3+pd1(j)*(m3-k0**2*l3)
s4(2*i-1,j)=k0**2*l4+pd1(j)*(m4-k0**2*l4)
s5(2*i-1,j)=k0**2*l5+pd1(j)*(m5-k0**2*l5)
s6(2*i-1,j)=k0**2*l6+pd1(j)*(m6-k0**2*l6)
s7(2*i-1,j)=k0**2*l7+pd1(j)*(m7-k0**2*l7)
1 continue
2 continue
c
c Eq. (2) of jasa 86, 1459-1464 in fluid layer.
c
do 4 i=ia-1,iz
m1=(-1.0/6.0)*(lamw(i)+2.0*lamw(i+1))/dz+
> (1.0/6.0)*(-lamw(i)+lamw(i+1))/dz
m2=0.0
m3=(1.0/6.0)*(lamw(i)-lamw(i+2))/dz+
> (1.0/3.0)*(-lamw(i)+lamw(i+2))/dz
m4=omega**2
m5=(1.0/6.0)*(lamw(i+2)+2.0*lamw(i+1))/dz+
> (1.0/6.0)*(lamw(i+2)-lamw(i+1))/dz
m6=0.0
m7=0.0
c
do 3 j=1,np
r1(2*i,j)=m1
r2(2*i,j)=m2
r3(2*i,j)=m3
r4(2*i,j)=m4
r5(2*i,j)=m5
r6(2*i,j)=m6
r7(2*i,j)=m7
s1(2*i,j)=0.0
s2(2*i,j)=0.0
s3(2*i,j)=0.0
s4(2*i,j)=0.0
s5(2*i,j)=0.0
s6(2*i,j)=0.0
s7(2*i,j)=0.0
3 continue
4 continue
c
c Eq. (4) of jasa 86, 1459-1464 in solid layer.
c
do 6 i=iz+1,ib+2
l1=0.0
l2=(lamb(i)+2.0*mub(i)+lamb(i+1)+2.0*mub(i+1))/12.0
l3=(2.0/6.0)*(-mub(i)+mub(i+1))/dz
l4=(lamb(i)+2.0*mub(i)+6.0*lamb(i+1)+12.0*mub(i+1)+
> lamb(i+2)+2.0*mub(i+2))/12.0
l5=(2.0/3.0)*(-mub(i)+mub(i+2))/dz
l6=(lamb(i+1)+2.0*mub(i+1)+lamb(i+2)+2.0*mub(i+2))/12.0
l7=(2.0/6.0)*(mub(i+2)-mub(i+1))/dz
m1=0.0
m2=(lamb(i+1)+2.0*mub(i+1))/dz**2+
> omega**2*(rhob(i)+rhob(i+1))/12.0+
> 0.5*(lamb(i)+2.0*mub(i)-lamb(i+1)-2.0*mub(i+1))/dz**2+
> 0.5*(lamb(i)-lamb(i+1))/dz**2
m3=(omega**2/6.0)*(-rhob(i)+rhob(i+1))/dz+
> 2.0*(-mub(i)+mub(i+1))/dz**3
m4=-2.0*(lamb(i+1)+2.0*mub(i+1))/dz**2+
> omega**2*(rhob(i)+6.0*rhob(i+1)+rhob(i+2))/12.0+
> 0.5*(2.0*lamb(i+1)+4.0*mub(i+1)-lamb(i)-2.0*mub(i)-
> lamb(i+2)-2.0*mub(i+2))/dz**2+
> 0.5*(lamb(i)-2.0*lamb(i+1)+lamb(i+2))/dz**2
m5=(omega**2/3.0)*(-rhob(i)+rhob(i+2))/dz+
> 2.0*(mub(i)-mub(i+2))/dz**3
m6=(lamb(i+1)+2.0*mub(i+1))/dz**2+
> omega**2*(rhob(i+1)+rhob(i+2))/12.0+
> 0.5*(lamb(i+2)+2.0*mub(i+2)-lamb(i+1)-2.0*mub(i+1))/dz**2+
> 0.5*(lamb(i+2)-lamb(i+1))/dz**2
m7=(omega**2/6.0)*(rhob(i+2)-rhob(i+1))/dz+
> 2.0*(mub(i+2)-mub(i+1))/dz**3
c
do 5 j=1,np
r1(2*i-1,j)=k0**2*l1+pd2(j)*(m1-k0**2*l1)
r2(2*i-1,j)=k0**2*l2+pd2(j)*(m2-k0**2*l2)
r3(2*i-1,j)=k0**2*l3+pd2(j)*(m3-k0**2*l3)
r4(2*i-1,j)=k0**2*l4+pd2(j)*(m4-k0**2*l4)
r5(2*i-1,j)=k0**2*l5+pd2(j)*(m5-k0**2*l5)
r6(2*i-1,j)=k0**2*l6+pd2(j)*(m6-k0**2*l6)
r7(2*i-1,j)=k0**2*l7+pd2(j)*(m7-k0**2*l7)
c
s1(2*i-1,j)=k0**2*l1+pd1(j)*(m1-k0**2*l1)
s2(2*i-1,j)=k0**2*l2+pd1(j)*(m2-k0**2*l2)
s3(2*i-1,j)=k0**2*l3+pd1(j)*(m3-k0**2*l3)
s4(2*i-1,j)=k0**2*l4+pd1(j)*(m4-k0**2*l4)
s5(2*i-1,j)=k0**2*l5+pd1(j)*(m5-k0**2*l5)
s6(2*i-1,j)=k0**2*l6+pd1(j)*(m6-k0**2*l6)
s7(2*i-1,j)=k0**2*l7+pd1(j)*(m7-k0**2*l7)
5 continue
6 continue
c
c Eq. (2) of jasa 86, 1459-1464 in solid layer.
c
do 8 i=iz+1,ib+2
l1=0.0
l2=(mub(i)+mub(i+1))/12.0
l3=0.0
l4=(mub(i)+6.0*mub(i+1)+mub(i+2))/12.0
l5=0.0
l6=(mub(i+1)+mub(i+2))/12.0
l7=0.0
m1=(-1.0/6.0)*(lamb(i)+mub(i)+2.0*lamb(i+1)+
> 2.0*mub(i+1))/dz+
> (1.0/6.0)*(-lamb(i)+lamb(i+1))/dz
m2=mub(i+1)/dz**2+
> omega**2*(rhob(i)+rhob(i+1))/12.0+
> (mub(i)-mub(i+1))/dz**2
m3=(1.0/6.0)*(lamb(i)+mub(i)-lamb(i+2)-mub(i+2))/dz+
> (1.0/3.0)*(-lamb(i)+lamb(i+2))/dz
m4=-2.0*mub(i+1)/dz**2+
> omega**2*(rhob(i)+6.0*rhob(i+1)+rhob(i+2))/12.0+
> (2.0*mub(i+1)-mub(i)-mub(i+2))/dz**2
m5=(1.0/6.0)*(lamb(i+2)+mub(i+2)+2.0*lamb(i+1)+
> 2.0*mub(i+1))/dz+
> (1.0/6.0)*(lamb(i+2)-lamb(i+1))/dz
m6=mub(i+1)/dz**2+
> omega**2*(rhob(i+1)+rhob(i+2))/12.0+
> (mub(i+2)-mub(i+1))/dz**2
m7=0.0
c
do 7 j=1,np
r1(2*i,j)=k0**2*l1+pd2(j)*(m1-k0**2*l1)
r2(2*i,j)=k0**2*l2+pd2(j)*(m2-k0**2*l2)
r3(2*i,j)=k0**2*l3+pd2(j)*(m3-k0**2*l3)
r4(2*i,j)=k0**2*l4+pd2(j)*(m4-k0**2*l4)
r5(2*i,j)=k0**2*l5+pd2(j)*(m5-k0**2*l5)
r6(2*i,j)=k0**2*l6+pd2(j)*(m6-k0**2*l6)
r7(2*i,j)=k0**2*l7+pd2(j)*(m7-k0**2*l7)
c
s1(2*i,j)=k0**2*l1+pd1(j)*(m1-k0**2*l1)
s2(2*i,j)=k0**2*l2+pd1(j)*(m2-k0**2*l2)
s3(2*i,j)=k0**2*l3+pd1(j)*(m3-k0**2*l3)
s4(2*i,j)=k0**2*l4+pd1(j)*(m4-k0**2*l4)
s5(2*i,j)=k0**2*l5+pd1(j)*(m5-k0**2*l5)
s6(2*i,j)=k0**2*l6+pd1(j)*(m6-k0**2*l6)
s7(2*i,j)=k0**2*l7+pd1(j)*(m7-k0**2*l7)
7 continue
8 continue
c
c Continuity conditions for a fluid-solid interface.
c
a11=lamw(iz)/lamw(iz+2)
a12=-2.0*dz*omega**2/lamw(iz+2)
c
a21=-dz*lamw(iz+1)/mub(iz+1)
a22=dz*lamb(iz+1)/mub(iz+1)
a23=1.0
c
a31=-2.0*(mub(iz+1)+mub(iz+2))*
> lamw(iz+1)/(mub(iz+1)*lamb(iz))
a32=2.0*(mub(iz+1)+mub(iz+2))*
> lamb(iz+1)/(mub(iz+1)*lamb(iz))
a33=2.0*rhob(iz+1)*dz*omega**2/lamb(iz)-
> 2.0*(mub(iz)+2.0*mub(iz+1)+mub(iz+2))/(dz*lamb(iz))
a34=lamb(iz+2)/lamb(iz)
a35=2.0*(mub(iz)+2.0*mub(iz+1)+mub(iz+2))/(dz*lamb(iz))
c
do 9 j=1,np
c
i=2*iz-1
r2(i,j)=r2(i,j)+a11*r6(i,j)
s2(i,j)=s2(i,j)+a11*s6(i,j)
r7(i,j)=a12*r6(i,j)
s7(i,j)=a12*s6(i,j)
r6(i,j)=0.0
s6(i,j)=0.0
c
i=2*iz
r1(i,j)=r1(i,j)+a11*r5(i,j)
s1(i,j)=s1(i,j)+a11*s5(i,j)
r6(i,j)=a12*r5(i,j)
s6(i,j)=a12*s5(i,j)
r5(i,j)=0.0
s5(i,j)=0.0
c
i=2*iz+1
r5(i,j)=r5(i,j)+a33*r2(i,j)
s5(i,j)=s5(i,j)+a33*s2(i,j)
r6(i,j)=r6(i,j)+a34*r2(i,j)
s6(i,j)=s6(i,j)+a34*s2(i,j)
r7(i,j)=r7(i,j)+a35*r2(i,j)
s7(i,j)=s7(i,j)+a35*s2(i,j)
r4(i,j)=r4(i,j)+a32*r2(i,j)
s4(i,j)=s4(i,j)+a32*s2(i,j)
r2(i,j)=a31*r2(i,j)
s2(i,j)=a31*s2(i,j)
c
r2(i,j)=r2(i,j)+a21*r3(i,j)
s2(i,j)=s2(i,j)+a21*s3(i,j)
r4(i,j)=r4(i,j)+a22*r3(i,j)
s4(i,j)=s4(i,j)+a22*s3(i,j)
r7(i,j)=r7(i,j)+a23*r3(i,j)
s7(i,j)=s7(i,j)+a23*s3(i,j)
r3(i,j)=0.0
s3(i,j)=0.0
c
i=2*iz+2
r3(i,j)=r3(i,j)+a32*r1(i,j)
s3(i,j)=s3(i,j)+a32*s1(i,j)
r4(i,j)=r4(i,j)+a33*r1(i,j)
s4(i,j)=s4(i,j)+a33*s1(i,j)
r5(i,j)=r5(i,j)+a34*r1(i,j)
s5(i,j)=s5(i,j)+a34*s1(i,j)
r6(i,j)=r6(i,j)+a35*r1(i,j)
s6(i,j)=s6(i,j)+a35*s1(i,j)
r1(i,j)=a31*r1(i,j)
s1(i,j)=a31*s1(i,j)
c
r1(i,j)=r1(i,j)+a21*r2(i,j)
s1(i,j)=s1(i,j)+a21*s2(i,j)
r3(i,j)=r3(i,j)+a22*r2(i,j)
s3(i,j)=s3(i,j)+a22*s2(i,j)
r6(i,j)=r6(i,j)+a23*r2(i,j)
s6(i,j)=s6(i,j)+a23*s2(i,j)
r2(i,j)=0.0
s2(i,j)=0.0
9 continue
c
c The matrix decomposition.
c
do 20 j=1,np
c
do 13 i=2*ia-3,2*iz
if(i.le.3)go to 10
r2(i,j)=r2(i,j)-r1(i,j)*r5(i-3,j)
r3(i,j)=r3(i,j)-r1(i,j)*r6(i-3,j)
r4(i,j)=r4(i,j)-r1(i,j)*r7(i-3,j)
10 if(i.le.2)go to 11
r3(i,j)=r3(i,j)-r2(i,j)*r5(i-2,j)
r4(i,j)=r4(i,j)-r2(i,j)*r6(i-2,j)
r5(i,j)=r5(i,j)-r2(i,j)*r7(i-2,j)
11 if(i.le.1)go to 12
r4(i,j)=r4(i,j)-r3(i,j)*r5(i-1,j)
r5(i,j)=r5(i,j)-r3(i,j)*r6(i-1,j)
r6(i,j)=r6(i,j)-r3(i,j)*r7(i-1,j)
12 r4(i,j)=1.0/r4(i,j)
r5(i,j)=r5(i,j)*r4(i,j)
r6(i,j)=r6(i,j)*r4(i,j)
r7(i,j)=r7(i,j)*r4(i,j)
13 continue
c
i1=2*nz-2
i2=2*nz-1
i3=2*nz
do 17 i=2*ib+4,2*iz+1,-1
if(i.ge.i1)go to 14
r6(i,j)=r6(i,j)-r7(i,j)*r3(i+3,j)
r5(i,j)=r5(i,j)-r7(i,j)*r2(i+3,j)
r4(i,j)=r4(i,j)-r7(i,j)*r1(i+3,j)
14 if(i.ge.i2)go to 15
r5(i,j)=r5(i,j)-r6(i,j)*r3(i+2,j)
r4(i,j)=r4(i,j)-r6(i,j)*r2(i+2,j)
r3(i,j)=r3(i,j)-r6(i,j)*r1(i+2,j)
15 if(i.ge.i3)go to 16
r4(i,j)=r4(i,j)-r5(i,j)*r3(i+1,j)
r3(i,j)=r3(i,j)-r5(i,j)*r2(i+1,j)
r2(i,j)=r2(i,j)-r5(i,j)*r1(i+1,j)
16 r4(i,j)=1.0/r4(i,j)
r3(i,j)=r3(i,j)*r4(i,j)
r2(i,j)=r2(i,j)*r4(i,j)
r1(i,j)=r1(i,j)*r4(i,j)
17 continue
c
c The six-by-six matrix near interface.
c
i0=2*iz-3
do 18 i=1,3
t4(i,j)=1.0
t5(i,j)=r5(i0+i,j)
t6(i,j)=r6(i0+i,j)
t7(i,j)=r7(i0+i,j)
18 continue
c
do 19 i=4,6
t1(i,j)=r1(i0+i,j)
t2(i,j)=r2(i0+i,j)
t3(i,j)=r3(i0+i,j)
t4(i,j)=1.0
t5(i,j)=0.0
t6(i,j)=0.0
t7(i,j)=0.0
19 continue
c
t2(4,j)=t2(4,j)-t1(4,j)*t5(1,j)
t3(4,j)=t3(4,j)-t1(4,j)*t6(1,j)
t4(4,j)=t4(4,j)-t1(4,j)*t7(1,j)
c
t3(4,j)=t3(4,j)-t2(4,j)*t5(2,j)
t4(4,j)=t4(4,j)-t2(4,j)*t6(2,j)
t5(4,j)=t5(4,j)-t2(4,j)*t7(2,j)
c
t4(4,j)=t4(4,j)-t3(4,j)*t5(3,j)
t5(4,j)=t5(4,j)-t3(4,j)*t6(3,j)
t6(4,j)=t6(4,j)-t3(4,j)*t7(3,j)
c
t5(4,j)=t5(4,j)/t4(4,j)
t6(4,j)=t6(4,j)/t4(4,j)
t4(4,j)=1.0/t4(4,j)
c
t2(5,j)=t2(5,j)-t1(5,j)*t5(2,j)
t3(5,j)=t3(5,j)-t1(5,j)*t6(2,j)
t4(5,j)=t4(5,j)-t1(5,j)*t7(2,j)
c
t3(5,j)=t3(5,j)-t2(5,j)*t5(3,j)
t4(5,j)=t4(5,j)-t2(5,j)*t6(3,j)
t5(5,j)=t5(5,j)-t2(5,j)*t7(3,j)
c
t4(5,j)=t4(5,j)-t3(5,j)*t5(4,j)
t5(5,j)=t5(5,j)-t3(5,j)*t6(4,j)
c
t5(5,j)=t5(5,j)/t4(5,j)
t4(5,j)=1.0/t4(5,j)
c
t2(6,j)=t2(6,j)-t1(6,j)*t5(3,j)
t3(6,j)=t3(6,j)-t1(6,j)*t6(3,j)
t4(6,j)=t4(6,j)-t1(6,j)*t7(3,j)
c
t3(6,j)=t3(6,j)-t2(6,j)*t5(4,j)
t4(6,j)=t4(6,j)-t2(6,j)*t6(4,j)
c
t4(6,j)=t4(6,j)-t3(6,j)*t5(5,j)
c
t4(6,j)=1.0/t4(6,j)
c
20 continue
c
return
end
c
c The heptadiagonal solver.
c
subroutine solve(mz,nz,mp,np,iz,u,v,r1,r2,r3,r4,r5,r6,r7,s1,s2,
> s3,s4,s5,s6,s7,t1,t2,t3,t4,t5,t6,t7,g0)
complex u(mz),v(mz),r1(mz,mp),r2(mz,mp),r3(mz,mp),r4(mz,mp),
> r5(mz,mp),r6(mz,mp),r7(mz,mp),s1(mz,mp),s2(mz,mp),s3(mz,mp),
> s4(mz,mp),s5(mz,mp),s6(mz,mp),s7(mz,mp),t1(6,mp),t2(6,mp),
> t3(6,mp),t4(6,mp),t5(6,mp),t6(6,mp),t7(6,mp),g0
eps=1.0e-30
c
do 6 j=1,np
c
c The right side.
c
i=1
v(i+2)=s2(i,j)*u(i)+s3(i,j)*u(i+1)+s4(i,j)*u(i+2)+
> s5(i,j)*u(i+3)+s6(i,j)*u(i+4)+s7(i,j)*u(i+5)+eps
do 1 i=2,2*nz
v(i+2)=s1(i,j)*u(i-1)+s2(i,j)*u(i)+s3(i,j)*u(i+1)+
> s4(i,j)*u(i+2)+s5(i,j)*u(i+3)+s6(i,j)*u(i+4)+
> s7(i,j)*u(i+5)+eps
1 continue
c
i1=2*iz-1
c
c The elimination steps.
c
i=1
v(i+2)=(v(i+2)-r2(i,j)*v(i)-r3(i,j)*v(i+1))*r4(i,j)+eps
do 2 i=2,2*iz
v(i+2)=(v(i+2)-r1(i,j)*v(i-1)-r2(i,j)*v(i)-
> r3(i,j)*v(i+1))*r4(i,j)+eps
2 continue
i=2*nz
v(i+2)=(v(i+2)-r6(i,j)*v(i+4)-r5(i,j)*v(i+3))*r4(i,j)+eps
do 3 i=2*nz-1,2*iz+1,-1
v(i+2)=(v(i+2)-r7(i,j)*v(i+5)-r6(i,j)*v(i+4)-
> r5(i,j)*v(i+3))*r4(i,j)+eps
3 continue
c
c The six-by-six system near the ocean bottom is solved.
c
v(4+i1)=(v(4+i1)-t1(4,j)*v(1+i1)-t2(4,j)*v(2+i1)-
> t3(4,j)*v(3+i1))*t4(4,j)+eps
v(5+i1)=(v(5+i1)-t1(5,j)*v(2+i1)-t2(5,j)*v(3+i1)-
> t3(5,j)*v(4+i1))*t4(5,j)+eps
u(6+i1)=(v(6+i1)-t1(6,j)*v(3+i1)-t2(6,j)*v(4+i1)-
> t3(6,j)*v(5+i1))*t4(6,j)+eps
c
u(5+i1)=v(5+i1)-t5(5,j)*u(6+i1)+eps
u(4+i1)=v(4+i1)-t5(4,j)*u(5+i1)-t6(4,j)*u(6+i1)+eps
u(3+i1)=v(3+i1)-t5(3,j)*u(4+i1)-t6(3,j)*u(5+i1)-
> t7(3,j)*u(6+i1)+eps
u(2+i1)=v(2+i1)-t5(2,j)*u(3+i1)-t6(2,j)*u(4+i1)-
> t7(2,j)*u(5+i1)+eps
u(1+i1)=v(1+i1)-t5(1,j)*u(2+i1)-t6(1,j)*u(3+i1)-
> t7(2,j)*u(4+i1)+eps
c
c The back substitution steps.
c
do 4 i=2*iz-3,1,-1
u(i+2)=v(i+2)-r5(i,j)*u(i+3)-r6(i,j)*u(i+4)-
> r7(i,j)*u(i+5)+eps
4 continue
do 5 i=2*iz+4,2*nz
u(i+2)=v(i+2)-r3(i,j)*u(i+1)-r2(i,j)*u(i)-
> r1(i,j)*u(i-1)+eps
5 continue
6 continue
c
do 7 i=1,nz
u(2*i+1)=g0*u(2*i+1)
u(2*i+2)=g0*u(2*i+2)
7 continue
c
return
end
c
c The rotated Pade coefficients [J. Acoust. Soc. Am. 101, 760-766
c (1997)].
c
subroutine rpade(mp,np,k0,dr,pd1,pd2,theta,g0)
c
complex ci,g0,tfact,rot0,rot1,rot2,pd1(mp),pd2(mp)
real k0
pi=4.0*atan(1.0)
ci=cmplx(0.0,1.0)
tfact=cexp(-ci*theta*pi/360.0)
den=float(2*np+1)
rot0=1.0
c
do 1 j=1,np
c
c The Pade coefficients.
c
pade1=(2.0/den)*sin(float(j)*pi/den)**2
pade2=cos(float(j)*pi/den)**2
c
c The rotated Pade coefficients.
c
rot1=tfact*pade1/(1.0+pade2*(tfact**2-1.0))**2
rot2=tfact**2*pade2/(1.0+pade2*(tfact**2-1.0))
rot0=rot0+pade1*(tfact**2-1.0)/(1.0+pade2*(tfact**2-1.0))
c
c The Crank-Nicolson coefficients.
c
pd1(j)=rot2+0.5*ci*k0*dr*rot1
pd2(j)=rot2-0.5*ci*k0*dr*rot1
c
1 continue
rot0=rot0/tfact
g0=cexp(ci*k0*dr*rot0)
c
return
end
c
c The coefficients of the rational approximation.
c
subroutine epade(mp,np,ns,ip,k0,c0,dr,pd1,pd2,irot,theta,g0,nu8)
c
implicit real*8 (a-h,o-z)
complex*16 ci,z1,z2,g,dg,dh1,dh2,dh3,a,b,nu
complex*8 ci8,g0,pd1(mp),pd2(mp),nu8
real*4 k0,c0,dr,theta
parameter (m=40)
dimension bin(m,m),a(m,m),b(m),dg(m),dh1(m),dh2(m),dh3(m),fact(m)
pi=4.0d0*datan(1.0d0)
ci=dcmplx(0.0d0,1.0d0)
ci8=cmplx(0.0,1.0)
sig=k0*dr
n=2*np
g0=cexp(ci8*k0*dr)
c
if((ip.eq.1).and.(irot.eq.1))then
call rpade(mp,np,k0,dr,pd1,pd2,theta,g0)
return
end if
c
if(ip.eq.1)then
nu=0.0d0
alp=0.0d0
c
else
nu=nu8
alp=-0.25d0
end if
c
c The factorials.
c
fact(1)=1.0d0
do 1 i=2,n
fact(i)=dfloat(i)*fact(i-1)
1 continue
c
c The binomial coefficients.
c
do 2 i=1,n+1
bin(i,1)=1.0d0
bin(i,i)=1.0d0
2 continue
do 4 i=3,n+1
do 3 j=2,i-1
bin(i,j)=bin(i-1,j-1)+bin(i-1,j)
3 continue
4 continue
c
do 6 i=1,n
do 5 j=1,n
a(i,j)=0.0d0
5 continue
6 continue
c
c The accuracy constraints.
c
call deriv(m,n,sig,alp,dg,dh1,dh2,dh3,bin,nu)
c
do 7 i=1,n
b(i)=dg(i+1)
7 continue
do 9 i=1,n
if(2*i-1.le.n)a(i,2*i-1)=fact(i)
do 8 j=1,i
if(2*j.le.n)a(i,2*j)=-bin(i+1,j+1)*fact(j)*dg(i-j+1)
8 continue
9 continue
c
c The stability constraints.
c
if(ns.ge.1)then
z1=-3.0d0
b(n)=-1.0d0
do 10 j=1,np
a(n,2*j-1)=z1**j
a(n,2*j)=0.0d0
10 continue
end if
c
if(ns.ge.2)then
z1=-1.5d0
b(n-1)=-1.0d0
do 11 j=1,np
a(n-1,2*j-1)=z1**j
a(n-1,2*j)=0.0d0
11 continue
end if
c
call gauss(m,n,a,b)
c
dh1(1)=1.0d0
do 12 j=1,np
dh1(j+1)=b(2*j-1)
12 continue
call fndrt(dh1,np,dh2,m)
do 13 j=1,np
pd1(j)=-1.0d0/dh2(j)
13 continue
c
dh1(1)=1.0d0
do 14 j=1,np
dh1(j+1)=b(2*j)
14 continue
call fndrt(dh1,np,dh2,m)
do 15 j=1,np
pd2(j)=-1.0d0/dh2(j)
15 continue
c
return
end
c
c The operator function.
c
function g(ci,sig,x,alp,nu)
complex*16 ci,g,nu
real*8 alp,sig,x
g=(1.0d0-nu*x)**2*cdexp(alp*dlog(1.0d0+x)+
> ci*sig*(-1.0d0+dsqrt(1.0d0+x)))
return
end
c
c The derivatives of the operator function at x=0.
c
subroutine deriv(m,n,sig,alp,dg,dh1,dh2,dh3,bin,nu)
implicit real*8 (a-h,o-z)
complex*16 ci,dg(m),dh1(m),dh2(m),dh3(m),nu
real*8 bin(m,m)
ci=dcmplx(0.0d0,1.0d0)
c
dh1(1)=0.5d0*ci*sig
exp1=-0.5d0
dh2(1)=alp
exp2=-1.0d0
dh3(1)=-2.0d0*nu
exp3=-1.0d0