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Copy pathFWI_solver.m
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FWI_solver.m
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function [pf,recf,C2,vg]=FWI_solver(nz,nx,dh,sz,sx,ome,sf,rz,rx,v,lp,d0,true_recfs)
% linearized wave equation
% dims frequency sample number
% Source: source signals in frequency domain
% lp: layers of PML
% d0: for PML, could be related to R (reflection coefficient)
%%
AA=sparse(nz*nx,nz*nx);
s=zeros(nz*nx,length(ome));
pf=zeros(nz*nx,length(ome));
recf=zeros(length(rx),length(ome));
% sampling operator
R=zeros(length(rx),nz*nx);
C=zeros(length(ome),1);
true_recfs2=true_recfs.';
vg=zeros(nz,nx);
%%
for i=1:size(R,1)
R(i,(rx(i)-1)*nz+rz(i))=1;
end
%%
for i=1:length(sx)
s(sz(i)+(sx(i)-1)*nz,:)=sf(:,i);
end
%%
for l=2:length(ome)
%% assign d
dx=zeros(nz,nx);
dz=zeros(nz,nx);
% top
for i=2:lp+1
for j=lp+2:nx-lp-1
dx(i,j)=0;
dz(i,j)=d0(i,j)*((lp+2-i)/lp)^2;
end
end
% bottom
for i=nz-lp:nz-1
for j=lp+2:nx-lp-1
dx(i,j)=0;
dz(i,j)=d0(i,j)*((i+1-(nz-lp))/lp)^2;
end
end
% left
for i=lp+2:nz-lp-1
for j=2:lp+1
dz(i,j)=0;
dx(i,j)=d0(i,j)*((lp+2-j)/lp)^2;
end
end
% right
for i=lp+1:nz-lp-1
for j=nx-lp:nx-1
dz(i,j)=0;
dx(i,j)=d0(i,j)*((j+1-(nx-lp))/lp)^2;
end
end
% upper left
for i=2:lp+1
for j=2:lp+1
dz(i,j)=d0(i,j)*((lp+2-i)/lp)^2;
dx(i,j)=d0(i,j)*((lp+2-j)/lp)^2;
end
end
% upper right
for i=2:lp+1
for j=nx-lp:nx-1
dz(i,j)=d0(i,j)*((lp+2-i)/lp)^2;
dx(i,j)=d0(i,j)*((j+1-(nx-lp))/lp)^2;
end
end
% lower left
for i=nz-lp:nz-1
for j=2:lp+1
dz(i,j)=d0(i,j)*((i+1-(nz-lp))/lp)^2;
dx(i,j)=d0(i,j)*((lp+2-j)/lp)^2;
end
end
% lower right
for i=nz-lp:nz-1
for j=nx-lp:nx-1
dz(i,j)=d0(i,j)*((i+1-(nz-lp))/lp)^2;
dx(i,j)=d0(i,j)*((j+1-(nx-lp))/lp)^2;
end
end
gx=1+dx/1i./ome(l);
gz=1+dz/1i./ome(l);
gx_x=diff(gx,1,2)/dh;
gz_z=diff(gz,1,1)/dh;
gx_x=[-gx_x(:,1:end-31),zeros([size(gx_x,1),1]),gx_x(:,end-30:end)];
gx_x(:,1)=0;
gx_x(:,end)=0;
gz_z=[-gz_z(1:end-31,:);zeros([1,size(gz_z,2)]);gz_z(end-30:end,:)];
gz_z(1,:)=0;
gz_z(end,:)=0;
%% domain interior
for i=lp+2:nz-1
for j=lp+2:nx-1
% i,j
AA((j-1)*nz+i,(j-1)*nz+i)=(ome(l)^2-2*v(i,j)^2/gx(i,j)^2/dh^2-2*v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i,j+1
AA((j-1)*nz+i,(j)*nz+i)=(-v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i,j-1
AA((j-1)*nz+i,(j-2)*nz+i)=(v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i+1,j
AA((j-1)*nz+i,(j-1)*nz+i+1)=(-v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i-1,j
AA((j-1)*nz+i,(j-1)*nz+i-1)=(v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
end
end
%% top local
for i=2:lp+1
for j=lp+2:nx-lp-1
% i,j
AA((j-1)*nz+i,(j-1)*nz+i)=(ome(l)^2-2*v(i,j)^2/gx(i,j)^2/dh^2-2*v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i,j+1
AA((j-1)*nz+i,(j)*nz+i)=(-v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i,j-1
AA((j-1)*nz+i,(j-2)*nz+i)=(v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i+1,j
AA((j-1)*nz+i,(j-1)*nz+i+1)=(v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i-1,j
AA((j-1)*nz+i,(j-1)*nz+i-1)=(-v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
end
end
%% left local
for i=lp+2:nz-lp-1
for j=2:lp+1
% i,j
AA((j-1)*nz+i,(j-1)*nz+i)=(ome(l)^2-2*v(i,j)^2/gx(i,j)^2/dh^2-2*v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i,j+1
AA((j-1)*nz+i,(j)*nz+i)=(v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i,j-1
AA((j-1)*nz+i,(j-2)*nz+i)=(-v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i+1,j
AA((j-1)*nz+i,(j-1)*nz+i+1)=(-v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i-1,j
AA((j-1)*nz+i,(j-1)*nz+i-1)=(v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
end
end
%% upper left
for i=2:lp+1
for j=2:lp+1
% i,j
AA((j-1)*nz+i,(j-1)*nz+i)=(ome(l)^2-2*v(i,j)^2/gx(i,j)^2/dh^2-2*v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i,j+1
AA((j-1)*nz+i,(j)*nz+i)=(v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i,j-1
AA((j-1)*nz+i,(j-2)*nz+i)=(-v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i+1,
AA((j-1)*nz+i,(j-1)*nz+i+1)=(v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i-1,j
AA((j-1)*nz+i,(j-1)*nz+i-1)=(-v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
end
end
%% upper right
for i=2:lp+1
for j=nx-lp:nx-1
% i,j
AA((j-1)*nz+i,(j-1)*nz+i)=(ome(l)^2-2*v(i,j)^2/gx(i,j)^2/dh^2-2*v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i,j+1
AA((j-1)*nz+i,(j)*nz+i)=(-v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i,j-1
AA((j-1)*nz+i,(j-2)*nz+i)=(v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i+1,j
AA((j-1)*nz+i,(j-1)*nz+i+1)=(v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i-1,j
AA((j-1)*nz+i,(j-1)*nz+i-1)=(-v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
end
end
%% lower left
for i=nz-lp:nz-1
for j=2:lp+1
% i,j
AA((j-1)*nz+i,(j-1)*nz+i)=(ome(l)^2-2*v(i,j)^2/gx(i,j)^2/dh^2-2*v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i,j+1
AA((j-1)*nz+i,(j)*nz+i)=(v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i,j-1
AA((j-1)*nz+i,(j-2)*nz+i)=(-v(i,j)^2/2/gx(i,j)^3/dh*gx_x(i,j)+v(i,j)^2/gx(i,j)^2/dh^2)/v(i,j)^2;
% i+1,j
AA((j-1)*nz+i,(j-1)*nz+i+1)=(-v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
% i-1,j
AA((j-1)*nz+i,(j-1)*nz+i-1)=(v(i,j)^2/2/gz(i,j)^3/dh*gz_z(i,j)+v(i,j)^2/gz(i,j)^2/dh^2)/v(i,j)^2;
end
end
%% boundary line
%% B2
i=1;
for j=2:nx-1
AA((j-1)*nz+i,(j-1)*nz+i)=-1/dh-1i*ome(l)/(v(i,j));
%AA((j-1)*nz+i,(j)*nz+i)=1/dh^2;
%AA((j-1)*nz+i,(j-2)*nz+i)=1/dh^2;
%AA((j-1)*nz+i,(j-1)*nz+i-1)=1/dh^2;
AA((j-1)*nz+i,(j-1)*nz+i+1)=1/dh;
end
%% B3
i=nz;
for j=2:nx-1
AA((j-1)*nz+i,(j-1)*nz+i)=-1/dh-1i*ome(l)/(v(i,j));
%AA((j-1)*nz+i,(j)*nz+i)=1/dh;
%AA((j-1)*nz+i,(j-2)*nz+i)=1/dh;
AA((j-1)*nz+i,(j-1)*nz+i-1)=1/dh;
%AA((j-1)*nz+i,(j-1)*nz+i+1)=1/dh;
end
%% B4
j=nx;
for i=2:nz-1
AA((j-1)*nz+i,(j-1)*nz+i)=-1/dh-1i*ome(l)/(v(i,j));
%AA((j-1)*nz+i,(j)*nz+i)=1/dh;
AA((j-1)*nz+i,(j-2)*nz+i)=1/dh;
%AA((j-1)*nz+i,(j-1)*nz+i-1)=1/dh;
%AA((j-1)*nz+i,(j-1)*nz+i+1)=1/dh;
end
%% B5
j=1;
for i=2:nz-1
AA((j-1)*nz+i,(j-1)*nz+i)=-1/dh-1i*ome(l)/(v(i,j));
AA((j-1)*nz+i,(j)*nz+i)=1/dh;
%AA((j-1)*nz+i,(j-2)*nz+i)=1/dh;
%AA((j-1)*nz+i,(j-1)*nz+i-1)=1/dh;
%AA((j-1)*nz+i,(j-1)*nz+i+1)=1/dh;
end
%% B6
j=nx;
i=1;
AA((j-1)*nz+i,(j-1)*nz+i)=-2/dh-2i*ome(l)/(v(i,j));
%AA((j-1)*nz+i,(j)*nz+i)=1/dh;
AA((j-1)*nz+i,(j-2)*nz+i)=1/dh;
%AA((j-1)*nz+i,(j-1)*nz+i-1)=1/dh;
AA((j-1)*nz+i,(j-1)*nz+i+1)=1/dh;
%% B7
j=1;
i=1;
AA((j-1)*nz+i,(j-1)*nz+i)=-2/dh-2i*ome(l)/(v(i,j));
AA((j-1)*nz+i,(j)*nz+i)=1/dh;
%AA((j-1)*nz+i,(j-2)*nz+i)=1/dh;
%AA((j-1)*nz+i,(j-1)*nz+i-1)=1/dh;
AA((j-1)*nz+i,(j-1)*nz+i+1)=1/dh;
%% B8
i=nz;
j=nx;
AA((j-1)*nz+i,(j-1)*nz+i)=-2/dh-2i*ome(l)/(v(i,j));
%AA((j-1)*nz+i,(j)*nz+i)=1/dh;
AA((j-1)*nz+i,(j-2)*nz+i)=1/dh;
AA((j-1)*nz+i,(j-1)*nz+i-1)=1/dh;
%AA((j-1)*nz+i,(j-1)*nz+i+1)=1/dh;
%% B9
j=1;
i=nz;
AA((j-1)*nz+i,(j-1)*nz+i)=-2/dh-2i*ome(l)/(v(i,j));
AA((j-1)*nz+i,(j)*nz+i)=1/dh;
%AA((j-1)*nz+i,(j-2)*nz+i)=1/dh;
AA((j-1)*nz+i,(j-1)*nz+i-1)=1/dh;
%AA((j-1)*nz+i,(j-1)*nz+i+1)=1/dh;
%% solve Ax=b
p0=AA\(-s(:,l));
pf(:,l)=p0;
%% sampling
recf(:,l)=R*p0;
%% cost
e=recf(:,l)-true_recfs2(:,l);
C(l)=norm(e);
%%
paf=AA'\(R'*e);
%% adjoint field
t2=ome(l)^2*(diag(pf(:,l)))'*paf;
%% gradient
vg=vg+reshape(t2,[size(vg,1),size(vg,2)]);
%%
fprintf('\n \t\t\t frequency number=%d/%d',l,length(ome));
end
vg(nz-lp+1:end,:)=0;
vg(1:lp,:)=0;
vg(:,1:lp)=0;
vg(:,nx-lp+1:end)=0;
vg=real(vg);
C2=mean(C);
end