├── F.m ├── GQN.m ├── README.md ├── exp1_refl.m ├── exp1_trans.m ├── exp2_refl.m ├── exp2_trans.m ├── getA.m ├── getG.m ├── getP.m ├── getoption.m ├── gradtest.m ├── license.txt └── misfit.m /F.m: -------------------------------------------------------------------------------- 1 | function S = F(m,Q,model) 2 | % Modeling operator based on 5-point discretization of Helmholtz operator 3 | % 4 | % S = PA(m)^{-1}Q 5 | % 6 | % use: 7 | % S = F(m,Q,model) 8 | % 9 | % input: 10 | % m - model [s^2/km^2] 11 | % Q - sources as 2D array (each column is a vector of source-weights, typically Q = I) 12 | % S - data as 3D array (receivers x sources x frequencies) 13 | % model.h - [dz,dx] gridspacing in z and x direction [m] 14 | % model.n - [nz,nx] number of gridpoints in z and x direction 15 | % model.f - frequencies 16 | % model.Is - source location indices Is{1} - z locations, Is{2} - x locations 17 | % model.Ir - receiver location indices 18 | % 19 | % ------------------------------------------------------------------------- 20 | % Copyright (C) 2013 Tristan van Leeuwen 21 | % Centrum Wiskunde & Informatica 22 | % Tristan.van.Leeuwen@cwi.nl 23 | % 24 | % This program is free software: you can redistribute it and/or modify 25 | % it under the terms of the GNU General Public License as published by 26 | % the Free Software Foundation, either version 3 of the License, or 27 | % (at your option) any later version. 28 | % 29 | % This program is distributed in the hope that it will be useful, 30 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 31 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 32 | % GNU General Public License for more details. 33 | % 34 | % You should have received a copy of the GNU General Public License 35 | % along with this program. If not, see . 36 | % ------------------------------------------------------------------------- 37 | 38 | % get sampling operators 39 | Ps = getP(model.n,model.Is{:}); 40 | Pr = getP(model.n,model.Ir{:}); 41 | 42 | S = zeros(size(Pr,1),size(Ps,1),length(model.f)); 43 | 44 | % loop over frequencies 45 | for k = 1:length(model.f) 46 | % get Helmholtz operator 47 | Ak = getA(model.f(k),m,model.h,model.n); 48 | 49 | % solve forward wave-equation and sample data 50 | Uk = Ak\Ps'*Q; 51 | S(:,:,k) = Pr*Uk; 52 | end -------------------------------------------------------------------------------- /GQN.m: -------------------------------------------------------------------------------- 1 | function [x,info] = GQN(fh,x0,options) 2 | % Simple Quasi/Gauss-Newton method for penalty method 3 | % 4 | % use: 5 | % [xn,info] = GQN(fh,x0,lambda,options) 6 | % 7 | % input: 8 | % fh - function handle to misfit of the form [f1,f2,g] = fh(x) 9 | % where f = f1 + lambda*f2 is the function value, g is the gradient of the same size 10 | % as the input vector x. 11 | % x0 - initial guess 12 | % 13 | % options.maxiter - max iterations [default 10] 14 | % options.tol - tolerance on 2-norm of gradient [1e-6] 15 | % options.M - history size [5] 16 | % options.fid - file id for output [1] 17 | % options.method - {'LBFGS','GN'} 18 | % options.cgtol - CG tolerance for GN subproblems 19 | % options.cgprec - precondition CG with LBFGS Hessian 20 | % 21 | % output: 22 | % xn - final estimate 23 | % info - iteration history (# iter, # eval, stepsize, f , ||g(x)||_2, cgiter) 24 | % 25 | % ------------------------------------------------------------------------- 26 | % Copyright (C) 2013 Tristan van Leeuwen 27 | % Centrum Wiskunde & Informatica 28 | % Tristan.van.Leeuwen@cwi.nl 29 | % 30 | % This program is free software: you can redistribute it and/or modify 31 | % it under the terms of the GNU General Public License as published by 32 | % the Free Software Foundation, either version 3 of the License, or 33 | % (at your option) any later version. 34 | % 35 | % This program is distributed in the hope that it will be useful, 36 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 37 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 38 | % GNU General Public License for more details. 39 | % 40 | % You should have received a copy of the GNU General Public License 41 | % along with this program. If not, see . 42 | % ------------------------------------------------------------------------- 43 | 44 | if nargin<3 45 | options = []; 46 | end 47 | 48 | maxiter = getoption(options,'maxiter',10); 49 | M = getoption(options,'M',5); 50 | tol = getoption(options,'tol',1e-6); 51 | fid = getoption(options,'fid',1); 52 | method = getoption(options,'method','LBFGS'); 53 | cgtol = getoption(options,'cgtol',1e-2); 54 | cgmax = getoption(options,'cgmax',100); 55 | cgprec = getoption(options,'cgprec',0); 56 | write = getoption(options,'write',0); 57 | 58 | % initialization 59 | fprintf(fid,'# iter, # eval, stepsize, f , ||g(x)||_2, cgiter\n'); 60 | outputstr = '%6d, %6d, %1.2e, %1.5e, %1.5e, %6d\n'; 61 | n = length(x0); 62 | converged = 0; 63 | iter = 0; 64 | x = x0; 65 | alpha0 = 1; 66 | cgiter = 0; 67 | S = zeros(n,M); 68 | Y = zeros(n,M); 69 | 70 | % initial evaluation 71 | 72 | [f,g,H] = fh(x); 73 | tol = tol*norm(g); 74 | nfeval = 1; 75 | 76 | % output 77 | info(1,:) = [iter,nfeval,alpha0,f,norm(g),cgiter]; 78 | fprintf(fid,outputstr,info(1,:)); 79 | if write 80 | dlmwrite(['x_' num2str(iter) '.dat'],x); 81 | end 82 | 83 | % main loop 84 | while ~converged 85 | alpha0 = 1; 86 | % compute search direction 87 | switch method 88 | case 'LBFGS'% L-BFGS 89 | if iter == 0; 90 | alpha0 = 1/norm(g); 91 | end 92 | 93 | s = B(-g,S,Y,[]); 94 | 95 | p = -(s'*g)/(g'*g); 96 | if (p <= 0) 97 | fprintf(fid,'Loss of descent, reset history\n'); 98 | S = zeros(n,M); 99 | Y = zeros(n,M); 100 | alpha0 = 1/norm(g); 101 | end 102 | 103 | case 'GN' 104 | 105 | if cgprec 106 | [s,flag,~,cgiter] = pcg(H,-g,cgtol,cgmax,@(x)B(x,S,Y,[])); 107 | else 108 | [s,flag,~,cgiter] = pcg(H,-g,cgtol,cgmax); 109 | end 110 | 111 | if flag 112 | fprintf(fid,'CG did not converge... (flag = %d)\n', flag); 113 | end 114 | 115 | p = -(s'*g)/(g'*g); 116 | if (p <= 0) 117 | fprintf(fid,'Loss of descent, resort to gradient-step\n'); 118 | s = -g; 119 | alpha0 = 1/norm(g); 120 | end 121 | 122 | end 123 | 124 | % linesearch 125 | [ft,gt,H,alpha,lsiter] = wWolfeLS(fh,x,f,g,s,alpha0); 126 | nfeval = nfeval + lsiter; 127 | 128 | % update 129 | xt = x + alpha*s; 130 | 131 | S = circshift(S,[0 -1]); 132 | Y = circshift(Y,[0 -1]); 133 | S = [S(:,1:end-1) xt - x]; 134 | Y = [Y(:,1:end-1) gt - g]; 135 | 136 | f = ft; 137 | g = gt; 138 | x = xt; 139 | 140 | iter = iter + 1; 141 | 142 | % output 143 | info(iter + 1,:) = [iter,nfeval,alpha,f,norm(g),cgiter]; 144 | fprintf(fid,outputstr,info(iter + 1,:)); 145 | if write 146 | dlmwrite(['x_' num2str(iter) '.dat'],x); 147 | end 148 | 149 | % check convergence 150 | converged = (iter>maxiter)||(norm(g) f0 + c1*alpha*g0'*s0 186 | nu = alpha; 187 | elseif gt'*s0 < c2*g0'*s0 188 | mu = alpha; 189 | else 190 | done = 1; 191 | end 192 | end 193 | end 194 | 195 | function z = B(x,S,Y,H0) 196 | % apply lbfgs inverse Hessian to vector 197 | % 198 | % Tristan van Leeuwen, 2011 199 | % tleeuwen@eos.ubc.ca 200 | % 201 | % use: 202 | % z = B(x,S,Y,b0,Binit) 203 | % 204 | % input: 205 | % x - vector of length n 206 | % S - history of steps in n x M matrix 207 | % Y - history of gradient differences in n x M matrix 208 | % 209 | % output 210 | % z - vector of length n 211 | % 212 | 213 | J = find(sum(abs(S),1)); 214 | S = S(:,J); 215 | Y = Y(:,J); 216 | M = size(S,2); 217 | n = length(x); 218 | 219 | if isempty(H0)&&(M>0) 220 | H0 = norm(Y(:,end))^2/(S(:,end)'*Y(:,end))*ones(n,1); 221 | else 222 | H0 = ones(n,1); 223 | end 224 | 225 | alpha = zeros(M,1); 226 | rho = zeros(M,1); 227 | for k = 1:M 228 | rho(k) = 1/(Y(:,k)'*S(:,k)); 229 | end 230 | q = x; 231 | % first recursion 232 | for k = M:-1:1 233 | alpha(k) = rho(k)*S(:,k)'*q; 234 | q = q - alpha(k)*Y(:,k); 235 | end 236 | 237 | % apply `initial' Hessian 238 | z = q./H0; 239 | % second recursion 240 | for k = 1:M 241 | beta = rho(k)*Y(:,k)'*z; 242 | z = z + (alpha(k) - beta)*S(:,k); 243 | end 244 | end 245 | 246 | 247 | 248 | 249 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | Variable-Projection-for-FWI 2 | =========================== 3 | 4 | Matlab code used to produce the results in the commentary on 5 | "Application of the variable projection scheme for frequency-domain full-waveform inversion" by Li et al. 2013, Geophysics 78(6). 6 | 7 | Tristan van Leeuwen 8 | Aleksandr Y. Aravkin 9 | Felix J. Herrmann 10 | 11 | 12 | -------------------------------------------------------------------------------- /exp1_refl.m: -------------------------------------------------------------------------------- 1 | % This script reproduces figure 2(b) from the paper 2 | % 3 | % ------------------------------------------------------------------------- 4 | % Copyright (C) 2013 Tristan van Leeuwen 5 | % Centrum Wiskunde & Informatica 6 | % Tristan.van.Leeuwen@cwi.nl 7 | % 8 | % This program is free software: you can redistribute it and/or modify 9 | % it under the terms of the GNU General Public License as published by 10 | % the Free Software Foundation, either version 3 of the License, or 11 | % (at your option) any later version. 12 | % 13 | % This program is distributed in the hope that it will be useful, 14 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 15 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 16 | % GNU General Public License for more details. 17 | % 18 | % You should have received a copy of the GNU General Public License 19 | % along with this program. If not, see . 20 | % ------------------------------------------------------------------------- 21 | 22 | % define velocity model 23 | n = [51 51]; 24 | h = [20 20]; 25 | z = [0:n(1)-1]'*h(1); 26 | x = [0:n(2)-1]*h(2); 27 | 28 | v0 = 2000*ones(n); 29 | dv = zeros(n); 30 | dv(11:21,11:21) = 100; 31 | dv(31:41,31:41) = -100; 32 | m = 1e6./(v0(:) + dv(:)).^2; 33 | m0 = 1e6./(v0(:)).^2; 34 | 35 | % grid 36 | model.h = h; 37 | model.n = n; 38 | 39 | % frequency 40 | model.f = [2.5 5 10]; 41 | 42 | % acquisition 43 | model.Is = {2,2:50}; 44 | model.Ir = {2,2:50}; 45 | 46 | % sources 47 | Q = speye(length(model.Is{1})*length(model.Is{2})); 48 | 49 | % data 50 | D = F(m,Q,model); 51 | 52 | % multiply with random amplitude factor 53 | s = RandStream('mt19937ar','Seed',1); 54 | RandStream.setGlobalStream(s); 55 | w = 5*abs(randn(1,size(D,2),size(D,3))); 56 | D = bsxfun(@times,w,D); 57 | 58 | % misfit 59 | fh = @(x)misfit(x,Q,D,model,true,false); 60 | fhc = @(x)misfit(x,Q,D,model,true,true); 61 | 62 | % perturbation 63 | [zz,xx] = ndgrid(z,x); 64 | mr = m0; 65 | alpha = linspace(-.2,.2,20); 66 | 67 | 68 | [fr,gr,Hr] = fh(mr); 69 | %dm = exp(-1e-4*((xx(:) - mean(x)).^2 + (zz(:) - mean(z)).^2))*mean(m0); 70 | dm = gr/norm(gr); 71 | %dm = m - m0; 72 | 73 | hr = Hr(dm); 74 | 75 | [frc,grc,Hrc] = fhc(mr); 76 | hrc = Hrc(dm); 77 | 78 | for k = 1:length(alpha) 79 | f(k) = fh(mr + alpha(k)*dm); 80 | q1(k) = fr + alpha(k)*dm'*gr + .5*alpha(k)^2*dm'*hr; 81 | q2(k) = fr + alpha(k)*dm'*grc + .5*alpha(k)^2*dm'*hrc; 82 | end 83 | 84 | % plot 85 | figure; 86 | plot(alpha,f,'k--',alpha,q1,'k',alpha,q2,'r','linewidth',2); 87 | legend('true','w/o correction','w correction','location','NorthWest'); 88 | xlabel('\alpha','fontsize',20);ylabel('misfit','fontsize',20); 89 | set(gca,'fontsize',20); 90 | 91 | % save 92 | print(1,'-depsc',mfilename); 93 | -------------------------------------------------------------------------------- /exp1_trans.m: -------------------------------------------------------------------------------- 1 | % This script reproduces figure 2(a) from the paper 2 | % 3 | % ------------------------------------------------------------------------- 4 | % Copyright (C) 2013 Tristan van Leeuwen 5 | % Centrum Wiskunde & Informatica 6 | % Tristan.van.Leeuwen@cwi.nl 7 | % 8 | % This program is free software: you can redistribute it and/or modify 9 | % it under the terms of the GNU General Public License as published by 10 | % the Free Software Foundation, either version 3 of the License, or 11 | % (at your option) any later version. 12 | % 13 | % This program is distributed in the hope that it will be useful, 14 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 15 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 16 | % GNU General Public License for more details. 17 | % 18 | % You should have received a copy of the GNU General Public License 19 | % along with this program. If not, see . 20 | % ------------------------------------------------------------------------- 21 | 22 | % define velocity model 23 | n = [51 51]; 24 | h = [20 20]; 25 | z = [0:n(1)-1]'*h(1); 26 | x = [0:n(2)-1]*h(2); 27 | 28 | v0 = 2000*ones(n); 29 | dv = zeros(n); 30 | dv(11:21,11:21) = 100; 31 | dv(31:41,31:41) = -100; 32 | m = 1e6./(v0(:) + dv(:)).^2; 33 | m0 = 1e6./(v0(:)).^2; 34 | 35 | % grid 36 | model.h = h; 37 | model.n = n; 38 | 39 | % frequency 40 | model.f = [2.5 5 10]; 41 | 42 | % acquisition 43 | model.Is = {2:50,2}; 44 | model.Ir = {2:50,50}; 45 | 46 | % sources 47 | Q = speye(length(model.Is{1})*length(model.Is{2})); 48 | 49 | % data 50 | D = F(m,Q,model); 51 | 52 | % multiply with random amplitude factor 53 | s = RandStream('mt19937ar','Seed',1); 54 | RandStream.setGlobalStream(s); 55 | w = 5*abs(randn(1,size(D,2),size(D,3))); 56 | D = bsxfun(@times,w,D); 57 | 58 | % misfit 59 | fh = @(x)misfit(x,Q,D,model,true,false); 60 | fhc = @(x)misfit(x,Q,D,model,true,true); 61 | 62 | % perturbation 63 | [zz,xx] = ndgrid(z,x); 64 | mr = m0; 65 | alpha = linspace(-.2,.2,20); 66 | 67 | 68 | [fr,gr,Hr] = fh(mr); 69 | %dm = exp(-1e-4*((xx(:) - mean(x)).^2 + (zz(:) - mean(z)).^2))*mean(m0); 70 | dm = gr/norm(gr); 71 | %dm = m - m0; 72 | 73 | hr = Hr(dm); 74 | 75 | [frc,grc,Hrc] = fhc(mr); 76 | hrc = Hrc(dm); 77 | 78 | for k = 1:length(alpha) 79 | f(k) = fh(mr + alpha(k)*dm); 80 | q1(k) = fr + alpha(k)*dm'*gr + .5*alpha(k)^2*dm'*hr; 81 | q2(k) = fr + alpha(k)*dm'*grc + .5*alpha(k)^2*dm'*hrc; 82 | end 83 | 84 | % plot 85 | figure; 86 | plot(alpha,f,'k--',alpha,q1,'k',alpha,q2,'r','linewidth',2); 87 | legend('true','w/o correction','w correction','location','NorthWest'); 88 | xlabel('\alpha','fontsize',20);ylabel('misfit','fontsize',20); 89 | set(gca,'fontsize',20); 90 | 91 | % save 92 | print(1,'-depsc',mfilename); 93 | 94 | 95 | -------------------------------------------------------------------------------- /exp2_refl.m: -------------------------------------------------------------------------------- 1 | % This script reproduces figure 5 from the paper 2 | % 3 | % ------------------------------------------------------------------------- 4 | % Copyright (C) 2013 Tristan van Leeuwen 5 | % Centrum Wiskunde & Informatica 6 | % Tristan.van.Leeuwen@cwi.nl 7 | % 8 | % This program is free software: you can redistribute it and/or modify 9 | % it under the terms of the GNU General Public License as published by 10 | % the Free Software Foundation, either version 3 of the License, or 11 | % (at your option) any later version. 12 | % 13 | % This program is distributed in the hope that it will be useful, 14 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 15 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 16 | % GNU General Public License for more details. 17 | % 18 | % You should have received a copy of the GNU General Public License 19 | % along with this program. If not, see . 20 | % ------------------------------------------------------------------------- 21 | 22 | % define velocity model 23 | n = [51 51]; 24 | h = [20 20]; 25 | z = [0:n(1)-1]'*h(1); 26 | x = [0:n(2)-1]*h(2); 27 | 28 | v0 = 2000*ones(n); 29 | dv = zeros(n); 30 | dv(11:21,11:21) = 100; 31 | dv(31:41,31:41) = -100; 32 | m = 1e6./(v0(:) + dv(:)).^2; 33 | m0 = 1e6./(v0(:)).^2; 34 | 35 | % grid 36 | model.h = h; 37 | model.n = n; 38 | 39 | % frequency 40 | model.f = [2.5 5 10]; 41 | 42 | % acquisition 43 | model.Is = {2,2:50}; 44 | model.Ir = {2,2:50}; 45 | 46 | % sources 47 | Q = speye(length(model.Is{1})*length(model.Is{2})); 48 | 49 | % data 50 | D = F(m,Q,model); 51 | 52 | % multiply with random amplitude factor 53 | s = RandStream('mt19937ar','Seed',1); 54 | RandStream.setGlobalStream(s); 55 | w = 5*abs(randn(1,size(D,2),size(D,3))); 56 | D = bsxfun(@times,w,D); 57 | 58 | % inversion 59 | fh = @(x)misfit(x,Q,D,model,true,false); 60 | fhc = @(x)misfit(x,Q,D,model,true,true); 61 | 62 | options.maxiter = 1000; 63 | options.tol = 1e-3; 64 | options.M = 5; 65 | options.fid = 1; 66 | options.cgtol = 1e-1; 67 | options.cgmax = 200; 68 | options.write = 1; 69 | 70 | tic 71 | options.method = 'LBFGS'; 72 | [m1,info1] = GQN(fh,m0,options); 73 | T1 = toc; 74 | cost1 = 2*info1(:,2)+ 3*cumsum(info1(:,6)); 75 | for k = 1:size(info1,1) 76 | mk = dlmread(['x_' num2str(k-1) '.dat']); 77 | error1(k) = norm(m - mk)/norm(m - m0); 78 | end 79 | delete('x*.dat'); 80 | display(['Elapsed time: ' num2str(T1) ' s, PDE solves: ' num2str(cost1(end))]); 81 | 82 | 83 | tic 84 | options.method = 'GN'; 85 | [m2,info2] = GQN(fh,m0,options); 86 | T2 = toc; 87 | cost2 = 2*info2(:,2)+ 3*cumsum(info2(:,6)); 88 | for k = 1:size(info2,1) 89 | mk = dlmread(['x_' num2str(k-1) '.dat']); 90 | error2(k) = norm(m - mk)/norm(m - m0); 91 | end 92 | delete('x*.dat'); 93 | display(['Elapsed time: ' num2str(T2) ' s, PDE solves: ' num2str(cost2(end))]); 94 | 95 | 96 | tic 97 | options.method = 'GN'; 98 | [m3,info3] = GQN(fhc,m0,options); 99 | T3 = toc; 100 | cost3 = 2*info3(:,2)+ 3*cumsum(info3(:,6)); 101 | for k = 1:size(info3,1) 102 | mk = dlmread(['x_' num2str(k-1) '.dat']); 103 | error3(k) = norm(m - mk)/norm(m - m0); 104 | end 105 | delete('x*.dat'); 106 | display(['Elapsed time: ' num2str(T3) ' s, PDE solves: ' num2str(cost3(end))]); 107 | 108 | 109 | % plot 110 | figure; 111 | imagesc(1e-3*x,1e-3*z,reshape(1./sqrt(m),n)); axis equal tight;caxis([min(1./sqrt(m)) max(1./sqrt(m))]); 112 | title('ground truth [km/s]','fontsize',20);colorbar; 113 | set(gca,'fontsize',20);xlabel('x [km]','fontsize',20);ylabel('z [km]','fontsize',20); 114 | 115 | figure; 116 | imagesc(1e-3*x,1e-3*z,reshape(1./sqrt(m1),n));axis equal tight;caxis([min(1./sqrt(m)) max(1./sqrt(m))]); title('LBFGS','fontsize',20); 117 | set(gca,'fontsize',20);xlabel('x [km]','fontsize',20);ylabel('z [km]','fontsize',20); 118 | 119 | figure; 120 | imagesc(1e-3*x,1e-3*z,reshape(1./sqrt(m2),n)); axis equal tight;caxis([min(1./sqrt(m)) max(1./sqrt(m))]);title('GN w/o correction','fontsize',20); 121 | set(gca,'fontsize',20);xlabel('x [km]','fontsize',20);ylabel('z [km]','fontsize',20); 122 | 123 | figure; 124 | imagesc(1e-3*x,1e-3*z,reshape(1./sqrt(m3),n)); axis equal tight;caxis([min(1./sqrt(m)) max(1./sqrt(m))]);title('GN w correction','fontsize',20); 125 | set(gca,'fontsize',20);xlabel('x [km]','fontsize',20);ylabel('z [km]','fontsize',20); 126 | 127 | figure; 128 | loglog(1+info1(:,1),info1(:,4),'k',1+info2(:,1),info2(:,4),'r',1+info3(:,1),info3(:,4),'b','linewidth',2); 129 | legend('LBFGS','GN w/o corr','GN w corr'); 130 | set(gca,'fontsize',20);xlabel('iteration','fontsize',20);ylabel('misfit','fontsize',20); 131 | 132 | figure; 133 | loglog(1+info1(:,1),info1(:,5),'k',1+info2(:,1),info2(:,5),'r',1+info3(:,1),info3(:,5),'b','linewidth',2); 134 | legend('LBFGS','GN w/o corr','GN w corr'); 135 | set(gca,'fontsize',20);xlabel('iteration','fontsize',20);ylabel('||g||_2','fontsize',20); 136 | 137 | figure; 138 | semilogx(1+info1(:,1),error1,'k',1+info2(:,1),error2,'r',1+info3(:,1),error3,'b','linewidth',2); 139 | legend('LBFGS','GN w/o corr','GN w corr'); 140 | set(gca,'fontsize',20);xlabel('iteration','fontsize',20);ylabel('reconstruction error','fontsize',20); 141 | 142 | for k = 1:7 143 | print(k,'-depsc',[mfilename '_' num2str(k)]); 144 | end 145 | 146 | -------------------------------------------------------------------------------- /exp2_trans.m: -------------------------------------------------------------------------------- 1 | % This script reproduces figure 4 from the paper 2 | % 3 | % ------------------------------------------------------------------------- 4 | % Copyright (C) 2013 Tristan van Leeuwen 5 | % Centrum Wiskunde & Informatica 6 | % Tristan.van.Leeuwen@cwi.nl 7 | % 8 | % This program is free software: you can redistribute it and/or modify 9 | % it under the terms of the GNU General Public License as published by 10 | % the Free Software Foundation, either version 3 of the License, or 11 | % (at your option) any later version. 12 | % 13 | % This program is distributed in the hope that it will be useful, 14 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 15 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 16 | % GNU General Public License for more details. 17 | % 18 | % You should have received a copy of the GNU General Public License 19 | % along with this program. If not, see . 20 | % ------------------------------------------------------------------------- 21 | 22 | % define velocity model 23 | n = [51 51]; 24 | h = [20 20]; 25 | z = [0:n(1)-1]'*h(1); 26 | x = [0:n(2)-1]*h(2); 27 | 28 | v0 = 2000*ones(n); 29 | dv = zeros(n); 30 | dv(11:21,11:21) = 100; 31 | dv(31:41,31:41) = -100; 32 | m = 1e6./(v0(:) + dv(:)).^2; 33 | m0 = 1e6./(v0(:)).^2; 34 | 35 | % grid 36 | model.h = h; 37 | model.n = n; 38 | 39 | % frequency 40 | model.f = [2.5 5 10]; 41 | 42 | % acquisition 43 | model.Is = {2:50,2}; 44 | model.Ir = {2:50,50}; 45 | 46 | % sources 47 | Q = speye(length(model.Is{1})*length(model.Is{2})); 48 | 49 | % data 50 | D = F(m,Q,model); 51 | 52 | % multiply with random amplitude factor 53 | s = RandStream('mt19937ar','Seed',1); 54 | RandStream.setGlobalStream(s); 55 | w = 5*abs(randn(1,size(D,2),size(D,3))); 56 | D = bsxfun(@times,w,D); 57 | 58 | % inversion 59 | fh = @(x)misfit(x,Q,D,model,true,false); 60 | fhc = @(x)misfit(x,Q,D,model,true,true); 61 | 62 | options.maxiter = 1000; 63 | options.tol = 1e-3; 64 | options.M = 5; 65 | options.fid = 1; 66 | options.cgtol = 1e-1; 67 | options.cgmax = 200; 68 | options.cgprec = 0; 69 | options.write = 1; 70 | 71 | tic 72 | options.method = 'LBFGS'; 73 | [m1,info1] = GQN(fh,m0,options); 74 | T1 = toc; 75 | cost1 = 2*info1(:,2)+ 3*cumsum(info1(:,6)); 76 | for k = 1:size(info1,1) 77 | mk = dlmread(['x_' num2str(k-1) '.dat']); 78 | error1(k) = norm(m - mk)/norm(m - m0); 79 | end 80 | delete('x*.dat'); 81 | display(['Elapsed time: ' num2str(T1) ' s, PDE solves: ' num2str(cost1(end))]); 82 | 83 | 84 | tic 85 | options.method = 'GN'; 86 | [m2,info2] = GQN(fh,m0,options); 87 | T2 = toc; 88 | cost2 = 2*info2(:,2)+ 3*cumsum(info2(:,6)); 89 | for k = 1:size(info2,1) 90 | mk = dlmread(['x_' num2str(k-1) '.dat']); 91 | error2(k) = norm(m - mk)/norm(m - m0); 92 | end 93 | delete('x*.dat'); 94 | display(['Elapsed time: ' num2str(T2) ' s, PDE solves: ' num2str(cost2(end))]); 95 | 96 | 97 | tic 98 | options.method = 'GN'; 99 | [m3,info3] = GQN(fhc,m0,options); 100 | T3 = toc; 101 | cost3 = 2*info3(:,2)+ 3*cumsum(info3(:,6)); 102 | for k = 1:size(info3,1) 103 | mk = dlmread(['x_' num2str(k-1) '.dat']); 104 | error3(k) = norm(m - mk)/norm(m - m0); 105 | end 106 | delete('x*.dat'); 107 | display(['Elapsed time: ' num2str(T3) ' s, PDE solves: ' num2str(cost3(end))]); 108 | 109 | 110 | % plot 111 | figure; 112 | imagesc(1e-3*x,1e-3*z,reshape(1./sqrt(m),n)); axis equal tight;caxis([min(1./sqrt(m)) max(1./sqrt(m))]); 113 | title('ground truth [km/s]','fontsize',20);colorbar; 114 | set(gca,'fontsize',20);xlabel('x [km]','fontsize',20);ylabel('z [km]','fontsize',20); 115 | 116 | figure; 117 | imagesc(1e-3*x,1e-3*z,reshape(1./sqrt(m1),n));axis equal tight;caxis([min(1./sqrt(m)) max(1./sqrt(m))]); title('LBFGS','fontsize',20); 118 | set(gca,'fontsize',20);xlabel('x [km]','fontsize',20);ylabel('z [km]','fontsize',20); 119 | 120 | figure; 121 | imagesc(1e-3*x,1e-3*z,reshape(1./sqrt(m2),n)); axis equal tight;caxis([min(1./sqrt(m)) max(1./sqrt(m))]);title('GN w/o correction','fontsize',20); 122 | set(gca,'fontsize',20);xlabel('x [km]','fontsize',20);ylabel('z [km]','fontsize',20); 123 | 124 | figure; 125 | imagesc(1e-3*x,1e-3*z,reshape(1./sqrt(m3),n)); axis equal tight;caxis([min(1./sqrt(m)) max(1./sqrt(m))]);title('GN w correction','fontsize',20); 126 | set(gca,'fontsize',20);xlabel('x [km]','fontsize',20);ylabel('z [km]','fontsize',20); 127 | 128 | figure; 129 | loglog(1+info1(:,1),info1(:,4),'k',1+info2(:,1),info2(:,4),'r',1+info3(:,1),info3(:,4),'b','linewidth',2); 130 | legend('LBFGS','GN w/o corr','GN w corr'); 131 | set(gca,'fontsize',20);xlabel('iteration','fontsize',20);ylabel('misfit','fontsize',20); 132 | 133 | figure; 134 | loglog(1+info1(:,1),info1(:,5),'k',1+info2(:,1),info2(:,5),'r',1+info3(:,1),info3(:,5),'b','linewidth',2); 135 | legend('LBFGS','GN w/o corr','GN w corr'); 136 | set(gca,'fontsize',20);xlabel('iteration','fontsize',20);ylabel('||g||_2','fontsize',20); 137 | 138 | figure; 139 | semilogx(1+info1(:,1),error1,'k',1+info2(:,1),error2,'r',1+info3(:,1),error3,'b','linewidth',2); 140 | legend('LBFGS','GN w/o corr','GN w corr'); 141 | set(gca,'fontsize',20);xlabel('iteration','fontsize',20);ylabel('reconstruction error','fontsize',20); 142 | 143 | for k = 1:7 144 | print(k,'-depsc',[mfilename '_' num2str(k)]); 145 | end 146 | 147 | -------------------------------------------------------------------------------- /getA.m: -------------------------------------------------------------------------------- 1 | function A = getA(f,m,h,n) 2 | % 5-point discretization of the 2D Helmholtz operator with absorbing 3 | % boundary conditions. 4 | % 5 | % use: 6 | % A = getA(f,m,h,n); 7 | % 8 | % input: 9 | % f - frequency [Hz] 10 | % m - model [s^2/km^2] 11 | % h - [dz,dx] gridspacing in z and x direction [m] 12 | % n - [nz,nx] number of gridpoints in z and x direction 13 | % 14 | % ouput: 15 | % A - sparse matrix 16 | % 17 | % ------------------------------------------------------------------------- 18 | % Copyright (C) 2013 Tristan van Leeuwen 19 | % Centrum Wiskunde & Informatica 20 | % Tristan.van.Leeuwen@cwi.nl 21 | % 22 | % This program is free software: you can redistribute it and/or modify 23 | % it under the terms of the GNU General Public License as published by 24 | % the Free Software Foundation, either version 3 of the License, or 25 | % (at your option) any later version. 26 | % 27 | % This program is distributed in the hope that it will be useful, 28 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 29 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 30 | % GNU General Public License for more details. 31 | % 32 | % You should have received a copy of the GNU General Public License 33 | % along with this program. If not, see . 34 | % ------------------------------------------------------------------------- 35 | 36 | % angular frequency (note factor 1e-3 because m is in [s^2/km^2]) 37 | omega = 1e-3*2*pi*f; 38 | 39 | % number of gridpoints 40 | N = prod(n); 41 | 42 | % Stiffness matrices 43 | D1 = spdiags(ones(n(1),1)*[1 -2 1]/h(1).^2,[-1:1],n(1),n(1)); D1(1,1:2) = [1 -1]/h(1); D1(end,end-1:end) = [-1 1]/h(1); 44 | D2 = spdiags(ones(n(2),1)*[1 -2 1]/h(2).^2,[-1:1],n(2),n(2)); D2(1,1:2) = [1 -1]/h(2); D2(end,end-1:end) = [-1 1]/h(2); 45 | 46 | S = kron(speye(n(2)),D1) + kron(D2,speye(n(1))); 47 | 48 | % Mass matrix including ABC's 49 | w = [0 ones(1,n(1)-2) 0]; 50 | w = w(:)*[0 ones(1,n(2)-2) 0]; 51 | w = w(:); 52 | 53 | M = omega^2*spdiags(w.*m,0,N,N) + 1i*omega*spdiags((1-w).*sqrt(m),0,N,N); 54 | 55 | % 56 | A = M + S; 57 | 58 | 59 | 60 | -------------------------------------------------------------------------------- /getG.m: -------------------------------------------------------------------------------- 1 | function G = getG(f,m,h,n) 2 | % Jacobian of A*u, see getA.m. 3 | % The Jacobian of A*u can be written as d(A*u)/dm = G*diag(m)*u 4 | % 5 | % use: 6 | % G = getG(f,m,h,n); 7 | % 8 | % input: 9 | % f - frequency [Hz] 10 | % m - model [s^2/km^2] 11 | % h - [dz,dx] gridspacing in z and x direction [m] 12 | % n - [nz,nx] number of gridpoints in z and x direction 13 | % 14 | % ouput: 15 | % G - sparse matrix 16 | % 17 | % ------------------------------------------------------------------------- 18 | % Copyright (C) 2013 Tristan van Leeuwen 19 | % Centrum Wiskunde & Informatica 20 | % Tristan.van.Leeuwen@cwi.nl 21 | % 22 | % This program is free software: you can redistribute it and/or modify 23 | % it under the terms of the GNU General Public License as published by 24 | % the Free Software Foundation, either version 3 of the License, or 25 | % (at your option) any later version. 26 | % 27 | % This program is distributed in the hope that it will be useful, 28 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 29 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 30 | % GNU General Public License for more details. 31 | % 32 | % You should have received a copy of the GNU General Public License 33 | % along with this program. If not, see . 34 | % ------------------------------------------------------------------------- 35 | 36 | % angular frequency (note factor 1e-3 because m is in [s^2/km^2]) 37 | omega = 1e-3*2*pi*f; 38 | N = prod(n); 39 | 40 | % 41 | w = [0 ones(1,n(1)-2) 0]; 42 | if n(2)>1 43 | w = w(:)*[0 ones(1,n(2)-2) 0]; 44 | end 45 | w = w(:); 46 | 47 | % construct sparse matrix; we ignore the contributions from the boundary for the time being to ensure 48 | % that all quantities are real. 49 | G = omega^2*spdiags(w,0,N,N); % + 1i*omega*spdiags((1-w)./(2*sqrt(m)),0,N,N); 50 | 51 | -------------------------------------------------------------------------------- /getP.m: -------------------------------------------------------------------------------- 1 | function P = getP(n,Iz,Ix) 2 | % sampling operator, selects gridpoints from 2D grid: 3 | % 4 | % use: 5 | % P = getP(n,Iz,Ix); 6 | % 7 | % input: 8 | % n - [nz,nx] number of gridpoints in z and x direction 9 | % Iz,Ix - indices 10 | % 11 | % ouput: 12 | % P - sparse matrix 13 | % 14 | % ------------------------------------------------------------------------- 15 | % Copyright (C) 2013 Tristan van Leeuwen 16 | % Centrum Wiskunde & Informatica 17 | % Tristan.van.Leeuwen@cwi.nl 18 | % 19 | % This program is free software: you can redistribute it and/or modify 20 | % it under the terms of the GNU General Public License as published by 21 | % the Free Software Foundation, either version 3 of the License, or 22 | % (at your option) any later version. 23 | % 24 | % This program is distributed in the hope that it will be useful, 25 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 26 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 27 | % GNU General Public License for more details. 28 | % 29 | % You should have received a copy of the GNU General Public License 30 | % along with this program. If not, see . 31 | % ------------------------------------------------------------------------- 32 | 33 | I1 = speye(n(1)); 34 | I2 = speye(n(2)); 35 | P = kron(I2(Ix,:),I1(Iz,:)); 36 | 37 | -------------------------------------------------------------------------------- /getoption.m: -------------------------------------------------------------------------------- 1 | function s = getoption(options,field,default) 2 | % parse option from struct 3 | % 4 | % use 5 | % s = getoption(options,field,default) 6 | % 7 | % input 8 | % options - struct with option fields 9 | % field - specify which field to parse 10 | % default - default value in case field does not exist 11 | % 12 | % output 13 | % s - value 14 | % 15 | % ------------------------------------------------------------------------- 16 | % Copyright (C) 2013 Tristan van Leeuwen 17 | % Centrum Wiskunde & Informatica 18 | % Tristan.van.Leeuwen@cwi.nl 19 | % 20 | % This program is free software: you can redistribute it and/or modify 21 | % it under the terms of the GNU General Public License as published by 22 | % the Free Software Foundation, either version 3 of the License, or 23 | % (at your option) any later version. 24 | % 25 | % This program is distributed in the hope that it will be useful, 26 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 27 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 28 | % GNU General Public License for more details. 29 | % 30 | % You should have received a copy of the GNU General Public License 31 | % along with this program. If not, see . 32 | % ------------------------------------------------------------------------- 33 | 34 | s = default; 35 | if isfield(options,field) 36 | s = getfield(options,field); 37 | end -------------------------------------------------------------------------------- /gradtest.m: -------------------------------------------------------------------------------- 1 | function [ep,em] = gradtest(fh,m,dm) 2 | % finite-difference gradient test based on Taylor expansion: 3 | % 4 | % f(m + alpha*dm) = f(m) + alpha*df(m)'*dm + O(alpha^2) 5 | % 6 | % The error e(alpha) = f(m + alpha*dm) - f(m) - alpha*df(m)'*dm 7 | % should be O(alpha^2) which can be verified by computing it for various 8 | % alpha. 9 | % 10 | % use: 11 | % [ep,em] = gradtest(fh,m,dm) 12 | % input: 13 | % fh - function handle [f,g] =fh(m) 14 | % m - reference model 15 | % dm - perturbation 16 | % 17 | % output 18 | % ep - error based on forward difference 19 | % em - error based on backward difference 20 | % 21 | % Copyright (C) 2013 Tristan van Leeuwen 22 | % Centrum Wiskunde & Informatica 23 | % Tristan.van.Leeuwen@cwi.nl 24 | % 25 | % This program is free software: you can redistribute it and/or modify 26 | % it under the terms of the GNU General Public License as published by 27 | % the Free Software Foundation, either version 3 of the License, or 28 | % (at your option) any later version. 29 | % 30 | % This program is distributed in the hope that it will be useful, 31 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 32 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 33 | % GNU General Public License for more details. 34 | % 35 | % You should have received a copy of the GNU General Public License 36 | % along with this program. If not, see . 37 | 38 | alpha = [1 .1 1e-2 1e-3 1e-4 1e-5 1e-6 1e-7 1e-8]; 39 | [f,g] = fh(m); 40 | for k = 1:length(alpha) 41 | fp = fh(m + alpha(k)*dm); 42 | fm = fh(m - alpha(k)*dm); 43 | 44 | ep(k) = f+alpha(k)*dm'*g - fp; 45 | em(k) = f-alpha(k)*dm'*g - fm; 46 | end 47 | -------------------------------------------------------------------------------- /license.txt: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /misfit.m: -------------------------------------------------------------------------------- 1 | function [f,g,H] = misfit(m,Q,D,model,srcest,corr) 2 | % least-squares misfit for FWI with source estimation 3 | % 4 | % use: 5 | % [f,g] = misfit(m,Q,D,model,srcest) 6 | % 7 | % input: 8 | % m - model [s^2/km^2] 9 | % Q - sources as 2D array 10 | % D - data as 3D array (receivers x sources x frequencies) 11 | % model.h - [dz,dx] gridspacing in z and x direction [m] 12 | % model.n - [nz,nx] number of gridpoints in z and x direction 13 | % model.f - frequencies 14 | % model.Is - source location indices 15 | % model.Ir - receiver location indices 16 | % srcest - source estimation on/off 17 | % corr - include correction term in reduced Hessian 18 | % 19 | % output: 20 | % f - misfit 21 | % g - gradient 22 | % 23 | % ------------------------------------------------------------------------- 24 | % Copyright (C) 2013 Tristan van Leeuwen 25 | % Centrum Wiskunde & Informatica 26 | % Tristan.van.Leeuwen@cwi.nl 27 | % 28 | % This program is free software: you can redistribute it and/or modify 29 | % it under the terms of the GNU General Public License as published by 30 | % the Free Software Foundation, either version 3 of the License, or 31 | % (at your option) any later version. 32 | % 33 | % This program is distributed in the hope that it will be useful, 34 | % but WITHOUT ANY WARRANTY; without even the implied warranty of 35 | % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 36 | % GNU General Public License for more details. 37 | % 38 | % You should have received a copy of the GNU General Public License 39 | % along with this program. If not, see . 40 | % ------------------------------------------------------------------------- 41 | 42 | % get sampling operators 43 | Ps = getP(model.n,model.Is{:}); 44 | Pr = getP(model.n,model.Ir{:}); 45 | 46 | % initialize misfit and gradient 47 | f = 0; 48 | g = 0*m; 49 | 50 | % loop over frequencies 51 | for k = 1:length(model.f) 52 | % get Helmholtz operator 53 | Ak = getA(model.f(k),m,model.h,model.n); 54 | Gk = getG(model.f(k),m,model.h,model.n); 55 | 56 | % solve forward wave-equation and sample data 57 | Uk = Ak\(Ps'*Q); 58 | Sk = Pr*Uk; 59 | 60 | % compute source-weight 61 | if srcest 62 | ck = real(sum(conj(Sk).*D(:,:,k),1)./sum(conj(Sk).*Sk,1))'; 63 | end 64 | Ck = spdiags(ck,0,size(Sk,2),size(Sk,2)); 65 | 66 | % solve adjoint wave-equation 67 | Vk = Ak'\(Pr'*(Sk*Ck - D(:,:,k))); 68 | 69 | % compute misfit and gradient 70 | f = f + .5*norm(Sk*Ck - D(:,:,k),'fro')^2; 71 | g = g - sum(conj(Uk*Ck).*(Gk'*Vk),2); 72 | end 73 | g = real(g); 74 | H = @(x)Hfun(x,m,Q,D,model,corr,srcest); 75 | 76 | function y = Hfun(x,m,Q,D,model,corr,srcest) 77 | 78 | % get sampling operators 79 | Ps = getP(model.n,model.Is{:}); 80 | Pr = getP(model.n,model.Ir{:}); 81 | 82 | y = 0*x; 83 | 84 | % loop over frequencies 85 | for k = 1:length(model.f) 86 | % get Helmholtz operator 87 | Ak = getA(model.f(k),m,model.h,model.n); 88 | Gk = getG(model.f(k),m,model.h,model.n); 89 | 90 | % solve forward wave-equation and sample data 91 | Uk = Ak\(Ps'*Q); 92 | Sk = Pr*Uk; 93 | 94 | % source-weight 95 | if srcest 96 | ck = real(sum(conj(Sk).*D(:,:,k),1)./sum(conj(Sk).*Sk,1))'; 97 | end 98 | Ck = spdiags(ck,0,size(Sk,2),size(Sk,2)); 99 | 100 | % dc/dm 101 | dck = real( bsxfun(@times,(conj(Uk).*(Gk'*(Ak'\(Pr'*(D(:,:,k) - 2*Sk*Ck))))),1./sum(abs(Sk).^2,1)) ); 102 | 103 | % dS = w*J*x + (dc'*x)*S 104 | dSk = (Pr*(Ak\(Gk*spdiags(x,0,length(x),length(x))*Uk)))*Ck; 105 | if corr 106 | dSk = dSk + Sk*spdiags((dck'*x),0,size(Sk,2),size(Sk,2)); 107 | end 108 | 109 | % g = w*J'*p + (s'*p)*dc 110 | g = sum(conj(Uk*Ck).*(Gk'*(Ak'\(Pr'*dSk))),2); 111 | if corr 112 | g = g + dck*transpose(sum(conj(Sk).*dSk,1)); 113 | end 114 | 115 | y = y + real(g); 116 | end --------------------------------------------------------------------------------