├── examples ├── aushield.mod ├── aushield.dat └── testbench.m ├── src ├── rms1.m ├── chi2.m ├── roughness1.m ├── plotlayer_log.m ├── bos2layer.m ├── plot1derr.m ├── jacob1.m ├── mt1dfwd10.m └── occam1dmt.m ├── README.md └── LICENSE /examples/aushield.mod: -------------------------------------------------------------------------------- 1 | #layerdepth LOG10(RES) 2 | 0.3680 2.00 3 | 0.5012 2.00 4 | 0.6383 2.00 5 | 1.1070 2.00 6 | 1.9199 2.00 7 | 3.3297 2.00 8 | 5.7748 2.00 9 | 10.0155 2.00 10 | 17.3701 2.00 11 | 30.1253 2.00 12 | 52.2471 2.00 13 | 90.6134 2.00 14 | 157.1531 2.00 15 | 272.5545 2.00 16 | 472.6981 2.00 17 | 819.8120 2.00 18 | 1421.8231 2.00 19 | 2456.9291 2.00 20 | 4276.6711 2.00 21 | 7414.1027 2.00 22 | 12864.1200 2.00 23 | 22319.2200 2.00 24 | 38693.4000 2.00 25 | 67105.2100 2.00 26 | 0.0000 2.00 -------------------------------------------------------------------------------- /src/rms1.m: -------------------------------------------------------------------------------- 1 | function rmsquare=rms1(obs,res,err) 2 | % script to calculate Root Mean Square 3 | % obs: observed 4 | % res: responsed 5 | % err: data *absolute* error 6 | switch nargin 7 | case 0 8 | error('not enough input arguments, 2 at least') 9 | case 1 10 | error('not enough input arguments, 2 at least') 11 | case 2 12 | err = ones(size(obs)); 13 | end 14 | misfit=abs(obs-res)./err; 15 | misfit=misfit.*misfit; 16 | N=length(obs); 17 | rmsquare=sqrt(sum(misfit)/N); 18 | return -------------------------------------------------------------------------------- /src/chi2.m: -------------------------------------------------------------------------------- 1 | function X2=chi2(d0,d,stderr) 2 | % script to calculate Chi square statistic 3 | %=========================================================================% 4 | % obs: observed 5 | % res: responsed 6 | % stderr: standard variance of the data 7 | % c2: chi square (normalized) 8 | % expected Chi square should be 2*length(obs) 9 | if size(d0)~=size(d) 10 | error('size of the input vectors must be identical') 11 | elseif size(d0)~=size(stderr) 12 | error('size of the input vectors must be identical') 13 | end 14 | W=diag(1./stderr); 15 | X2=sum((W*(d0'-d')).^2); 16 | return 17 | -------------------------------------------------------------------------------- /src/roughness1.m: -------------------------------------------------------------------------------- 1 | function r=roughness1(sigma,opt) 2 | % description of parameters: 3 | % 4 | % sigma: conductivity 1/res 5 | % opt: option of outputing the forward, could be 1 for L1 and 2 for L2 6 | %===============================================================% 7 | % version 0.2 8 | % DONG Hao 9 | %====================checking parameters========================% 10 | switch nargin 11 | case 0 12 | error('not enough input arguments, 1 at least') 13 | case 1 14 | opt = 1; % use L1 roughness 15 | end 16 | n=length(sigma); %number of layers 17 | switch opt 18 | case 1 %gradient 19 | r=abs(sigma(1:n-1)-sigma(2:n)); 20 | r=r'*r; 21 | case 2 %laplacian 22 | r=abs(-sigma(1:n-2)+2*sigma(2:n-1)-1*sigma(3:n)); 23 | r=r'*r; 24 | end 25 | return -------------------------------------------------------------------------------- /examples/aushield.dat: -------------------------------------------------------------------------------- 1 | # periods rho erho phs ephs 2 | 0.020 0.712 0.0434 20.50 4.50 3 | 0.050 0.813 0.1406 22.00 19.98 4 | 0.079 0.964 0.1338 19.28 16.08 5 | 0.126 1.148 0.1609 20.14 11.03 6 | 0.199 1.030 0.3046 35.65 23.57 7 | 0.316 1.238 0.4880 34.44 20.66 8 | 0.501 1.503 0.3244 39.71 16.60 9 | 0.794 1.520 0.4772 41.90 23.40 10 | 1.258 1.675 0.4306 47.28 16.60 11 | 1.995 1.944 0.4412 42.78 16.77 12 | 3.162 1.844 0.4081 48.50 18.41 13 | 5.011 1.737 0.5236 56.87 18.05 14 | 7.943 1.614 0.1995 58.80 17.63 15 | 12.580 1.592 0.4780 58.88 18.10 16 | 19.950 1.336 0.2228 61.44 18.31 17 | 31.620 1.358 0.4471 61.90 16.58 18 | 50.110 1.345 0.3524 51.48 25.52 19 | 79.430 0.935 0.1041 65.50 10.51 20 | 125.800 0.821 0.0961 58.20 23.53 21 | 199.500 0.535 0.4000 54.00 20.00 22 | 794.299 0.312 0.4000 86.00 30.00 23 | 1258.000 -0.107 0.5561 41.00 32.00 24 | 1995.000 0.390 0.8612 58.66 21.63 -------------------------------------------------------------------------------- /src/plotlayer_log.m: -------------------------------------------------------------------------------- 1 | function plotlayer_log(res,layer,ccode) 2 | % a barn door function to plot 1d Layered model 3 | % in logrithm scale 4 | % res --> array of resistivity 5 | % layer --> array of layer depth 6 | % ccode --> color code to specify the line style and color 7 | if nargin<3 8 | ccode='k'; 9 | end 10 | if (size(res,2)~=size(layer,2)) 11 | if length(res)==length(layer) 12 | layer=layer'; 13 | else 14 | disp('please check the input parametres ') 15 | error('res and layer should have the same size'); 16 | end 17 | end 18 | NL=length(layer); 19 | depth=0.1; 20 | h=gca; 21 | for i=1:NL-1 22 | plot(h,[res(i) res(i)],[depth depth+layer(i)],ccode,'linewidth',2) 23 | hold(h,'on'); 24 | plot(h,[res(i) res(i+1)],[depth+layer(i) depth+layer(i)],ccode,'linewidth',2) 25 | depth=depth+layer(i); 26 | end 27 | plot(h,[res(NL) res(NL)],[depth depth+layer(NL)*10],ccode,'linewidth',2); 28 | hold(h,'off'); 29 | set(h, 'ydir', 'reverse'); 30 | set(h, 'xlim', [0 4]) 31 | set(h, 'ylim', [10 depth+layer(NL)*10]); 32 | xlabel(h,'log_{10} Resistivity(\Omega m)'); 33 | ylabel(h,'Depth(m)'); 34 | set(h,'yscale','log'); 35 | grid on; 36 | -------------------------------------------------------------------------------- /src/bos2layer.m: -------------------------------------------------------------------------------- 1 | function [z0,res0]=bos2layer(rho0,phs0,freq) 2 | % a silly function to guess a initial model from bostick transformation for 3 | % 1D layered inversion 4 | % first set up resistivity gauge 5 | rmax=10;rmin=-10; 6 | [resb,depth]=bostick(rho0,phs0,1./freq,'Bostick'); 7 | resb(end+1)=resb(end); 8 | % try removing unpratical resistivity values first... 9 | resb(resb>rmax)=rmax; 10 | resb(resbz0(n-1),1); 27 | if isempty(idx) 28 | idx=length(depth); 29 | else 30 | idx=find(depth>z0(n),1); 31 | end 32 | end 33 | res0(n)=mean(resb(idx)); 34 | end 35 | idx=find(z0==0,1); 36 | z0(1:idx-2)=z0(2:idx-1); 37 | z0(idx-1:end)=[]; 38 | res0(1:idx-2)=res0(2:idx-1); 39 | res0(idx-1:end)=[]; 40 | return -------------------------------------------------------------------------------- /src/plot1derr.m: -------------------------------------------------------------------------------- 1 | function plot1derr(freq,var1,var2,opt,ccode,err1,err2) 2 | % a barn door function to plot 1d apparent resistivities and impedance 3 | % phases... 4 | % 5 | switch nargin 6 | case 0 7 | error('not enough input arguments, 3 at least') 8 | case 1 9 | error('not enough input arguments, 3 at least') 10 | case 2 11 | error('not enough input arguments, 3 at least') 12 | case 3 13 | ccode='b-'; 14 | opt = 'rho'; 15 | case 4 16 | ccode='b-'; 17 | case 6 18 | error('not enough input arguments, need 2 vars and 2 errs') 19 | end 20 | mju0=4.0*pi*1E-7; 21 | omega=2*pi*freq; 22 | if strcmp(opt,'imped')==1 23 | rho=(var1.^2+var2.^2)*mju0.*omega'; 24 | rho = log(rho); 25 | phase=atan2(var2,var1); 26 | rhoe=2*sqrt(err1.^2+err2.^2)./((var1.^2+var2.^2).^0.5); 27 | rhoe = 0.434*rhoe; 28 | phse=sqrt(err1.^2+err2.^2)./((var1.^2+var2.^2).^0.5); 29 | else % in log10 domain 30 | rho=var1; 31 | phase=var2; 32 | if nargin > 5 33 | rhoe=err1; 34 | phse=err2; 35 | end 36 | end 37 | a1=subplot(2,1,1); 38 | if nargin <= 5 39 | plot(a1,freq,rho,ccode); 40 | else 41 | errorbar(a1,freq,rho,rhoe,ccode); 42 | end 43 | % loglog(a1,freq,rho,mark); 44 | hold(a1,'on'); 45 | set(a1,'xdir','reverse') 46 | set(a1,'xscale','log'); 47 | set(a1,'xgrid','on','ygrid','on') 48 | a2=subplot(2,1,2); 49 | phase=phase/pi*180; 50 | if nargin <= 5 51 | plot(a2,freq,phase,ccode); 52 | else 53 | phse = phse/pi*180; 54 | errorbar(a2,freq,phase,phse,ccode); 55 | end 56 | hold(a2,'on'); 57 | set(a2,'xdir','reverse','ylim',[0 90],'ytick',[0 15 30 45 60 75 90]); 58 | set(a2,'xscale','log'); 59 | set(a2,'xgrid','on','ygrid','on') 60 | return -------------------------------------------------------------------------------- /examples/testbench.m: -------------------------------------------------------------------------------- 1 | % simple testbench script, for 1D MT (magnetotellurics) occam inversion 2 | % DONG Hao 3 | % 2011/06/25 4 | % Golmud 5 | % ======================================================================= % 6 | clear 7 | addpath(genpath('..'),'-end'); 8 | % some settings here 9 | % terminating RMS misfit 10 | Trms=1.0; 11 | % number of maximum iteration 12 | Niter=20; 13 | % read a 24-layered model file 14 | fid = fopen('aushield.mod'); 15 | tmp = textscan(fid,'%f %f','CommentStyle', '#'); 16 | fclose(fid); 17 | res0 = tmp{2}'; 18 | l = tmp{1}(1:end-1)'; 19 | % depth of each layer INTERFACE, note that z1=0 20 | z = cumsum([0 l]); 21 | nz = length(l); 22 | % read a MT sounding data file, which are the 23 | % 1D MT data published by Cull (1984) 24 | fid = fopen('aushield.dat'); 25 | tmp = textscan(fid,'%f %f %f %f %f','CommentStyle', '#'); 26 | fclose(fid); 27 | % the data are (literally) copied from Table 5 of the Constable 1987 paper 28 | % see: 29 | % 30 | % Constable, S. C., Parker, R. L., & Constable, C. G. (1987). Occam’s 31 | % inversion: A practical algorithm for generating smooth models from 32 | % electromagnetic sounding data. Geophysics, 52(3), 289–300. 33 | % 34 | % Cull, J. P. (1985). Magnetotelluric soundings over a Precambrian contact 35 | % in Australia. Geophys. J. Roy. Astr. Sot., 80, 661-675. 36 | period = tmp{1}'; 37 | rho0 = tmp{2}'; 38 | erho = tmp{3}'; 39 | phs0 = tmp{4}'; 40 | ephs = tmp{5}'; 41 | 42 | % a little setup to convert periods to frequency, and degree to rad 43 | freq = 1./period; 44 | phs0 = phs0/180*pi; 45 | ephs = ephs/180*pi; 46 | [resi,rho,phs]=occam1dmt(z,res0,freq,rho0,erho,phs0,ephs,Trms,Niter); 47 | figure(1); 48 | l(end+1)=l(end)*1.5; 49 | plotlayer_log(res0, l,'r'); 50 | hold on; 51 | plotlayer_log(resi,l,'b'); 52 | figure(2); 53 | plot1derr(freq,rho0,phs0,'rho','rx',erho,ephs); 54 | plot1derr(freq,rho,phs,'rho','b-'); 55 | legend('obs','rsp'); 56 | % hasta la vista(? -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # occam1dmt 2 | Toy Occam inversion code for 1D MT (Magnetotellurics) method in Matlab 3 | 4 | ## DATA 5 | The example data are (literally) copied from Table 5 of the Constable 1987 paper, which is in turn from Cull (1985). See: 6 | 7 | Constable, S. C., Parker, R. L., & Constable, C. G. (1987). Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics, 52(3), 289–300. 8 | 9 | Cull, J. P. (1985). Magnetotelluric soundings over a Precambrian contact in Australia. Geophys. J. Roy. Astr. Sot., 80, 661-675. 10 | 11 | see also my toy occam code for DC resistivity: 12 | 13 | [https://github.com/dong-hao/occam1ddc] 14 | 15 | and a RJMCMC version of a student's project: 16 | 17 | [https://github.com/NyankoSong/rjMCMC_MT_1D_Inversion] 18 | 19 | ## USAGE 20 | See example/testbench.m for a simple demonstration on how to load the data and call the inversion code. 21 | 22 | ## something like a disclaimer 23 | 24 | This was one of many toy codes I fiddled with when I was a student - I hope this could be useful to our students nowadays in the EM community. 25 | Those who want to try this script are free to use it on academic/educational cases. But of course, I cannot guarantee the script to be working properly and calculating correctly (although I wish so). Have you any questions or suggestions, please feel free to contact me (but don't you expect that I will reply quickly!). 26 | 27 | ## HOW TO GET IT 28 | ``` 29 | git clone https://github.com/dong-hao/occam1dmt/ your_local_folder 30 | ``` 31 | 32 | ## UNITS 33 | The internal scale here is log10(Ohmm) - instead of linear scale for both resistivity and apparent resistivity. The layer depth is in (linear) metres, while the impedance phase is in rads. 34 | 35 | ## ERRORS 36 | Currently the internal error here is standard deviation. 37 | 38 | ## HOW TO GET UPDATED 39 | ``` 40 | cd to_you_local_folder 41 | git pull 42 | ``` 43 | 44 | ## Contact 45 | 46 | DONG Hao – donghao@cugb.edu.cn 47 | 48 | China University of Geosciences, Beijing 49 | 50 | Distributed under the GPL v3 license. See ``LICENSE`` for more information. 51 | 52 | [https://github.com/dong-hao/occam1dmt] 53 | 54 | ## Contributing 55 | 56 | Those who are willing to contribute are welcomed to try - but I probably won't have the time to review the commits frequently (not that I would expect there will be any). 57 | 58 | 1. Fork it () 59 | 2. Create your feature branch (`git checkout -b feature/somename`) 60 | 3. Commit your changes (`git commit -am 'Add some features'`) 61 | 4. Push to the branch (`git push origin feature/somename`) 62 | 5. Create a new Pull Request - lather, rinse, repeat 63 | -------------------------------------------------------------------------------- /src/jacob1.m: -------------------------------------------------------------------------------- 1 | function [J1,J2,C]=jacob1(freq,res,z) 2 | % rewritten Jacobian calculation script for 1D MT 3 | % for drhoa/dlog10(res) and dphs/dlog10(res) 4 | % used for occam 1D MT inversion 5 | % DONG Hao 6 | % 2011/06/25 7 | % Golmud 8 | res=10.^res; 9 | mju0=4.0*pi*1E-7; % Vacuum Permeability, unit is Vs/(Am) in SI. 10 | sigma=1./res; % conductivity 11 | omega=2*pi*freq; % angular frequency 12 | NL=length(z); % number of layers 13 | NF=length(freq); % number of freqs 14 | J1=ones(NF, NL); % jacobian for df/dx at each layer and each frequency. 15 | J2=J1; 16 | l=diff(z); 17 | C=zeros(NF, NL); 18 | dCdr=ones(NF, NL); 19 | q=zeros(NF,1); 20 | for ifreq=1:NF % looping through frequencies 21 | % calculating the last layer first 22 | IOM=1j*omega(ifreq)*mju0; 23 | q(NL)=sqrt(IOM*sigma(NL)); 24 | C(ifreq,NL)=1/q(NL); 25 | dCdr(ifreq,NL)=1/(q(NL)/sigma(NL)*2); % Note this is right only at the last layer 26 | if NL==1 % in the case of a homogenous half space 27 | continue 28 | end 29 | for ilayer=NL-1:-1:1 30 | q(ilayer)=sqrt(IOM*sigma(ilayer)); 31 | qp1 = C(ifreq,ilayer+1)*q(ilayer)+1; 32 | qm1 = C(ifreq,ilayer+1)*q(ilayer)-1; 33 | theexp=exp(-2.*q(ilayer)*l(ilayer)); 34 | opexp = theexp*qm1/qp1; 35 | omexp = 1 - opexp; 36 | opexp = 1 + opexp; 37 | dCdr(ifreq,ilayer) = 2*theexp*(C(ifreq,ilayer+1)/(qp1.^2)-(l(ilayer)*qm1/qp1))/omexp.^2; 38 | dCdr(ifreq,ilayer) = opexp/omexp/q(ilayer)/2 - dCdr(ifreq,ilayer); 39 | dCdr(ifreq,ilayer) = dCdr(ifreq,ilayer)*sigma(ilayer); 40 | % calculating dCi/dCi+1 41 | dciip1 = 4*theexp/(qp1*omexp).^2; 42 | % now calculating C using Wait's Recursion Formula 43 | C(ifreq,ilayer)=opexp/omexp/q(ilayer); 44 | % cumproduction for dCi/dCi+1 45 | for klayer=ilayer+1:NL 46 | % note the J(dCdr) here is "drhoa/dlog10(res)" 47 | dCdr(ifreq,klayer)=dCdr(ifreq,klayer)*dciip1; 48 | end 49 | end 50 | end 51 | for ifreq=1:NF 52 | for ilayer=1:NL 53 | % for log10(res) (logrithum) 54 | J1(ifreq,ilayer)=2*res(ilayer)*(real(C(ifreq,1))*real(dCdr(ifreq,ilayer))+... 55 | imag(C(ifreq,1))*imag(dCdr(ifreq,ilayer)))/abs(C(ifreq,1)).^2; 56 | % for res (linear) 57 | % J1(ifreq,ilayer)=omega(ifreq)*mju0*(real(C(ifreq,1))*real(dCdr(ifreq,ilayer))+... 58 | % imag(C(ifreq,1))*imag(dCdr(ifreq,ilayer))); 59 | % for phs (linear in RAD) 60 | J2(ifreq,ilayer)=(real(C(ifreq,1))*imag(dCdr(ifreq,ilayer))-... 61 | imag(C(ifreq,1))*real(dCdr(ifreq,ilayer)))/abs(C(ifreq,1)).^2; 62 | end 63 | end 64 | return 65 | -------------------------------------------------------------------------------- /src/mt1dfwd10.m: -------------------------------------------------------------------------------- 1 | function [o1,o2] = mt1dfwd10(freq,res,z,opt) 2 | % a barn door 1d layered forward routine for 3 | % magnetotelluric 1D inversion 4 | % log10 version (both the input resisitivity and output app. res will be in 5 | % log10 space. 6 | % apparent resistivity and phase are generated from layerd model. 7 | % DONG Hao 8 | % 2011/06/25 9 | % Golmud 10 | %=========================================================================% 11 | % description of parameters: 12 | % 13 | % freq: array of frequency for response to be generated 14 | % sigma: conductivity 1/res 15 | % res: array of resistivity of each layer 16 | % z: array of layer DEPTH of each layer INTERFACE, z1=0(surface of 17 | % the earth) 18 | % mju0: absolute magnetic permeability of vacuum 19 | % omega: Angular frequency (2 pi frequency) 20 | % opt: option of outputing the forward, could be "rho" or "imped" 21 | %=========================================================================% 22 | % version 0.2 23 | % DONG Hao 24 | %=========================checking parameters=============================% 25 | switch nargin 26 | case 0 27 | error('not enough input arguments, 2 at least') 28 | case 1 29 | error('not enough input arguments, 2 at least') 30 | case 2 31 | z = 0; % treat the earth as half space 32 | case 3 33 | opt = 'imped'; 34 | end 35 | if (size(z)~=size(res)) 36 | if length(z)==length(res) 37 | res=res'; 38 | else 39 | disp('please check the input parametres ') 40 | error('res and z should have same size'); 41 | end 42 | end 43 | %=========================================================================% 44 | mju0=4.0*pi*1E-7; % unit is Wb/(A��m) in SI. 45 | omega=2*pi*freq; 46 | sigma=1./(10.^res); 47 | NL=length(z); % number of layers 48 | NF=length(freq); % number of freqs 49 | C=zeros(NL, NF);% transfer function for each layer and each frequency. 50 | q=zeros(1,NL); 51 | for ifreq=1:NF 52 | q(NL)=sqrt(1j*omega(ifreq)*mju0*sigma(NL)); % calculating the last layer first 53 | C(NL,ifreq)=1/q(NL); % Note this is right only at the last layer 54 | if NL==1 % in the case of a homogenous half space 55 | continue 56 | end 57 | for ilayer=NL-1:-1:1 58 | q(ilayer)=sqrt(1j*omega(ifreq)*mju0*sigma(ilayer)); 59 | l=z(ilayer+1)-z(ilayer); % depth of the layer 60 | % using Wait's Recursion Formula 61 | C(ilayer,ifreq)=1/q(ilayer)*(q(ilayer)*C(ilayer+1,ifreq)+tanh(q(ilayer)*l))... 62 | /(1+q(ilayer)*C(ilayer+1,ifreq)*tanh(q(ilayer)*l)); 63 | end 64 | end 65 | rho=C(1,:).*conj(C(1,:)).*omega*mju0; % only take the transfer function from the first layer 66 | phase=atan2(real(C(1,:)),-imag(C(1,:))); 67 | switch opt 68 | case 'rho' 69 | o1=log10(rho); 70 | o2=phase; 71 | case 'imped' 72 | Z=C(1,:).*(1j*omega); 73 | o1=real(Z); 74 | o2=-imag(Z); 75 | end 76 | return -------------------------------------------------------------------------------- /src/occam1dmt.m: -------------------------------------------------------------------------------- 1 | function [res,rho,phs]=occam1dmt(z,res0,freq,rho0,erho,phs0,ephs,Trms,Niter) 2 | % main function of occam inversion 3 | % for inversion of log10 app. resistivity and impedance phase. 4 | % please note APPARENT RESISTIVITY AND PHASE ARE INVERTED simutanously 5 | % for now. 6 | % DONG Hao 7 | % 2011/06/25 8 | % Golmud 9 | % Note: this is but a toy scheme (no one is using 1D anymore, oh wait...) 10 | % when I was fiddling my 3D inversion code for my PhD. Thesis. 11 | % Although this sort of worked (as far as I recall), I have not extensively 12 | % tested this script - use with caution. 13 | %=========================================================================% 14 | % input and output parametres: 15 | % 16 | % z: array of layer DEPTH of each layer INTERFACE of the starting 17 | % model, z1=0(surface of the earth) 18 | % res0: array of resistivity of each layer of the starting model 19 | % freq: array of input data frequency 20 | % rho0: array of input data apparent resistivity 21 | % erho: array of input resistivity (absolute) error 22 | % phs0: array of input data impedance phase 23 | % ephs: array of input phase (absolute) error 24 | % Trms: target Root Mean Square misfit for iteration 25 | % Niter: maximum number of iteration 26 | %=========================================================================% 27 | % other parametres that might be useful: 28 | % 29 | % Nz: number of model layers 30 | % Nf: number of frequencies 31 | % sigma: conductivity, 1/res 32 | % lambda0: initial lagrange multiplier 33 | %=========================================================================% 34 | % setup some parametres 35 | M=length(res0); 36 | res=res0; 37 | lambda=10; 38 | dlambda=10; 39 | %==================initialize inversion iteration=========================% 40 | for iter=1:Niter 41 | % the outer loop 42 | disp(['================ iteration ', num2str(iter),' ==================']) 43 | % plot_along(res,z,iter); 44 | lambdar=lambda*dlambda; % lambda on the right 45 | lambdal=lambda/dlambda; % lambda on the left 46 | [J1,J2,C]=jacob1(freq,res,z); 47 | rho=log10(7.8957E-06*abs(C(:,1)').^2.*freq); 48 | phs=atan2(real(C(:,1)),-imag(C(:,1)))'; 49 | imrms=rms1([rho,phs],[rho0,phs0],[erho,ephs]); % starting rms 50 | fprintf('! previous RMS = %5.3f \n', imrms); 51 | drho=rho0-rho; 52 | dphs=phs0-phs; 53 | resm=occam(drho,erho,dphs,ephs,res,lambda,J1,J2); % resistivity in the middle 54 | resr=occam(drho,erho,dphs,ephs,res,lambdar,J1,J2); % resistivity on the right 55 | resl=occam(drho,erho,dphs,ephs,res,lambdal,J1,J2); % resistivity on the left 56 | [rhom,phsm] = mt1dfwd10(freq,resm,z,'rho'); 57 | [rhor,phsr] = mt1dfwd10(freq,resr,z,'rho'); 58 | [rhol,phsl] = mt1dfwd10(freq,resl,z,'rho'); 59 | [ChiSl,ChiSm,ChiSr]=dispfit([rho0,phs0], [rhom,phsm], [rhol,phsl], ... 60 | [rhor,phsr],[erho,ephs], resl ,resm, resr, lambdal ,lambda, lambdar); 61 | for ifind=1:10 62 | % the inner loop to sweep through different Lagrange multipliers 63 | % (or mu in origin Occam theory) 64 | if ChiSm <= ChiSl && ChiSm <= ChiSr 65 | disp('=============regional minimum found=============') 66 | break 67 | elseif ChiSm > ChiSl && ChiSr > ChiSl 68 | disp('================<< searching <<================') 69 | if lambdal/dlambda<=0.0001 70 | disp('! lambda too small, restart iteration') 71 | break 72 | end 73 | resr=resm; 74 | lambdar=lambda; 75 | resm=resl; 76 | lambda=lambdal; 77 | lambdal=lambda/dlambda; 78 | resl=occam(drho,erho,dphs,ephs,res,lambdal,J1,J2); % resistivity on the left 79 | elseif ChiSr < ChiSm && ChiSr < ChiSl 80 | disp('================>> searching >>================') 81 | if lambda*dlambda>=100000 82 | disp('! lambda too large, restart iteration') 83 | break 84 | end 85 | resl=resm; 86 | lambdal=lambda; 87 | resm=resr; 88 | lambda=lambdar; 89 | lambdar=lambda*dlambda; 90 | resr=occam(drho,erho,dphs,ephs,res,lambdar,J1,J2); % resistivity on the right 91 | else 92 | % seems not converging 93 | % use the middle value for next inversion iteration 94 | disp('=============cannot find a local minimum=============') 95 | break 96 | end 97 | [rhom,phsm] = mt1dfwd10(freq,resm,z,'rho'); 98 | [rhol,phsl] = mt1dfwd10(freq,resl,z,'rho'); 99 | [rhor,phsr] = mt1dfwd10(freq,resr,z,'rho'); 100 | [ChiSl,ChiSm,ChiSr]=dispfit([rho0,phs0], [rhom,phsm], [rhol,phsl], ... 101 | [rhor,phsr],[erho,ephs], resl ,resm, resr, lambdal ,lambda, lambdar); 102 | end 103 | res=resm'; 104 | rho=rhom; 105 | phs=phsm; 106 | irms=rms1([rho, phs],[rho0, phs0],[erho, ephs]); 107 | if irms<=imrms 108 | % So far so good... 109 | disp(['! finishing iteration # ' num2str(iter)]); 110 | else 111 | % Huston, we have a convergence problem. 112 | disp(['! warning: iteration # ' num2str(iter) ' not converged']); 113 | disp('! try stablizing inversion with smaller lambda...'); 114 | disp(['! RMS= ' num2str(irms)]); 115 | lambda=lambda/dlambda.^2; 116 | continue 117 | end 118 | if irms<=Trms 119 | % check if the desired rms is reached 120 | fprintf('! current RMS = %5.3f \n', irms); 121 | fprintf('! target rms (%5.3f given by user) reached \n', Trms); 122 | disp('! try to find a smoothest model that fit the data') 123 | rmsr=irms; 124 | drho=rho0-rho; 125 | dphs=phs0-phs; 126 | while rmsr<=Trms 127 | disp('================>> searching >>================') 128 | lambda=lambda*sqrt(dlambda); 129 | resr=occam(drho,erho,dphs,ephs,res,lambda,J1,J2); 130 | [rhor,phsr] = mt1dfwd10(freq,resr,z,'rho'); 131 | rmsr=rms1([rho0,phs0],[rhor,phsr],[erho,ephs]); 132 | fprintf('! evaluating a smoother model with Lambda = %5.3f \n', lambda); 133 | fprintf('! current RMS = %5.3f \n', rmsr); 134 | if rmsr < Trms 135 | fprintf('! model accepted \n'); 136 | rhom = rhor; 137 | phsm = phsr; 138 | resm = resr; 139 | else 140 | fprintf('! model rejected \n'); 141 | end 142 | end 143 | disp('! picking up the smoothest model...') 144 | disp('! exiting...') 145 | res=resm'; 146 | break 147 | end 148 | if ifind>=10 149 | disp('! maximum lambda search reached, start next search...') 150 | end 151 | end 152 | % =========== end main iterations here =========== % 153 | if iter>=Niter 154 | disp('! iteration limit reached, stop...') 155 | else 156 | rho = rhom; 157 | phs = phsm; 158 | end 159 | frms=rms1([rho,phs],[rho0,phs0],[erho,ephs]); 160 | ChiS=chi2([rho,phs],[rho0,phs0],[erho,ephs]); 161 | if ChiS<=2*M 162 | disp('! WARNING: desired chi2 reached. '); 163 | disp('! You might be over fitting the data! ') 164 | end 165 | fprintf('! final RMS = %5.3f \n', frms); 166 | return 167 | 168 | function res=occam(dpara1,epara1,dpara2,epara2,res,lambda,J1,J2) 169 | % main function for a single (multi-variance) occam iteration 170 | % the dpara and epara be the diff and variance of the parameter, which can 171 | % be apparent res, impedance phase, and (real or imag) part of complex 172 | % impedance C. 173 | N=size(J1,2);% number of model 174 | D=mk_pmat(N); % roughness matrix 175 | % penalty matrix 176 | W1=diag(1./epara1); 177 | W2=diag(1./epara2); 178 | a=lambda*(D)'*D;%+diag(ones(N,1)); 179 | % construct weighted Jacobian premultiplied by TRANS(W.J) 180 | wjtwj=(W1*J1)'*(W1*J1)+(W2*J2)'*(W2*J2); 181 | % construct weighted translated data premultiplied by TRANS(W.J) 182 | wjtwd=(W1*J1)'*(W1*(dpara1'+J1*res'))+(W2*J2)'*(W2*(dpara2'+J2*res')); 183 | res=(a+wjtwj)\wjtwd; 184 | return 185 | 186 | function DEL=mk_pmat(N) 187 | % constructing roughness matrix 188 | % for L1 norm 189 | DEL=diag(ones(N,1),0)-diag(ones(N-1,1),-1); 190 | DEL(1,1)=0; 191 | return 192 | 193 | function [ChiSl,ChiSm,ChiSr]=dispfit(co, cm, cl, cr, stderr, resl, resm, resr, lambdal ,lambda, lambdar) 194 | ChiSm=chi2(co,cm,stderr); 195 | ChiSl=chi2(co,cl,stderr); 196 | ChiSr=chi2(co,cr,stderr); 197 | roughl=roughness1(resl,1); 198 | roughm=roughness1(resm,1); 199 | roughr=roughness1(resr,1); 200 | fprintf(' L M R\n'); 201 | fprintf('Chi Square : %8.4e %8.4e %8.4e \n',... 202 | ChiSl, ChiSm, ChiSr); 203 | fprintf('Roughness(L1): %8.4e %8.4e %8.4e \n',... 204 | roughl, roughm, roughr); 205 | fprintf('Lambda : %8.5g %8.5g %8.5g \n',... 206 | lambdal, lambda, lambdar ); 207 | return -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU AFFERO GENERAL PUBLIC LICENSE 2 | Version 3, 19 November 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 Affero General Public License is a free, copyleft license for 11 | software and other kinds of works, specifically designed to ensure 12 | cooperation with the community in the case of network server software. 13 | 14 | The licenses for most software and other practical works are designed 15 | to take away your freedom to share and change the works. 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No Surrender of Others' Freedom. 529 | 530 | If conditions are imposed on you (whether by court order, agreement or 531 | otherwise) that contradict the conditions of this License, they do not 532 | excuse you from the conditions of this License. If you cannot convey a 533 | covered work so as to satisfy simultaneously your obligations under this 534 | License and any other pertinent obligations, then as a consequence you may 535 | not convey it at all. For example, if you agree to terms that obligate you 536 | to collect a royalty for further conveying from those to whom you convey 537 | the Program, the only way you could satisfy both those terms and this 538 | License would be to refrain entirely from conveying the Program. 539 | 540 | 13. Remote Network Interaction; Use with the GNU General Public License. 541 | 542 | Notwithstanding any other provision of this License, if you modify the 543 | Program, your modified version must prominently offer all users 544 | interacting with it remotely through a computer network (if your version 545 | supports such interaction) an opportunity to receive the Corresponding 546 | Source of your version by providing access to the Corresponding Source 547 | from a network server at no charge, through some standard or customary 548 | means of facilitating copying of software. This Corresponding Source 549 | shall include the Corresponding Source for any work covered by version 3 550 | of the GNU General Public License that is incorporated pursuant to the 551 | following paragraph. 552 | 553 | Notwithstanding any other provision of this License, you have 554 | permission to link or combine any covered work with a work licensed 555 | under version 3 of the GNU General Public License into a single 556 | combined work, and to convey the resulting work. The terms of this 557 | License will continue to apply to the part which is the covered work, 558 | but the work with which it is combined will remain governed by version 559 | 3 of the GNU General Public License. 560 | 561 | 14. Revised Versions of this License. 562 | 563 | The Free Software Foundation may publish revised and/or new versions of 564 | the GNU Affero General Public License from time to time. Such new versions 565 | will be similar in spirit to the present version, but may differ in detail to 566 | address new problems or concerns. 567 | 568 | Each version is given a distinguishing version number. If the 569 | Program specifies that a certain numbered version of the GNU Affero General 570 | Public License "or any later version" applies to it, you have the 571 | option of following the terms and conditions either of that numbered 572 | version or of any later version published by the Free Software 573 | Foundation. If the Program does not specify a version number of the 574 | GNU Affero General Public License, you may choose any version ever published 575 | by the Free Software Foundation. 576 | 577 | If the Program specifies that a proxy can decide which future 578 | versions of the GNU Affero General Public License can be used, that proxy's 579 | public statement of acceptance of a version permanently authorizes you 580 | to choose that version for the Program. 581 | 582 | Later license versions may give you additional or different 583 | permissions. However, no additional obligations are imposed on any 584 | author or copyright holder as a result of your choosing to follow a 585 | later version. 586 | 587 | 15. Disclaimer of Warranty. 588 | 589 | THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY 590 | APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT 591 | HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY 592 | OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, 593 | THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 594 | PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM 595 | IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF 596 | ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 597 | 598 | 16. Limitation of Liability. 599 | 600 | IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING 601 | WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS 602 | THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY 603 | GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE 604 | USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF 605 | DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD 606 | PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), 607 | EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF 608 | SUCH DAMAGES. 609 | 610 | 17. Interpretation of Sections 15 and 16. 611 | 612 | If the disclaimer of warranty and limitation of liability provided 613 | above cannot be given local legal effect according to their terms, 614 | reviewing courts shall apply local law that most closely approximates 615 | an absolute waiver of all civil liability in connection with the 616 | Program, unless a warranty or assumption of liability accompanies a 617 | copy of the Program in return for a fee. 618 | 619 | END OF TERMS AND CONDITIONS 620 | 621 | How to Apply These Terms to Your New Programs 622 | 623 | If you develop a new program, and you want it to be of the greatest 624 | possible use to the public, the best way to achieve this is to make it 625 | free software which everyone can redistribute and change under these terms. 626 | 627 | To do so, attach the following notices to the program. It is safest 628 | to attach them to the start of each source file to most effectively 629 | state the exclusion of warranty; and each file should have at least 630 | the "copyright" line and a pointer to where the full notice is found. 631 | 632 | 633 | Copyright (C) 634 | 635 | This program is free software: you can redistribute it and/or modify 636 | it under the terms of the GNU Affero General Public License as published 637 | by the Free Software Foundation, either version 3 of the License, or 638 | (at your option) any later version. 639 | 640 | This program is distributed in the hope that it will be useful, 641 | but WITHOUT ANY WARRANTY; without even the implied warranty of 642 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 643 | GNU Affero General Public License for more details. 644 | 645 | You should have received a copy of the GNU Affero General Public License 646 | along with this program. If not, see . 647 | 648 | Also add information on how to contact you by electronic and paper mail. 649 | 650 | If your software can interact with users remotely through a computer 651 | network, you should also make sure that it provides a way for users to 652 | get its source. For example, if your program is a web application, its 653 | interface could display a "Source" link that leads users to an archive 654 | of the code. There are many ways you could offer source, and different 655 | solutions will be better for different programs; see section 13 for the 656 | specific requirements. 657 | 658 | You should also get your employer (if you work as a programmer) or school, 659 | if any, to sign a "copyright disclaimer" for the program, if necessary. 660 | For more information on this, and how to apply and follow the GNU AGPL, see 661 | . 662 | --------------------------------------------------------------------------------