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GaussianMixMCMC_metropolis.m
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GaussianMixMCMC_metropolis.m
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function [ INVERSION ] = GaussianMixMCMC_metropolis(real_seismic, theta, signal2noise, wavelet, PRIOR, n_it, prob_map, P, facies_initial)
tic
%% burn in period
it_conv = min([10000 round(n_it*0.1)]);
n_angles = length(theta);
I = size(real_seismic,1);
n_facies = length(PRIOR);
real_seismic = real_seismic(1:end-1,:);
G = elasticForwardModel(exp(PRIOR(1).MU(1))*ones(I,1), exp(PRIOR(1).MU(2))*ones(I,1), wavelet, theta);
%% Seismic noise covariance matrix
d = reshape(real_seismic,size(real_seismic,2)*size(real_seismic,1),1);
var_sismica_final = var(real_seismic)./(signal2noise);
var_d = repmat(var_sismica_final,size(d,1)/n_angles,1);
C_d = diag(var_d(:));
I_d = size(d,1);
%% INITIAL CONFIGURATION - RANDOM MODEL
if nargin < 9
facies_initial = simulate_markov_chain(P, I, randi(n_facies), 1, prob_map);
end
%% Compute the prior distribution of elastic properties along the traces given de facies configuration
facies_samples(:,1) = facies_initial;
if length(fieldnames(PRIOR))==2
[mu_mlpi_previous , C_mlpi_previous ] = construct_prior_facies(PRIOR,facies_initial,2.5);
else
for facie=1:length(PRIOR)
trendVp(:,facie) = PRIOR(facie).VP.trend;
trendVs(:,facie) = PRIOR(facie).VS.trend;
trendRhob(:,facie) = PRIOR(facie).RHOB.trend;
end
[mu_mlpi_previous , C_mlpi_previous ] = construct_prior_trend_correlated_v2(PRIOR, trendVp, trendVs, trendRhob, facies_initial, 2.5);
end
%% MCMC METHOD
model_samples = zeros(3*I,n_it);
sy_seisimic = zeros(I_d,n_it);
C_samples = zeros(3*I,n_it);
log_likelihood = zeros(1,n_it);
log_likelihood(1) = nan;
OPERATOR = C_mlpi_previous*G'*inv(G*C_mlpi_previous*G' + C_d);
model_samples(:,1) = mu_mlpi_previous + OPERATOR*(d - G*mu_mlpi_previous);
C_samples (:,1) = diag(C_mlpi_previous - OPERATOR*G*C_mlpi_previous);
for it = 2:n_it
100*it/n_it
%% From the previous facies configuration, drawn a facies samples from the prior
window_size = round(4 + rand*6);
window_size = window_size - 1;
index_sort = 1 + round(rand*(I-window_size-1)) + window_size/2;
indexes_sort = index_sort-window_size/2:window_size/2+index_sort;
indexes_sort_not = [1:(indexes_sort(1)-1) (indexes_sort(end)+1):I ];
if indexes_sort(1) == 1 || indexes_sort(end) == I
if indexes_sort(1) == 1
initial_facies = randi(n_facies);
final_facies = facies_samples(indexes_sort(end)+1,it-1);
else
initial_facies = facies_samples(indexes_sort(1)-1,it-1);
final_facies = randi(n_facies);
end
else
initial_facies = facies_samples(indexes_sort(1),it-1);
final_facies = facies_samples(indexes_sort(end),it-1);
end
prob_map_window = prob_map(indexes_sort,:,:);
facies_proposal_window = simulate_markov_chain_finalDefined(P, window_size+1, initial_facies, final_facies, 1, prob_map_window);
facies_proposal = facies_samples(:,it-1);
facies_proposal(indexes_sort) = facies_proposal_window ;
%% Construct prior mean an covariance of the the prior p(m|pi)
if length(fieldnames(PRIOR))==2
[mu_mlpi, C_mlpi] = construct_prior_facies(PRIOR,facies_proposal,2.5);
else
[mu_mlpi, C_mlpi] = construct_prior_trend_correlated_v2(PRIOR, trendVp, trendVs, trendRhob, facies_proposal,2.5);
end
%% Metropolis Acceptance rule
GAMMA_dlpi = log( mvnpdf(d, G*mu_mlpi, G*C_mlpi*G' + C_d ) ) ;
GAMMA_dlpi_previous = log( mvnpdf(d, G*mu_mlpi_previous, G*C_mlpi_previous*G' + C_d ) ) ;
%%% Accept or reject the facies proposal
if rand<exp(GAMMA_dlpi-GAMMA_dlpi_previous)
facies_samples(:,it) = facies_proposal;
OPERATOR = C_mlpi*G'*inv(G*C_mlpi*G' + C_d);
mu_mlpid = mu_mlpi + OPERATOR*(d - G*mu_mlpi);
C_mlpid = C_mlpi - OPERATOR*G*C_mlpi;
C_samples(:,it) = diag(C_mlpid );
% If you want to take a sample of the continuous properties:
model_samples(:,it) = mvnrnd(mu_mlpid, C_mlpid)';
% else, just define it as the mean
%model_samples(:,it) = mu_mlpid;
sy_seisimic(:,it) = G*model_samples(:,it);
mu_mlpi_previous = mu_mlpi;
C_mlpi_previous = C_mlpi;
log_likelihood(it) = -0.5*(d-sy_seisimic(:,it))'*inv(C_d)*(d-sy_seisimic(:,it));
else
facies_samples(:,it) = facies_samples(:,it-1);
model_samples(:,it) = model_samples(:,it-1);
sy_seisimic(:,it) = sy_seisimic(:,it-1);
log_likelihood(it) = log_likelihood(it-1);
end
end
%% Write results in the INVERSION structure
C_samples = reshape(C_samples,I,3,n_it);
INVERSION.VP.Csamples(:,:) = C_samples(:,1,it_conv:end);
INVERSION.VS.Csamples(:,:) = C_samples(:,2,it_conv:end);
INVERSION.RHOB.Csamples(:,:) = C_samples(:,3,it_conv:end);
model_samples = reshape(model_samples,I,3,n_it);
INVERSION.VP.MUsamples(:,:) = model_samples(:,1,it_conv:end);
INVERSION.VS.MUsamples(:,:) = model_samples(:,2,it_conv:end);
INVERSION.RHOB.MUsamples(:,:) = model_samples(:,3,it_conv:end);
INVERSION.VP.mean = mean(INVERSION.VP.MUsamples(:,it_conv:end)')';
INVERSION.VS.mean = mean(INVERSION.VS.MUsamples(:,it_conv:end)')';
INVERSION.RHOB.mean = mean(INVERSION.RHOB.MUsamples(:,it_conv:end)')';
[INVERSION.VP.probability,INVERSION.VP.axis,INVERSION.VP.map] = calculate_posterior_probability(INVERSION.VP.MUsamples,INVERSION.VP.Csamples,1);
[INVERSION.VS.probability,INVERSION.VS.axis,INVERSION.VS.map] = calculate_posterior_probability(INVERSION.VS.MUsamples,INVERSION.VS.Csamples,1);
[INVERSION.RHOB.probability,INVERSION.RHOB.axis,INVERSION.RHOB.map] = calculate_posterior_probability(INVERSION.RHOB.MUsamples,INVERSION.RHOB.Csamples,1);
INVERSION.FACIES.samples = facies_samples;
facies_samples_chain = facies_samples(:,it_conv:end);
for facie = 1:length(PRIOR)
indicator = zeros(size(facies_samples_chain(:)));
indicator(facies_samples_chain==facie) = 1;
indicator = reshape(indicator,size(facies_samples_chain));
facies_prob(:,facie) = sum(indicator,2)./size(indicator,2);
end
INVERSION.FACIES.prob = facies_prob;
[ ~, INVERSION.FACIES.likely] = max(facies_prob,[],2);
INVERSION.SYNTHETIC = sy_seisimic(:,2:end);
INVERSION.erro = sum( (repmat(d,1,size(INVERSION.SYNTHETIC,2)) - INVERSION.SYNTHETIC).^2,1);
INVERSION.log_likelihood = log_likelihood;
INVERSION.processing_time = toc;
end
function [mu, C] = construct_prior_facies(PRIOR, facies_sample,L)
I = length(facies_sample);
corr_12 = zeros(I,1);
corr_23 = zeros(I,1);
corr_13 = zeros(I,1);
mask_byfacies = zeros(I,I);
var_ = zeros(3*I,1);
A = covariance_matrix_exp(ones(I,1),L,1);
for facies=1:length(PRIOR)
nu = covariance2correlation(PRIOR(facies).C);
index = find(facies_sample==facies);
mu(index,1) = PRIOR(facies).MU(1);
mu(index+I,1) = PRIOR(facies).MU(2);
mu(index+2*I,1) = PRIOR(facies).MU(3);
var_(index) = sqrt(PRIOR(facies).C(1,1));
var_(index+I) = sqrt(PRIOR(facies).C(2,2));
var_(index+2*I) = sqrt(PRIOR(facies).C(3,3));
corr_12(index) = sqrt(nu(1,2));
corr_23(index) = sqrt(nu(2,3));
corr_13(index) = sqrt(nu(1,3));
mask_byfacies(index,index) = 1;
end
mask_byblock = ones(I,I);
abs_diff = abs(diff(facies_sample));
boundary_positions = find(abs_diff>0);
for block = 1:length(boundary_positions)
mask_byblock(1:boundary_positions(block),boundary_positions(block)+1:end) = 0;
end
mask_byblock = mask_byblock.*mask_byblock';
GAMMA = [ A diag(corr_12)*A*diag(corr_12) diag(corr_13)*A*diag(corr_13);
diag(corr_12)*A*diag(corr_12) A diag(corr_23)*A*diag(corr_23);
diag(corr_13)*A*diag(corr_13) diag(corr_23)*A*diag(corr_23) A ];
C = diag(var_)*GAMMA*diag(var_);
% Without the mask, the properties from different facies will be
% spaially correlated, which is not correct
% Option 1: Allow spatial correlation within the same facies
%mask = mask_byfacies;
% Option 2: Allow spatial correlation within the same facies and within the
% same "block/layer"
mask = mask_byblock;
C = C.*[mask mask mask;
mask mask mask;
mask mask mask];
end
function [mu, C] = construct_prior_trend_correlated_v2(PRIOR, trendVp, trendVs, trendRhob, facies_sample, L)
I = length(facies_sample);
mu = zeros(3*I,1);
var_ = zeros(3*I,1);
corr_12 = zeros(I,1);
corr_23 = zeros(I,1);
corr_13 = zeros(I,1);
mask_byfacies = zeros(I,I);
A = covariance_matrix_exp(ones(I,1),L,1);
for facies=1:length(PRIOR)
nu = covariance2correlation(PRIOR(facies).C);
index = find(facies_sample==facies);
mu(index,1) = trendVp(index,facies);
mu(index+I,1) = trendVs(index,facies);
mu(index+2*I,1) = trendRhob(index,facies);
var_(index) = sqrt(PRIOR(facies).C(1,1));
var_(index+I) = sqrt(PRIOR(facies).C(2,2));
var_(index+2*I) = sqrt(PRIOR(facies).C(3,3));
corr_12(index) = sqrt(abs(nu(1,2)));
corr_23(index) = sqrt(abs(nu(2,3)));
corr_13(index) = sqrt(abs(nu(1,3)));
mask_byfacies(index,index) = 1;
end
GAMMA = [ A diag(corr_12)*A*diag(corr_12) diag(corr_13)*A*diag(corr_13);
diag(corr_12)*A*diag(corr_12) A diag(corr_23)*A*diag(corr_23);
diag(corr_13)*A*diag(corr_13) diag(corr_23)*A*diag(corr_23) A ];
C = diag(var_)*GAMMA*diag(var_);
mask = mask_byfacies;
C = C.*[mask mask mask;
mask mask mask;
mask mask mask];
end