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final_synthetic_model2.m
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% ***************************************************************
% *** Matlab function for synthetic model-2 with fixed density is a part of SPoDEA programe that includes a set of *.m files to compute basement depth of the complex sedimentary basin.
% *** Source Code is mainly written for research purposes. The codes are
% *** having copyrights and required proper citations whenever it is used.
% *** Originated by:
% *** Mr. Arka Roy (email: [email protected])
% *** Dr. Chandra Prakash Dubey (email:[email protected])
% *** Mr. M. Prasad (email:[email protected])
% *** Crustal Processes Group, National Centre for Earth Science Studies,
% *** Ministry of Earth Sciences, Government of India
% *** Thiruvanthapuram, Kerala, India
% ****************************************************************
%Code for synthetic model(2) having fixed density for inversion of gravity data
clear all
close all
%%%%%%%%%%%%%%%%%%%%%%%% Part 1. %%%%%%%%%%%%%%%%%%%%%%%%
%%%% Creation of Synthetic Model and finding Gravity anomaly %%%%
%synthetic function for depth of the basin
f=@(x,mu,sigma) (exp(-(x-mu).^2)/(2*sigma^2));
%synthetic function for depth of the basin
xx=linspace(5.5,11.5,100);
yy=1000*(f(xx,7,2)+f(xx,10,1.5));
%t and c are Legendre Gaussian quadrature points for numerical integration
[t_leg,c_leg]=lgwt(10,0,1);
%synthetic depth and observation points
x_obs=linspace(0,5000,100);
depth=yy;
%plotting of x_obs vs. depth
figure(1)
subplot(2,1,2)
plot(x_obs,depth)
set(gca,'Ydir','reverse')
xlabel('Distance in meter')
ylabel('Depth in meter')
title('Depth profile of the synthetic basin (Model 2)')
box on
%Finding Gravity field of the basin for density -800 kg/m^3
density=-800;
z_obs=0; %height of observation point is in ground
%polygonic profile of the basin
xx1=[x_obs x_obs(end) 0];
yy1=[depth 0 0];
%Gravity anomaly for fixed density model
zz1=poly_gravity(x_obs,z_obs,xx1,yy1,density,t_leg,c_leg);
zz2=zz1;
%adding noise to anomaly having 0 mean and sqrt(0.05) standard deviation
zz1 = zz1+sqrt(0.05).*randn(size(zz1))+0;
%for model without noise comment line 36.
%Plotting the Gravity field anomaly
figure(1)
subplot(2,1,1)
plot(x_obs,zz1)
xlabel('Observation points in meter')
ylabel('Gravity anomaly in mGal')
title('Gravity anomaly for Model 2 ')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% Part 2. %%%%%%%%%%%%%%%%%%%%%%%%
%%%% Inversion of Gravity anomaly using Differential Evolution %%%%
data=zz1; %data is gravity field from Part 1.
%Finding inversion of the same gravity profile for fixed density
%Parameters for DE algorithm
Cross_rate=0.8; %Crossover Rate
nVar = 10; %Number of Genes
K= 0.5; F=.8; %Scalling and Combinition Factor
%Parameters for b-Spline
n=9; % number of knots
k=4; % order of polynomial
%Population size
nPoP = 30; %Number of Chromosome
%Total number of Generations
MaxIt = 1000; %Number of Generations
%%Problem Definition
%Objective Function
CostFunction =@(x,data) myCostFunction(x,data,n,k,t_leg,c_leg)+1000*(constrained1(x,n,k)+constrained2(x,n,k));
VarSize = [1 nVar]; %Matrix size of Decision variables
VarMin= -500*ones(1,nVar); %Lower Bound of Unknown variable
VarMax= 500*ones(1,nVar); %Upper Bound of Unknown variable
%%Initialization
Empty.Particle.Position =[]; %creating Empty vectors for each Cromosome
Empty.Particle.Cost =[]; %creating Empty vectors for cost of Cromosome
Particle=repmat(Empty.Particle, nPoP,1); %Empty Matrix for all cromosome
Vector=Particle; %Cromosome
Next_Particle=Particle;
%Loop for initialization of Population
for i=1:nPoP
%initialize position with random number from VarMin and VarMax
for j=1:nVar
Particle(i).Position(j) =unifrnd(VarMin(j),VarMax(j));
end
%initial cost function for each Chromosome
Particle(i).Cost = CostFunction(Particle(i).Position,data);
end
%Loop for generations
for jj=1:MaxIt
%% Mutation of DE
%loop for all Chromosome
for it=1:nPoP
%Any random Chromosome which is not same as i
ss=rand_spl(nPoP,i);
%Choosing 3 random population
for jt=1:length(ss)
temp_Particle(jt)=Particle(ss(jt));
end
%Step for Mutation
Vector(it).Position=Particle(i).Position+K.*([temp_Particle(1).Position]-[Particle(i).Position])+...
F.*([temp_Particle(2).Position]-[temp_Particle(3).Position]);
end
%% Crossover of DE
%loop for all Chromosome
for i=1:nPoP
%loop for all genes
for j=1:nVar
%Parent Chromosome is mixing with mutated Chromosome
if rand(1)>Cross_rate && j~=randperm(nVar,1)
New_Particle(i).Position(j)=Particle(i).Position(j);
else
New_Particle(i).Position(j)=Vector(i).Position(j);
end
end
%After mixing the objective function value
New_Particle(i).Cost= CostFunction(New_Particle(i).Position,data);
end
%% Selection of DE
for i=1:nPoP
%Offspring process for next generation
if New_Particle(i).Cost<=Particle(i).Cost
New_Offspring(i)= New_Particle(i);
else
New_Offspring(i)= Particle(i);
end
end
%Selection of best offsprings for next generation
[x,idx]=sort([New_Offspring.Cost],'ascend'); %Sorting in ascending order
fmin=New_Offspring(idx(1)).Cost; %minimum value of objective function
best_var=New_Offspring(idx(1)).Position; %best parameters from optimization
Particle=New_Offspring; %all Chromosomes after optimization
%printing the best solution for each generation
fprintf('Cost Function=%.7f\n',fmin)
%value for error energy plot
error_energy(jj)=fmin.^2;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% Part 3. %%%%%%%%%%%%%%%%%%%%%%%%
%%%% Plotting the results obtained form Differential Evolution %%%%
xx=x_obs; %observation points
x=best_var; %optimized parameters of gravity inversion
%bSpline method for finding depth from parameters
t=linspace(0,1,100);
%All basis functions for b-Spline
N_final=b_spline_basis(t,n,k);
%loop for finding depth profile from optimized parameters
yy_eval=0;
for i=1:n+1
%Depth profile
yy_eval=yy_eval+x(i)*N_final(:,i);
end
%Horizontal position of depth profile
xx1=linspace(min(xx),max(xx),30);
%Vertical position of depth profile
yy1=spline(xx,yy_eval,xx1);
%End point adjustment for closed form
x1=[xx1 x_obs(end) 0];
y1=[yy1 0 0];
%Gravity anomaly for optimized depth profile
zz1=poly_gravity(xx,0,x1,y1,density,t_leg,c_leg);
% Plotting of observed and inverted Gravity anomaly
figure(2)
subplot(2,1,1)
hold on
%Observed
plot(xx,data,'-o','linewidth',1.25)
%Inverted
plot(xx,zz1,'linewidth',1.25)
%Axis lebeling
xlabel('Observation points in meter')
ylabel('Gravity anomaly in mGal')
title('Gravity anomaly for Model 2 ')
legend('Observed Data','Inverted Data','location','southeast')
box on
%RMSE for gravity
N_g=length(data); %Number of Observation points
%RMSE of given model
RMSE_g=sqrt((sum((data-zz1).^2))/N_g)/(max(data(:))-min(data(:)));
%RMSE of True model
RMSE_true=sqrt((sum((data-zz2).^2))/N_g)/(max(data(:))-min(data(:)));
% Plotting of depth profile
figure(2)
subplot(2,1,2)
hold on
%plot(xx,yy_eval)
box on
yy2=spline(x_obs,depth,xx1);
N_d=length(xx1);
% RMSE error for depth profile
RMSE_d=sqrt((sum((yy2-yy1).^2))/N_d)/(max(yy2(:))-min(yy2(:)));
%Printing the RMSE error for depth and gravity profile
fprintf('RMSE in gravity field=%f, and in depth profile=%f\n',RMSE_g,RMSE_d)
fprintf('True RMSE in gravity field=%f\n',RMSE_true)
%plotting of true model and optimized model
zd=density*ones(size(depth));
%true model
patch(xx,depth,zd)
colormap winter
%optimized model
plot(xx1,yy1,'linewidth',1.25,'color','r')
set(gca,'Ydir','reverse')
c = colorbar('location','southoutside');
c.Label.String = 'Density in kg/m^3';
xlabel('Distance in meter')
ylabel('Depth in meter')
legend('True Model','Best Model','location','southeast')
title('Depth profile of synthetic basin (Model 2)')
%plotting of residuals
figure(3)
subplot(2,1,1)
%plotting for gravity residual
plot(xx,data-zz1)
xlabel('Observation points in meter')
ylabel('Gravity anomaly in mGal')
title('Residual of Gravity anomaly for Model 2 ')
figure(3)
subplot(2,1,2)
%plotting for depth residual
plot(xx,yy_eval'-depth)
set(gca,'Ydir','reverse')
xlabel('Distance in meter')
ylabel('Depth in meter')
title('Residual in depth profile of synthetic basin (Model 2)')
%error energy plot
figure(4)
semilogy(1:length(error_energy),error_energy)
xlabel('Generation')
ylabel('Error Energy in mGal^2')
title('Error Energy plot Model 2')
legend('Without Noise')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% Part 4. %%%%%%%%%%%%%%%%%%%%%%%%
%%%% Objective functions and Constraints %%%%
function val=myCostFunction(x,data,n,k,t_leg,c_leg)
%%inputs are
%x= parameters for DE algorithm
%data= observed gravity field
%n= number of knots
%k= order of polynomial for bspline
%t_leg and c_leg are Legendre Gaussian quadrature points for numerical integration
% subroutine for t_leg and c_leg evaluation is given in lgwt.m file
%observation points having 100 linearly spaced data points
xx=linspace(0,5*10^3,100);
%100 linearly spaced data points from 0 to 1 for bspline basis
t=linspace(0,1,100);
%all bspline basis for given knots and polynomial order
N_final=b_spline_basis(t,n,k);
%loop for depth profile reconstruction using bspline basis
yy=0;
for i=1:n+1
yy=yy+x(i)*N_final(:,i);
end
%close polygonal form of depth profile
xx1=linspace(min(xx),max(xx),20);
yy1=spline(xx,yy,xx1);
x1(1:22)=[xx1 5000 0];
y1(1:22)=[yy1 0 0];
%gravity field for given depth profile
zz1=poly_gravity(xx,0,x1,y1,-800,t_leg,c_leg);
%misfit functional for observed and inverted gravity anomaly
val=norm(data-zz1);
end
%Constraints for upper bound
function val=constrained1(x,n,k)
%%inputs are
%x= parameters for DE algorithm
%n= number of knots
%k= order of polynomial for bspline
%100 linearly spaced data points from 0 to 1 for bspline basis
t=linspace(0,1,100);
%all bspline basis for given knots and polynomial order
N_final=b_spline_basis(t,n,k);
%loop for depth profile reconstruction using bspline basis
yy=0;
for i=1:n+1
yy=yy+x(i)*N_final(:,i);
end
%minimum of depth
m_yy=min(yy);
%penalty barrier method for minimum bound
gg=(-m_yy+0);
val=(max(0,gg))^2;
end
function val=constrained2(x,n,k)
%%inputs are
%x= parameters for DE algorithm
%n= number of knots
%k= order of polynomial for bspline
%100 linearly spaced data points from 0 to 1 for bspline basis
t=linspace(0,1,100);
%all bspline basis for given knots and polynomial order
N_final=b_spline_basis(t,n,k);
%loop for depth profile reconstruction using bspline basis
yy=0;
for i=1:n+1
yy=yy+x(i)*N_final(:,i);
end
%maximum of depth
m_yy=max(yy);
%penalty barrier method for minimum bound
gg=(m_yy-1500);
val=(max(0,gg))^2;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%