Matlab Scripts for artificial noise

Matlab Noise Analysis/CorrelatedNoiseSphericalGrid.m

+% Generate noise with spatial correlation in poloidal, radial and toroidal
+% directions
+% Sampling on spherical grid with spatial correlation
+close all
+clear all
+clc
+
+Rmax = 500;
+rmin = 100;
+rstart = 10;
+
+L = 2*pi*Rmax;
+% Parameters of the noise distribution
+mu    = 0;     % mean of the noise (spatially constant)
+sigma = 1;     % standard deviation for the noise (spatially constant)
+Lr     = rmin/5;  % correlation length in radial direction
+Lp     = 2*pi*rmin/10; % Correlation "length" in poloidal direction
+Lt    = 2*pi*Rmax/2; % Correlation "length" in toroidal direction
+
+
+nr = 10; % number of grid points in x-direction
+np = 5; % number of grid points in y-direction
+nt = 10;
+
+r = (rstart/rmin:0.9/(nr-1):1)*rmin;
+p = (0:1/(np-1):1)*2*pi;
+[P,R] = meshgrid(p,r);
+
+% Vector of mean at each grid point
+Mu = mu*ones(nr*np,1);
+
+% Construct covariance matrices
+Sigma_corr    = zeros(nr*np);
+Sigma_no_corr = zeros(nr*np);
+
+for j = 1:np
+    for i = 1:nr
+        
+        k = i + (j-1)*nr;
+        
+        % Main diagonal: variance
+        Sigma_no_corr(k,k) = sigma^2;
+        Sigma_corr(k,k)    = sigma^2;
+        
+        % Off-diagonal: spatial correlation
+        for n = 1:np
+            for m = 1:nr
+                l = m + (n-1)*nr;
+                if l ~= k
+                    Sigma_corr(k,l) = sigma^2*exp(-sqrt((R(m,n)-R(i,j))^2)/Lr)*exp(-(abs(P(m,n)-P(i,j))*(R(m,n)+R(i,j))/2)/Lp);
+                end
+            end
+        end
+    end
+end
+
+b = issymmetric(Sigma_corr)
+d = eig(Sigma_corr)
+% Generate the noise from multivariate normal distributions
+noise_no_corr = mvnrnd(Mu,Sigma_no_corr);
+noise_corr    = mvnrnd(Mu,Sigma_corr);
+
+% Reshape onto mesh
+noise_no_corr = reshape(noise_no_corr,nr,np);
+noise_corr = reshape(noise_corr,nr,np);
+
+% Plot
+figure;surf(R,P,noise_no_corr);title('Independent');
+figure;surf(R,P,noise_corr); title('Spatial correlation');
+
+% Also check toroidal
+
+t = (0:1/(nt-1):1)*2*pi;
+p = (0:1/(np-1):1)*2*pi;
+[P,T] = meshgrid(p,t);
+
+% Vector of mean at each grid point
+Mu = mu*ones(nt*np,1);
+
+% Construct covariance matrices
+Sigma_corr    = zeros(nt*np);
+Sigma_no_corr = zeros(nt*np);
+
+for j = 1:np
+    for i = 1:nt
+        
+        k = i + (j-1)*nt;
+        
+        % Main diagonal: variance
+        Sigma_no_corr(k,k) = sigma^2;
+        Sigma_corr(k,k)    = sigma^2;
+        
+        % Off-diagonal: spatial correlation
+        for n = 1:np
+            for m = 1:nt
+                l = m + (n-1)*nt;
+                if l ~= k
+                    Sigma_corr(k,l) = sigma^2*exp(-(abs(T(m,n)-T(i,j))*(Rmax+R(m,n)*cos(P(m,n))+Rmax+R(i,j)*cos(P(i,j)))/2)/Lt) ...
+                        *exp(-(abs(P(m,n)-P(i,j))*(R(m,n)+R(i,j))/2)/Lp);
+                end
+            end
+        end
+    end
+end
+
+b = issymmetric(Sigma_corr)
+d = eig(Sigma_corr)
+% Generate the noise from multivariate normal distributions
+noise_no_corr = mvnrnd(Mu,Sigma_no_corr);
+noise_corr    = mvnrnd(Mu,Sigma_corr);
+
+% Reshape onto mesh
+noise_no_corr = reshape(noise_no_corr,nt,np);
+noise_corr = reshape(noise_corr,nt,np);
+
+% Plot
+figure;surf(T,P,noise_no_corr);title('Independent');
+figure;surf(T,P,noise_corr); title('Spatial correlation');
+

Matlab Noise Analysis/CorrelatedNoiseSphericalGrid3D.m

+% Generate noise with spatial correlation in poloidal, radial and toroidal
+% directions
+% Sampling on spherical grid with spatial correlation
+% 3D case
+close all
+clear all
+clc
+
+Rmax = 500;
+rmin = 100;
+rstart = 10;
+
+L = 2*pi*Rmax;
+% Parameters of the noise distribution
+mu    = 0;     % mean of the noise (spatially constant)
+sigma = 1;     % standard deviation for the noise (spatially constant)
+Lr     = rmin/10;  % correlation length in radial direction
+Lp     = 2*pi*rmin/10; % Correlation "length" in poloidal direction
+Lt    = 2*pi*Rmax/10; % Correlation "length" in toroidal direction
+
+
+nr = 5; 
+np = 10; 
+nt = 2;
+
+r = (rstart/rmin:0.9/(nr-1):1)*rmin;
+p = (0:1/(np-1):1)*2*pi;
+t = (0:1/(nt-1):1)*2*pi;
+[P,R,T] = meshgrid(p,r,t);
+
+% Vector of mean at each grid point
+Mu = mu*ones(nr*np*nt,1);
+
+% Construct covariance matrices
+Sigma_corr    = zeros(nr*np*nt);
+Sigma_no_corr = zeros(nr*np*nt);
+
+for h = 1:nt
+    for j = 1:np
+        for i = 1:nr
+        
+            k = i + (j-1)*nr + (h-1)*nr*np;
+        
+            % Main diagonal: variance
+            Sigma_no_corr(k,k) = sigma^2;
+            Sigma_corr(k,k)    = sigma^2;
+        
+         % Off-diagonal: spatial correlation
+            for p = 1:nt
+                for n = 1:np
+                    for m = 1:nr
+                        l = m + (n-1)*nr + (p-1)*nr*np;
+                        if l ~= k
+                            Sigma_corr(k,l) = sigma^2*exp(-sqrt((R(m,n,p)-R(i,j,h))^2)/Lr)...
+                                *exp(-(abs(P(m,n,p)-P(i,j,h))*(R(m,n,p)+R(i,j,h))/2)/Lp) ...
+                                *exp(-(abs(T(m,n,p)-T(i,j,h))*(Rmax+R(m,n,p)*cos(P(m,n,p))+Rmax+R(i,j,h)*cos(P(i,j,h)))/2)/Lt);
+                        end    
+                    end
+                end
+            end
+        end
+    end
+end
+
+b = issymmetric(Sigma_corr)
+d = eig(Sigma_corr)
+% Generate the noise from multivariate normal distributions
+noise_no_corr = mvnrnd(Mu,Sigma_no_corr);
+noise_corr    = mvnrnd(Mu,Sigma_corr);
+
+% To extract the noise, store in same format as BFSTREN (1 noise value for
+% every grid point) => then shift to cell centers
+
+
+% Reshape onto mesh
+noise_no_corr = reshape(noise_no_corr,nr,np,nt);
+noise_corr = reshape(noise_corr,nr,np,nt);
+
+% Plot
+%figure;surf(R,P,T);title('Independent');
+%figure;surf(R,P,T); title('Spatial correlation');
\ No newline at end of file

Matlab Noise Analysis/Sampling_spatial_correlation.m

+% Sampling on cartesian grid with spatial correlation
+close all
+clear all
+clc
+
+
+% Parameters of the noise distribution
+mu    = 0;     % mean of the noise (spatially constant)
+sigma = 1;     % standard deviation for the noise (spatially constant)
+L     = 0.25;  % correlation length (spatially constant)
+
+
+% Grid to sample on
+Lx = 1; % length of domain in x-direction
+Ly = 1; % length of domain in y-direction
+
+nx = 10; % number of grid points in x-direction
+ny = 10; % number of grid points in y-direction
+
+x = (0:1/(nx-1):1)*Lx;
+y = (0:1/(ny-1):1)*Ly;
+[X,Y] = meshgrid(x,y);
+
+
+% Vector of mean at each grid point
+Mu = mu*ones(nx*ny,1);
+
+% Construct covariance matrices
+Sigma_corr    = zeros(nx*ny);
+Sigma_no_corr = zeros(nx*ny);
+
+for j = 1:ny
+    for i = 1:nx
+        
+        k = i + (j-1)*ny;
+        
+        % Main diagonal: variance
+        Sigma_no_corr(k,k) = sigma^2;
+        Sigma_corr(k,k)    = sigma^2;
+        
+        % Off-diagonal: spatial correlation
+        for n = 1:ny
+            for m = 1:nx
+                l = m + (n-1)*ny;
+                if l ~= k
+                    Sigma_corr(k,l) = sigma^2*exp(-((X(m,n)-X(i,j))^2 + (Y(m,n)-Y(i,j))^2)^0.5/L);
+                end
+            end
+        end
+    end
+end
+
+% Generate the noise from multivariate normal distributions
+noise_no_corr = mvnrnd(Mu,Sigma_no_corr);
+noise_corr    = mvnrnd(Mu,Sigma_corr);
+
+% Reshape onto mesh
+noise_no_corr = reshape(noise_no_corr,nx,ny);
+noise_corr = reshape(noise_corr,nx,ny);
+
+% Plot
+figure;surf(X,Y,noise_no_corr);title('Independent');
+figure;surf(X,Y,noise_corr); title('Spatial correlation');
\ No newline at end of file