%function [Sig,Omg,tau,Sigc,Omgc,sdT,sdS,corTS]= build_cov(sdT,sdS,corTS,tau);
%function [Sig,Sigc,Omg,Omgc,sdT,sdS,corTS,tau]= build_cov(sdT,sdS,corTS,tau);
%function [Sig,Sigc,Omg,Omgc,sdT,sdS,corTS,tau]= build_cov;
%
% INPUTS
%   SDT    is/are the standard deviation(s) for TEMPERATURE
%   SDS    is/are the standard deviation(s) for FUEL SUPPLY
%   CORTS  is the correlation for the kronecker product (must be  -1< x < 1)
%   TAU    observation variance = tau*(Ensemble variance).
%
% OUTPUTS:  SAVED IN INIT.mat (append!)
%   Sig, Sigc     Variance of Observations (and chol. decomp)
%                 tau*diag(Omg)
%   Omg, Omgc     Prior Variance (and chol. decomp)
%   sdT           input value (sd. devs for Temp: default = 20)
%   sdS           input value (sd. devs. for Supply: default = 0.1)
%   corTS         input value (correlation between Temp and Supply)
%                 (default to -.25)
%   tau           Variance inflation factor
%

if(~exist('sdT','var'));   sdT   = 20;   1, end
if(~exist('sdS','var'));   sdS   = .1;    end
if(~exist('corTS','var')); corTS = -.25;  end
if(~exist('tau','var'));   tau   =  1;    end; 
if(~exist('rng','var'));   rng   = .2;    end; 
global PARAMS;

 % Correlation and Covariance for ENSEMBLE and DATA
m    = PARAMS.mesh.m;
msh  = PARAMS.mesh.x;
dum  = msh*ones(1,m);
cmat = exp(-abs(dum-dum')./rng);   % Correlation Matrix over space
cmat(1:5,1:5)

if(length(sdT)==1)                % Covariance of TEMP 
  sigT = sdT^2*cmat;
  crss = sdT*cmat;
else  
  sigT = diag(sdT)*cmat*diag(sdT);
  crss = diag(sdT)*cmat;
end

if(length(sdS)==1)                % Covariance of SUPPLY
  sigS = sdS^2*cmat;
  crss = crss*sigS;
else  
  sigS = diag(sdS)*cmat*diag(sdS);
  crss = crss*diag(sdS);
end

% kronecker products preserve Positive Definite-ness.
Omg  = [[sigT,corTS*crss];[corTS*crss',sigS]]; 
Omgc = chol(Omg)';

Sig  = diag(tau*Omg);             % Observation Variances
Sigc = diag(sqrt(Sig));           % Standard deviations for observations

save INIT Sig Sigc Omg Omgc sdT sdS corTS tau -append;

clear crss dum cmat;