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MianLT
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MianLT
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%% this code is simulation of Lorenz96 using Runge-Kutta method
% this code together with Lorenz96RK_LT.m file are dedicated for
% <Data Assimilation> Class as a final project test withou any
% conflict of interest; Supervisor: GuoCan Wu, GCESS of BNU
% Author: Teng Li@20160702; Contact: [email protected]
%% construct initial value and parameter set
clear;
global nx dt;
nx = 40; % 40 variable distributed in a cycle
dt = 0.05; % time steps to integral in R_K function
Ftrue = 8; % to generate true state
Ffore = 6; % to forecast in assimilation
size = 30; % ensemble size in EnKF filter
initX = zeros(40, 1);
initX(1:40) = Ftrue; % initial condition
initX(20) = 1.001*Ftrue;
pertb = 0.2; % forecast pertubation's std
obseError = 1; % observation error's std
finalTime = 2000; % the last time step 2000
obseFreq = 1/4; % observe once for every 4 forecast
Rspatial = zeros(40); % spatial correlation matrix to localization
for ii = 1:40
for jj = 1:40
tt = min(abs(ii-jj),40 - abs(ii-jj));
Rspatial(ii,jj) = 0.5^tt;
end
end
clear ii jj tt
% imagesc(Rspatial)
trueState = zeros(nx, finalTime);
foreEnse = zeros(nx, finalTime, size);
foreEnseInf = foreEnse;
foreMean = trueState;
foreMeanInf = trueState;
obse = zeros(nx, finalTime*obseFreq);
assiEnse = zeros(nx, finalTime*obseFreq, size);
assiEnseInf = assiEnse;
P = zeros(nx, nx, finalTime*obseFreq);
PInf = P;
diff = obse;
diffij = assiEnse;
diffInf = obse;
diffijInf = assiEnse;
L = zeros(2, finalTime*obseFreq); % Max Likelihood with lamda = 1 or min
lamdaMin = zeros(500,1);
%% constuct true state and observation with error;
% 1st method to generate obseError, plus RspatialRoot*obseError*randn(40,1)
% separately every obseTime, Matrix Rooting using SVD of lamda rooting.
% RspatialRoot = sqrtm(Rspatial);
% 2nd method to generate obseError, plus multi-normal random matrix
% holistically, MengLiu told me he used normrnd function?
obseErrorAll = mvnrnd(zeros(1,40),Rspatial,500)';
for time = 1:finalTime
if time == 1
trueState(:,time) = LorenzRK_LT(Ftrue, initX);
else
trueState(:,time) = LorenzRK_LT(Ftrue,trueState(:,time-1));
if mod(time, 4) == 0
obse(:,time*obseFreq) = trueState(:, time) + obseErrorAll(:, time*obseFreq);
end
end
end
%% EnKF iteration begin! 1 st part is about forecast without assimilation
for time = 1:finalTime
if time == 1
% forePertb = pertb*randn(nx, size);
% foreTemp = LorenzRK_LT(Ffore, initX)';
foreEnse(:,time,:) = bsxfun(@plus,pertb*randn(nx, size), LorenzRK_LT(Ffore, initX));
foreMean(:,time) = mean(squeeze(foreEnse(:,time,:)),2);
foreEnseInf(:,time, :) = foreEnse(:,time,:);
foreMeanInf(:,time) = foreMean(:,time);
else
foreEnse(:,time,:) = bsxfun(@plus,pertb * randn(nx, size), ...
LorenzRK_LT(Ffore, foreMean(:, time-1)));
foreMean(:,time) = mean(squeeze(foreEnse(:,time,:)),2);
foreEnseInf(:, time, :) = bsxfun(@plus,pertb * randn(nx, size), ...
LorenzRK_LT(Ffore, foreMeanInf(:, time-1)));
foreMeanInf(:, time) = mean(squeeze(foreEnseInf(:,time,:)),2);
% these disp()s are for debugging, which can be deleted them if you want.
% disp(time);
% disp(norm(trueState(:,time) - foreMean(:,time)));
% disp(norm(trueState(:,time) - foreMeanInf(:,time)));
% disp(norm(squeeze(foreEnseInf(:,time, :) - foreEnse(:,time,:))));
%% Here comes the true essense of EnKF：2nd part is inplementation of diagram.
if mod(time, 4) == 0
diff(:,time*obseFreq) = obse(:,time*obseFreq) - foreMean(:,time);
diffij(:,time*obseFreq,:) = bsxfun(@minus,obse(:,time*obseFreq),...
squeeze(foreEnse(:, time,:))) + obseError*randn(40, 30);
diffInf(:,time*obseFreq) = obse(:,time*obseFreq) - foreMeanInf(:,time);
diffijInf(:,time*obseFreq,:) = bsxfun(@minus,obse(:,time*obseFreq),...
squeeze(foreEnseInf(:, time,:))) + obseError*randn(40, 30);
foreError = bsxfun(@minus, squeeze(foreEnse(:,time,:)),foreMean(:, time));
P(:,:,time*obseFreq) = foreError * foreError'/ (size -1);
foreErrorInf = bsxfun(@minus, squeeze(foreEnseInf(:,time,:)),foreMeanInf(:, time));
PInf(:,:,time*obseFreq) = foreErrorInf *foreErrorInf' / (size-1);
%% without inflation which means lamda = 1
L(1, time*obseFreq) = log(det(P(:,:,time*obseFreq) + Rspatial))...
+ diff(:,time*obseFreq)'* inv(P(:,:,time*obseFreq) + Rspatial)*diff(:,time*obseFreq);
temp1 = P(:,:,time*obseFreq);
assiEnse(:,time*obseFreq,:) = squeeze(foreEnse(:,time,:)) + ...
temp1 * inv(temp1 +Rspatial) * squeeze(diffij(:,time*obseFreq,:));
foreMean(:,time) = mean(squeeze(assiEnse(:,time*obseFreq,:)),2);
% with inflation requite optimize lamda
FunOptm = @(lamda)(log(det(lamda * PInf(:,:,time*obseFreq) + Rspatial)) + ...
diffInf(:,time*obseFreq)' * inv(lamda * PInf(:,:,time*obseFreq) + Rspatial) * diffInf(:,time*obseFreq));
[lamdaMin(time*obseFreq), L(2, time*obseFreq)] = fminbnd(FunOptm, 0,100);
% Question: how can i define the search inteval of optimazed
% function to find the exact lamda without priori knowledge?
temp2 = lamdaMin(time*obseFreq) * PInf(:,:,time*obseFreq);
assiEnseInf(:, time*obseFreq,:) = squeeze(foreEnseInf(:,time,:)) + ...
temp2 * inv(temp2 + Rspatial) * squeeze(diffijInf(:, time*obseFreq,:));
foreMeanInf(:,time) = mean(squeeze(assiEnseInf(:,time*obseFreq,:)),2);
% temp2 - temp1
end
end
end
%% validation RMSE and plot
rmse1 = sqrt(sum((trueState - foreMean).^2)/nx);
rmse2 = sqrt(sum((trueState - foreMeanInf).^2)/nx);
plot(rmse1)
hold on
plot(rmse2)
script1 = ['Mean RMSE of EnKF with inflation is ',num2str(mean(rmse2))];
script2 = ['comparing with that of ',num2str(mean(rmse1)),' without inflation'];
text(300,8,script1);
text(300,7.5,script2)
xlabel('TimeSteps from 1 to 2000')
ylabel('RMSE')
title('EnKalman Filter RMSE with & without inflation')
legend('No-Inf','With Inf')
figure(2)
plot(L(1,:))
% semilogy(L(1,:));
hold on
plot(L(2,:));
% semilogy(L(2,:));
script1 = ['Mean Likelihood of EnKF with inflation is ',num2str(mean(L(2,:)))];
script2 = ['comparing with that of ', num2str(mean(L(1,:))), 'without inflation'];
text(70,2700,script1);
text(70,2500,script2)
xlabel('TimeSteps from 1 to 500')
ylabel('Max Likelihood')
title('-2*ln(Likelihood) with & wiouth inflation')
legend('No-Inf','With Inf')