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## Copyright (C) 2006-2018 Alexander Barth <barth.alexander@gmail.com>
##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Loadable Function} {[@var{fi},@var{vari}] = } optiminterp2(@var{x},@var{y},@var{f},@var{var},@var{lenx},@var{leny},@var{m},@var{xi},@var{yi})
## Performs a local 2D-optimal interpolation (objective analysis).
##
## Every elements in @var{f} corresponds to a data point (observation)
## at location @var{x},@var{y} with the error variance @var{var}.
##
## @var{lenx} and @var{leny} are correlation length in x-direction
## and y-direction respectively.
## @var{m} represents the number of influential points.
##
## @var{xi} and @var{yi} are the data points where the field is
## interpolated. @var{fi} is the interpolated field and @var{vari} is
## its error variance.
##
## The background field of the optimal interpolation is zero.
## For a different background field, the background field
## must be subtracted from the observation, the difference
## is mapped by OI onto the background grid and finally the
## background is added back to the interpolated field.
## The error variance of the background field is assumed to
## have a error variance of one.
## @end deftypefn
function [fi,vari] = optiminterp2(x,y,f,var,lenx,leny,m,xi,yi)
[fi,vari] = optiminterpn(x,y,f,var,lenx,leny,m,xi,yi);
endfunction
%!test
%! # grid of background field
%! [xi,yi] = ndgrid(linspace(0,1,30));
%! fi_ref = sin( xi*6 ) .* cos( yi*6);
%!
%! # grid of observations
%! [x,y] = ndgrid(linspace(0,1,20));
%! x = x(:);
%! y = y(:);
%!
%! on = numel(x);
%! var = 0.01 * ones(on,1);
%! f = sin( x*6 ) .* cos( y*6);
%!
%! m = 30;
%!
%! [fi,vari] = optiminterp2(x,y,f,var,0.1,0.1,m,xi,yi);
%!
%! rms = sqrt(mean((fi_ref(:) - fi(:)).^2));
%!
%! assert (rms <= 0.005, "unexpected large difference with reference field");
%!test
%! # grid of background field
%! [xi,yi] = ndgrid(linspace(0,1,30));
%!
%! fi_ref(:,:,1) = sin( xi*6 ) .* cos( yi*6);
%! fi_ref(:,:,2) = cos( xi*6 ) .* sin( yi*6);
%!
%! # grid of observations
%! [x,y] = ndgrid(linspace(0,1,20));
%!
%! on = numel(x);
%! var = 0.01 * ones(on,1);
%! f(:,:,1) = sin( x*6 ) .* cos( y*6);
%! f(:,:,2) = cos( x*6 ) .* sin( y*6);
%!
%! m = 30;
%!
%! [fi,vari] = optiminterp2(x,y,f,var,0.1,0.1,m,xi,yi);
%!
%! rms = sqrt(mean((fi_ref(:) - fi(:)).^2));
%!
%! assert (rms <= 0.005, "unexpected large difference with reference field");
%!test
%! # grid of background field
%! [xi,yi] = ndgrid(linspace(0,1,30));
%! fi_ref = sin( xi*6 ) .* cos( yi*6);
%!
%! # grid of observations
%! [x,y] = ndgrid(linspace(0,1,6));
%! x = x(:);
%! y = y(:);
%!
%! on = numel(x);
%! var = 0.01 * ones(on,1);
%! f = sin( x*6 ) .* cos( y*6);
%!
%! len = 0.1;
%! m = min(30,on);
%!
%! # covariance function
%! # gaussian
%! bcovar2 = @(d2) exp(-d2/len^2) ;
%! # diva
%! #bcovar2 = @(d2) max(sqrt(d2)/len,eps) .* besselk(1,max(sqrt(d2)/len,eps));
%!
%! # P: covariance between grid points (xi,yi) and grid points (xi,yi)
%! P = zeros(numel(xi),numel(xi));
%!
%! for j=1:numel(xi)
%! for i=1:numel(xi)
%! P(i,j) = (xi(i) - xi(j))^2 + (yi(i) - yi(j))^2;
%! end
%! end
%! P = bcovar2(P);
%!
%! # HPH: covariance between observation points (x,y) and observation points (x,y)
%! HPH = zeros(numel(x),numel(x));
%!
%! for j=1:numel(x)
%! for i=1:numel(x)
%! HPH(i,j) = (x(i) - x(j))^2 + (y(i) - y(j))^2;
%! end
%! end
%! HPH = bcovar2(HPH);
%!
%! # PH: covariance between grid points (xi,yi) and observation points (x,y)
%! PH = zeros(numel(xi),numel(x));
%!
%! for j=1:numel(x)
%! for i=1:numel(xi)
%! PH(i,j) = (xi(i) - x(j))^2 + (yi(i) - y(j))^2;
%! end
%! end
%! PH = bcovar2(PH);
%!
%! R = diag(var);
%!
%! # call optiminterp
%! [fi,vari] = optiminterp2(x,y,f,var,len,len,m,xi,yi);
%!
%! # Kalman gain
%! K = PH * inv(HPH + R);
%!
%! # analysis
%! fi2 = K * f;
%!
%! # error field
%! vari2 = diag(P - K * PH');
%!
%! # transform vectors into 2d-arrays
%! fi2 = reshape(fi2,size(fi));
%! vari2 = reshape(vari2,size(fi));
%!
%! rms = sqrt(mean((fi2(:) - fi(:)).^2));
%!
%! assert (rms <= 1e-4, "unexpected large RMS difference (analysis)");
%!
%! rms = sqrt(mean((vari2(:) - vari(:)).^2));
%!
%! assert (rms <= 1e-4, "unexpected large RMS difference (error field)");
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