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## Copyright (C) 2007 Aida Alvera-Azcárate <aalvera@marine.usf.edu>
## Copyright (C) 2007, 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}] = } optiminterp4(@var{x},@var{y},@var{z},@var{t},@var{f},@var{var},@var{lenx},@var{leny},@var{lenz},@var{lent},@var{m},@var{xi},@var{yi},@var{zi},@var{ti})
## Performs a local 4D-optimal interpolation (objective analysis).
##
## Every elements in @var{f} corresponds to a data point (observation)
## at location @var{x}, @var{y}, @var{z}, @var{t} with the error variance var
##
## @var{lenx},@var{leny},@var{lenz} and @var{lent} are correlation length in x-,y-,z-direction and time,
## respectively.
## @var{m} represents the number of influential points.
##
## @var{xi},@var{yi},@var{zi} and @var{ti} 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] = optiminterp4(x,y,z,t,f,var,lenx,leny,lenz,lent,m,xi,yi,zi,ti)
[fi,vari] = optiminterpn(x,y,z,t,f,var,lenx,leny,lenz,lent,m,xi,yi,zi,ti);
endfunction
%!test
%! # grid of background field
%! [xi,yi,zi,ti] = ndgrid(linspace(0,1,5));
%! fi_ref = sin(6*xi) .* cos(6*yi) .* sin(6*zi) .* cos(6*ti);
%!
%! # grid of observations
%! [x,y,z,t] = ndgrid(linspace(0,1,10));
%! x = x(:);
%! y = y(:);
%! z = z(:);
%! t = t(:);
%!
%! on = numel(x);
%! var = 0.01 * ones(on,1);
%! f = sin(6*x) .* cos(6*y) .* sin(6*z) .* cos(6*t);
%!
%! m = 20;
%!
%! [fi,vari] = optiminterp4(x,y,z,t,f,var,0.1,0.1,0.1,0.1,m,xi,yi,zi,ti);
%!
%! rms = sqrt(mean((fi_ref(:) - fi(:)).^2));
%!
%! assert (rms <= 0.04, "unexpected large difference with reference field");
%!test
%! # grid of background field
%! [xi,yi,zi,ti] = ndgrid(linspace(0,1,5));
%!
%! fi_ref(:,:,:,:,1) = sin(6*xi) .* cos(6*yi) .* sin(6*zi) .* cos(6*ti);
%! fi_ref(:,:,:,:,2) = cos(6*xi) .* sin(6*yi) .* cos(6*zi) .* sin(6*ti);
%!
%! # grid of observations
%! [x,y,z,t] = ndgrid(linspace(0,1,10));
%! x = x(:);
%! y = y(:);
%! z = z(:);
%! t = t(:);
%!
%! on = numel(x);
%! var = 0.01 * ones(on,1);
%! f(:,:,:,:,1) = sin(6*x) .* cos(6*y) .* sin(6*z) .* cos(6*t);
%! f(:,:,:,:,2) = cos(6*x) .* sin(6*y) .* cos(6*z) .* sin(6*t);
%!
%! m = 20;
%!
%! [fi,vari] = optiminterp4(x,y,z,t,f,var,0.1,0.1,0.1,0.1,m,xi,yi,zi,ti);
%!
%! rms = sqrt(mean((fi_ref(:) - fi(:)).^2));
%!
%! assert (rms <= 0.04, "unexpected large difference with reference field");
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