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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}] = } optiminterp3(@var{x},@var{y},@var{z},@var{f},@var{var},@var{lenx},@var{leny},@var{lenz},@var{m},@var{xi},@var{yi},@var{zi})
## Performs a local 3D-optimal interpolation (objective analysis).
##
## Every elements in @var{f} corresponds to a data point (observation)
## at location @var{x}, @var{y}, @var{z} with the error variance var
##
## @var{lenx},@var{leny} and @var{lenz} are correlation length in x-,y- and z-direction
## respectively.
## @var{m} represents the number of influential points.
##
## @var{xi},@var{yi} and @var{zi} 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] = optiminterp3(x,y,z,f,var,lenx,leny,lenz,m,xi,yi,zi)
[fi,vari] = optiminterpn(x,y,z,f,var,lenx,leny,lenz,m,xi,yi,zi);
endfunction
%!test
%! # grid of background field
%! [xi,yi,zi] = ndgrid(linspace(0,1,15));
%! fi_ref = sin(6*xi) .* cos(6*yi) .* sin(6*zi);
%!
%! # grid of observations
%! [x,y,z] = ndgrid(linspace(0,1,10));
%! x = x(:);
%! y = y(:);
%! z = z(:);
%!
%! on = numel(x);
%! var = 0.01 * ones(on,1);
%! f = sin(6*x) .* cos(6*y) .* sin(6*z);
%!
%! m = 20;
%!
%! [fi,vari] = optiminterp3(x,y,z,f,var,0.1,0.1,0.1,m,xi,yi,zi);
%!
%! 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] = ndgrid(linspace(0,1,15));
%!
%! fi_ref(:,:,:,1) = sin(6*xi) .* cos(6*yi) .* sin(6*zi);
%! fi_ref(:,:,:,2) = cos(6*xi) .* sin(6*yi) .* cos(6*zi);
%!
%! # grid of observations
%! [x,y,z] = ndgrid(linspace(0,1,10));
%!
%! on = numel(x);
%! var = 0.01 * ones(on,1);
%! f(:,:,:,1) = sin(6*x) .* cos(6*y) .* sin(6*z);
%! f(:,:,:,2) = cos(6*x) .* sin(6*y) .* cos(6*z);
%!
%! m = 20;
%!
%! [fi,vari] = optiminterp3(x,y,z,f,var,0.1,0.1,0.1,m,xi,yi,zi);
%!
%! rms = sqrt(mean((fi_ref(:) - fi(:)).^2));
%!
%! assert (rms <= 0.04, "'unexpected large difference with reference field");
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