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function varargout=xyz2spl(flatlon,lat,lon,latp,lonp,method,pars)
% flatlonp=xyz2spl(flatlon,lat,lon,latp,lonp,method,pars)
%
% Spline interpolation of irregularly sampled spherical functions
%
% INPUT:
%
% fthph Function values at given lat and lon
% lat Given latitudes, in degrees
% lon Given longitudes, in degrees
% latp Desired latitudes, in degrees
% lonp Desired longitudes, in degrees
% method 'abelpoisson' [default]
% pars The parameter, e.g. h for Abel-Poisson [default: 0.9]
%
% OUTPUT:
%
% flatlonp Function values at desired lat and lon
%
% EXAMPLES:
%
% XYZ2SPL('demo1')
% XYZ2SPL('demo2')
%
% Should also do an example with the competing method
%
% Last modified by fjsimons-at-alum.mit.edu, 05/16/2008
defval('flatlon',0)
if ~isstr(flatlon)
% Specify what kind of spline and the parameters involved
defval('lat',0)
defval('lon',0)
defval('method','abelpoisson')
defval('pars',0.90)
if sum(flatlon.*lat.*lon)==0
% Bogus example
[lon,lat]=randsphere(100);
flatlon=rand(100,1);
lonp=lon; latp=lat;
end
% Make sure they are all column vectors
flatlon=flatlon(:);
% So, what we need is the Gram matrix with the relevant spline kernel
% Convert to colatitude and longitude in radians
[TH,THp]=meshgrid((90-lat(:))*pi/180);
[PH,PHp]=meshgrid(lon(:)*pi/180);
% Make the matrix with the epicentral distances of all the combinations
cosDelta=cos(TH).*cos(THp)+sin(TH).*sin(THp).*cos(PH-PHp);
% And of the desired locations with the old locations, so no longer square
[TH,THPp]=meshgrid((90-latp(:))*pi/180,(90-lat(:))*pi/180);
[PH,PHPp]=meshgrid(lonp(:)*pi/180,lon(:)*pi/180);
cosDeltaP=cos(TH).*cos(THPp)+sin(TH).*sin(THPp).*cos(PH-PHPp);
switch method
case 'abelpoisson'
h=pars;
% Calculate the Abel-Poisson kernel for these pairwise points
K=(1-h^2)/4/pi./(1+h^2-2*cosDelta*h).^(3/2);
KP=(1-h^2)/4/pi./(1+h^2-2*cosDeltaP*h).^(3/2);
end
% Now find the expansion coefficients for the spline
ak=pinv(K)*flatlon;
% Return the function interpolated at the desired longitudes etc
% Interpolation to the same points would be identical
flatlonp=[ak(:)'*KP]';
% Provide output
vars={flatlonp};
varargout=vars(1:nargout);
elseif strcmp(flatlon,'demo1')
% Inspired from PLM2XYZ('demo2')
% We make up a function, sample randomly from it, and see how it
% approximates the underlying function while fitting the sampled points
% exactly
lmax=10; L=10;
[m,l,mzero]=addmon(lmax);
c=randn(addmup(lmax),2).*([l l].^(-1));
c(1)=3; c(mzero,2)=0; lmcosi=[l m c];
[r,lon,lat]=plm2xyz(lmcosi,180/sqrt(L*(L+1)));
tol=length(lon)*length(lat);
defval('degres',0.4)
fra=degres;
unform=2;
[LON,LAT]=meshgrid(lon,lat);
% Really uniform on the sphere
[lonr,latr]=randsphere(100);%ceil(fra*tol));
indo=sub2ind(size(r),ceil(scale(latr,[1 length(lat)])),...
ceil(scale(lonr,[1 length(lon)])));
latr=LAT(indo);
% Decide on the width of an Abel-Poisson kernel by choosing h
h=rand(1);
% Now use the function here for r(indo) and reconstruct it on a whole
% fine grid... from lon(INDO) back to lon
flatlonp=xyz2spl(r(indo),LAT(indo),LON(indo),LAT(:),LON(:),...
[],h);
rp=reshape(flatlonp,length(lat),length(lon));
% Predecide on an y-axis range
ylix=1.1*minmax([r(:) rp(:)]);
% Now we verify that the ones that we provided are fit EXACTLY
difer(flatlonp(indo)-r(indo))
% And we judge visually how we've smoothed around the unknown locations
clf
[ah,ha]=krijetem(subnum(2,2));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
axes(ah(1))
% Plot the original function in Cartesian projection
plotplm(r,lon*pi/180,lat*pi/180,4)
% Plot the sample locations
hold on
% [x,y]=mollweide(LON(indo)*pi/180,LAT(indo)*pi/180);
[x,y]=deal(LON(indo),LAT(indo));
p1=plot(x,y,'o');
xl(1)=xlabel('longitude');
yl(1)=ylabel('latitude');
title('Original')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
axes(ah(2))
% Plot the newly interpolated function
plotplm(rp,lon*pi/180,lat*pi/180,4)
% Plot the sample locations
hold on
% [x,y]=mollweide(LON(indo)*pi/180,LAT(indo)*pi/180);
[x,y]=deal(LON(indo),LAT(indo));
p2=plot(x,y,'o');
xl(2)=xlabel('longitude');
title('Interpolated')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
axes(ah(3))
% Plot the selection of data
plot(r(indo),'o-'); hold on
% And their exact fit by interpolation
plot(rp(indo),'r+-'); hold on
axis tight
title('Original and interpolated values [sampled]')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
axes(ah(4))
% Plot the entire data set
plot(r(:),'-'); hold on
% And their inexact approximation by interpolation
plot(rp(:),'r-'); hold on
axis tight
title('Original and interpolated values [all]')
% Cosmetics
longticks(ah)
set(ah(1:2),'clim',minmax([get(ah(1),'clim') get(ah(2),'clim')]))
shrink(ah(3:4),1,1.25)
nolabels(ha(3:4),2)
set(ah(3),'xtick',unique([1 get(ah(3),'xtick')]))
set(ah(4),'xtick',unique([1 get(ah(4),'xtick')]))
set(ah(3:4),'ylim',ylix)
set([p1 p2],'MarkerE','k','MarkerF','k','MarkerS',2)
movev(ah(1:2),-.1)
supertit(ah(1:2),...
sprintf('Abel-Poisson interpolation with h = %4.2f',h))
fig2print(gcf,'landscape')
elseif strcmp(flatlon,'demo2')
% We make up a function, sample randomly from it, and see how it
% approximates the underlying function while fitting the sampled points
% exactly
lmax=20;
[m,l,mzero]=addmon(lmax);
c=randn(addmup(lmax),2).*([l l].^(-1));
c(1)=0; c(mzero,2)=0; lmcosi=[l m c];
[r,lon,lat]=plm2xyz(lmcosi,1);
tol=length(lon)*length(lat);
defval('degres',0.4)
fra=degres;
unform=2;
[LON,LAT]=meshgrid(lon,lat);
% Really uniform on the sphere
[lonr,latr]=randsphere(50);%ceil(fra*tol));
indo=sub2ind(size(r),ceil(scale(latr,[1 length(lat)])),...
ceil(scale(lonr,[1 length(lon)])));
latr=LAT(indo);
% Decide on the width of an Abel-Poisson kernel by choosing h
h=rand(3);
h=[0.50 0.65 0.80];
% And we judge visually how we've smoothed around the unknown locations
clf
[ah,ha]=krijetem(subnum(2,2));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
axes(ah(1))
% Plot the original function in Cartesian projection
plotplm(r,lon*pi/180,lat*pi/180,4)
% Plot the sample locations
hold on
% [x,y]=mollweide(LON(indo)*pi/180,LAT(indo)*pi/180);
[x,y]=deal(LON(indo),LAT(indo));
p(1)=plot(x,y,'o');
xl(1)=xlabel('longitude');
yl(1)=ylabel('latitude');
title('Original')
% Do the different kinds of interpolation
for in=1:length(h)
% Now use the function here for r(indo) and reconstruct it on a whole
% fine grid... from lon(INDO) back to lon
flatlonp=xyz2spl(r(indo),LAT(indo),LON(indo),LAT(:),LON(:),...
[],h(in));
rp=reshape(flatlonp,length(lat),length(lon));
% Now we verify that the ones that we provided are fit EXACTLY
difer(flatlonp(indo)-r(indo))
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
axes(ah(1+in))
% Plot the newly interpolated function
plotplm(rp,lon*pi/180,lat*pi/180,4)
% Plot the sample locations
hold on
% [x,y]=mollweide(LON(indo)*pi/180,LAT(indo)*pi/180);
[x,y]=deal(LON(indo),LAT(indo));
p(in+1)=plot(x,y,'o');
xl(1+in)=xlabel('longitude');
yl(1+in)=ylabel('latitude');
title(sprintf('Abel-Poisson interpolation with h = %4.2f',h(in)))
end
% Cosmetics
longticks(ah)
% seemax(ah,3)
set(ah,'clim',minmax(r(:)))
nolabels(ha(3:4),2)
delete(yl([2 4]))
set(p,'MarkerE','k','MarkerF','w','MarkerS',4)
movev(ah(1:2),-.125)
movev(ah,.1)
fig2print(gcf,'landscape')
cb=colorbarf('hor',12,'Helvetica',[0.5703 0.1100 0.3347 0.0256]);
layout(cb,0.5,'m','x')
movev(cb,0.05); longticks(cb);
kelicol
end
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