sdwcapupLatvar.m

function [E,V,N,th,C,ngl1,ngl2,unc,com,sdl,K]=sdwcapupLatvar(TH,L,m,nth,vcut,grd,method,rplanet,rsatfun,savename)
% [E,V,N,th,C,ngl1,ngl2,unc,com,sdl,K]=SDWCAPUPLATVAR(TH,L,m,nth,vcut,grd,method,rplanet,rsatfun,savename)
%
% This function is designed for the radial component of the gradient
% at satellite altitude
% with a satellite radial position that varies with latitude, given as
% a function handle rsatfun
%  
% UPWARD CONTINUED AND DERIVED VERSION OF:
% Scalar spherical Slepian function for upward continued spherical hemonics
% for a polar cap.
%
% INPUT:
%
% TH          Angular extent of the spherical cap, in degrees
%             OR: Angles of two spherical caps and we want the ring between 
%             them [TH1 TH2] 
% L           Bandwidth (maximum angular degree), or passband (two degrees)
% m           Angular order of the required data window, -l<m<l
% nth         Number of colatitudes to evaluate (none if 0) [default: 720]
% vcut        Cut-off eigenvalue [default= eps*10]
%             If vcut==-1 automatically recalculates everything
% grd         1 Colatitudes only; returns matrix E
%             2 Colatitude/Longitude; returns cell E
% method      'gl' for Gauss-Legendre via LEGENDREPRODINT [default]
%             'paul' for Paul via PAUL 
% rplanet     planetary radial position
% rsatfun     function handle for satellite radial position depending on cos(colat)
% savename    unique name you want to use for this function handle to
%             save and restore the calculated kernels  
%             on sin(latitude) or cos(colatitude)
%
% OUTPUT:
%
% E           Optimally concentrated tapers, expanded to space domain 
% V           Eigenvalues, sorted
% N           Sum of ALL the eigenvalues (Shannon number for ALL orders)
% th          Colatitudes at which the functions are evaluated, in degrees
% C           Optimally concentrated tapers, spherical harmonics
%             coefficients, normalized to unity on the unit sphere
% ngl1, ngl2  Number of GL points on unit sphere and domain, respectively
% unc         Uncertainty product
% com         Center of mass
% sdl         Spherical harmonics standard deviation
% K           The matrix that is diagonalized to produce the eigenfunctions
%
% See also SDWCAP, GRUNBAUM, KERNELC
%
% Last modified by plattner-at-princeton.edu, 7/12/2018


defval('TH',40)
defval('L',18)
defval('m',0)
defval('nth',720)
defval('vcut',eps*10)
defval('grd',1)
defval('method','gl')
defval('rint',[1 2])

% Work with the absolute value of m
mor=m;
m=abs(m);

if(m>max(L))
  error('Order cannot exceed degree')
end

% Figure out if it's lowpass or bandpass
lp=length(L)==1;
bp=length(L)==2;
maxL=max(L);

% The spherical harmonic dimension
ldim=(L(2-lp)+1)^2-bp*L(1)^2;

% Filename of saved data
dirname=fullfile(getenv('IFILES'),'SDWCAPUPLATVAR');
if length(TH)==1
    if lp
      fnpl=fullfile(dirname,sprintf(...
          'SDWUPLATVAR-%g-%i-%i-%i-%g-%s.mat',TH,L,nth,m,rplanet,savename));
    elseif bp
      fnpl=fullfile(dirname,sprintf(...
          'SDWBLUPLATVAR-%g-%i-%i-%i-%i-%g-%s.mat',TH,L(1),L(2),nth,m,...
          rplanet,savename));
    else
      error('The degree range should be either one or two numbers')
    end
elseif length(TH)==2
    if lp
      fnpl=fullfile(dirname,sprintf(...
          'SDWUPLATVAR-%g-%g-%i-%i-%i-%g-%s.mat',...
          max(TH),min(TH),L,nth,m,rplanet,savename));
    elseif bp
      fnpl=fullfile(dirname,sprintf(...
          'SDWBLUPLATVAR-%g-%g-%i-%i-%i-%i-%g-%s.mat',...
          max(TH),min(TH),L(1),L(2),nth,m,...
          rplanet,savename));
    else
      error('The degree range should be either one or two numbers')
    end
    
else
    error('Bad choice for TH')
end
    

% In 7.0.0.19901 (R14), in very rare cases (*-180-0-0) this
% save was buggy, and could not be subsequently loaded.
% Thus used  ~(L==180 & m==0) as an extra condition, now gone.
% Used matzerofix to remediate this.
if exist(fnpl,'file')==2 && (vcut>0) & 1==3
  load(fnpl)
  disp(sprintf('%s loaded by SDWCAPUPLATVAR',fnpl))
else
  % Get the full Shannon number
  N=ldim*(1-cos(max(TH)/180*pi))/2;
  
  % Approximate Nyquist degree is $\pi/d\theta$
  if nth < (maxL+1) & nth~=0
    error('Sample finer to avoid aliasing')
  end
  
  % Convert to radians
  TH=TH*pi/180;

  % Initialize kernel
  %nKs=length(TH);
  % Order TH by absolute value. First the larger cap, then the smaller one
  if length(TH)==2
    TH=[max(abs(TH)) min(abs(TH))];
  end
  
  %K=zeros(maxL+1-max(m,bp*min(L)),maxL+1-max(m,bp*min(L)));
  for whichK=1:length(TH) % Everything for both K until we get eigenfunctions
  Kst{whichK}=zeros(maxL+1-max(m,bp*min(L)),maxL+1-max(m,bp*min(L)));
  

  % Construct kernel: integral of Ylm over the cap
  lmin=max(m,bp*min(L));
  
%   rtop=max(rint);
%   rbot=min(rint);

  switch method
   case 'gl'
    tic
    for lr=lmin:maxL
      for lc=lr:maxL
        % Orthonormalization of Ylm is to unity over unit sphere
        % When TH=180, K should be the identity matrix
        % The pi*(1+(m==0)) is the longitudinal integral
        % Note that this line ALSO would work with 'paul' but then it
        % would be very inefficient in not reusing the database
        Kst{whichK}(lr+1-lmin,lc+1-lmin)=...
            legendreprodintRadLatvar(lr,m,lc,m,cos(TH(whichK)),method,rplanet,rsatfun)...
            *sqrt(2*lr+1)*sqrt(2*lc+1)/(4*pi)*pi*(1+(m==0));       
        % Symmetrize
        Kst{whichK}(lc+1-lmin,lr+1-lmin)=...
            Kst{whichK}(lr+1-lmin,lc+1-lmin);
      end
    end
    toc
  end
  toc
  end % Until here everything for both K
  if length(TH)==2
    K=Kst{1}-Kst{2};
  else
      K=Kst{1};
  end
  

   
  % In order to avoid numerical desymmetrification:
  K=(K+K')/2;
  
  % Calculate eigenvalues and eigenvectors; C'*C=I
  [C,V]=eig(K);
  
  
  if lp && vcut<=0
    % Check the partial Shannon number
%    difer(sum(diag(V))-indeks(nsubm(N,m,3,L),'end'))
  elseif bp
    disp('Still need to fix NSUBM for bandpass functions')
  end

  % Fill in the missing degrees in case it was bandpass
  C=[zeros(lmin-m,size(C,2)) ; C];


  % Order eigenvalues and eigenfunctions downgoing
  [V,isrt]=sort(sum(V,1),'descend');
  % V=fliplr(V); C=C(:,fliplr(isrt));
  C=C(:,isrt);
%   unc=fliplr(unc); sdl=fliplr(sdl); com=fliplr(com); sdv=fliplr(sdv); 

  % Only return nonzero "useful" eigenvalues
  C=C(:,V>vcut); 
  V=V(V>vcut);

  % Compute spatial functions, colatitudinal part only
  if nth~=0
    % Zonal functions only 
    if m==0
      % Make spatial functions
      % This is SDW (2005) equation (5.10) combined with the sqrt(2-dom) of
      % (5.12) already included!
      [E,th]=pl2th(C,nth,1);
      th=th*180/pi;
      nlon=2*nth-1;
    else
      % This is SDW (2005) equation (5.10) combined with the sqrt(2-dom) of
      % (5.12) already included!
      [E,nlon,lat]=plm2th(C,nth,m,1);
      th=linspace(0,180,size(E,1));
    end    
    % Make E start with a positive lobe and ajust C too
    % Don't take first sample as for m~=0 it is 0
    for index=1:size(E,2)
      C(:,index)=C(:,index)*sign(E(2,index));
      E(:,index)=E(:,index)*sign(E(2,index));
    end
  else
    E=0;
    th=0;
    nlon=0;
  end
  % In 7.0.0.19901 (R14), in very rare cases (3-180-0-0) this
  % save is buggy, and cannot be subsequently loaded
  % Output in degrees
  try
    save(fnpl,...
	       'E','V','L','N','TH','C','th','nlon','K','-v7.3')
  catch
    save(fnpl,...
	       'E','V','L','N','TH','C','th','nlon','K')
  end
end

if nth~=0 & grd==2
  % Output on full grid; watch the sign of m
  % Note that sqrt(2-dom) is already part of the harmonic
  if mor<=0
    EE=E; clear E
    for index=1:size(EE,2)
      E{index}=EE(:,index)*cos(m*linspace(0,2*pi,nlon));
    end
  end
  if mor>0
    EE=E; clear E
    for index=1:size(EE,2)
      E{index}=EE(:,index)*sin(m*linspace(0,2*pi,nlon));
    end
  end
end