# csdms-contrib/slepian_delta

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 function varargout=plm2avg(lmcosi,dom) % [Int,A,miniK,XY]=PLM2AVG(lmcosi,dom) % % Computes the integral and average value of a spherical harmonic % function (lmcosi) within a specific area (dom). This is done by % creating an integration vector and multiplying by the unwrapped % coefficients of the field (from lmcosi). % % INPUT: % % lmcosi Standard-type real spherical harmonic expansion coefficients % % dom A string name with an approved region such as 'africa', OR % XY its coordinates (such as supplied from 'africa' etc) % % OUTPUT: % % Int Integral of the function within the region % A Average value of the function within the region % miniK The integration vector (also the first row/column from Klmlmp in KERNELC) % XY The coordinates of the region we integrated over % % % EXAMPLE: % % plm2avg('demo1') Integrate the output from GEOBOXCAP over a region and % see that you get something close to the region's area % plm2avg('demo2') Compare this integration to summing the field which has % been expanded onto an equal area Fibonacci grid % % SEE ALSO: % KERNELC, SPHAREA, FIBONACCI_GRID % % Last modified by charig-at-princeton.edu, 06/27/2012 defval('dom','greenland') defval('lmcosi',[0 0 0 0]) if ~isstr(lmcosi) % Highest degree (bandwidth of the expansion) Lmax=lmcosi(end,1); % Get the coordinates if isstr(dom) % Specify the region by name defval('pars',10); eval(sprintf('XY=%s(%i);',dom,pars)); else % Specify the region by the coordinates of the bounding curve XY=dom; end % Calculate northernmost and southernmost colatitude thN=90-max(XY(:,2)); thN=thN*pi/180; thS=90-min(XY(:,2)); thS=thS*pi/180; % Introduce and dimensionalize variables and arrays [dems,dels,mz,lmc,mzi,mzo,bigm,bigl,rinm,ronm]=addmon(Lmax); dimK=(Lmax+1)^2; lenm=length(dems); % Calculate some Gauss-Legendre points intv=cos([thS thN]); % Degree of Gauss-Legendre integration ngl=200; nGL=min(ngl,size(XY,1)/2); % These are going to be the required colatitudes - forget XY [w,x,N]=gausslegendrecof(nGL,[],intv); disp(sprintf('%i Gauss-Legendre points and weights calculated',N)) % Some arrays needed in a bit % Note that dimK==sum(dubs) dubs=repmat(2,lenm,1); dubs(mz)=1; comdi=[]; % Calculate all the Legendre functions themselves Xlm=repmat(NaN,length(x),lenm); ind=0; for l=0:Lmax Xlm(:,ind+1:ind+l+1)=[legendre(l,x(:)','sch')*sqrt(2*l+1)]'; ind=ind+l+1; end %disp('Legendre functions calculated') % No need for this to be longer, or to loop over anything % Normally it would help take out the first dimension of redundancy between % XlmXlmp and Klmlmp comdi=dubs'; % In our ordering, the -1 precedes 1 and stands for the cosine term % This creates an indexing vector into Xlm that enlarges it from [0 01 012] % to [0 0-11 0-11-22] % Since we only have Xlm and not XlmXlm, we stop with coss. In Kernelc, % this was again expanded into bigo to account for this redundancy in two % dimensions comdex=[1:lenm]; coss=gamini(comdex,comdi); bigo=coss; % Get the longitudinal integration info for the domain defval('Nk',10); % Now we may have multiple pairs phint=dphregion(acos(x)*180/pi,Nk,dom); phint=phint*pi/180; % No need to initialize miniK since we can make it all at once % Calculate the longitudinal integrals for each l,m combination for lm1dex=1:dimK m1=bigm(lm1dex); if m1<=0 I(:,lm1dex)=coscos(acos(x),m1,0,phint); elseif m1>0 I(:,lm1dex)=sincos(acos(x),m1,0,phint); end end % coscos was reused here even though one of the m values is 0 because it is % useful for doing odd geographics in matrix form (i.e. if the continent % looks like a circle with hole in it) % Make miniK, really the first row/column of Klmlmp from KERNELC miniK = w(:)'*(Xlm(:,bigo).*I); % To make this exactly equivalent to Tony's \ylm, i.e. undo what we % did above here, taking the output of YLM and multiplying miniK=miniK/4/pi; % Compare with KERNELC defval('xver',0) if xver==1 KK=rindeks(kernelc(Lmax,dom),1); end % This then makes miniK(1) the fractional area on the sphere % Check by comparing to output from spharea, if you want % Note: SPHAREA defaults to 17 abcissas and weights, while this code uses 101. % So differences to be expected when continents are squiggly. % Check the first term which should equal the area on the unit sphere % A1=spharea(XY); A2=areaint(XY(:,2),XY(:,1)); %disp(sprintf('Area check... PLM2AVG A: %6.7f ; SPHAREA A: %6.7f ; AREAINT A: %6.7f',... % miniK(1),A1,A2)) % Now take the vector miniK and multiply with the vector for the function % you want (lmcosi) and you will get the integral of that field within the % region of interest thecofs=lmcosi(:,3:4); theINT=miniK*thecofs(mzo); % To get the average value of the function in the region, divide by the % area, which is likely most accurate in miniK (i.e. more accurate than SPHAREA) A=theINT/miniK(1); % Provide output where requested varns={theINT,A,miniK,XY}; varargout=varns(1:nargout); elseif strcmp(lmcosi,'demo1') % Integrate the output from GEOBOXCAP which puts 1 in the region and 0 % elsewhere. This should integrate to the area of the region. In % practice, since the SH expansion of the GEOBOXCAP mask has rings, this % comparison has a good amount of error. % This is FRACTIONAL area on the unit sphere. dom='australia'; L=40; degres=[]; [Bl,dels,r,lon,lat,lmcosi]=geoboxcap(L,dom,[],degres,1); [Int,A,miniK,XY]=plm2avg(lmcosi,dom); maps=plotplm(lmcosi,[],[],4,degres); colorbar % Really dumb thing %sum(maps(:))*2*pi/size(maps,2)*pi/size(maps,1)/4/pi disp(' '); A1=spharea(XY); disp('Integration check... This should equal the area of the region with error mostly from GEOBOXCAP \n'); disp(sprintf('PLM2AVG Int: %6.7f ; SPHAREA A: %6.7f ; ERROR: %6.2f%%',... Int,spharea(XY),(spharea(XY)-Int)/spharea(XY)*100)) % Provide output where requested varns={Int,A,miniK,XY}; varargout=varns(1:nargout); elseif strcmp(lmcosi,'demo2') % Test the integration against expanding on a Fibonacci grid and summing dom='australia'; Lmax=20; pars=10; disp(['Testing integration of EGM2008 (Lmax=' num2str(Lmax) ') over ' dom]); % Get coordinates eval(sprintf('XY=%s(%i);',dom,pars)); % Make sure the coordinates make sense XY(:,1)=XY(:,1)-360*[XY(:,1)>360]; % Use the first Lmax data from EGM2008 as a field of interest v=fralmanac('EGM2008_ZeroTide','SHM'); % Note that gravity does not start at zero % Geoid = 3 % Free-air gravity anomaly = 2 v=plm2pot(v(1:addmup(Lmax)-addmup(v(1)-1),:),[],[],[],3); % Create a Fibonacci grid [lonF,latF] = Fibonacci_grid(30000); % Expand EGM2008 on the Fib Grid [rF,lon,lat,Plm] = plm2xyz(v(:,1:4),latF,lonF); % Now decide if we're inside or outside of the region, and set outside % points to zero rF(~inpolygon(lonF,latF,XY(:,1),XY(:,2)))=0; % Now compute the integration which can be % discretized due to the equal area Fibonacci grid IntF = sum(rF)/length(rF); % Do the same integration with plm2avg [Int,A]=plm2avg(v(:,1:4),dom); % Now check the averages. indx = find(rF); AF=sum(rF(indx))/length(indx); disp(' '); disp('Integration check... PLM2AVG should equal the Fib Grid for good resolution'); disp(sprintf('PLM2AVG Int: %6.7f ; FIB GRID Int: %6.7f ; ERROR: %6.2f%%',... Int,IntF,(Int-IntF)/Int*100)) disp(' '); disp('Avg value check... PLM2AVG should equal the Fib Grid for good resolution'); disp(sprintf('PLM2AVG Avg: %6.7f ; FIB GRID Avg: %6.7f ; ERROR: %6.2f%%',... A,AF,(A-AF)/A*100)) end