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SHExpandLSQ
Expand a set of irregularly sampled data points into spherical harmonics using a least squares inversion.
Usage
call SHExpandLSQ (cilm
, d
, lat
, lon
, nmax
, lmax
, norm
, chi2
, csphase
, exitstatus
)
Parameters
cilm
: output, real*8, dimension (2,lmax
+1,lmax
+1)- The real spherical harmonic coefficients of the function. The coefficients
C1lm
andC2lm
refer to the cosine (Clm
) and sine (Slm
) coefficients, respectively, withClm=cilm(1,l+1,m+1)
andSlm=cilm(2,l+1,m+1)
. d
: input, real*8, dimension (nmax
)- The value of the function at the coordinates (
lat
,lon
). lat
: input, real*8, dimension (nmax
)- The latitude in DEGREES corresponding to the value in
d
. lon
: input, real*8, dimension (nmax
)- The longitude in DEGREES corresponding to the value in
d
. nmax
: input, integer- The number of data points.
lmax
: input, integer- The maximum spherical harmonic degree of the output coefficients
cilm
. norm
: input, optional, integer, default = 1- 1 (default) = Geodesy 4-pi normalized harmonics; 2 = Schmidt semi-normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.
chi2
: output, optional, real*8- The residual sum of squares misfit for an overdetermined inversion.
csphase
: input, optional, integer, default = 1- 1 (default) = do not apply the Condon-Shortley phase factor to the associated Legendre functions; -1 = append the Condon-Shortley phase factor of (-1)^m to the associated Legendre functions.
exitstatus
: output, optional, integer- If present, instead of executing a STOP when an error is encountered, the variable exitstatus will be returned describing the error. 0 = No errors; 1 = Improper dimensions of input array; 2 = Improper bounds for input variable; 3 = Error allocating memory; 4 = File IO error.
Description
SHExpandLSQ
will expand a set of irregularly sampled data points into spherical harmonics by a least squares inversion. When there are more data points than spherical harmonic coefficients (i.e., nmax>(lmax+1)**2
), the solution of the overdetermined system will be determined. If there are more coefficients than data points, then the solution of the underdetermined system that minimizes the solution norm will be determined. See the LAPACK documentation concerning DGELS for further information.
The employed spherical harmonic normalization and Condon-Shortley phase convention can be set by the optional arguments norm
and csphase
; if not set, the default is to use geodesy 4-pi normalized harmonics that exclude the Condon-Shortley phase of (-1)^m.
See also
makegriddh
, shexpanddh
, makegriddhc
, shexpanddhc
, makegridglq
, shexpandglq
, makegridglqc
, shexpandglqc
, dgels(1)
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