Expand a set of irregularly sampled data points into spherical harmonics using a weighted least squares inversion.
Usage
cilm, chi2 = SHExpandWLSQ (d, w, lat, lon, lmax, [norm, csphase])
Returns
- cilm : float, dimension (2, lmax+1, lmax+1)
- The real spherical harmonic coefficients of the function. The coefficients C0lm and C1lm refer to the cosine (Clm) and sine (Slm) coefficients, respectively, with Clm=cilm[0,l,m] and Slm=cilm[1,l,m].
- chi2 : float
- The residual sum of squares misfit for an overdetermined inversion.
Parameters
- d : float, dimension (nmax)
- The value of the function at the coordinates (lat, lon).
- w : float, dimension (nmax)
- The weights used in the weighted least squares inversion.
- lat : float, dimension (nmax)
- The latitude in DEGREES corresponding to the value in d.
- lon : float, dimension (nmax)
- The longitude in DEGREES corresponding to the value in d.
- lmax : integer
- The maximum spherical harmonic degree of the output coefficients cilm.
- norm : optional, integer, default = 1
- 1 (default) = Geodesy 4-pi normalized harmonics; 2 = Schmidt semi-normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.
- csphase : 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.
Description
SHExpandWLSQ will expand a set of irregularly sampled data points into spherical harmonics by a weighted least squares inversion. For this problem, there must be more data points than spherical harmonic coefficients (i.e., nmax>(lmax+1)**2). It is assumed that each measurement is statistically independent (i.e., the weighting matrix is diagonal), and the inversion is performed using the LAPACK routine DGGGLM.
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.
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