shlsq()

Determine the spherical harmonic coefficients of an irregularly sampled function using a (weighted) least squares inversion, optionally with a precomputed data kernel matrix.

Usage

coeffs, chi2 = shlsq (data, latitude, longitude, lmax, [weights, g, normalization, csphase, kind, degrees])

Returns

coeffs : float, dimension (2, lmax+1, lmax+1)

The real spherical harmonic coefficients of the function, where coeffs[0, :, :] and coeffs[1, :, :] refer to the cosine and sine terms, respectively.

chi2 : float

The (weighted) residual sum of squares misfit.

Parameters

data : float, dimension (nmax)

The value of the function at the latitude and longitude coordinates.

latitude : float, dimension (nmax)

The latitude in degrees of the data.

longitude : float, dimension (nmax)

The longitude in degrees of the data.

lmax : integer

The maximum spherical harmonic degree of the output coefficients.

weights : float, dimension (nmax)

The weights used for a weighted least squares inversion.

g : float, dimension(nmax, (lmax+1)**2)

The precomputed data kernel matrix G obtained from LSQ_G.

normalization : str, optional, default = ‘4pi’

‘4pi’, ‘ortho’, ‘schmidt’, or ‘unnorm’ for geodesy 4pi normalized, orthonormalized, Schmidt semi-normalized, or unnormalized spherical harmonic functions, respectively.

csphase : integer, optional, default = 1

If 1 (default), the Condon-Shortley phase will be excluded. If -1, the Condon-Shortley phase of (-1)^m will be appended to the spherical harmonic functions.

kind : str, optional, default = ‘real’

‘real’ or ‘complex’ spherical harmonic coefficients.

degrees : bool, optional, default = True

If True, latitude and longitude are in degrees, otherwise they are in radians.

Notes

When the number of data points is greater or equal to the number of 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.

When weigths are present, they should be set equal to the inverse of the data variance. It is assumed explicitly that each measurement is statistically independent (i.e., the weighting matrix is diagonal). The weighted least squares inversion must be overdetermined (i.e., nmax>(lmax+1)**2).

The inversions are performed using the LAPACK routine DGELS. If this routine is used several times with the same latitude and longitude coordinates, the data kernel matrix G can be precomputed using LSQ_G.