cross_spectrum()

Return the cross-spectrum of the spherical harmonic coefficients as a function of spherical harmonic degree.

Usage

array = cross_spectrum(clm1, clm2, [normalization, degrees, lmax, convention, unit, base])

Returns

array : ndarray, shape (len(degrees)) 1-D ndarray of the spectrum.

Parameters

clm1 : ndarray, shape (2, lmax + 1, lmax + 1) ndarray containing the first set of spherical harmonic coefficients.

clm2 : ndarray, shape (2, lmax + 1, lmax + 1) ndarray containing the second set of spherical harmonic coefficients.

normalization : str, optional, default = ‘4pi’ ‘4pi’, ‘ortho’, ‘schmidt’, or ‘unnorm’ for geodesy 4pi normalized, orthonormalized, Schmidt semi-normalized, or unnormalized coefficients, respectively.

lmax : int, optional, default = len(clm[0,:,0]) - 1. Maximum spherical harmonic degree to output.

degrees : ndarray, optional, default = numpy.arange(lmax+1) Array containing the spherical harmonic degrees where the spectrum is computed.

convention : str, optional, default = ‘power’ The type of spectrum to return: ‘power’ for power spectrum, ‘energy’ for energy spectrum, and ‘l2norm’ for the l2-norm spectrum.

unit : str, optional, default = ‘per_l’ If ‘per_l’, return the total contribution to the spectrum for each spherical harmonic degree l. If ‘per_lm’, return the average contribution to the spectrum for each coefficient at spherical harmonic degree l. If ‘per_dlogl’, return the spectrum per log interval dlog_a(l).

base : float, optional, default = 10. The logarithm base when calculating the ‘per_dlogl’ spectrum.

Notes

This method returns either the cross-power spectrum, cross-energy spectrum, or l2-cross-norm spectrum of two functions expressed in spherical harmonics. Total cross-power is defined as the integral of the first function times the conjugate of the second function over all space, divided by the area the functions span. If the means of the two functions are zero, this is equivalent to the covariance of the two functions. The total cross-energy is the integral of the first function times the conjugate of the second function over all space and is 4 pi times the total power. For normalized coefficients (‘4pi’, ‘ortho’, or ‘schmidt’), the l2-cross norm is the sum of the magnitude of the coefficients of the first function multiplied by the conjugate of the coefficients of the second function.

The output spectrum can be expresed using one of three units. ‘per_l’ returns the contribution to the total power, energy or l2-norm from all angular orders at degree l. ‘per_lm’ returns the average contribution to the total power, energy or l2-norm from a single coefficient at degree l, and is equal to the ‘per_l’ spectrum divided by (2l+1). ‘per_dlogl’ returns the contribution to the total power, energy or l2-norm from all angular orders over an infinitessimal logarithmic degree band. The contrubution in the band dlog_a(l) is spectrum(l, ‘per_dlogl’)*dlog_a(l), where a is the base, and where spectrum(l, ‘per_dlogl) is equal to spectrum(l, ‘per_l’)*l*log(a).