Real Spherical Harmonics

"Geodesy" 4π-normalized real spherical harmonics

Any real square-integrable function can be expressed in spherical harmonics as

where f_lm is the spherical harmonic coefficient, Y_lm is the spherical harmonic function, Ω represents position on the sphere in terms of co-latitude θ and longitude φ, and l and m are the spherical harmonic degree and order, respectively. (It is common to refer to the positive order coefficient as C_lm, and the negative order coefficient as S_lm, in reference to the "cosine" and "sine" coefficients, respectively.) The real spherical harmonics are defined as

where the normalized associated Legendre functions, as used in geodesy and gravity applications (calculated by the routine PlmBar), are given by

with the following definition for the unnormalized associated Legendre functions:

The normalized associated Legendre functions are orthogonal for a given value of m,

The normalized Legendre functions (including the orthonormalized and Schmidt semi-normalized functions) are efficiently calculating using the algorithm of Holmes and Featherstone (2002, J. Geodesy, 76, 279-299) that is accurate to approximately degree 2800. Note that the above definition of the associated Legendre functions does not include the Condon-Shortley phase of (-1)^m that is often employed in the physics and seismology communities (e.g., Varsalovich et al., 1988; Dahlen and Tromp, 1998). This can be included by specifying the optional argument CSPHASE=-1 in most of the Legendre and spherical harmonic routines. (Alternatively, the default phase convention can be modified during compilation of the SHTOOLS archive.)

The spherical harmonics are orthogonal over both l and m with the normalization

Using this relationship, it is straightforward to show that the spherical harmonic coefficients of a function can be calculated by the integral

The cross-power of two functions f and g is equal to

where the "cross-power spectrum" of the two functions is given by

If the functions f and g have a zero mean, then S_fg is the contribution to the covariance as a function of spherical harmonic degree.

Other real normalization conventions

The above 4π normalization scheme is the default for all routines in SHTOOLS with the exception of MakeMagGrid2D, which uses the standard geomagnetism Schmidt semi-normalization. By specifying an optional parameter norm, the default normalization can be changed in most of the routines (e.g., PreCompute, MakeGridGLQ, SHExpandGLQ, SHExpandDH, MakeGrid2D, MakeGridPoint, SHMultiply and SHExpandLSQ). The following normalizations are permitted:

norm=1: Geodesy 4π normalization (default, unless stated otherwise).
norm=2: Schmidt semi-normalization.
norm=3: Unnormalized.
norm=4: Orthonormalized.

Explicit expressions for the Legendre functions and real spherical harmonic normalizations are the following:

Geodesy 4π

Schmidt semi-normalized