SHTOOLS --- Tools for working with spherical harmonics

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Complex spherical harmonics

Real Spherical Harmonics

"Geodesy" 4π-normalized real spherical harmonics

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

f.gif

where flm is the spherical harmonic coefficient, Ylm 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 Clm, and the negative order coefficient as Slm, in reference to the "cosine" and "sine" coefficients, respectively.) The real spherical harmonics are defined as

ylm.gif

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

plmbar.gif

with the following definition for the unnormalized associated Legendre functions:

plmdef.gif.

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

plmnorm.gif.

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

ylmnorm.gif.

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

flm.gif.

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

totalpower.gif,

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

crosspower.gif.

If the functions f and g have a zero mean, then Sfg 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:

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

Geodesy 4π

plmbar.gif
plmnorm.gif
ylmnorm.gif

Schmidt semi-normalized

plmschmidt.gif
plmschmidtnorm.gif
ylmschmidt.gif

Unnormalized

plmunnormalized.gif
plmunnormnorm.gif
ylmunnorm.gif

Orthonormalized

plmon.gif
plmonnorm.gif
ylmorthonorm.gif

Further information

Further information regarding spherical harmonic functions can be found at the following web sites:

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Institut de Physique du Globe de Paris University of Sorbonne Paris Cité © 2014 Mark Wieczorek