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Real spherical harmonics |
spherical harmonics software package, spherical harmonic transform, legendre functions, multitaper spectral analysis, fortran, Python, gravity, magnetic field |
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real-spherical-harmonics.html |
SHTOOLS uses by default 4π-normalized spherical harmonic functions that exclude the Condon-Shortley phase factor. Schmidt semi-normalized, orthonormalized, and unnormalized harmonics can be employed in most routines by specifying optional parameters. |
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Any real square-integrable function can be expressed as a series of spherical harmonic functions
\begin{equation} f\left(\theta,\phi\right) = \sum_{l=0}^{\infty} \sum_{m=-l}^l f_{lm} , Y_{lm}\left(\theta,\phi \right), \label{eq:f} \end{equation}
where
$$ \begin{equation}
Y_{lm}(\theta,\phi) = \left \lbrace \begin{array}{ll} \bar{P}{lm}(\cos
\theta) \cos m \phi & \mbox{if $m \ge 0$} \
\bar{P}{l|m|}(\cos \theta) \sin |m| \phi & \mbox{if
where the normalized associated Legendre functions for use with the
$$ \begin{eqnarray} \bar{P}{lm}(\mu) = \sqrt{\left(2-\delta{m0}\right) \left(2l+1\right)\frac{(l-m)!}{(l+m)!}}, P_{lm}(\mu) \end{eqnarray} $$
and where
\begin{equation} P_l(\mu) = \frac{1}{2^l l!}\frac{d^l}{d\mu^l}\left(\mu^2-1\right)^l. \end{equation}
The normalized associated Legendre functions are orthogonal for a given value of
$$ \begin{equation} \int_{-1}^{1} \bar{P}{lm}(\mu) ,\bar{P}{l'm}(\mu) = 2\left(2-\delta_{0m}\right) \delta_{ll'}, \end{equation} $$
and the spherical harmonic functions are orthogonal for all degrees
\begin{equation}
\int_\Omega Y_{lm}(\theta,\phi) , Y_{l'm'}(\theta,\phi) , d\Omega = 4 \pi , \delta_{ll'}, \delta_{mm'},
\end{equation}
where
Parseval's theorem in Cartesian geometry relates the integral of a function squared to the sum of the squares of the function's Fourier coefficients. This relation is easily extended to spherical geometry using the orthogonality properties of the spherical harmonic functions. Defining power to be the integral of the function squared divided by the area it spans, the total power of a function is equal to a sum over its power spectrum
\begin{equation}
\frac{1}{4\pi} \int_\Omega f^2(\theta,\phi) , d\Omega
= \sum_{l=0}^{\infty} S_{ff}(l),
\end{equation}
where the power spectrum
Similarly, the cross power of two functions
The power spectrum is unmodified by a rotation of the coordinate system. Furthermore, the numerical values of the power spectrum are independent of the normalization convention used for the spherical harmonic functions (though the mathematical formulae will be different, as given below). If the functions
The above definitions of the Legendre functions and spherical harmonic functions do not include the Condon-Shortley phase factor of
csphase = 0
: exclude the Condon-Shortley phase factor (default)csphase = 1
: append the Condon-Shortley phase factor to the Legendre functions.
The choice of the Condon-Shortley phase factor does not affect the numerical value of the power spectrum.
SHTOOLS supports the use of norm
in the Fortran 95 routines or normalization
in the Python routines:
-
norm = 1
,normalization = '4pi'
:$$4\pi$$ normalized (default, unless stated otherwise) -
norm = 2
,normalization = 'schmidt'
: Schmidt semi-normalized -
norm = 3
,normalization = 'unnorm'
: Unnormalized -
norm = 4
,normalization = 'ortho'
: Orthonormalized.
Each of these normalizations has slightly different definitions for the normalized Legendre functions, the orthogonality conditions of the Legendre functions and spherical harmonic functions, and the power spectrum. These equations are provided below.
| $$ \displaystyle \bar{P}{lm}(\mu) = \sqrt{\left(2-\delta{m0}\right) \left(2l+1\right)\frac{(l-m)!}{(l+m)!}}, P_{lm}(\mu) $$ |
| $$\displaystyle \int_{-1}^{1} \bar{P}{lm}(\mu) ,\bar{P}{l'm}(\mu)= 2\left(2-\delta_{0m}\right) , \delta_{ll'}$$ |
|
| $$\displaystyle \bar{P}{lm}(\mu) = \sqrt{\left(2-\delta{m0}\right) \frac{(l-m)!}{(l+m)!}}, P_{lm}(\mu) $$ |
| $$\displaystyle \int_{-1}^{1} \bar{P}{lm}(\mu) ,\bar{P}{l'm}(\mu)= \frac{2\left(2-\delta_{0m}\right)}{(2l+1)} , \delta_{ll'}$$ |
|
| $$\displaystyle \bar{P}{lm}(\mu) = \sqrt{\frac{\left(2-\delta{0m}\right) \left(2l+1\right)}{4 \pi} \frac{(l-m)!}{(l+m)!}}, P_{lm}\left(\mu\right)$$ |
| $$\displaystyle \int_{-1}^{1} \bar{P}{lm}(\mu) ,\bar{P}{l'm}(\mu)= \frac{\left(2-\delta_{0m}\right)}{2 \pi} , \delta_{ll'}$$ |
|
| $$\displaystyle \bar{P}{lm}(\mu) = P{lm}(\mu)$$ |
| $$\displaystyle \int_{-1}^{1} \bar{P}{lm}(\mu) ,\bar{P}{l'm}(\mu)= \frac{2}{(2l+1)} \frac{(l+m)!}{(l-m)!} , \delta_{ll'}$$ |
|
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Dahlen, F. A. and J. Tromp, "Theoretical Global Seismology," Princeton University Press, Princeton, New Jersey, 1025 pp., 1998.
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Varshalovich, D. A., A. N. Moskalev, and V. K. Khersonskii, "Quantum theory of angular momentum," World Scientific, Singapore, 1988.