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DCL:MATH2:SHTLIB: Spherical Harmonic Functions
4.1 Summary
This is a package of subroutines that performs spectral (spherical harmonic
function) transformations, and converts a spherical harmonic function into grid
data by expansion, or the vice versa by inverse transformation.
The package design is
especially suited for data analysis, and a special feature of the package is
that it can handle equal-interval grid data.
Furthermore, to enhance its
capability in spectral data analysis, it is equipped with a wide variety of
inverse transformation routines.
The FFTLIB subroutine is used within this package.
A spectral inversion with cut-off wavenumber of M (triangular truncation) can be expressed as follows:
|
(4.1) |
Or, by using the inverse Legendre transformation:
|
(4.2) |
(4.1) can be expressed as a product of an inverse
Legendre transformation and an inverse Fourier transformation.
|
(4.3) |
Here, &lambda and &phi are latitude and longitude,
respectively.
Furthermore, Pmn(&mu) is an
associated Legendre function normalized to 2, and is defined as follows:
|
(4.4) |
|
(4.5) |
The inverse spectral transformation can also be expressed as follows:
|
(4.6) |
As in the case of the inverse transformation, by using
the forward Fourier transformation:
|
(4.7) |
(4.6) can be expressed as a product of a forward Fourier
transformation and a forward Legendre transformation:
|
(4.8) |
If we assume that G(&lambda, &phi)
is a floating-point number, then Smn and
Wm(&phi)
must satisfy the relationship below.
|
(4.9) |
¡¡
Here, {}*
represents a complex conjugate. Therefore, Wm(sin&phi)
and ??? needs only to be determined for m¡æ0.
Furthermore, from the above restrictions,
Wm(sin&phi)
and S0n will be floating-point
numbers.
This library consists of
a group of routines that performs inverse transformation from spectral data (Smn) into
wave data in an equal-interval meridional plane (Wm(&phij))
and into an equal-interval grid data (G(&phij)) based
on Eq. (1)-(3);
a group of routines that performs forward transformation from an
equal-interval grid data (G(&phij))
into
wave data in an equal-interval meridional plane (Wm(&phij))
and into spectral data (Smn) based on
Eq. (6)-(8);
and a group of other auxiliary routines.
Here, it is assumed that the the latitude (&lambdai) and longitude (&phij)
of the grid points can be represented as follows using the partition numbers I and J:
|
(4.11) |
|
(4.12) |
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DCL:MATH2:SHTLIB: Spherical Harmonic Functions