pyharm.integ

Module to compute the following integrals:

• a product of two fully-normalized associated Legendre functions and a sine of a co-latitude,

  \[\int\limits_{\theta_1}^{\theta_2} \bar{P}_{n_1,m_1}(\cos\theta) \, \bar{P}_{n_2,m_2}(\cos\theta) \, \sin(\theta) \, \mathrm{d}\theta{,} \quad \theta_1 \leq \theta_2{,}\]

• a product of two 4pi fully-normalized surface spherical harmonic functions over a rectangular cell on the unit sphere,

  \[\int\limits_{\theta_1}^{\theta_2} \int\limits_{\lambda_1}^{\lambda_2} \bar{Y}_{n_1,m_1}(\theta, \lambda) \, \bar{Y}_{n_2,m_2}(\theta, \lambda) \, \mathrm{d}\lambda \, \sin(\theta) \, \mathrm{d}\theta{,} \quad \theta_1 \leq \theta_2{,} \quad \lambda_1 \leq \lambda_2{.}\]

Note

This documentation is written for double precision version of PyHarm.

pyharm.integ.pn1m1pn2m2(cltmin, cltmax, n1, m1, n2, m2, pnmj)

Analytically computes the integral

\[\mathrm{ip} = \int\limits_{\theta_{\mathrm{min}}}^{\theta_{\mathrm{max}}} \bar{P}_{n_1, m_1}(\cos\theta) \, \bar{P}_{n_2, m_2}(\cos\theta) \, \sin\theta \, \mathrm{d}\theta{,} \quad \theta_{\mathrm{min}} \leq \theta_{\mathrm{max}}{.}\]

The computation is based on the Fourier coefficients of the associated Legendre functions (see Eq. 33 of Pail and Plank, 2001).

References:

• Pail, R., Plank, G., Schuh, W.-D. (2001) Spatially restricted data distributions on the sphere: the method of orthonormalized functions and applications. Journal of Geodesy 75:44–56

Parameters:

• cltmin (floating point) – Minimum co-latitude in radians

• cltmax (floating point) – Maximum co-latitude in radians

• n1 (integer) – Harmonic degree of the first Legendre function

• m1 (integer) – Harmonic order of the first Legendre function

• n2 (integer) – Harmonic degree of the second Legendre function

• m2 (integer) – Harmonic order of the second Legendre function

• pnmj (Pnmj) – Fourier coefficients of Legendre functions up to degree max(n1, n2).

Returns:

out – The output value ip of the integral.

Return type:

floating point

pyharm.integ.yi1n1m1yi2n2m2(cltmin, cltmax, lonmin, lonmax, i1, n1, m1, i2, n2, m2, pnmj)

Analytically computes the integral

\[\mathrm{iy} = \int\limits_{\theta_{\mathrm{min}}}^{\theta_{\mathrm{max}}} \int\limits_{\lambda_{\mathrm{min}}}^{\lambda_{\mathrm{max}}} \bar{Y}_{i_1,n_1,m_1}(\theta, \lambda) \, \bar{Y}_{i_2,n_2,m_2}(\theta, \lambda) \, \mathrm{d} \lambda \, \sin(\theta) \, \mathrm{d}\theta{,} \quad \theta_{\mathrm{min}} \leq \theta_{\mathrm{max}}{,} \quad \lambda_{\mathrm{min}} \leq \lambda_{\mathrm{max}}{,}\]

where

\[\begin{split}\bar{Y}_{i,n,m}(\theta, \lambda) = \begin{cases} \bar{P}_{nm}(\cos\theta) \, \cos(m \, \lambda) \quad &\textrm{if} \quad i = 0{,}\\ \bar{P}_{nm}(\cos\theta) \, \sin(m \, \lambda) \quad &\textrm{if} \quad i = 1{.} \end{cases}\end{split}\]

Parameters:

• cltmin (floating point) – Minimum co-latitude in radians

• cltmax (floating point) – Maximum co-latitude in radians

• lonmin (floating point) – Minimum longitude in radians

• lonmax (floating point) – Maximum longitude in radians

• i1 (integer) – 0 if the first spherical harmonic function is of the cos type, 1 for the sin type

• n1 (integer) – Harmonic degree of the first spherical harmonic function

• m1 (integer) – Harmonic order of the first spherical harmonic function

• i2 (integer) – The same as i1 but for the second spherical harmonic function

• n2 (integer) – Harmonic degree of the second spherical harmonic function

• m2 (integer) – Harmonic order of the second spherical harmonic function

• pnmj (Pnmj) – Fourier coefficients of Legendre functions at least up to degree max(n1, n2).

Returns:

out – The value iy of the integral.

Return type:

floating point