Polynomial approximation for fast generation of associated Legendre functions

Abstract

Today high-speed computers have simplified many computational problems, but fast techniques and algorithms are still relevant. In this study, the Hermitian polynomial approximation is used for fast evaluation of the associated Legendre functions (ALFs). It has lots of applications in geodesy and geophysics. This method approximates the ALFs instead of computing them by recursive formulae and generate them several times faster. The approximated ALFs by the Newtonian polynomials are compared with Hermitian ones and their differences are discussed. Here, this approach is applied for computing a global geoid model point-wise from EGM08 to degree and order 2160 and in propagating the orbit of a low Earth orbiting satellite. Our numerical results show that the CPU-time decreases at least two times for orbit propagation, and five times for geoid computation comparing to the case where recursive formulae for generation of ALFs are used. The approximation error in the orbit computation is at a sub-millimeter level over two weeks and that the computed geoid 0.01 mm, with a maximum of 1 mm.

Appendices

Appendix A

The Earth’s gravity acceleration in the Earth-centered inertial frame is computed using (Keller and Sharifi 2005)

$$\begin{aligned} \mathbf r _g= & {} \left[ \begin{array}{ccc} \cos (\varphi )\cos (\lambda ) &{} {-\sin (\varphi )\cos (\lambda ) \over r} &{} {-\sin (\lambda ) \over r\cos (\varphi )} \\ \cos (\varphi )\sin (\lambda ) &{} {-\sin (\varphi )\sin (\lambda ) \over r} &{} {\cos (\lambda ) \over r\cos (\varphi )} \\ \sin (\varphi ) &{} {\cos (\varphi ) \over r} &{} 0 \\ \end{array} \right] \nonumber \\&\times {GM\over r}\sum _{n=0}^{\infty }\left( {R\over r}\right) ^n \sum _{m=0}^n \left[ \begin{array}{c} -{n+1\over r} \left( \overline{C}_{nm}\cos (m\lambda ) + \overline{S}_{nm}\sin (m\lambda )\right) \overline{P}_{nm}(\sin (\varphi )) \\ \left( \bar{C}_{nm}\cos (m\lambda ) + \bar{S}_{nm}\sin (m\lambda )\right) \bar{P}'_{nm}(\sin (\varphi )) \\ \left( -m\bar{C}_{nm}\sin (m\lambda ) + m\bar{S}_{nm}\cos (m\lambda )\right) \bar{P}_{nm}(\sin (\varphi )) \\ \end{array}\right] \end{aligned}$$

(19)

In this equation $\bar{P}'_{nm}$ is the first order derivative of the associated Legendre function ($\hbox {degree}=n$ and $\hbox {order}=m$) with respect to the latitude ($\varphi$). The first order derivative of the ALFs could be computed based on the ALFs themselves (Robin 1957), and $\bar{C}_{nm}$ and $\bar{S}_{nm}$ are the degree-n and order-m Stokes coefficients (e.g. Hofmann-Wellenhof and Moritz 2006).

$$\begin{aligned} \bar{P}'_{nm}={\partial \bar{P}_{nm} \over \partial \varphi }=K_{nm}\bar{P}_{n,m+1}-m\tan (\varphi )\bar{P}_{nm} \end{aligned}$$

(20)

where

$$\begin{aligned} K_{nm} = \sqrt{{(2-\delta _{m,0})(n-m)(n+m+1) \over 2}} \end{aligned}$$

(21)

Matlab code description

There are one main function needed for the ALFs computation and two related program files which have been used to create the required grid data. The main source functions and scripts are listed bellow.

• 1. ${\mathtt {APnmH.m}}$: This sub-routine is used for the fast computation of the ALFs using the Hermite polynomial approximation based on Eq. (13).

• 2. ${\mathtt {PnmGrid.m}}$: This script is used for generating the needed grid-data.

• 3. ${\mathtt {P.m}}$: This function is used to compute the ALFs and their first- and second-derivatives.

• 4. ${\mathtt {example.m}}$: In this script, the fast computation of the ALFs has been described for more details.

In order to use this source codes, please follow the steps: (i) Run $\mathtt {PnmGrid.m}$ to produce and save the grid file, the latitude interval of $0.5^{\circ }$ is suggested for the fast orbit propagation. (ii) At the first of the computation, load the grid file. Once loaded, nothing more loading is needed. (iii) Run $\mathtt {APnmH.m}$ for any desired latitude. As an example, run $\mathtt {example.m}$ after producing grid file.