Associated Legendre Functions Research Articles

Construction of regional geoid using a virtual spherical harmonics model

Abstract The spherical cap harmonics (SCH) method can be used in regional geoid modeling. The core of this approach is the computation of its associated Legendre functions (ALF) with non-integer degree. However, it is unlikely to obtain a large number of zero-root values for the non-integer ALF. To overcome this problem, a new approach called virtual spherical harmonics (VSH) is proposed in this paper to transform the cap range into the whole sphere so that unlimited numbers of zero-root values can be obtained. The new approach was tested using four cap ranges with the radii of 30 ∘ {30^{\circ }} , 15 ∘ {15^{\circ }} , 10 ∘ {10^{\circ }} and 5 ∘ {5^{\circ }} , and geoid undulations for each of the regions are calculated from EGM2008. For each of the regions, the geoid undulations were used to construct three models with three different degrees of 20, 30 and 40. Numerical results showed that with the increase in the degree of the VSH model, the value of the maximum error decreases; and the maximum error of the model was less than 1 mm while the maximum degree is 40.

Ultra-high-Degree Surface Spherical Harmonic Analysis Using the Gauss–Legendre and the Driscoll/Healy Quadrature Theorem and Application to Planetary Topography Models of Earth, Mars and Moon

In geodesy and geophysics, spherical harmonic techniques are popular for modelling topography and potential fields with ever-increasing spatial resolution. For ultra-high-degree spherical harmonic modelling, i.e. degree 10,000 or more, classical algorithms need to be extended to avoid under- or overflow problems associated with the computation of associated Legendre functions (ALFs). In this work, two quadrature algorithms—the Gauss–Legendre (GL) quadrature and the quadrature following Driscoll/Healy (DH)—and their implementation for the purpose of ultra-high (surface) spherical harmonic analysis of spheroid functions are reviewed and modified for application to ultra-high degree. We extend the implementation of the algorithms in the SHTOOLS software package (v2.8) by (1) the X-number (or Extended Range Arithmetic) method for accurate computation of ALFs and (2) OpenMP directives enabling parallel processing within the analysis. Our modifications are shown to achieve feasible computation times and a very high precision: a degree-21,600 band-limited (=frequency limited) spheroid topographic function may be harmonically analysed with a maximum space-domain error of \(3 \times 10^{-5}\) and \(5 \times 10^{-5}\) m in 6 and 17 h using 14 CPUs for the GL and for the DH quadrature, respectively. While not being inferior in terms of precision, the GL quadrature outperforms the DH algorithm in terms of computation time. In the second part of the paper, we apply the modified quadrature algorithm to represent for—the first time—gridded topography models for Earth, Moon and Mars as ultra-high-degree series expansions comprising more than 2 billion coefficients. For the Earth’s topography, we achieve a resolution of harmonic degree 43,200 (equivalent to ~500 m in the space domain), for the Moon of degree 46,080 (equivalent to ~120 m) and Mars to degree 23,040 (equivalent to ~460 m). For the quality of the representation of the topographic functions in spherical harmonics, we use the residual space-domain error as an indicator, reaching a standard deviation of 3.1 m for Earth, 1.9 m for Mars and 0.9 m for Moon. Analysing the precision of the quadrature for the chosen expansion degrees, we demonstrate limitations in the implementation of the algorithms related to the determination of the zonal coefficients, which, however, do not exceed 3, 0.03 and 1 mm in case of Earth, Mars and Moon, respectively. We investigate and interpret the planetary topography spectra in a comparative manner. Our analysis reveals a disparity between the topographic power of Earth’s bathymetry and continental topography, shows the limited resolution of altimetry-derived depth (Earth) and topography (Moon, Mars) data and detects artefacts in the SRTM15 PLUS data set. As such, ultra-high-degree spherical harmonic modelling is directly beneficial for global inspection of topography and other functions given on a sphere. As a general conclusion, our study shows that ultra-high-degree spherical harmonic modelling to degree ~46,000 has become possible with adequate accuracy and acceptable computation time. Our software modifications will be freely distributed to fill a current availability gap in ultra-high-degree analysis software.

Exact solutions to a class of differential equation and some new mathematical properties for the universal associated-Legendre polynomials

The exact solutions to a class of differential equation are studied. Some special cases are discussed for the central potentials, single ring-shaped potential and the angular Teukolsky equation. A new expression to the associated Legendre polynomials is found. Some new properties of the universal associated-Legendre polynomials (UALPs) including the generating function, Rodrigues’ formula, parity, some special values and the recurrence relations are presented. A new different Rodrigues’ formula of associated Legendre polynomials is also obtained.

The Origin and Mathematical Characteristics of the Super-Universal Associated-Legendre Polynomials

We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. We find that there exists the even or odd parity for the polar angular wave functions when the parameter η is taken to be positive integer, while there exist both even and odd parities when η is taken as positive non-integer real values. The relations among the super-universal associated-Legendre polynomials, the hypergeometric polynomials, and the Jacobi polynomials are also established.

Exact Field Expressions for Circular Loop Antennas Using Spherical Functions Expansion

A new technique is introduced for evaluating the vector potential of circular loop antennas (CLAs) with arbitrary current distribution and dimensions, which can be used to obtain the exact electromagnetic fields in near and far regions of the antenna. The technique is based on the spherical Bessel functions (SBFs) and associated Legendre polynomials (ALPs) expansion. This expansion is used to decouple the azimuth variables of the integrand of the vector potential integrals from the other variables in a separate functions that depend exponentially on azimuth variables. This technique reduces greatly the mathematical operations for evaluation of the vector potential to only one simple integration. The technique is used to obtain exact electromagnetic fields expressions for a thin-wire CLA of arbitrary current distribution and dimension, which is valid everywhere except on the loop. The well known far-fields for the CLA of arbitrary current distribution as well as near and far-fields for uniform current loop and small loop are obtained from the general field expressions presented in this paper as an special cases.

Numerical computation of spherical harmonics of arbitrary degree and order by extending exponent of floating point numbers

By extending the exponent of floating point numbers with an additional integer as the power index of a large radix, we compute fully normalized associated Legendre functions (ALF) by recursion without underflow problem. The new method enables us to evaluate ALFs of extremely high degree as 232 = 4,294,967,296, which corresponds to around 1 cm resolution on the Earth’s surface. By limiting the application of exponent extension to a few working variables in the recursion, choosing a suitable large power of 2 as the radix, and embedding the contents of the basic arithmetic procedure of floating point numbers with the exponent extension directly in the program computing the recurrence formulas, we achieve the evaluation of ALFs in the double-precision environment at the cost of around 10% increase in computational time per single ALF. This formulation realizes meaningful execution of the spherical harmonic synthesis and/or analysis of arbitrary degree and order.

Non-singular expressions for the vector and the gradient tensor of gravitation in a geocentric spherical frame

The traditional expressions of the gravitational vector (GV) and the gravitational gradient tensor (GGT) have complicated forms depending on the first- and the second-order derivatives of associated Legendre functions (ALF), and also singular terms when approaching the poles. This article presents alternative expressions for the GV and GGT, which are independent of the derivatives, and are also non-singular. By using such expressions, it suffices to compute the ALF to two additional degrees and orders, instead of computing the first and the second derivatives of all the ALF. Therefore, the formulas are suitable for computer programming. Matlab software as well as an output of a numerical computation around the North Pole is also presented based on the derived formulas.

Algorithms for Computing Ultra‐High‐Degree Disturbing Potential Elements

AbstractThe fast development of modern spatial gravimetric technology makes it possible and more necessary to establish the ultra‐high‐degree global potential model. The numerical characters of the ultra‐high‐degree fully normalized associated Legendre functions (ALFs) and their 1st, 2nd derivatives of the disturbing potential spherical harmonic expansions have been studied from the standard forward column recursions. Then, the abovementioned standard forward column methods are modified to avoid overflow. Finally, some practical algorithms for evaluating the ultra‐high‐degree disturbing potential elements by personal computer are presented.

The overlap integral of three associated Legendre polynomials

A closed formula with a double sum is obtained for the overlap integral of three associated Legendre polynomials (ALPs). The result is applicable to integral involving the ALP with arbitrary degree l and order m. Special overlap integrals, including the cases m 3 = m 1 + m 2 or | m 1 – m 2|, are presented. A general formula for the overlap integral of an arbitrary number of ALPs is also developed.

A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions

Spherical harmonic expansions form partial sums of fully normalised associated Legendre functions (ALFs). However, when evaluated increasingly close to the poles, the ultra-high degree and order (e.g. 2700) ALFs range over thousands of orders of magnitude. This causes existing recursion techniques for computing values of individual ALFs and their derivatives to fail. A common solution in geodesy is to evaluate these expansions using Clenshaw's method, which does not compute individual ALFs or their derivatives. Straightforward numerical principles govern the stability of this technique. Elementary algebra is employed to illustrate how these principles are implemented in Clenshaw's method. It is also demonstrated how existing recursion algorithms for computing ALFs and their first derivatives are easily modified to incorporate these same numerical principles. These modified recursions yield scaled ALFs and first derivatives, which can then be combined using Horner's scheme to compute partial sums, complete to degree and order 2700, for all latitudes (except at the poles for first derivatives). This exceeds any previously published result. Numerical tests suggest that this new approach is at least as precise and efficient as Clenshaw's method. However, the principal strength of the new techniques lies in their simplicity of formulation and implementation, since this quality should simplify the task of extending the approach to other uses, such as spherical harmonic analysis.

An Efficient Spectral Dynamical Core for Distributed Memory Computers

Abstract The practical question of whether the classical spectral transform method, widely used in atmospheric modeling, can be efficiently implemented on inexpensive commodity clusters is addressed. Typically, such clusters have limited cache and memory sizes. To demonstrate that these limitations can be overcome, the authors have built a spherical general circulation model dynamical core, called BOB (“Built on Beowulf”), which can solve either the shallow water equations or the atmospheric primitive equations in pressure coordinates. That BOB is targeted for computing at high resolution on modestly sized and priced commodity clusters is reflected in four areas of its design. First, the associated Legendre polynomials (ALPs) are computed “on the fly” using a stable and accurate recursion relation. Second, an identity is employed that eliminates the storage of the derivatives of the ALPs. Both of these algorithmic choices reduce the memory footprint and memory bandwidth requirements of the spectral transf...

Comment on `A single-sum expression for the overlap integral of two associated Legendre polynomials'

The overlap integral of two associated Legendre polynomials (ALPs) presented by Mavromatis (Mavromatis H A 1999 J. Phys. A: Math. Gen. 32 2601) is revised. The results are valid for any product of two ALPs with arbitrary degree l and order m.

EXACT SOLUTION OF SCHR?DINGER EQUATION WITH REFLECTIONLESS POTENTIAL WELL

Using a transformation of hyperbolic function, the Schr?dinger equation with reflection-less potential well is transformed into an associated-Legendre equation. Then both bound and scattering state eigenfunctions are expressed in terms of associated-Legendre polynomials and functions respectively. The exact solutions obtained in this paper are more general and systematic than some asymptotic solutions or solutions of reflectionless potential with special parameters in literatures. The normalization of the scattering state is discussed in detail.

ON THE PRODUCT OF TWO ASSOCIATED LEGENDRE FUNCTIONS

ON THE PRODUCT OF TWO ASSOCIATED LEGENDRE FUNCTIONS Get access W. N. BAILEY W. N. BAILEY Manchester Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume os-11, Issue 1, 1940, Pages 30–35, https://doi.org/10.1093/qmath/os-11.1.30 Published: 01 January 1940 Article history Published: 01 January 1940 Received: 12 August 1940