Construction of regional geoid using a virtual spherical harmonics model

2019 ◽  
Vol 13 (2) ◽  
pp. 151-158 ◽  
Author(s):  
Jianqiang Wang ◽  
Keqiang Wu
Keyword(s):  
Spherical Harmonics ◽  
Maximum Degree ◽  
Maximum Error ◽  
Numerical Results ◽  
Legendre Functions ◽  
Spherical Cap ◽  
Associated Legendre Functions ◽  
New Approach ◽  
The Core ◽  
Geoid Undulations

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^{\circ }}, {15^{\circ }}, {10^{\circ }} and {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.

scholarly journals Derivatives of addition theorems for Legendre functions

1995 ◽  
Vol 37 (2) ◽  
pp. 212-234 ◽  
Author(s):  
D.E. Winch ◽  
P.H. Roberts
Keyword(s):  
Spherical Harmonics ◽  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Addition Theorems ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Derivatives Of

AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

scholarly journals The Asymptotic Expansions of the Spherical Harmonics

1922 ◽  
Vol 41 ◽  
pp. 82-93
Author(s):  
T. M. MacRobert
Keyword(s):  
Real Axis ◽  
Spherical Harmonics ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
The Real ◽  
Image Position

Associated Legendre Functions as Integrals involving Bessel Functions. Let,where C denotes a contour which begins at −∞ on the real axis, passes positively round the origin, and returns to −∞, amp λ=−π initially, and R(z)>0, z being finite and ≠1. [If R(z)>0 and z is finite, then R(z±)>0.] Then if I−m (λ) be expanded in ascending powers of λ, and if the resulting expression be integrated term by term, it is found that

Mathematical Aspects of Spectral Models

1998 ◽  
Author(s):  
T. N. Krishnamurti ◽  
H. S. Bedi ◽  
V. M. Hardiker
Keyword(s):  
Spherical Harmonics ◽  
Basis Functions ◽  
Spectral Model ◽  
Water Model ◽  
Data Sets ◽  
Legendre Functions ◽  
Global Spectral Model ◽  
Associated Legendre Functions ◽  
Spectral Models ◽  
Legendre Transforms

In this chapter we provide an introduction to the topic of spherical harmonics as basis functions for a global spectral model. The spherical harmonics are made up of trigonometric functions along the zonal direction and associated Legendre functions in the meridional direction. A number of properties of these functions need to be understood for the formulation of a spectral model. This chapter describes some useful properties that will be used to illustrate the procedure for the representation of data sets over a sphere with spherical harmonics as basis functions. The calculations of Fourier and Legendre transforms and their inverse transforms are an important part of global spectral modeling, and these are covered in some detail in this chapter. Finally, this chapter addresses the formulation of two simple spectral models. One of these is a single-level barotropic model, and the other is a shallow-water model.

The Analysis of the Associated Legendre Functions with Non-Integral Degree

2011 ◽  
Vol 130-134 ◽  
pp. 3001-3005
Author(s):  
Jian Qiang Wang ◽  
Hao Yuan Chen ◽  
Yin Fu Chen
Keyword(s):  
Spherical Harmonic ◽  
Legendre Function ◽  
Scalar Potential ◽  
Regional Model ◽  
Legendre Functions ◽  
Spherical Cap ◽  
Associated Legendre Functions ◽  
Core Content ◽  
Local Gravity ◽  
Integral Degree

Spherical cap harmonic (SCH) theory has been widely used to format regional model of fields that can be expressed as the gradient of a scalar potential. The functions of this method consist of trigonometric functions and associated Legendre functions with integral-order but non-integral degree. Evidently, the constructing and computing of Legendre functions are the core content of the spherical cap functions. In this paper,the approximated calculation method of the normalized association Legendre functions with non-integral degree is introduced and an analysis of the entire order of associated non-Legendre function calculation is presented. Besides, we use the Muller method to search out for all intrinsic values. The results showed that the highest order of spherical harmonic function for constructing regional model of fields is limited, thus high-resolution spherical harmonic structure of local gravity field need to be improved.

scholarly journals Simplification of Geopotential Perturbing Force Acting on A Satellite

2008 ◽  
Vol 43 (2) ◽  
pp. 45-64 ◽  
Author(s):  
M. Eshagh ◽  
M. Abdollahzadeh ◽  
M. Najafi-Alamdari
Keyword(s):  
Maximum Degree ◽  
Legendre Functions ◽  
Orbital Elements ◽  
Associated Legendre Functions ◽  
Geopotential Models ◽  
Orbit Integration ◽  
Vector Differential Equation ◽  
Derivatives Of ◽  
Geopotential Coefficients ◽  
Short Arc

Simplification of Geopotential Perturbing Force Acting on A SatelliteOne of the aspects of geopotential models is orbit integration of satellites. The geopotential acceleration has the largest influence on a satellite with respect to the other perturbing forces. The equation of motion of satellites is a second-order vector differential equation. These equations are further simplified and developed in this study based on the geopotential force. This new expression is much simpler than the traditional one as it does not derivatives of the associated Legendre functions and the transformations are included in the equations. The maximum degree and order of the geopotential harmonic expansion must be selected prior to the orbit integration purposes. The values of the maximum degree and order of these coefficients depend directly on the satellite's altitude. In this article, behaviour of orbital elements of recent geopotential satellites, such as CHAMP, GRACE and GOCE is considered with respect to the different degree and order of geopotential coefficients. In this case, the maximum degree 116, 109 and 175 were derived for the Earth gravitational field in short arc orbit integration of the CHAMP, GRACE and GOCE, respectively considering millimeter level in perturbations.

scholarly journals GENERALIZED COHERENT STATES FOR THE SPHERICAL HARMONICS $Y_{m}^{m}(\theta,\phi)$

2010 ◽  
Vol 25 (10) ◽  
pp. 2165-2170 ◽  
Author(s):  
H. FAKHRI ◽  
B. MOJAVERI
Keyword(s):  
Spherical Harmonics ◽  
Generating Functions ◽  
Coherent States ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Rodrigues Formula ◽  
Generalized Coherent States ◽  
Infinite Sequences

The associated Legendre functions [Formula: see text] for a given l-m, may be taken into account as the increasing infinite sequences with respect to both indices l and m. This allows us to construct the exponential generating functions for them in two different methods by using Rodrigues formula. As an application then we present a scheme to construct generalized coherent states corresponding to the spherical harmonics [Formula: see text].

scholarly journals Problems and methods of calculating the Legendre functions of arbitrary degree and order

2016 ◽  
Vol 65 (2) ◽  
pp. 283-312 ◽  
Author(s):  
Elena Novikova ◽  
Alexander Dmitrenko
Keyword(s):  
Maximum Degree ◽  
Absolute Zero ◽  
Legendre Functions ◽  
Numerical Accuracy ◽  
Arbitrary Degree ◽  
Associated Legendre Functions ◽  
Calculation Time ◽  
Computational Speed

Abstract The known standard recursion methods of computing the full normalized associated Legendre functions do not give the necessary precision due to application of IEEE754-2008 standard, that creates a problems of underflow and overflow. The analysis of the problems of the calculation of the Legendre functions shows that the problem underflow is not dangerous by itself. The main problem that generates the gross errors in its calculations is the problem named the effect of “absolute zero”. Once appeared in a forward column recursion, “absolute zero” converts to zero all values which are multiplied by it, regardless of whether a zero result of multiplication is real or not. Three methods of calculating of the Legendre functions, that removed the effect of “absolute zero” from the calculations are discussed here. These methods are also of interest because they almost have no limit for the maximum degree of Legendre functions. It is shown that the numerical accuracy of these three methods is the same. But, the CPU calculation time of the Legendre functions with Fukushima method is minimal. Therefore, the Fukushima method is the best. Its main advantage is computational speed which is an important factor in calculation of such large amount of the Legendre functions as 2 401 336 for EGM2008.

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