method is known as HEALPix, while Crittenden has countered with IGLOO. Lukatela's method is based on Voronoi's tesselations. These are illustrated below.
Optimization of Computations in Spherical Geopotential Field Applications J.A.R. Blais, D.A. Provins and C.J.K. Tan* Department of Geomatics Engineering, University of Calgary2500 University Drive NW, Calgary, Alberta T2N 1N4 *High Performance Computing Centre, University of Reading, Reading, United Kingdom RG6 6AY
Colombo’s Method For an equiangular sampled dataset, a function F may be expanded as
Introduction Geopotential field applications generally involve the analysis of large quantities of data. They are generally sampled irregularly, but are re-sampled for in depth study. Of the many methods of analysis, Legendre transformation stands out as the classical method of determining the spectral components of geopotential data, or of the many other global datasets that are being collected today.
^ α
C nm =
Over the years, many methods have been developed for performing the necessary numerical calculations. Several of these are illustrated in the following presentation. The early methods took advantage of fast techniques such as the Fast Fourier transform. Later methods have explored other techniques.
N −1 2 N −1 −
∑ ∑P i =0 j =0
nm
cos (cosθ i ) (mj∆λ ) f (θ i , λ j ) ∆ ij î sin
4π
^ α
In fact, with the advances in space technology, global datasets have become increasingly common. Researchers are now designing measurement systems that acquire datasets whose size was unimaginable only a few years ago. Over the life of current and planned projects, it is estimated that quantities of the order of tens of thousands of terabytes will be collected.
1 4π
Substituting X inm = 1 P− nm (cosθ i ) ∆ ij , then C nm =
N / 2−1
∑ i =0
nm 2 N −1cos X i ∑ ( mj∆) f (θ i , λ j ) + î j=0 î sin
2 N −1cos (−1) n −m X inm ∑ (mj∆) f (θ n −1−i , λ j ) j=0 î sin
Like other methods, a Fourier transform is employed and a weighted sum of coefficients determines the final Legendre coefficients [Colombo 1981] ^ α
^ α ( i −1)
C nm = C nm
a a + K + ( − 1) n − m î b î b i m i m
N −1− i m N −1− i m
Mohlenkamp’s Method The method uses Gaussian nodes and weights as only N points can achieve the accuracy of a polynomial of degree 2N-1.
Power Spectrum of the EGM96 Model
To facilitate the calculation of the spherical harmonic coefficients with O( N 5 / 2 log , aN ) local cosine basis is used for sparse matrices [Mohlenkamp 1999]. The following figure shows the associated Legendre function P6020 (cosθ ) weighted by sin θ .
π
nm X i
Processing Results
Methods of Analysis Several methods computing the Legendre transform of data on the sphere have been developed over the past 30 years. Of note are those of Ricardi and Burrows in 1972, Colombo in 1981, Driscoll and Healy in 1994 and most recently, Mohlenkamp in 1999. These and others have exploited the availability of the FFT to reduce the number of operations from O(N4) to O(N3logN) at the very least.
Experimentation with various Legendre transform codes, including an implementation from Ottawa, an in-house development of the Driscoll and Healy code and the publicly available SpherePack code is illustrated in the GEMT1 and EGM96 models below:
Data Structures Various data structures have been proposed to facilitate the handling and the analysis of global datasets. Among these are those proposed by Gorski, Crittenden and locally by Lukatela. Gorski’s method is known as HEALPix, while Crittenden has countered with IGLOO. Lukatela’s method is based on Voronoi’s tesselations. These are illustrated below.
The newest methods have reduced the number of operations even further. For example, that proposed by Driscoll and Healy is of order O(N2(log N) 3) for a data structure of O(N2). They find an exact analysis of the band-limited data described by N2 points, where N is a power of 2. Their inverse is said to be O(N3). Mohlenkamp describes a method of analysing N2 data values in O(N5/2log N), or with somewhat less accuracy, in O((NlogN)2). Unlike the method of Driscoll and Healy, Mohlenkamp uses Gaussian nodes for his points of analysis, much like the publicly available SpherePack software.
Spherical Harmonics The data are given on a grid where, for polar coordinates 0 ≤ θ ≤ π and 0 ≤ λ < 2π , Analysis m −
f n,m = ∫ f (θ , λ)Y n dσ σ
(2n + 1)(n − m)! = f (θ , λ )Pnm (cosθ )eimλ dσ 4π (n + m)! ∫
Geodetic Formulation The data are given on a grid where, for polar coordinates 0 ≤ θ ≤ π and 0 ≤ λ < 2π,
Various derivatives of the Earth’s potential are illustrated below.
Synthesis f (θ , λ ) =
∞
−
∑ ∑ C
f (θ , λ ) =
∑∑f n =0 |m|≤ n
n ,m
where
Y (θ , λ )
and
Y nm (θ , λ ) = ( − 1) m
1 4π
σ
1 4π
−
∫
∫ f (θ , λ ) cos mλ P
nm
σ
2(2n + 1)(n − m)! Pnm (cosθ ) (n + m)!
−
Other Approaches Monte Carlo methods provide the opportunity to solve integrals in an approximate, but rapid manner. Quadrature methods, such as those described earlier, imply an error of O(N-1). Quasi-Monte Carlo methods approach this error limit asymptotically, and the computational effort is simply linear in terms of the function evaluation. For additional background, see [Press et al., 1992].
(cosθ )dσ
−
f (θ , λ ) sin mλ P nm (cosθ )dσ
−
Pn (cosθ ) = 2n + 1Pn (cosθ )
and
Pnm (cosθ ) = ( −1) m Pnm (cosθ )
−
−
Pnm (cosθ ) =
( 2 n + 1)( n − m )! m Pn (cos θ ) e im λ 4π ( n + m )!
− cos m λ + S nm sin m λ Pnm (cos θ )
C nm = S nm =
m n
where
nm
n = 0 |m | ≤ n
Synthesis ∞
Field Derivatives
Assuming that fast Fourier transforms are employed for the longitudinal components, then quasi-Monte Carlo approximations may be employed for some partition of the latitude axis. Various strategies may be employed, and the resulting algorithms can be run in parallel very efficiently. The FFT operation leaves N identical integrals in latitude to be evaluated, all of which may be run in parallel. As Mohlenkamp has indicated, given “p” processors, the computational effort would be reduced by “p”.
See [Blais et al. 2000]
[Varshalovish et al. 1988]
Driscoll and Healy Formulation The data are given on a regular grid where, for 2b = 2 N, πj 2b πk λk = b
θj =
j, k = 0,...,2b − 1
Ricardi and Burrows Formulation The data are given on an equiangular grid where, for polar coordinates 0 ≤ θ ≤ π and 0 ≤ λ < 2,πa Fourier transform over the m coordinate results in
f
e ,o m
(θ ) = ∑ a pnm (cosθ ) e, o nm
n
Analysis f n ,m aj =
2π = 2b
2 b −1 2 b −1
∑∑a j = 0 k =0
−m
j
f (θ j , λ k ) Y n (θ j , λ k )
2 πj b −1 1 πj sin ∑ sin (2h + 1) b 2b 2b b = 0 2 h + 1
Ap A second Fourier transform determines and establishes that the coefficients are e ,o anm =
2n + 1 (n − m)! ∑I nm, p Ap 2 (n + m)! p
for values of n=m, m+1,… The I nm, p are π
Synthesis b −1
f (θ j , λk ) = ∑ ∑ f n .m Ynm (θ j , λ k ) n =0 |m|≤ n
See [Driscoll and Healy 1994]
I nm , p = ∫ cos pθ Pnm (cos θ ) sin θ d θ
m
even
m
odd
0
References John C. Adams and Paul N. Swarztrauber, SPHEREPACK 2.0: A Model Development Facility, http://www.scd.ucar.edu/softlib/SPHERE.html (September 1997). J.A.R. Blais and D.A. Provins, Spherical Harmonic Analysis and Synthesis for Multiresolution Applications, Presented at the 25th Annual Meeting of the Canadian Geophysical Union, Banff, Canada (May, 2000). O.L. Colombo, "Numerical Methods for Harmonic Analysis on the Sphere," Report No. 310, Department of Geodetic Science, Ohio State University, Columbus (March 1981). Robert G. Crittenden and Neil G. Turok, Exactly Azimuthal Pixelizations of the Sky, http://xxx.lanl.gov/list/astro-ph/9806[374] (i.e. article 374 in the 9806 directory) (June 1998).
π
I nm , p = ∫ sin pθ Pnm (cos θ ) sin θ d θ
The I nm, pmay be calculated via recursion [Ricardi and Burrows, 1972] 0
Krzysztof M.Gorski, Eric Hivon and Benjamin D. Wandelt,
References "Analysis Issues for Large CMB Data Sets" in Proceedings:
James R.Driscoll and Dennis M. Healy, Jr ., "Computing Fourier Transforms and Convolutions on the 2-Sphere," Advances in Applied Mathematics, 15, pp .202-250 (1994).
Evolution of Large Scale Structure, Garching (August 1998). Hrvoje Lukatela, "Hipparchus Geopositioning Model: An Overview” in AUTO-CARTO 8,http://www.geodyssey.com (March 1987). Martin J. Mohlenkamp, "A Fast Transform for Spherical Harmonics," The Journal of Fourier Analysis and Applications, 5,2/3, pp.159-184, http://amath-www.colorado.edu/appm/faculty/mjm (1999). W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, numerical Recipes = The Art of Scientific Computing, Second Edition, Cambridge Press (1992). L.J. Ricardi and M.L. Burrows, "A Recurrence Technique for Expanding a Function in Spherical Harmonics,” IEEE Transactions on Computers, pp. 583-585 (June 1972). D.A. Varshalovich , A.N. Moskalev , and V.K. Khersonskii, Quantum Theory of Angular Momentum,World Scientific, Singapore (1988).
