On the computation and approximation of ultra-high-degree spherical harmonic series

Jekeli, Christopher; Lee, Jong Ki; Kwon, Jae-Sool

doi:10.1007/s00190-006-0123-z

Cited by 57 publications

(33 citation statements)

References 18 publications

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“…Furthermore, there are several strategies with recursions in n and m and these are far from being numerically equivalent (see e.g. [1,3,14]). The geodetically normalized associated Legendre functions nm P (cos ) θ are computed as a lower triangular matrix with the rows corresponding to the degrees n and the columns corresponding to the orders m. With the initialization for degrees and orders 0 and 1, 00 10 P (cos ) 1, P (cos ) 3 cos θ = θ = θ and 11 P (cos ) 3sin , θ = θ the diagonal terms nn n 1,n 1 P (cos ) (2n 1) / 2n sin P (cos ) [18], based on [10,15].…”

Section: Numerical Preconditioning and Optimizationmentioning

confidence: 99%

“…This strategy of biasing the EXPONENT is independent of the colatitude θ which is very important in this context. The literature on computing very high degree spherical harmonics contains numerous strategies such as using the Clenshaw summation approach [13], preconditioning the variable with 10 280 sinθ in SHTools [19], modifying the recursion to avoid high powers of sinθ [14], and others. These modifications have been reported to achieve degrees around 2700-2800 in synthesis computations [13,14] and in synthesis and analysis [19].…”

Section: Numerical Preconditioning and Optimizationmentioning

confidence: 99%

“…The literature on computing very high degree spherical harmonics contains numerous strategies such as using the Clenshaw summation approach [13], preconditioning the variable with 10 280 sinθ in SHTools [19], modifying the recursion to avoid high powers of sinθ [14], and others. These modifications have been reported to achieve degrees around 2700-2800 in synthesis computations [13,14] and in synthesis and analysis [19]. Notice that in synthesis only computations, it is only necessary to avoid underflows and set the corresponding incremental contributions to zero while in synthesis and analysis computations, the numerical recovery of the input spectral coefficients is expected in addition to avoiding possible underflows.…”

Section: Numerical Preconditioning and Optimizationmentioning

confidence: 99%

“…With proper normalization such as the geodetic one used in the following computations, one can achieve degrees and orders to around 1800 in REAL*8 or double precision arithmetic [4] and over 3600 in REAL*16 or quadruple precision arithmetic [5,6]. With latitudebased preconditioning, [13,14,19] achieved about 2700. With Clenshaw's approach, similar results can be achieved for appropriately decreasing spectral coefficients such as with the EGM06 model coefficients of degree and order 2190 [7,13].…”

Section: Introductionmentioning

confidence: 99%

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Discrete Spherical Harmonic Transforms: Numerical Preconditioning and Optimization

Blais

2008

Computational Science – ICCS 2008

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Abstract. Spherical Harmonic Transforms (SHTs) which are essentially Fourier transforms on the sphere are critical in global geopotential and related applications. Among the best known strategies for discrete SHTs are Chebychev quadratures and least squares. The numerical evaluation of the Legendre functions are especially challenging for very high degrees and orders which are required for advanced geocomputations. The computational aspects of SHTs and their inverses using both quadrature and least-squares estimation methods are discussed with special emphasis on numerical preconditioning that guarantees reliable results for degrees and orders up to 3800 in REAL*8 or double precision arithmetic. These numerical results of spherical harmonic synthesis and analysis using simulated spectral coefficients are new and especially important for a number of geodetic, geophysical and related applications with ground resolutions approaching 5 km.

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Section: Numerical Preconditioning and Optimizationmentioning

confidence: 99%

Section: Numerical Preconditioning and Optimizationmentioning

confidence: 99%

Section: Numerical Preconditioning and Optimizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Discrete Spherical Harmonic Transforms: Numerical Preconditioning and Optimization

Blais

2008

Computational Science – ICCS 2008

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“…No tests with 'real' spherical harmonic coefficients were presented, but are in Holmes (2003). Jekeli et al (2007) have proposed a new approach for the computation of ALFs. It is mainly based on the observation that ALFs show a very strong attenuation in the degree-andorder domain for specific orders as function of the degree and the co-latitude.…”

Section: Introductionmentioning

confidence: 99%

Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic

Wittwer

Klees

Seitz

et al. 2007

J Geod

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We present software for spherical harmonic analysis (SHA) and spherical harmonic synthesis (SHS), which can be used for essentially arbitrary degrees and all co-latitudes in the interval (0 • , 180 • ). The routines use extended-range floating-point arithmetic, in particular for the computation of the associated Legendre functions. The price to be paid is an increased computation time; for degree 3, 000, the extended-range arithmetic SHS program takes 49 times longer than its standard arithmetic counterpart. The extended-range SHS and SHA routines allow us to test existing routines for SHA and SHS. A comparison with the publicly available SHS routine GEOGFG18 by Wenzel and HARMONIC SYNTH by Holmes and Pavlis confirms what is known about the stability of these programs. GEOGFG18 gives errors <1 mm for latitudes [−89 • 57.5 , 89 • 57.5 ] and maximum degree 1, 800. Higher degrees significantly limit the range of acceptable latitudes for a given accuracy. HARMONIC SYNTH gives good results up to degree 2, 700

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Topographic gravity modeling for global Bouguer maps to degree 2160: Validation of spectral and spatial domain forward modeling techniques at the 10 microGal level

2016

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Over the past years, spectral techniques have become a standard to model Earth's global gravity field to 10 km scales, with the EGM2008 geopotential model being a prominent example. For some geophysical applications of EGM2008, particularly Bouguer gravity computation with spectral techniques, a topographic potential model of adequate resolution is required. However, current topographic potential models have not yet been successfully validated to degree 2160, and notable discrepancies between spectral modeling and Newtonian (numerical) integration well beyond the 10 mGal level have been reported. Here we accurately compute and validate gravity implied by a degree 2160 model of Earth's topographic masses. Our experiments are based on two key strategies, both of which require advanced computational resources. First, we construct a spectrally complete model of the gravity field which is generated by the degree 2160 Earth topography model. This involves expansion of the topographic potential to the 15th integer power of the topography and modeling of short‐scale gravity signals to ultrahigh degree of 21,600, translating into unprecedented fine scales of 1 km. Second, we apply Newtonian integration in the space domain with high spatial resolution to reduce discretization errors. Our numerical study demonstrates excellent agreement (8 μGgal RMS) between gravity from both forward modeling techniques and provides insight into the convergence process associated with spectral modeling of gravity signals at very short scales (few km). As key conclusion, our work successfully validates the spectral domain forward modeling technique for degree 2160 topography and increases the confidence in new high‐resolution global Bouguer gravity maps.

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On the computation and approximation of ultra-high-degree spherical harmonic series

Cited by 57 publications

References 18 publications

Discrete Spherical Harmonic Transforms: Numerical Preconditioning and Optimization

Discrete Spherical Harmonic Transforms: Numerical Preconditioning and Optimization

Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic

Topographic gravity modeling for global Bouguer maps to degree 2160: Validation of spectral and spatial domain forward modeling techniques at the 10 microGal level

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