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associated legendre polynomials

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Journal Article
Published: 01 April 1971
Bulletin of the Seismological Society of America (1971) 61 (2): 375–381.
...Ralph A. Wiggins; Masanori Saito abstract An interval arithmetic that consists of tracing the number of significant figures during each calculation is applied to computational algorithms for the Associated Legendre Polynomial, P n m (cos ϑ). The results indicate that the interval arithmetic scheme...
Journal Article
Published: 01 April 1964
Bulletin of the Seismological Society of America (1964) 54 (2): 755–776.
... poly- nomials or the associated Legendre polynomials at the epicentra] distance is the simplest test which will definitely show if the spectral peaks do not have amplitudes similar to polynomials of a given degree m. Therefore the zero point curves for PJ and P~/of Sat6 and Usami (1963), who proposed...
Journal Article
Journal: Geophysics
Published: 19 June 2017
Geophysics (2017) 82 (4): R259–R268.
... based on the distribution of the input variable according to Table  1 . Table 1. Orthogonal polynomials associated to different distributions along with their standard support. Distribution ( ρ ( X ) ) Polynomial ( ψ ) Support ( Ω ) Normal Hermite...
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Journal Article
Journal: Geophysics
Published: 29 January 2016
Geophysics (2016) 81 (1): G13–G26.
... | ) are extracted as candidates for the test. Figure 1. Numerical stability test. Relative error growth due to increasing dimensionless target distance for the Gauss-FFT method applied to (a) 2D and (b) 3D general polynomial density models of different orders. The ordinary Gauss-Legendre numerical...
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Journal Article
Published: 01 February 1999
Bulletin of the Seismological Society of America (1999) 89 (1): 332–334.
... be of interest to compare our results for the realistic 3D propagation problems (Komatitsch and Vilotte, 1996) with their method. The spectral element method, including the formulation based on Legendre polynomials and Gauss-Lobatto Legen- dre quadrature, has become aclassical high-order variational formulation...
Journal Article
Journal: Geophysics
Published: 01 January 2003
Geophysics (2003) 68 (6): 1909–1916.
...Juan García-Abdeslem Abstract A method is developed for 2D forward modeling and nonlinear inversion of gravity data. The forward modeling calculates the gravity anomaly caused by a 2D source body with an assumed depth-dependent density contrast given by a cubic polynomial. The source body...
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Journal Article
Journal: Geophysics
Published: 11 July 2016
Geophysics (2016) 81 (5): F41–F48.
... ∑ i = 1 N W i f ( x i ) , (11) where N is the order of the quadrature, i.e., the number of points used in the GLQ. The points x i are called the quadrature nodes. They are the roots of the N th-order Legendre polynomial P N ( x...
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Journal Article
Journal: Geophysics
Published: 20 February 2000
Geophysics (2000) 65 (4): 1251–1260.
...-Legendre (GLL) points which are the ( N + 1) roots of (4) where P ′ N ( ξ ) is the derivative of the Legendre polynomial of degree N . On the reference domain Λ, the restriction of a given function u N to the element Ω e can be expressed as (5) where h p ( ξ ) denotes the p th 1-D...
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Journal Article
Published: 01 March 2013
Russ. Geol. Geophys. (2013) 54 (3): 263–271.
... the equation ( Plotkin, 2004 ) The coefficients k and k ′ ( k ) in Table 1 , for each spherical harmonic (SH), correspond to numbers in ( 5 ), so that k ′ coincides with the subscript of the associated Legendre polynomial and is the harmonic degree. Taking into account ( 4 )–( 5 ), from ( 2...
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Journal Article
Journal: Geophysics
Published: 15 November 2018
Geophysics (2018) 83 (6): U79–U88.
..., we assume the velocities to be uniformly distributed within the intervals [ V NMO min , V NMO max ] and [ V hor min , V hor max ] . Using the Legendre polynomials as the orthogonal basis for the PC expansion, we have to first select a proper change...
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Journal Article
Published: 01 May 2008
Environmental and Engineering Geoscience (2008) 14 (2): 121–131.
...-selected points of integration and their associated weights for the Gauss-Legendre rule. Two, three, and four different points are shown. Values can be readily obtained from mathematical tables. Essentially one is approximating a series of rectangle areas in an interval and then summing them...
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Journal Article
Published: 01 April 1998
Bulletin of the Seismological Society of America (1998) 88 (2): 368–392.
... the good stability and ap- by the discretization pair (nel, IV). Each element integral in- proximation properties of the polynomial spaces and from the accuracy associated with the Gauss-Lobatto Legendre volved in the variational formulation, defined over the do- quadrature and interpolation. The discrete...
Journal Article
Journal: Geophysics
Published: 22 January 2021
Geophysics (2021) 86 (1): T61–T74.
... ( Priolo et al., 1994 ; Seriani and Priolo, 1994 ). Nonetheless, Legendre polynomials, together with a Gauss-Lobatto (GL) quadrature rule, have become the standard for SEM because this formulation yields a mass matrix that is inherently diagonal ( Faccioli et al., 1996 ; Komatitsch and Vilotte, 1998...
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Journal Article
Journal: Geophysics
Published: 17 November 1999
Geophysics (2000) 65 (2): 623–631.
...) where P ′ N (ξ) is the derivative of the Legendre polynomial of degree N . These ( N +1) points can be computed by numerical resolution of equation ( 12 ) ( Canuto et al., 1988 , p. 61). We subsequently choose the set of basis functions to be products of the ( N +1) 1-D Lagrange interpolants h...
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Journal Article
Journal: Geophysics
Published: 21 October 2020
Geophysics (2020) 85 (6): S343–S355.
... GLL points and Legendre basis functions perform well in suppressing numerical dispersion. Moreover, the Legendre polynomials are orthogonal under the L 2 inner product and have simple recursion formulas. Based on these analysis, we use the basis functions formed by the tensor product...
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Journal Article
Journal: Geophysics
Published: 19 December 2019
Geophysics (2020) 85 (1): T33–T43.
.... Compared to classic low-degree finite elements, Legendre SEM is based on a tensorised high-degree polynomial approximation per element combined with a precise numerical quadrature associated with the so-called Gauss-Lobatto-Legendre (GLL) points. It has a spectral convergence with the element polynomial...
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Journal Article
Journal: Geophysics
Published: 20 October 2021
Geophysics (2021) 86 (6): S417–S430.
... the standard polynomial basis throughout the text, we note that other choices are available. Orthogonal or orthonormal polynomial basis (e.g., Legendre or Hermite polynomials) have the same range as the basis we used and therefore cannot improve accuracy, but they have analytical formulas for computing...
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Journal Article
Published: 01 December 2011
Bulletin of the Seismological Society of America (2011) 101 (6): 2855–2865.
... of the Legendre polynomial of degree N computed numerically (e.g., Canuto et al. , 1987 ). In the reference cube Λ , the restriction of a given function u e (e.g., a component of the displacement) to the element Ω e can be expressed using a product of 1D Lagrange interpolants, a property...
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Journal Article
Published: 01 January 2020
Russ. Geol. Geophys. (2020) 61 (1): 110–118.
... are the Legendre polynomials; m = 0 and n ≠ 0 are the associated Legendre functions). The D functions have two important properties. First, D m n ( l ) at constant l is a unitary matrix: D m n ( l ) ( u ) D m k ( l ) ( u ) = δ n k , (17...
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Journal Article
Journal: Geophysics
Published: 02 July 2019
Geophysics (2019) 84 (4): T299–T311.
... with Taylor-series, Chebyshev, Hermite, and Legendre polynomial expansions or any other expansion for the cosine function and used for seismic modeling, reverse time migration, and inverse problems. Extension of this method to the solution of elastic and anisotropic wave equations is also straightforward. We...
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