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Gaussian quadrature

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Series: Society of Exploration Geophysicists Geophysical Monograph Series
Published: 01 January 2001
DOI: 10.1190/1.9781560802112.ch2
EISBN: 9781560802112
.... The integrals expressed by isoparametric elements usually cannot be calculated through analytical methods because they are rather complicated. Thus, numerical methods must be used. This chapter introduces the isoparametric element, followed by the Gaussian quadrature method of numerical integration (Rao, 1982...
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Journal Article
Journal: Geophysics
Published: 01 April 1977
Geophysics (1977) 42 (3): 610–622.
...C. C. Ku Abstract A computational method, which combines the Gaussian quadrature formula for numerical integration and a cubic spline for interpolation in evaluating the limits of integration, is employed to compute directly the gravity and magnetic anomalies caused by 2-dimensional and 3...
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Table 2. <span class="search-highlight">Gaussian</span> <span class="search-highlight">quadrature</span> spreadsheet for the first stratigraphic layer ...
Published: 01 May 2008
Table 2. Gaussian quadrature spreadsheet for the first stratigraphic layer of lacustrine clay before and after deformation. Column A shows the partition number. For example, the deformed layer was divided into three partitions. Columns B and C show the limits of integration, a and b. Column D shows
Journal Article
Journal: Geophysics
Published: 05 August 2014
Geophysics (2014) 79 (5): G59–G68.
... the trapezoidal rule with smaller steps. We tested a high-precision Fourier-domain forward method based on a combined use of shift-sampling technique and Gaussian quadrature theory. The trapezoidal rule applied by the standard FFT algorithm to evaluate the continuous Fourier transform was modified by introducing...
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Journal Article
Journal: Geophysics
Published: 13 May 2022
Geophysics (2022) E205–E217.
... used. Nevertheless, the new method is capable of reducing the memory requirement and computational time in the mixed domain, and the 1D governing equations are highly parallel among different wavenumbers. For each of the 1D equations, the two- or four-node Gaussian quadrature rule can be used in both...
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Journal Article
Journal: Geophysics
Published: 01 April 1988
Geophysics (1988) 53 (4): 546–552.
... quadrature formula is used to calculate the integral. In this way, the integral equation is converted into a set of linear equations. The potential field value on the ground surface is obtained by solving the linear equation system with Gaussian elimination.This method uses a special triangular element mesh...
Journal Article
Journal: Geophysics
Published: 19 July 2018
Geophysics (2018) 83 (5): E293–E302.
... functions are assumed in the tetrahedral and triangular elements to obtain the final system of linear equations. In the final system matrix, singularity-free analytical expressions are developed for entries arising from volume integrals over tetrahedral elements and Gaussian quadrature formulas are used...
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Journal Article
Journal: Geophysics
Published: 29 November 2017
Geophysics (2018) 83 (1): G1–G13.
... method for complicated target structures by comparing the results with those from a high-order Gaussian quadrature approach. 11 4 2017 30 8 2017 10 10 2017 2018 Society of Exploration Geophysicists density contrast horizontal and vertical variations closed-form solutions...
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Journal Article
Published: 01 May 2008
Environmental & Engineering Geoscience (2008) 14 (2): 121–131.
...Table 2. Gaussian quadrature spreadsheet for the first stratigraphic layer of lacustrine clay before and after deformation. Column A shows the partition number. For example, the deformed layer was divided into three partitions. Columns B and C show the limits of integration, a and b. Column D shows...
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Journal Article
Published: 01 February 1966
American Mineralogist (1966) 51 (1-2): 159–167.
...Charles W. Burnham Abstract A method of computing transmission factors using Gaussian quadrature numerical integration for crystals whose shape can be described by plane faces has been programmed specifically for equi-inclination Weissenberg diffraction geometry. This program is readily adaptable...
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Comparison of results between the 14-point <span class="search-highlight">quadrature</span> and the Eisenburg-Mal...
Published: 03 June 2011
for the 14-point Gaussian quadrature without significant loss of accuracy.
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Table 3. A comparison of initial and final layer areas obtained from Gaussi...
Published: 01 May 2008
Table 3. A comparison of initial and final layer areas obtained from Gaussian quadrature spreadsheets in units squared. Areas for Layers 1, 2, and 3 (numbered in Figure 6 ) before and after deformation are well balanced. The special case for Layer Number 4 is discussed in the text.
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Figure 6. Digitization of the four stratigraphic layers before and after sl...
Published: 01 May 2008
Figure 6. Digitization of the four stratigraphic layers before and after slip movements. The coordinates are exported to an Excel spreadsheet, in which Gaussian quadrature is applied to compute the area of each entire layer (A) before deformation and (B) after deformation. The dots locate
Journal Article
Journal: Geophysics
Published: 06 November 2020
Geophysics (2020) 85 (6): T315–T329.
... induced by moving loads. Zhou et al. (2009) develop a revised version of SEM, i.e., the “Gaussian quadrature grid” (GQG) method for 2.5D seismic wave modeling in heterogeneous elastic anisotropic media ( Zhou et al., 2012a ). Doyon and Giroux (2014) model viscoelastic waves in heterogeneous isotropic...
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The inversion results for the hydrocarbon trap model with 5% <span class="search-highlight">Gaussian</span> noise...
Published: 22 November 2013
Figure 12. The inversion results for the hydrocarbon trap model with 5% Gaussian noise in the data. The in-phase conductivity of the model is still well-recovered, with the general structure clearly present and defined. The inversion recovers the high values of the quadrature conductivity
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Several synthetic data time series and the corresponding spectrum following...
Published: 08 October 2015
Figure B-2. Several synthetic data time series and the corresponding spectrum following demodulation for the case in which T 2 = 150     ms , which allows the signal to be recorded for a duration of approximately 0.5 s. (a) The real time-domain signal, (b) the quadrature time
Journal Article
Journal: Geophysics
Published: 08 August 2012
Geophysics (2012) 77 (5): ID1–ID8.
... using grids based on maximally sparse cubature integration rules. The N + 1 (degree-two) and 2 N (degree-three) cubature rules exist to exactly integrate symmetric Gaussian distributions in N stochastic dimensions. These represent the two sparsest integration rules possible and thus provide the most...
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Journal Article
Journal: Geophysics
Published: 01 January 2001
Geophysics (2001) 66 (4): 1110–1115.
.... Gravity terrain modeling by Gaussian quadrature integration has also been reported by Mateskon and von Frese (1985) . Terrain corrections for airborne gravimetry have also been treated by Hammer (1974 , 1976 ) and von Frese et al. (1991) . Substitution of the lower limit in the integrand...
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Journal Article
Journal: Geophysics
Published: 15 October 2008
Geophysics (2008) 73 (6): I43–I50.
... boundary has the added advantage that the interval [0, 1] for t is adapted easily for Gaussian quadrature. Thus, equation 15 becomes (31) g z ( x i , 0 ) = − 2 G ∑ k = 1 M ∫ z k z k + 1 Δ ρ ( z ) arctan ( x − x i z ) d z...
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Journal Article
Journal: Geophysics
Published: 19 June 2017
Geophysics (2017) 82 (4): R259–R268.
... will be the most time-consuming part of the gPC calculation. Thus, we have to look for the integration schemes that need minimum evaluations of g ( x ) . That is why we need Gaussian quadrature methods (Appendix  B ). One interesting feature of PC expansion is the possibility to calculate...
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