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Cauchy transform

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Journal Article
Published: 01 April 2003
American Mineralogist (2003) 88 (4): 694–700.
...Manuel Muñoz; Pierre Argoul; François Farges Abstract To better understand the extended X-ray absorption fine structure (EXAFS) spectroscopic information obtained for complex materials such as those encountered in Earth materials, we propose to use the Continuous Cauchy Wavelet Transform (CCWT...
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Series: Society of Exploration Geophysicists Geophysics Reprint Series, Society of Exploration Geophysicists Geophysics Reprint Series
Published: 01 January 2016
EISBN: 9781560803188
... is the derivation of boundary integral equations in terms of reflection/transmission phenomena. Such equations can be obtained by means of a special transformation of the Cauchy data in surface integrals (reflection/transmission transform —RTT) that takes into account the boundary conditions. The representation...
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Journal Article
Journal: Geophysics
Published: 01 December 1991
Geophysics (1991) 56 (12): 2027–2035.
... of the scale parameter in the Cauchy norm is given.We compare the least-squares and Cauchy error criteria by inverting synthetic data corrupted by random noise and weather noise. The data are transformed to the frequency-wavenumber domain before the inversion starts. The numerical examples show...
Journal Article
Journal: Geophysics
Published: 01 January 2000
Geophysics (2000) 65 (4): 1154–1161.
... transform by using Bayes' rule to combine prior information with the data likelihood and modeling the prior distribution of the spectral amplitudes by means of the “long-tailed” Cauchy probabilistic density function. The result is a sparse distribution of the Fourier coefficients, which suggests...
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Journal Article
Journal: Geophysics
Published: 01 June 1984
Geophysics (1984) 49 (6): 780–786.
... on the Y component. As for the previously developed 2-D case, it is shown that in 3-D the vertical and horizontal derivatives are the Hilbert transforms of each other. The 2-D Cauchy-Riemann relations between a potential function and its Hilbert transform are generalized for the 3-D case.--Modified journal...
Journal Article
Published: 01 December 1986
Bulletin of the Seismological Society of America (1986) 76 (6): 1755–1775.
... with respect to k is performed exactly using Cauchy residue theory. The remaining integration with respect to ω is then carried out by use of the Fast Fourier transform. The method is simply to apply, accurate, and computationally efficient. There are no spurious arrivals, and provided the number of points...
Series: Society of Exploration Geophysicists Geophysics Reprint Series, Society of Exploration Geophysicists Geophysics Reprint Series
Published: 01 January 2007
DOI: 10.1190/1.9781560801931.ch4c
EISBN: 9781560801931
... infinitely small paths [ functions]) [1, 1st ser., I, 135, 288–303, 402–406; 2nd ser., I, 335–339; IV, 145–150; XII, 409–469]. Cauchy made much use of discontinuous factors [1, 1st ser., XII, 79–94] and of the Fourier transform (see under Differential Equations). Cauchy also invented what is now called...
Journal Article
Journal: Geophysics
Published: 22 April 2009
Geophysics (2009) 74 (3): S47–S56.
... weights derived from the non-Cartesian geometry. Generalized-coordinate ADCIGs can be calculated directly using Radon-based offset-to-angle approaches for coordinate systems satisfying the Cauchy-Riemann differentiability criteria. The canonical examples of tilted-Cartesian, polar, and elliptical...
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Journal Article
Journal: Geophysics
Published: 01 May 1989
Geophysics (1989) 54 (5): 609–620.
.... There are no contributions from branch line integrals. The integrations with respect to k are performed exactly using Cauchy residue theory. The remaining integrations with respect to omega are then carried out partly by using the fast Fourier transform (FFT) and partly by using another numerical method. Provided...
Journal Article
Published: 01 October 1987
Bulletin of the Seismological Society of America (1987) 77 (5): 1804–1822.
...-plane, the integration with respect to k in the well-known integrals representing the exact solution is performed exactly using Cauchy residue theory. The remaining integration (with respect to ω) is then performed approximately by use of the Fast Fourier transform and by other numerical integration...
Journal Article
Journal: Geophysics
Published: 01 August 1987
Geophysics (1987) 52 (8): 1099–1105.
... to k are performed exactly using Cauchy residue theory. The remaining integrations with respect to omega are then carried out using the fast Fourier transform (FFT). The complete solution, including all possible body waves, is expressed simply as a superposition of modes without any contributions from...
Journal Article
Journal: Geophysics
Published: 15 March 2016
Geophysics (2016) 81 (2): S69–S77.
... , ω > 0 0 , ω = 0 − 1 , ω < 0 , (7) where u ˜ ( x , ω ) is the Fourier transform of u ( x , t ) with respect to t . Equation  7 defines the CWF because it is constructed from the Cauchy condition of the wave equation  1 . Note...
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Journal Article
Published: 31 May 2016
Bulletin of the Seismological Society of America (2016) 106 (4): 1628–1636.
... . Zheng G. Rice J. R. ( 1998 ). Conditions under which velocity‐weakening friction allows a self‐healing versus a cracklike mode of rupture , Bull. Seismol. Soc. Am. 88 , no.  6 , 1466–1483 . On the other hand, Cauchy’s principal value has a logarithmic singularity if Δ u...
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Journal Article
Published: 28 June 2019
Journal of Environmental and Engineering Geophysics (2019) 24 (2): 215–224.
... transform is performed on the signal to obtain frequency domain electromagnetic data Z ( i ω), including E x ( i ω) and H z ( i ω). This method differs from the conventional controlled source electromagnetic method in that two sets of time and frequency data are acquired in one acquisition, thereby...
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Journal Article
Journal: Geophysics
Published: 02 May 2011
Geophysics (2011) 76 (3): R43–R55.
.... This promoted the Univariate Cauchy probability distribution for high-resolution Radon transform ( Sacchi and Ulrych, 1995 ) and AVO inversion ( Downton, 2005 ; Theune et al., 2010 ). However, the Univariate Cauchy probability distribution does not allow including the correlation information in the three-term...
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Journal Article
Journal: Interpretation
Published: 05 December 2016
Interpretation (2017) 5 (1): T107–T119.
...-Cauchy distribution obtained from the statistical analysis of petrophysics data. To demonstrate the effectiveness of our algorithm, we applied our inversion strategy to a 2D synthetic model and a real seismic data set, and an uncertainty assessment was also performed. The joint inversion method can...
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Journal Article
Journal: Geophysics
Published: 11 October 2006
Geophysics (2006) 71 (6): V145–V152.
... (2001 , 2005) show that the (aliased) crossline summation can be replaced by an inversion procedure, similar to a hyperbolic Radon transform, parameterized by intercept time, apex position, and curvature. This time-domain approach leads to few, but large, systems of linear equations. Numerically...
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Journal Article
Journal: Geophysics
Published: 05 September 2007
Geophysics (2007) 72 (6): R109–R114.
... and dispersion are bonded together by the Hilbert transform which is central in causal systems. We are concerned with poststack trace-by-trace deabsorption processing. In reality, a Q curve is required for each trace and, consequently, a Q section is the ultimate desideratum. Many techniques for Q...
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Journal Article
Journal: Geophysics
Published: 12 December 2008
Geophysics (2009) 74 (1): L7–L15.
... weighting of the solution. Sparseness is imposed by using the Cauchy norm on model parameters. The inverse problem is posed in the data space, leading to a linear system of equations with dimensions based on the number of data, N . This contrasts with the standard least-squares solution, derived through...
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Journal Article
Published: 01 August 2011
Bulletin of the Seismological Society of America (2011) 101 (4): 1807–1821.
... with power law behaviors controlled by α ∼1. The analysis of the PDF s of the random variables associated with the slip spatial distribution of two source models suggests that α ∼1.45, though the Cauchy law ( α =1) provides a reasonable fit of the same PDF s. * Now at Earth Research Institute...
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