Two-and-One-Half-Dimensional Inversion

Stockwell, John W.; Cohen, Jack K.

doi:10.1007/978-1-4613-0001-4_6

Cited by 1 publication

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“…The derivation starts with the Kirchhoff approximation for the upward scattered wave from a single reflector. This representation can be found in many sources; we use equation (5.3.1) of Bleistein et al . (1998 ):…”

Section: Derivation Of a 3d Kirchhoff Data Mapping Formulamentioning

confidence: 99%

“…This volume integral representation of Kirchhoff‐approximate forward modelling could be used to derive an inversion formula for β, following the generalized Radon transform method introduced by Beylkin (1985) for this problem or by the asymptotic Fourier approach of Bleistein et al . (1998 ).…”

Section: Derivation Of a 3d Kirchhoff Data Mapping Formulamentioning

confidence: 99%

“…The correspondence of elements is as follows, with constituents of equation (5.1.7) of Bleistein et al . (1998 ) appearing on the left and constituents of (5) appearing on the right:…”

Section: Derivation Of a 3d Kirchhoff Data Mapping Formulamentioning

confidence: 99%

“…In practice, it is easiest first to apply the stationary phase analysis in ξ I2 and then in x 2 as iterated integrals, rather than to do them together as a two‐dimensional stationary phase calculation. The analysis has already been carried out by Bleistein et al . (1998 ) and only the results will be reported here.…”

Section: 5d Kirchhoff Data Mappingmentioning

confidence: 99%

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A platform for Kirchhoff data mapping in scalar models of data acquisition

Bleistein

Jaramillo

2000

Geophysical Prospecting

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Kirchhoff data mapping (KDM) is a procedure for transforming data from a given input source/receiver configuration and background earth model to data corresponding to a different output source/receiver configuration and background model. The generalization of NMO/DMO, datuming and offset continuation are three examples of KDM applications. This paper describes a ‘platform’ for KDM for scalar wavefields. The word, platform, indicates that no calculations are carried out in this paper that would adapt the derived formula to any one of a list of KDMs that are presented in the text. Platform formulae are presented in 3D and in 2.5D. For the latter, the validity of the platform equation is verified — within the constraints of high‐frequency asymptotics — by applying it to a Kirchhoff approximate representation of the upward scattered data from a single reflector and for an arbitrary source/receiver configuration. The KDM formalism is shown to map this Kirchhoff model data in the input source/receiver configuration to Kirchhoff data in the output source/receiver configuration, with one exception. The method does not map the reflection coefficient. Thus, we verify that, asymptotically, the ray theoretical geometrical spreading effects due to propagation and reflection (including reflector curvature) are mapped by this formalism, consistent with the input and output modelling parameters, while the input reflection coefficient is preserved. In this sense, this is a ‘true‐amplitude’ formalism. As with earlier Kirchhoff inversion, a slight modification of the kernel of KDM provides alternative integral operators for estimating the specular reflection angle, both in the input configuration and in the output configuration, thereby providing a basis for amplitude‐versus‐angle analysis of the data.

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Section: Derivation Of a 3d Kirchhoff Data Mapping Formulamentioning

confidence: 99%

Section: Derivation Of a 3d Kirchhoff Data Mapping Formulamentioning

confidence: 99%

“…The correspondence of elements is as follows, with constituents of equation (5.1.7) of Bleistein et al . (1998 ) appearing on the left and constituents of (5) appearing on the right:…”

Section: Derivation Of a 3d Kirchhoff Data Mapping Formulamentioning

confidence: 99%

Section: 5d Kirchhoff Data Mappingmentioning

confidence: 99%

See 2 more Smart Citations