An Approach to Inverse Filtering of Near‐surface Layer Effects From Seismic Records

Goupillaud, Pierre L.

doi:10.1190/1.1438951

Cited by 197 publications

(78 citation statements)

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“…The above focusing result can be used for any number of interfaces in the 1D model. In the frequency domain, the reflection and transmission responses for any layered medium can be written in the fractional form used above and their denominators are always the same (Goupillaud, 1961). This means that the focusing wavefield for a medium with interfaces from z 0 to z i can be written aŝ …”

Section: Focusing a Wavefield Just Below A Layered Mediummentioning

confidence: 99%

“…In the frequency domain, the corresponding reflectionRðz 0 ; ωÞ and transmission T þ ðz 2 ; z 0 ; ωÞ responses generated by the unit amplitude plane wave are given by (Goupillaud, 1961) Rðz 0 ; ωÞ ¼ r 0 þ r 1 e −2iωt 1 þ r 2 e −2iωðt 1 þt 2 Þ þ r 0 r 1 r 2 e −2iωt 2 1 þ r 0 r 1 e −2iωt 1 þ r 0 r 2 e −2iωðt 1 þt 2 Þ þ r 1 r 2 e −2iωt 2 ;…”

Section: Appendix a Wavefield Focusing And Green's Function Representmentioning

confidence: 99%

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Seismic reflector imaging using internal multiples with Marchenko-type equations

et al. 2014

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We present an imaging method that creates a map of reflection coefficients in correct one-way time with no contamination from internal multiples using purely a filtering approach. The filter is computed from the measured reflection response and does not require a background model. We demonstrate that the filter is a focusing wavefield that focuses inside a layered medium and removes all internal multiples between the surface and the focus depth. The reflection response and the focusing wavefield can then be used for retrieving virtual vertical seismic profile data, thereby redatuming the source to the focus depth. Deconvolving the upgoing by the downgoing vertical seismic profile data redatums the receiver to the focus depth and gives the desired image. We then show that, for oblique angles of incidence in horizontally layered media, the image of the same quality as for 1D waves can be constructed. This step can be followed by a linear operation to determine velocity and density as a function of depth. Numerical simulations show the method can handle finite frequency bandwidth data and the effect of tunneling through thin layers.

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Section: Focusing a Wavefield Just Below A Layered Mediummentioning

confidence: 99%

Section: Appendix a Wavefield Focusing And Green's Function Representmentioning

confidence: 99%

Seismic reflector imaging using internal multiples with Marchenko-type equations

et al. 2014

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“…Apart from the scaling factor (τ 0 τ 1 ) −1 the upgoing part contains the correct local reflection coefficient r 1 at t = t 1 . This result can be generalized to any number of layers as was already shown before [12]. The downgoing part of the focusing wavefield is the inverse of the impulse transmission response and for a layered medium with reflectors from z 0 to z i it will have 2 i number of events in the time window…”

Section: The Character Of the Focusing Wavefieldmentioning

confidence: 99%

Data-driven inversion of GPR surface reflection data for lossless layered media

Slob

Wapenaar

2014

The 8th European Conference on Antennas and Propagation (EuCAP 2014)

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Abstract-Two wavefields can be retrieved from the measured reflection response at the surface. One is the Green's function at a chosen virtual receiver depth level in a layered model generated by a source at the surface. The other wavefield consists of the upgoing and downgoing parts of a wavefield that focuses at the virtual receiver depth level. From the upgoing part of the focusing wavefield an image can be computed at one-way vertical travel time and with correct amplitudes of the local reflection coefficients as a function of incidence angle. These reflection coefficient values can be used to invert for electric permittivity and magnetic permeability. From these values and the known image times the layer thickness values can be obtained for each layer. This method renders the full waveform inversion problem for horizontally layered media a linear problem.

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“…Therefore, the interface matrix can be written in the simple form (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) This interface matrix is interesting because it is completely analogous to the simpler.case of compressional waves at normal incidence derived by Goupillaud (1961), Sherwood and Trorey (1966) and others.…”

Section: A21mentioning

confidence: 99%

“…By ingeniously constraining all layers to have transit times which are integer multiples of a small time increment A' Wuenschel showed that for impulsive sources, the vertical motion and stress at each -2 S interface could be expressed as a ratio of polynomials in e which can be expanded into an infinite series in integer powers of Later this solution was expressed in terms of up and down travelling waves in each layer by Goupillaud (1961), Sherwood andTrorey (1965), andTreitel (1966). This solution is also summarized by Claerbout (1968) in connection with an inverse problem solved by Kunetz (1962), in which the layer impedances are recovered from the upgoing waves recorded at the free surface of a layered halfspace.…”

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confidence: 99%

Discrete Time Solution of Plane P‐sv Waves in a Plane Layered Medium

Frasier

1970

GEOPHYSICS

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For plane waves at normal incidence to a layered, elastic medium both the forward and inverse discrete time problems have been previously solved. Here, the forward problem of calculating the waves in a medium of plane, homogeneous, isotropic layers is extended to P and SV body waves at non-normal incidence, where the horizontal phase velocity of each wave is greater than the shear and compressional waves of each layer.Vertical travel times for P and SV waves through each layer are rounded off to unequal integer multiples of a small time increment This gives a 4 x 4 layer matrix analagous to the 2 x 2 layer matrix for normal incidence obtained by previous authors.Reflection and transmission responses for layered media are derived as matrix series in integer powers of a Fourier transform variable z=e D 7 . These responses are generated recursively by polynomial division and include all multiply reflected P and SV waves with mode conversions.For a layered halfspace, the reflection response matrix for a source at the free surface equals the positive time part of the autocorrelation matrix of the transmission response matrix for a deep source. This can be used to convert surface records of teleseismic events to reflection seismograms for mapping the crust. For a known crustal structure, the reverberations contaminating a teleseismic event can be removed in time by a simple convolution, rather than by dividing spectra.Time domain transmission responses for two crustal models under LASA, and reflection responses for several core-mantle boundary models 3.are calculated as examples of the method. These responses are useful for studying the first motion and window length of transition layer responses.Finally, the method is extended to media containing any arrangement of solid and fluid layers.

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An Approach to Inverse Filtering of Near‐surface Layer Effects From Seismic Records

Cited by 197 publications

References 1 publication

Seismic reflector imaging using internal multiples with Marchenko-type equations

Seismic reflector imaging using internal multiples with Marchenko-type equations

Data-driven inversion of GPR surface reflection data for lossless layered media

Discrete Time Solution of Plane P‐sv Waves in a Plane Layered Medium

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