Free-surface Multiples Research Articles

Non-iterative multiple-attenuation methods: linear inverse solutions to non-linear inverse problems - II. BMG approximation

The classical linear solutions to the problem of multiple attenuation, like predictive deconvolution, τ−p filtering, or F-K filtering, are generally fast, stable, and robust compared to non-linear solutions, which are generally either iterative or in the form of a series with an infinite number of terms. These qualities have made the linear solutions more attractive to seismic data-processing practitioners. However, most linear solutions, including predictive deconvolution or F-K filtering, contain severe assumptions about the model of the subsurface and the class of free-surface multiples they can attenuate. These assumptions limit their usefulness. In a recent paper, we described an exception to this assertion for OBS data. We showed in that paper that a linear and non-iterative solution to the problem of attenuating free-surface multiples which is as accurate as iterative non-linear solutions can be constructed for OBS data. We here present a similar linear and non-iterative solution for attenuating free-surface multiples in towed-streamer data. For most practical purposes, this linear solution is as accurate as the non-linear ones.

Interferometric/daylight seismic imaging

SUMMARY Claerbout’s daylight imaging concept is generalized to a theory of interferometric seismic imaging (II). Interferometric seismic imaging is defined to be any algorithm that inverts correlated seismic data for the reflectivity or source distribution. As examples, we show that II can image reflectivity distributions by migrating ghost reflections in passive seismic data and generalizes the receiver-function imaging method used by seismologists. Interferometric seismic imaging can also migrate free-surface multiples in common depth point (CDP) data and image source distributions from passive seismic data. Both synthetic and field data examples are used to illustrate the different possibilities of II. The key advantage of II is that it can image source locations or reflectivity distributions from passive seismic data where the source position or wavelet is unknown. In some cases it can mitigate defocusing errors as a result of statics or an incorrect migration velocity. The main drawback with II is that severe migration artefacts can be created by partial focusing of virtual multiples.

Relations between reflection and transmission responses of three-dimensional inhomogeneous media

Relations between reflection and transmission responses of horizontally layered media were formulated by Claerbout in 1968 and by many others. In this paper we derive similar relations for 3-D inhomogeneous media. As the starting point for these derivations, we make use of two types of propagation invariants, based on one-way reciprocity theorems of the convolution type and of the correlation type. We obtain relations between reflection and transmission responses, including their codas, due to internal multiple scattering. These relations can be used for deriving the reflection response from transmission measurements (which is useful for seismic imaging of the subsurface, using passive recordings of noise sources in the subsurface, also known as acoustic daylight imaging) as well as for deriving the transmission coda from the reflection measurements (which is useful for seismic imaging schemes that take internal multiple scattering into account). Furthermore, following the same approach, we obtain mutual relations between reflection responses with and without free-surface multiples. The convolution-type relations are similar to those used by Berkhout and others for surface-related multiple elimination, whereas the correlation-type relations resemble Schuster's relations for seismic interferometry. Last, but not least, we obtain expressions for the reflection response at a boundary below an inhomogeneous medium, which may be useful for imaging the medium ‘from below’. The main text of this paper deals with the acoustic situation; the Appendices provide extensions to the elastodynamic situation.

Noniterative multiple attenuation methods: Linear inverse solutions to nonlinear inverse problems

It is now well established that the relationship between data without free-surface multiples (i.e., primaries) and data containing free-surface multiples (i.e., the actual recorded data) is nonlinear. As with most nonlinear inverse problems, solutions to the problem of constructing primaries from recorded data (or attenuating free-surface multiples) are either iterative or in the form of a series with an infinite number of terms. The drawbacks of these solutions are that they are computationally very expensive and often unstable especially when compared to linear solutions like predictive deconvolution for f-k filtering. However, most linear solutions contain unrealistic assumptions about the model of the subsurface and the class of free-surface multiples they can attenuate. These assumptions limit their usefulness. Fortunately, there are exceptions to this assertion. We are going to present a linear and noniterative solution to the problem of attenuating free-surface multiples which is as accurate as the...

Velocity modelling of Bengal Basin refraction data—refinement using multiples

Free-surface multiples that are generated due to high-velocity gradients in the sedimentary layer sequence above the basement of the West Bengal Basin were modeled to constrain a 2-D velocity–depth section of the southern part of the basin. Modelling of the velocity structure utilizing free-surface multiples with the asymptotic ray method produced synthetic record sections that could replicate the observed seismic phases better than those generated by matching only the primary refractions.

Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion

By specifying a discrete matrix formulation for the frequency–space modelling problem for linear partial differential equations (‘FDM’ methods), it is possible to derive a matrix formalism for standard iterative non-linear inverse methods, such as the gradient (steepest descent) method, the Gauss–Newton method and the full Newton method. We obtain expressions for each of these methods directly from the discrete FDM method, and we refer to this approach as frequency-domain inversion (FDI). The FDI methods are based on simple notions of matrix algebra, but are nevertheless very general. The FDI methods only require that the original partial differential equations can be expressed as a discrete boundary-value problem (that is as a matrix problem). Simple algebraic manipulation of the FDI expressions allows us to compute the gradient of the misfit function using only three forward modelling steps (one to compute the residuals, one to backpropagate the residuals, and a final computation to compute a step length). This result is exactly analogous to earlier backpropagation methods derived using methods of functional analysis for continuous problems. Following from the simplicity of this result, we give FDI expressions for the approximate Hessian matrix used in the Gauss–Newton method, and the full Hessian matrix used in the full Newton method. In a new development, we show that the additional term in the exact Hessian, ignored in the Gauss–Newton method, can be efficiently computed using a backpropagation approach similar to that used to compute the gradient vector. The additional term in the Hessian predicts the degradation of linearized inversions due to the presence of first-order multiples (such as free-surface multiples in seismic data). Another interpretation is that this term predicts changes in the gradient vector due to second-order non-linear effects. In a numerical test, the Gauss–Newton and full Newton methods prove effective in helping to solve the difficult non-linear problem of extracting a smooth background velocity model from surface seismic-reflection data.

Multiple diving waves and high-velocity gradients in the Bengal sedimentary basin

SUMMARY Seismic refraction and wide-angle reflection studies were carried out in the Bengal basin, India. Trace normalized record sections were produced from the digital data recorded in this sedimentary basin. The most significant feature observed in these seismograms is the presence of high-energy later arrivals which appear in arcuate shapes with their curvatures concave downwards. These arrivals are identified as free-surface reflected refractions (or multiple diving waves), which belong to the family of free-surface multiples. The nature and generation of these waves are demonstrated with the help of synthetic examples. It is found that the presence of high-velocity gradients in the top layers is a necessary prerequisite for the generation of these high-energy multiple diving waves. These later arrival multiples, along with the other primary phases present on the observed record section from the Bengal basin, have been used to prepare a well-constrained velocity model. The results bring out the presence of very high-velocity gradients (∼0.3–1.0 km s−1 km−1) in the sedimentary layers of the Bengal basin, which are an order of magnitude higher than those obtained in other sedimentary basins of India.

Velocity and Q structure of the Great Valley, California, based on synthetic seismogram modeling of seismic refraction data

The modified reflectivity method is used here to interpret seismic refraction data from the west side of the Great Valley, California, in terms of a onedimensional compressional wave velocity model of the crust. In addition, the apparent attenuation, Qp has been estimated for the 6-km-thick sedimentary section. The reflectivity method was used to calculate the prominent free-surface multiples that propagate in this sedimentary section. Modeling of the normalized amplitudes of these multiples yielded average Qp values of 100 from 0 to 2.83 km depth and 200 from 2.93 to 6.20 km depth. The velocity structure in the crust is complicated. It is characterized by positive velocity gradients within the sediment to a depth of 4.8 km and by a 1.4-kin-thick, low-velocity zone below this depth. The sediment overlies a basement with a velocity of 5.7 km/sec at its top, which increases to 6.3 km/sec at 12 km depth. Ambiguity of the data in critical regions leads to two equally valid interpretations of the lower crust at depths of 16 to 27 km: simple step increases or a more complex structure consisting of alternating high- and low-velocity zones.

Reconciliation of Measurement Differences Observed from Seismic, Well Logs, and VSP: ABSTRACT

The reconciliation of measurement differences between seismic and well-log derived data is a critical issue in exploration. Both data sets contain apparent ground truth information, but neither is free from errors. Because well logs can be considered as in-situ measurements and seismics are considered as surface sensing measurements, the major physical differences between the two are: (1) the volume of rocks the data represent, and (2) the presence of wave propagation effects for the seismic method. The conventional method of reconciliation is through the use of the synthetic seismogram. The geophysicist simulates the seismic signature of the given earth model defined by sonic and density logs, and attempts to match the synthetic with the seismic trace. In practice, the character and travel-time of the events from the seismic trace usually do not match those from synthetic seismograms. In addition to the physical differences between the measurements, the mismatch can be attributed to : (1) one-dimensional model used to generate the synthetic seismogram, (2) use of an approximate solution to the wave equations in synthetic seismogram calculation, and (3) inability to simulate seismic processing effects in the synthetic seismogram generation. Some of the reconciliation methods show that (1) free surface multiples and interbed multiples remaining on seismic data can be identified by using VSP-processed data; (2) observations that seismic times are longer than integrated sonic times are reconciled by determining and comprehending multiples, Q, and dispersion factors; and (3) full waveform offset-dependent modeling is essential to comprehend and understand VSP, seismic, and integrated sonic-log derived differences. End_of_Article - Last_Page 292------------