1 

SplitStep Fourier Migration

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2 

The Critical Reflection Theorem  Reply

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3 

3D Migration Using the Causality Principle

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4 

Seismic Vibrator Modeling

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5 

The Critical Reflection Theorem

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6 

The Critical Reflection Theorem

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7 

Introduction to the supplement on seismic interferometry

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8 

General representations for wavefield modeling and inversion in geophysics
Acoustic, electromagnetic, elastodynamic, poroelastic, and electroseismic waves are all governed by a unified matrixvector wave equation. The matrices in this equation obey the same symmetry properties for each of these wave phenomena. This implies that the wave vectors for each of these phenomena obey the same reciprocity theorems. By substituting Green's matrices in these reciprocity theorems, unified wavefield representations are obtained. Analogous to the wellknown acoustic wavefield representations, these unified representations find applications in geophysical modeling, migration, inversion, multiple elimination, and interferometry.

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9 

Passive seismic interferometry by multidimensional deconvolution
We introduce seismic interferometry of passive data by multidimensional deconvolution (MDD) as an alternative to the crosscorrelation method. Interferometry by MDD has the potential to correct for the effects of source irregularity, assuming the first arrival can be separated from the full response. MDD applications can range from reservoir imaging using microseismicity to crustal imaging with teleseismic data.

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10 

A new empirical complex electrical resistivity model
Macroscopic measurements of electrical resistivity require frequencydependent effective models that honor the microscopic effects observable in macroscopic measurements. Effective models based on microscopic physics exist alongside with empirical models. We adopted an empirical model approach to modify an existing physical model. This provided a description of electrical resistivity as a function of not only frequency, but also water saturation. We performed twoelectrode laboratory measurements of the complex resistivity on a number of fine and mediumgrained unconsolidated sand packs saturated with water of three different salinities. For frequencies between 0.1 and 1 MHz, the data were fitted with the new model and compared to fits with Archie’s law. Our model described the relaxation times and DC resistivity values as negative exponential functions with increasing water saturation. All data could be accurately described as a function of frequency and water saturation with nine parameters.

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11 

Focusing the wavefield inside an unknown 1D medium: Beyond seismic interferometry
With seismic interferometry one can retrieve the response to a virtual source inside an unknown medium, if there is a receiver at the position of the virtual source. Using inverse scattering theory, we demonstrate that, for a 1D medium, the requirement of having an actual receiver inside the medium can be circumvented, going beyond seismic interferometry. In this case, the wavefield can be focused inside an unknown medium with independent variations in velocity and density using reflection data only.

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12 

Blended acquisition with dispersed source arrays
Blended source arrays are historically configured with equal source units, such as broadband vibrators (land) and broadband airgun arrays (marine). I refer to this concept as homogeneous blending. I have proposed to extend the blending concept to inhomogeneous blending, meaning that a blended source array consists of different source units. More specifically, I proposed to replace in blended acquisition the traditional broadband sources by narrowband versions — imagine coded single air guns with different volumes or coded single narrowband vibrators with different central frequencies — together representing a dispersed source array (DSA). Similar to what we see in today's audio systems, the DSA concept allows the design of dedicated narrowband source elements that do not suffer from the low versus high frequency compromise. In addition, the DSA concept opens the possibility to use source depths and spatial sampling intervals that are optimum for the low, mid, and highfrequency sources (multiscale shooting grids). DSAs are considered to be an important step in robotizing the seismic acquisition process.

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13 

Spurious multiples in seismic interferometry of primaries
Seismic interferometry is a technique for estimating the Green's function that accounts for wave propagation between receivers by correlating the waves recorded at these receivers. We present a derivation of this principle based on the method of stationary phase. Although this derivation is intended to be educational, applicable to simple media only, it provides insight into the physical principle of seismic interferometry. In a homogeneous medium with one horizontal reflector and without a free surface, the correlation of the waves recorded at two receivers correctly gives both the direct wave and the singly reflected waves. When more reflectors are present, a product of the singly reflected waves occurs in the crosscorrelation that leads to spurious multiples when the waves are excited at the surface only. We give a heuristic argument that these spurious multiples disappear when sources below the reflectors are included. We also extend the derivation to a smoothly varying heterogeneous background medium.

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14 

Focal transformation, an imaging concept for signal restoration and noise removal
Interpolation of data beyond aliasing limits and removal of noise that occurs within the seismic bandwidth are still important problems in seismic processing. The focal transform is introduced as a promising tool in data interpolation and noise removal, allowing the incorporation of macroinformation about the involved wavefields. From a physical point of view, the principal action of the forward focal operator is removing the spatial phase of the signal content from the input data, and the inverse focal operator restores what the forward operator has removed. The strength of the method is that in the transformed domain, the focused signals at the focal area can be separated from the dispersed noise away from the focal area. Applications of particular interest in preprocessing are interpolation of missing offsets and reconstruction of signal beyond aliasing. The latter can be seen as the removal of aliasing noise.

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15 

On the relation between seismic interferometry and the migration resolution function
Seismic interferometry refers to the process of retrieving new seismic responses by crosscorrelating seismic observations at different receiver locations. Seismic migration is the process of forming an image of the subsurface by wavefield extrapolation. Comparing the expressions for backward propagation known from migration literature with the Green's function representations for seismic interferometry reveals that these seemingly distinct concepts are mathematically equivalent. The frequencydomain representation for the resolution function of migration is identical to that for the Green's function retrieved by seismic interferometry (or its square, in the case of double focusing). In practice, they differ because the involved Green's functions in seismic interferometry are all defined in the actual medium, whereas in migration one of the Green's functions is defined in a background medium.

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16 

Recursive prestack depth migration using CFP gathers
The commonfocuspoint technology (CFP) describes prestack migration by focusing in two steps: emission and detection. The output of the first focusing step represents a CFP gather. This gather defines a shot record that represents the subsurface response resulting from a focused source wavefield. We propose applying the recursive shotrecord, depthmigration algorithm to the CFP gathers of a seismic data volume and refer to this process as CFPgather migration. In the situation of complex geology and/or low signaltonoise ratio, CFPbased image gathers are easier to interpret for nonalignment than the conventional image gathers. This makes the CFPbased image gathers better suited for velocity analysis. This important property is illustrated by examples on the Marmousi model.

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17 

Seismic processing in the inverse data space
Until now, seismic processing has been carried out by applying inverse filters in the forward data space. Because the acquired data of a seismic survey is always discrete, seismic measurements in the forward data space can be arranged conveniently in a data matrix (P). Each column in the data matrix represents one shot record. If we represent seismic data in the temporal frequency domain, then each matrix element consists of a complexvalued number. Considering the dominant role of multiple scattering in seismic data, it is proposed to replace data matrix P by its inverse P–1 before starting seismic processing. Making use of the feedback model for seismic data, multiple scattered energy is mapped onto the zero time axis of the inverse data space. The practical consequence of this remarkable property may be significant: multiple elimination in the inverse data space simplifies to removing data at zero time only. Moving to the inverse data space may cause a fundamental change in the way we preprocess and image seismic data.

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18 

3D surfacerelated multiple prediction: A sparse inversion approach
The theory of iterative surfacerelated multiple elimination holds for 2D as well as 3D wavefields. The 3D prediction of surface multiples, however, requires a dense and extended distribution of sources and receivers at the surface. Since current 3D marine acquisition geometries are very sparsely sampled in the crossline direction, the direct Fresnel summation of the multiple contributions, calculated for those surface positions at which a source and a receiver are present, cannot be applied without introducing severe aliasing effects. In this newly proposed method, the regular Fresnel summation is applied to the contributions in the densely sampled inline direction, but the crossline Fresnel summation is replaced with a sparse parametric inversion. With this procedure, 3D multiples can be predicted using the available input data. The proposed method is demonstrated on a 3D synthetic data set as well as on a 3D marine data set from offshore Norway.

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19 

Removal of internal multiples with the commonfocuspoint (CFP) approach. Part 1: Explanation of the theory
Removal of surface and internal multiples can be formulated by removing the influence of downwardscattering boundaries and downwardscattering layers. The involved algorithms can be applied in a modeldriven or a datadriven way. A unified description is proposed that relates both types of algorithms based on wave theory. The algorithm for the removal of surface multiples shows that muted shot records play the role of multichannel prediction filters. The algorithm for the removal of internal multiples shows that muted CFP gathers play the role of multichannel prediction filters. The internal multiple removal algorithm is illustrated with numerical examples. The conclusion is that the layerrelated version of the algorithm has significant practical advantages.

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20 

Timedomain modeling of electromagnetic diffusion with a frequencydomain code
We modeled timedomain EM measurements of induction currents for marine and land applications with a frequencydomain code. An analysis of the computational complexity of a number of numerical methods shows that frequencydomain modeling followed by a Fourier transform is an attractive choice if a sufficiently powerful solver is available. A recently developed, robust multigrid solver meets this requirement. An interpolation criterion determined the automatic selection of frequencies. The skin depth controlled the construction of the computational grid at each frequency. Tests of the method against exact solutions for some simple problems and a realistic marine example demonstrate that a limited number of frequencies suffice to provide timedomain solutions after piecewisecubic Hermite interpolation and a fast Fourier transform.

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