Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

The problem of interpretation of seismic reflection data can be posed with sufficient generality using the concepts of inverse theory. In its roughest formulation, the inverse problem consists of obtaining the Earth model for which the predicted data best fit the observed data. If an adequate forward model is used, this best model will give the best images of the Earth’s interior. Three parameters are needed for describing a perfectly elastic, isotropic, Earth: the density ρ(x) and the Lame parameters λ(x) and μ(x), or the density ρ(x) and the P-wave and S-wave velocities α(x) and β(x). The choice of parameters is not neutral, in the sense that although theoretically equivalent, if they are not adequately chosen the numerical algorithms in the inversion can be inefficient. In the long (spatial) wavelengths of the model, adequate parameters are the P-wave and S-wave velocities, while in the short (spatial) wavelengths, P-wave impedance, S-wave impedance, and density are adequate. The problem of inversion o...

A strategy for nonlinear elastic inversion of seismic reflection data

Migration of an observed zero-offset wavefield can be performed as the solution of a boundary value problem in which the data are extrapolated backward in time. This concept is implemented through a finite-difference solution of the two-dimensional acoustic wave equation. All depths are imaged simultaneously at time 0 (the imaging condition), and all dips (right up to vertical) are correctly migrated. Numerical examples illustrate this technique in both constant and variable velocity media.

Migration by extrapolation of time‐dependent boundary values*

The major amount of multiple energy in seismic data is related to the large reflectivity of the surface. A method is proposed for the elimination of all surface-related multiples by means of a process that removes the influence of the surface reflectivity from the data. An important property of the proposed multiple elimination process is that no knowledge of the subsurface is required. On the other hand, the source signature and the surface reflectivity do need to be provided. As a consequence, the proposed process has been implemented adaptively, meaning that multiple elimination is designed as an inversion process where the source and surface reflectivity properties are estimated and where the multiple-free data equals the inversion residue. Results on simulated data and field data show that the proposed multiple elimination process should be considered as one of the key inversion steps in stepwise seismic inversion.

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Adaptive surface-related multiple elimination

Preface 1. Introduction to rock physics 2. Rock physics interpretation of texture, lithology and compaction 3. Statistical rock physics: combining rock physics, information theory, and statistics to reduce uncertainty 4. Common techniques for quantitative seismic interpretation 5. Case studies: lithology and pore-fluid prediction from seismic data 6. Workflows and guide lines 7. Hands-on References Index.

Quantitative Seismic Interpretation: Applying Rock Physics Tools to Reduce Interpretation Risk

The acoustics in auditoria are determined by the properties of both the direct sound and the later arriving reflections. If electroacoustic means are used to repair disturbing deficiencies in the acoustics, one has to cope with unfavorable side effects such as localization problems and artificial impressions of the reverberant field (electronic flavor). To avoid those side effects, the concept of electroacoustic wave front synthesis is introduced. The underlying theory is based on the Kirchhoff–Helmholtz integral. In this new concept the wave fields of the sound sources on stage are measured by directive microphones; next they are electronically extrapolated away from the stage, and finally they are re‐emitted in the hall by one or more loudspeaker arrays. The proposed system aims at emitting wave fronts that are as close as possible to the real wave fields. Theoretically, there need not be any differences between the electronically generated wave fields and the real wave fields. By using the image source concept, reflections can be generated in the same way as direct sound.

Acoustic control by wave field synthesis

A review has been given of the surface-related multiple problem by making use of the so-called feedback model. From the resulting equations it has been concluded that the proposed solution does not require any properties of the subsurface. However, source-detector and reflectivity properties of the surface need be specified. Those properties have been quantified in a surface operator and this operator is estimated as part of the multiple removal problem. The surface-related multiple removal algorithm has been formulated in terms of a Neumann series and in terms of an iterative equation. The Neumann formulation requires a nonlinear optimization process for the surface operator; while the iterative formulation needs a number of linear optimizations. The iterative formulation also has the advantage that it can be integrated easily with another multiple removal method. An algorithm for the removal of internal multiples has been proposed as well. This algorithm is an extension of the surface-related method. Removal of internal multiples requires knowledge of the macro velocity model between the surface and the upper boundary of the multiple generating layer. In Part II (also published in this issue) the success of the proposed algorithms has been demonstrated on numerical experiments and field data examples.

Estimation of multiple scattering by iterative inversion, Part I: Theoretical considerations

Seismic acquisition surveys are designed such that the time intervals between shots are sufficiently large to avoid the tail of the previous source response interfering with the next one (zero overlap in time). To economize on survey time and processing effort, the current compromise is to keep the number of shots to some acceptable minimum. The result is that in current practice the source domain is poorly sampled.

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Changing the mindset in seismic data acquisition

A surface-related multiple-elimination method can be formulated as an iterative procedure: the output of one iteration step is used as input for the next iteration step (part I of this paper). In this paper (part II) it is shown that the procedure can be made very efficient if a good initial estimate of the multiple-free data set can be provided in the first iteration, and in many situations, the Radon-based multiple-elimination method may provide such an estimate. It is also shown that for each iteration, the inverse source wavelet can be accurately estimated by a linear (least-squares) inversion process. Optionally, source and detector variations and directivity effects can be included, although the examples are given without these options. The iterative multiple elimination process, together with the source wavelet estimation, are illustrated with numerical experiments as well as with field data examples. The results show that the surface-related multiple-elimination process is very effective in time gates where the moveout properties of primaries and multiples are very similar (generally deep data), as well as for situations with a complex multiple-generating system.

A. J. Berkhout

Papers

Acoustic control by wave field synthesis

Adaptive surface-related multiple elimination

Estimation of multiple scattering by iterative inversion, Part I: Theoretical considerations

Changing the mindset in seismic data acquisition

Estimation of multiple scattering by iterative inversion; Part II, Practical aspects and examples