Conditioning numbers

Conditioning numbers as a function of the maximum scattering angle. (a) Perturbation of impedance and velocity. (b) Perturbation of compressibility and specific volume. W(θ) is the radiation-pattern vector associated with each perturbation. This vector is used in the computation of the Hessian matrices.

let us recall that we have selected to parameterize our acoustic perturbed medium by the acoustic impedance and velocity based on the goal of selecting parameters that are as independent as possible. The variations of acoustic impedance affect mostly small angles (i.e., small offsets), and the variations of velocity affect mostly large angles (i.e., large offsets). Although it is physically a clear and interesting choice, we did not provide a quantitative justification of this choice. The concept of the condition number of the Hessian matrix and the analysis of eigenvectors associated with the eigenvalues of this matrix provide us this quantification justification. Figures 11-34 and 11-35 show various the condition number associated with the Hessian matrices corresponding to various possible parameterizations of an acoustic perturbed medium. In this computation, we have assumed that θ_min is null.

The condition numbers show that one of the two eigenvalues contains almost the entire energy of the signal for scattering angles under about 50 degrees. The energy of the second eigenvalue is truly negligible for scattering angles below 30 degrees. Only when the maximum scattering angle becomes greater than 100 degrees can the significant value for the second eigenvalue be obtained. For angles larger than about 110 degrees, the parameterizations in (ΔI, ΔV) and in (ΔK, Δσ) begin getting significantly better conditioned [i.e., we begin being capable of reconstructing the two parameters through the inversion of the linear system, as N_d is now smaller than 10] than the parameterizations in (ΔI, Δσ) and in (ΔK, ΔV). From these remarks, we deduce the following conclusions if no a priori information is available:

• For small angles we can estimate only the linear combination of parameters associated with the first eigenvalue.

• The inversion of the linear system for reconstructing two parameters requires a wider angle, at least 100 degrees.

• The parameterizations of acoustic perturbed media by (ΔI, ΔV) or by (ΔK, Δσ) are some of the best possible choices.

Other Images in Chapter 11

An example of the sailing path of a marine vessel in a towed-streamer survey

Images - Chapter 8 An example of the sailing path of a marine vessel in a towed-streamer survey. Note that the time for turning from one sailing line to another is about nine hours for vessels carrying streamers that are 10 km long. The dotted line indicates the turning legs of the sailing path.

This figure illustrates a typical sailing path of a 3D survey; the vessel travels back and forth, shooting and collecting data along many parallel lines, resulting in seismic data generated along lines 25 to 50 meters apart. Note that it takes about nine hours to turn from one sailing line to another for a vessel carrying 10-km-long streamers. Today, data are recorded even when the vessel is turning.

A shot diagram

Images - Chapter 7 A display of source and receiver distribution of a 2D seismic line in the so-called shot diagram. The rows corresponding to common-shot gathers and columns to common-receiver gathers. The diagonal is the zero-offset section, and all the other lines parallel to the diagonal are common-offset gathers (also known as common-offset sections). The lines perpendicular to the diagonal are the CMP gathers.

Another illustration of towed streamer acquisition

Images - Chapter 7 Another illustration of towed streamer acquisition

Interference noise

Images - Chapter 7 An illustration of interference noise in seismic data before and after. This figure shows the stack of seismic interference noise contaminated shots from another line in the Gulf of Mexico. Interference noise is clearly visible. Attenuation of seismic interference noise can be achieved by the use f-x prediction filters. Courtesy of Western Geco.

Other examples of structural traps

Images - Chapter 1 Structural traps. (A) Tilted fault blocks in an extensional regime. The seals are overlying mudstones and cross-fault juxtaposition against mudstones. (B) Rollover anticline on thrust. Petroleum accumulations may occur on both the hanging wall and the footwall. The hanging wall accumulation is dependent on a subthrust fault seal, whereas at least part of the hanging wall trap is likely to be a simple, four-way, dip-closed structure. (C) Lateral seal of a trap against a salt diapir and compactional drape trap over the diapir crest. (D) Diapiric mudstone associated trap with lateral seal against mud wall. Diapiric mud associated traps share many common features with that of salt. In this diagram, the diapiric mud wall developed at the core of a compressional fold. (E) Compactional drape over a basement block commonly creates enormous low-relief traps. (F) Gravity-generated trapping commonly occurs in deltaic sequences. Sediment loading causes gravity-driven failure and produces convex-down (listric) faults. The hanging wall of the fault rotates, creating space for sediment accumulation adjacent to the fault planes. The marker beds (grey) illustrate the form of the structure that has many favourable sites for petroleum accumulation. Adapted from Gluyas JG and Swarbrick RE (2003) Petroleum Geoscience. Oxford: Blackwell Science.