Windowed Fourier transform

The magnitude of the windowed Fourier transform of a signal using (a) a Gaussian window of width 16 ms, (b) a Gaussian window of width 32 ms, (c) a Gaussian window of width 64 ms, (d) a Gaussian window of width 128 ms, (e) a Gaussian window of width 256 ms, and (f) a Gaussian window of width 512 ms. The jet colorscale is also used here [going from blue (minimum value) to red (maximum value)].

Let us now look at examples of WFT representations. We will use the examples of cosine waves with impulses, cosine waves with shutdown time, and quadratic chirp signals. Let us start with the example of cosine waves with impulses. This signal consists of the sum of two cosine waves and two impulses. This figure shows the WFT of this signal for various window widths [16 ms (i.e., 8 samples), 32 ms (i.e., 16 samples), 64 ms (32 samples), 128 ms (64 samples), 256 ms (128 samples), and 512 ms (256 samples)]. We have used the Gaussian window for the signal g(t) (i.e., g(t) = exp [- a x²] for a convenient value a> 0) for the computations of the WFT in these figures and actually for all the WFT figures in this chapter. The following observations can be made from these figures:

• In (a) we see that two impulses were detected with a good localization in time. The two cosine waves have also been detected, but the localization of their frequencies is not accurate.

• In (b) we have improved the localization of the two cosine frequencies, but they are still not clearly distinguishable.

• In (c) and (d) we have a localization of the two impulses in time and a good localization of the cosine wave frequencies. Notice that if the two impulses were close together, we would not be able to distinguish them because a small time spread is still visible around the locations of these impulses.

• In (e) and (f), we can see that as the window g(t) get wider, the WFT representation converges toward the classical Fourier transform representation.

Other Images in Chapter 4

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.