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Fermat principle

Images - Chapter 3 There are several possible paths connecting point A to point B. The raypath is the path normal to the wavefronts. In this case the straight line between A and B is optimum raypath. The Fermat principle also allows us to select the raypath among the different possibilities. It states that in the wave propagation, the wavepath between any two fixed points is that along which the traveltime is the least of all possible paths, (t0=200 ms; t1=400 ms; t=600 ms). The jet colorscale displayed here will be used to displayed snapshots throughout this chapter.

Snapshots of wave propagation

Images - Chapter 3 Snapshots of wave propagation in a model made of two homogenous acoustic half-spaces. The properties of the top half-space are VP = 1500m/s and ρ = 1.0g/cc, and those of the bottom half-space are VP = 2000 m/s and ρ = 2.25g/cc. The waves were generated by an explosive. The physical quantity displayed here is the pressure. (i indicates the incident wave, r indicates the reflected wave, and t indicates the transmitted wave).

Wavefronts

Images - Chapter 3 Some of the wavefronts of the wave propagation through two homogeneous acoustic half-spaces.

Let us now consider a heterogeneous model consisting of two infinitely homogeneous and isotropic media separated by a horizontal surface. This model is also known as a ?two-half-space model,? in which each homogeneous medium represents a half-space. Assume that an explosive source generates a P-wave which propagates in the top half-space. When the wave reaches the interface between the half-spaces, part of its energy returns to the half-space from which it came. This process is called reflection. The remaining energy enters the second medium. This process is called transmission.

In this figure, the phenomena of reflection and transmission, described by the snapshots, are now displayed by the wavefronts. Having superimposed the corresponding raypaths to these wavefronts, we can sometimes abandon the complexity of snapshots and wavefronts to use rays alone.

Head wave (refracted wave)

Images - Chapter 3 An illustration of the head wave (refracted wave). Notice that the head wave propagates in the incident half-space with the velocity of the bottom half-space. The properties of the top half-space are VP = 1850 m/s and ρ = 2.0 g/cc, and those of the bottom half-space are VP = 4500 m/s and ρ = 3.0 g/cc. The waves were generated by an explosive. The physical quantity displayed here is the pressure. (i indicates the wave, r indicates the reflected wave, t indicates the transmitted wave, and s indicates the source position).

Note that if θiθicp, no seismic energy can penetrate into the bottom half-space, and it is consequently reflected into the top half-space. If VP2 ≤ VP1, there is no critical angle.

Snapshots of wave propagation

Images - Chapter 3 Snapshots of wave propagation in a model made of two homogenous half-spaces. (a) The properties of the top half-space are VP = 2000 m/s, VS = 900 m/s, and ρ = 2.0 g/cc; and those of the bottom half-space are VP = 3000 m/s, VS = 1600 m/s, and ρ = 2.65 g/cc. (b) The properties of the top half-space are VP = 1500 m/s, VS = 0 m/s, and ρ = 1.0 g/cc; and those of the bottom half-space are VP = 2000 m/s, VS = 1000 m/s, and ρ = 2.0 g/cc. The waves were generated by an explosive. The physical quantity displayed here is the stress component τzz. (iP indicates the incident P-wave, rP indicates the reflected P-wave, rS indicates the reflected P-wave, tP indicates the transmitted P-wave, and tS indicates the transmitted S-wave).

Possible refraction (head-wave) paths

Images - Chapter 3 Possible refraction (head-wave) paths from the source to the receiver for incident P- and S-wave. The top half-space has velocities VP1, and VS1, and the bottom half-space has velocities VP2, and VS2. Notice that if VP1 > VS2, the modes P1S2S1 and P1S2P1 are not possible.

Refracted waves (also known as head waves) occur after the critical angle. As the elastic case that we have just described includes several possibilities of critical angles, several head waves are possible at the interface of two half-spaces. This figure shows the raypaths of four possible head waves for the case of an incident P-wave. These head waves are also illustrated in the snapshot plots in this figure. We can see that there are five possible head waves with an incident P-wave. Actually, five head waves is the maximum possible number for an incident P-wave; four are in the upper medium, one in the lower medium. This maximum number of five head waves is possible only if VP2 > VS2 > VP1 > VS1.

Snapshots of wave propagation

Images - Chapter 3 (a) Snapshots of wave propagation in a model made of two homogeneous elastic half-spaces. The properties of the top half-space are VP = 1850 m/s, VS = 1000 m/s and ρ = 2.0 g/cc; and those of the bottom half-space are VP = 4500 m/s, VS = 2750 m/s, and ρ = 3.0 g/cc. The waves were generated by an explosive. The physical quantity displayed here is the normal stress τzz. (iP indicates the incident P-wave, rP indicates the reflected P-wave, rS indicates the reflected P-wave, tP indicates the transmitted P-wave, and tS indicates the transmitted S-wave. Based on the nomenclature in Figure 3-11, hP2P1 indicates the head wave P1P2P1, hP2S1 indicates the head wave P1P2S1, hS2P1 indicates the head wave P1S2P1, hS2S1 indicates the head wave P1S2S1, and hP2S2 indicates the head wave P1P2S2.

Snapshots of wave propagation (Cont'd)

Images - Chapter 3 (b) Possible critical angles and head waves associated with an interface between two elastic half-spaces.

Snapshots of wave propagation

Images - Chapter 3 (a) Snapshots of wave propagation in a model made of two homogenous half-spaces sandwiched by a homogeneous acoustic layer (fluid) [VP = 1500$ m/s, VS = 0 m/s, and ρ = 1 g/cc]. The properties of the top half-space are [VP = 360 m/s, VS = 0 m/s, and ρ = 0.012 g/cc]. Those of the bottom half-space are [VP = 2500 m/s, VS = 1000.0 m/s, and ρ = 2.25 g/cc].

(b) Snapshots of wave propagation in a model made of two homogenous half-spaces sandwiched by a homogeneous acoustic layer (fluid) [VP = 1500 m/s, VS = 900 m/s, and ρ = 1.8 g/cc]. The properties of the top half-space are [VP = 360 m/s, VS = 0.0 m/s, and ρ = 0.012 g/cc]. Those of the bottom half-space are [VP = 3000 m/s, VS = 1700.0 m/s, and ρ = 2.75 g/cc].

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