Home »
Images »
**Chapter 3**

Contribute Image
##
Images in: Chapter 3

**1** of **5** Next»
### Fermat principle

### Snapshots of wave propagation

### Wavefronts

### Head wave (refracted wave)

### Snapshots of wave propagation

### Possible refraction (head-wave) paths

### Snapshots of wave propagation

### Snapshots of wave propagation (Cont'd)

### Snapshots of wave propagation

**1** of **5** Next»

Contribute Image

Fermat principle
Snapshots of wave propagation
Wavefronts
Head wave (refracted wave)
Snapshots of wave propagation
Possible refraction (head-wave) paths
Possible refraction (head-wave) paths (Cont'd)
Snapshots of wave propagation
Snapshots of wave propagation (Cont'd)
Snapshots of wave propagation
Traveltime curves
P-S reflection
P-S reflection (Cont'd)
Reflection and transmission coefficients
Reflection and transmission coefficients (Cont'd)
Reflection and transmission coefficients
Reflection and transmission coefficients (Cont'd)
Reflection and transmission coefficients
Reflection and transmission coefficients (Cont'd)
Reflection and transmission coefficients
Reflection and transmission coefficients (Cont'd)
Reflection at the free surface
Particle motions
Rayleigh wave
Scholte surface-wave velocity
Zoeppritz' equations
Reflection coefficients
Zoeppritz equation
Zoeppritz' equations
A and B crossplotting of AVO
A diffraction tail
Snapshot of diffraction
Snapshots of diffraction
Raypaths
A reflected ray
Diffraction and reflection patterns
Diffraction and reflection patterns (Cont'd)
Diffraction and reflection patterns (Cont'd)
Diffraction and reflection patterns (Cont'd)
Postcritical reflection
P-P reflection
Wavefront pattern

Images - Chapter 3
There are several possible paths connecting point ** A** to point

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 V_{P} = 1500m/s and ρ = 1.0g/cc, and those of the bottom half-space are V_{P} = 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).

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.

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 V_{P} = 1850 m/s and ρ = 2.0 g/cc, and those of the bottom half-space are V_{P} = 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 *θ _{i}* ≥

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 V_{P} = 2000 m/s, V_{S} = 900 m/s, and ρ = 2.0 g/cc; and those of the bottom half-space are V_{P} = 3000 m/s, V_{S} = 1600 m/s, and ρ = 2.65 g/cc. (b) The properties of the top half-space are V_{P} = 1500 m/s, V_{S} = 0 m/s, and ρ = 1.0 g/cc; and those of the bottom half-space are V_{P} = 2000 m/s, V_{S} = 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 τ_{zz}. (**i**_{P} indicates the incident P-wave, **r**_{P} indicates the reflected P-wave, **r**_{S} indicates the reflected P-wave, **t**_{P} indicates the transmitted P-wave, and **t**_{S} indicates the transmitted S-wave).

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 V_{P1}, and V_{S1}, and the bottom half-space has velocities V_{P2}, and V_{S2}. Notice that if V_{P1} > V_{S2}, the modes P_{1}S_{2}S_{1} and P_{1}S_{2}P_{1} 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 V_{P2} > V_{S2} > V_{P1} > V_{S1}.

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 V_{P} = 1850 m/s, V_{S} = 1000 m/s and ρ = 2.0 g/cc; and those of the bottom half-space are V_{P} = 4500 m/s, V_{S} = 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 τ_{zz}. (**i**_{P} indicates the incident P-wave, **r**_{P} indicates the reflected P-wave, **r**_{S} indicates the reflected P-wave, **t**_{P} indicates the transmitted P-wave, and **t**_{S} indicates the transmitted S-wave. Based on the nomenclature in Figure 3-11, **h _{P2P1}** indicates the head wave P

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

Images - Chapter 3
(a) Snapshots of wave propagation in a model made of two homogenous half-spaces sandwiched by a homogeneous acoustic layer (fluid) [V_{P} = 1500$ m/s, V_{S} = 0 m/s, and ρ = 1 g/cc]. The properties of the top half-space are [V_{P} = 360 m/s, V_{S} = 0 m/s, and ρ = 0.012 g/cc]. Those of the bottom half-space are [V_{P} = 2500 m/s, V_{S} = 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) [V_{P} = 1500 m/s, V_{S} = 900 m/s, and ρ = 1.8 g/cc]. The properties of the top half-space are [V_{P} = 360 m/s, V_{S} = 0.0 m/s, and ρ = 0.012 g/cc]. Those of the bottom half-space are [V_{P} = 3000 m/s, V_{S} = 1700.0 m/s, and ρ = 2.75 g/cc].