Global Seismology and Seismic Waves
4.4 Finding the path of a ray through the Earth
4.4.1 Refraction: Snell’s law
When wave fronts cross obliquely into a rock with a higher seismic velocity, they speed up, which causes them to slew around and change direction (Fig.
4.10a), just as a line of people, arms linked, would slew if each person went faster once they had crossed a line. This bending is called refraction.
How much is the change of direction? First we note that a long way from the source a small part of a wave front will be nearly a flat plane, so two suc-cessive wave fronts – or sucsuc-cessive positions of the same wave front – will be parallel, and rays will be straight, for they are perpendicular to wave fronts.
The time between successive wave fronts, AB and A′B′ in Figure 4.10b, remains unchanged, so the wavelength must increase in the second rock, in pro-portion to the increase in velocity. From geometry,
Eq. 4.2
By trigonometry,
Eq. 4.3 Therefore
Eq. 4.4
As BB′ and AA′ are in proportion to the velocities v1
and v2(Eq. 4.2), this can be rearranged to give
Eq. 4.5
Figure 4.9 Travel-times for a uniform and the actual Earth.
This equation, Snell’s Law, is essentially the same as the one used for the refraction of light, but this form is more useful for seismology (for light, the equation is rearranged with the velocities replaced by their ratio, which is called the refractive index).
Since it is more convenient to use rays, which show the direction of propagation, than wave fronts, i1and i2are measured as the angles between the rays and the perpendicular, or normal, to the interface between the two rock types (Fig. 4.10c);
these are called the angles of incidence and refrac-tion. They have the same values as the angles between the wave fronts and the interface.
As an example, and referring to Figure 4.11, at what angle would the ray leave the interface if the angle of incidence is 37°?
After refraction the ray leaves the interface at 48.8°.
Now that Snell’s Law has been introduced, most diagrams will be simplified to show only rays. Snell’s Law also applies to reflection, as will be shown in Section 4.5.2.
4.4.2 Tracing rays through the Earth:
The ray parameter, p
If there are several parallel and uniform layers (Fig.
4.12), a ray meets the next interface at the angle at which it left the last one, that is, i2= i1′, and so on.
Applying Snell’s Law at each interface, we have
Eq. 4.6 As i1′ = i1, i2′ = i2, and so on
Eq. 4.7 The ratio (sin i/v) remains unchanged, or constant, along the ray path.
In the Earth, however, the layers are curved, so it is not true that i2 = i2′, and so on (Fig. 4.13). The differences between these angles depend not on the velocities of the layers but only on the geometry of triangle ABO (the relationship will not be derived).
Snell’s Law determines how the angle of a ray changes on crossing an interface, while geometry determines the change of angle between interfaces.
These can be combined to give
Eq. 4.8
Figure 4.10 Refraction of a wave front.
i2
37° v1 = 4 km/s
v2 = 5 km/s
Figure 4.11 Example of refraction.
4.4 Finding the path of a ray through the Earth ✦ 31
The constant p is known as the ray parameter and has the same value all along the path of any given ray, provided all three quantities, v, i, and r, are mea-sured at the same place. For refraction at any single interface, r is the same on both sides and the p para-meter simplifies to Snell’s Law; thus the p parapara-meter includes Snell’s Law.
4.4.3 Ray tracing and the Earth’s velocity–depth structure
If the variation of the seismic velocity with depth (or distance, r, from the centre) were known, then ray paths through the Earth could be deduced using the ray parameter, p. The Earth would be divided into a large number of thin shells (Fig. 4.14a), which would take account of any changes of seismic veloc-ity, whether abruptly between real layers or grada-tionally within layers. Then a ray could be chosen that dives into the Earth at some angle i, the take-off angle (Fig. 4.14a), and its path through the layers could be followed using the ray parameter equation, Eq. 4.8 (Fig. 4.14b). The times to travel along each of the little sections within shells could be calculated and added together to give the total travel-time. By repeating this for take-off angles from horizontal to vertical, a travel-time versus distance, t–Δ, diagram could be worked out (Fig. 4.14c).
However, our problem is the inverse. We are confined to the surface and can only measure travel-times from a seismic source to detectors at different
distances; from these, we have to determine how velocity changes with depth. This is an example of the inversion problem, introduced in Section 2.4.
Generally, it is much harder to solve the inverse than the forward problem. The mathematics needed to invert travel-times into velocity versus depth were solved by the pioneers of global seismology but are beyond the level of this book. Nowadays, we can adopt a different approach using computers: Start-ing with some estimate of how the velocity depends on depth, such as earlier solutions, travel-times are calculated for the distances to actual seismic receivers and compared with the observed times. If there is a discrepancy, the velocity–depth curve is adjusted to minimise it. This is repeated for millions of seismic records of thousands of earthquakes obtained from hundreds of seismometers all over the world.
One result is shown in Figure 4.15a, a solution published by Kennett and Engdahl in 1991, and known as model iasp91. It is an average solution, treating the Earth as being perfectly spherically sym-metrical (after allowance for the equatorial bulge), and ignoring small, lateral variations.
Why there are two curves and what they reveal about the Earth is discussed in the following section.
i1'
v1
v3 > v2
v4 > v3 i2
i2'
i3
i3'
i4 i1
v2 > v1 surface
Figure 4.12 Refractions at parallel interfaces.
i2
'
i2
i1
'
i3 v1
v2
v3
r2 B A
r3
0 i1
Figure 4.13 Ray path through layered Earth.