Seismology Part II: Body Waves and Ray Theory

Some definitions: Body Waves: Waves that propagrate through the "body" of a medium (in 3 dimensions) WRONG!

Wavefront: The solution to the wave equation a contant time. Rays: Normals to the wavefront, or in direction of maximum change in time. Direction of wave propagation.. We think of energy traveling along rays, but it is important to remember that the reality is waves, not rays. Rays are an approximation (high f), but they are so useful that we tend to make use of them whenever possible.

Two useful ideas from Classical Optics: Huygen's Principle: An expanding wavefront is generated as the sum of contributions of individual point sources. Christian Huygens

Fermat's principle: (1) Energy will follow an extremal time path (usually, but not always, minimum) and (2) 1st order deviations in ray path result in 2nd order deviations in travel time. Now, if all we had to deal with was a homogeneous, isotropic space, we would be done. But we would not have anything very useful to apply to the Earth. Pierre de Fermat

Let's start relaxing these assumptions by allowing heterogeneity. We can do this in two ways: 1. Allow for small gradients in, , and  2. Allow for large localized gradients (i.e., interfaces) The first option is very common in seismology, but a good explanation is a bit cumbersome. It is gone over in detail in the next several (and optional) “green slides”. The second one is a bit easier to follow and we’ll come back to it in a bit.

Allowing for gradients in, , and  Recall that the homogeneity assumption allowed us to take and  out from the spatial derivatives, for example in:

If we let and  be a function of space, but require that their gradients are much less than du/dx, then we can still be approximately right in ignoring them. du/dx is related to the wavelength of the wave, so basically this means that the wavelength is short compared to the variations in elastic moduli. We recall that for scalar wave potential equation for a plane wave is: with the solution: and the wavespeed is:

If, , and  are functions of position, then so is  and k. We also should allow the amplitude A to be a function of position as well. Let  o be the mean wavespeed, so that k o is a mean wavenumber and k o =  /  o and k = W(x)k o. We attempt to find a solution analogous to the plane wave solution: where so

Substitution into the wave equation above gives, for d/dx 1 : And similarly for the other two directions.

If we add all three equations together, we get for the real part: or

and for the imaginary part: or

Rearrange the real part: or Let's define a reference wavelength as: Then

We seek the conditions where the right side of this equation is small and so can be neglected. The original form suggests this should be true at high frequencies. But we can be more precise. The imaginary part of the solution shows that so

If the term on the right is small, then so and Hence, requiring the right hand side to be small is equivalent to satisfying:

If we estimate the gradient in wavespeed over one In other words, the change in wavespeed gradient over a distance of a wavelength is small compared to the wavespeed. To the extent that this is true (and note that it will be more precise for short wavelength/high f waves), then or

which is a form of the eikonal equation and forms the basis for ray theory. Recall that so and the eikonal equation becomes

We can think of the components of wavespeed in these directions as the rate that travel time changes along the coordinate axes: This is known as “apparent” slowness. Thus Which is a form that commonly appears in various texts and has proven to be very useful in solving the problem of calculating travel times in three dimensional media.

Now, we can use this result to tell us about the ray path (where the ray goes in space). Note that the following form of the eikonal equation: or So, the eikonal equation is really about the direction of wave propagation, parallel to k, the components of which are (k 1, k 2, k 3 ) are proportional to direction cosines (by 2  ) for angles between the propagation direction and the Cartesian frame.

If we consider a raypath moving a distance ds in the k direction, then so Also, we can write these components of ds as direction cosines:

is the index of refraction n: We can figure out what the ray path is by examining how the direction cosines change as we progress along the ray (s): Change order of integration and use chain rule:

Then If you skipped the green slides, the above equation relates a short segment of a ray path (ds) in one of the cardinal directions (x 1 – the other two are x 2 and x 3 ). The dx/ds derivatives are direction cosines, and n is the index of refraction: which is the general raypath equation.  o is the mean wavespeed, and  (x) is the variation of wavespeed with position. Combining the above with the other two components gives:

It will be useful to consider what happens when the wavespeed changes in one direction (one-dimensional Earth), because most of the variation in elastic moduli is vertical. Let’s suppose that wavespeed changes only in the x 3 direction: The fact that the direction cosines in the 1 and 2 directions are constant means that those angles are constant and hence there is no change in orientation of the ray relative to those axes. The ray this therefore confined to a plane that is perpendicular to the (1,2) plane).

Let’s suppose that that plane is parallel to the 1 axis. Along any point of the ray, we have: where i is called the angle of incidence. Then Which means that and p is called the ray parameter. It is a characteristic of the entire ray. It is also a statement of Snell’s law, which is a consequence of Fermat’s principle.

If we look at the cosine part: So

or, again So, if speed increases with depth, so does the angle w.r.t. the x 3 axis, which means that the ray is curving up. Likewise, if the speed decreases with depth, the ray curves down.

We can calculate the time and distance for a ray (surface to surface). First, let’s derive a relation between dx 1 and dx 3 :

So, to calculate the total distance traveled by the ray, integrate: And the total travel time is We can combine the two to give: where . Note then that Which means that we can deduce the ray parameter from surface observations, as p determines the rate of change of T.

1.The raypath is characterized by the ray parameter “p”, which is the ratio of the sin of the incidence angle to the local wavespeed 2.We can deduce the ray parameter from surface observations, because p determines the rate of change of T. In fact, you can think of p as the apparent “slowness” of the wave. 3. If speed increases with depth, so does the angle w.r.t. the x 3 axis, which means that the ray is curving up. Likewise, if the speed decreases with depth, the ray curves down. To reiterate the important findings from these derivations - In a one- dimensional medium, the following are true: