EAS 557
Introduction to Seismology
Robert L. Nowack
Lecture 14B
Ray Theory in a Vertically Varying Medium
The ray equations can be written as , . Solving these equations gives positions and the ray tangent along the ray.
Assuming a vertically varying medium, , then,
where and are constants along the ray. Also,
We will assume that and the ray is in the (x1,x3) plane. Then, = constant, , and
Also from the ray equations,
resulting in
resulting in
resulting in
Once the ray is found, the travel time can be written as an integral along the ray.
Since p1 is a constant and depends on the initial take off angle of the ray, it can be used to index the ray (we could also use the initial takeoff angle i(s0)). We will call the horizontal slowness p1, the ray parameter. This is sometimes just written as p.
We will now assume a “common source gather” geometry with the source at the origin and receivers along the surface.
The recorded data will be the travel times of the seismic arrivals at the surface receivers. Let’s first consider the travel time for a given ray indexed by the ray parameter p1 (or just p).
X(p) is the horizontal distance of the ray and Z(p) is the maximum depth of the ray. At the maximum depth, the tangent to the ray is horizontal. Therefore, at this point
or
Thus, the ray parameter p not only indexes the ray, but it also gives the speed at the bottoming point.
Let’s draw a couple of local ray diagrams
From the figure, . The total horizontal distance of the ray is
This can also be written as
Recalling that , then
The travel time along the ray can be written
or
where can be written from the diagrams above , then
Thus, for the vertically varying case, the distance and travel time for the ray can be obtained as twice the integrals from the surface to the bottoming point of the ray. The bottoming point is the maximum depth of the ray where the ray becomes horizontal and .
We now investigate the case of a radially varying spherical Earth. From the figure below, is the angular distance of the ray in degrees, is the radius of the Earth and is the radius of the bottoming point of the ray. The average radius of the Earth is = 6371 km.
A ray is shown below in a model with three radial shells.
Applying Snell’s law at point A, then
and
From the figure, , thus
In general, is a constant along a given ray in a radically varying spherical model. The ray parameter in the radically varying case is then
Now,
then, and . The angular distance of the ray can be written as
Using
this results in
Now, the travel time can be written
Using,
this results in
Thus, the angular distance and travel times in a radial Earth can be written as twice integrals from the bottoming point of the ray to the surface. At the bottoming point of the ray, the ray tangent is horizontal and or .
The results for the spherical earth can also be gotten from the flat earth case by the following transformation. This is called an earth flattening transformation.
Cartesian |
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Spherical |
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X(p) |
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x3 |
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and |
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v(x3) |
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p |
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For example,
can be written for the radial Earth as
Now, is the same as . Also, Z(p) corresponds to rp. Then,
In order to compute ray integrals in a radial Earth using a computer program in a layered model, then
1) Specify .
2) determines the specific ray where .
3) Transform these quantitites to .
4) Solve for in the flat earth.
5) Convert and to and .
As an example for a homogeneous spherical Earth, the ray path is straight. But, in the equivalent flat Earth, an increasing velocity is needed to turn the ray in the flat Earth back to the surface.
Since ray calculations can be done in flat Earth and transformed to give results in a spherical Earth, we will just investigate the flat Earth case in the next section.