Purdue University

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

 

Spherical

 

 

X(p)

 

 

x3

and

v(x3)

 

 

p

 

 

 

 

 

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.