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GG 450 SNELL's LAW Lab Mar 13, 2006
DUE Mar 20
Snell's Law is one of the basic principles of seismology and any science
that deals with waves.
PURPOSE:
Get acquainted with Snell's Law and how to apply it to simple model problems
PROBLEM:
How can we predict how a seismic wave will travel through the earth?
PROCEDURES:
Using the method explained below, construct ray diagrams for the models shown
on the accompanying work sheets.
TOOLS NEEDED: Compass and straight edge - or proficiency with a good
drawing program.
OBSERVATIONS and ANALYSIS:
Answer the questions pertaining to the models on each work sheet.
CONCLUSIONS:
This is more an exercise than an experiment, so no conclusions are expected, but any
comments on how to improve the exercise are always appreciated.
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Snell's Law
A mathematical formulation for Snell's Law in an isotropic homogeneous elastic medium
where velocity changes only with depth is:
sin  
p
vz
Where  is the angle of incidence, v (z) is the propagation velocity, z is the depth, and
p is the ray parameter.
• The ray parameter is a constant which depends only on the velocity at depth z
and on the ray's angle with the vertical (  ).
  • A major implication of Snell's Law is that, if you know the ray parameter, you
can find the ray angle at any velocity, and, if you know how velocity changes
with depth, you can use this information to calculate the shape of the ray.
 have the exactly the same shape in a
• All rays with the same ray parameter
region where the velocity-depth function is the same.
• A ray traveling vertically (  =0°), either has a ray parameter equal to zero, or
the velocity at that depth is zero.
• A ray traveling horizontally (  =90°), is traveling at a depth where the velocity
is equal to 1/p, since sin(90°)=1.

Ray curvature:

• A ray is a line perpendicular
to a wave front. In isotropic material, the ray
points in the direction of energy propagation. Ray theory assumes that the
wavelength is much shorter than the wavelength of changes in the medium. If
this not true, then there are other effects, and the predictions of ray theory
break down.
• A ray traveling in a material where the velocity is constant will move on a
straight line.
• A ray traveling in a material where the velocity is changing will bend in the
direction where the velocity is lower.
• A ray traveling in a material where the velocity changes linearly with depth:
v(z)  v0  kz, where v0 and k are constants,
will curve in circular paths, where the horizontal distance the ray will travel
is given by the equation below (where the zero subscript denotes the value at
the start of the ray): (These equations are for your information, they are not
needed for the lab)
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This last point – that rays passing through regions with LINEAR VELOCITY
GRADIENTS – are circular segments is the basis for the work done in this
lab.
x
1
(cos  0  cos ), where x is the distance along the ground from the start
pk
of the ray at  0 to where the ray has an angle of , k is the velocity gradient,
p is the ray parameter.
the time it takes to travel this distance is
:
1 tan  /2 
t  log 
, and
k tan  0 /2 
the radius of curvature of the ray is
1 v 0  1
r
  
.
pk  k sin  0
:
The formulas (from Shearer) show how easily distances and travel times of
seismic waves can be calculated if the velocity-depth model can be
approximated by layers with linear velocity gradients.

This is a FORWARD MODEL concept. Conclusions are based on MODELS –
generating “synthetic data” for later comparison with “real” DATA.
The figure below shows the type of graph used in this exercise. The grid on the
left shows how the seismic propagation velocity changes with depth and the
graph on the right shows a cross section through the earth on which seismic
rays can be drawn.
dep th where velocity is z ero
0
0
v elocit y , km/ s
4
8
0
10
Dist ance, km
20
30
40
50
5
r
10
15
20
25
x
At what depth is the center of the circle that forms the ray?
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and
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It must be at the depth where the velocity is zero. This depth will not be in the region
where the velocity model works, but if it were, the ray would be vertical at that
depth.
Where is the center of the circle if k=0? (Constant velocity)
It is at infinity, and the ray is a straight line.
Armed with the above information, we can construct ray paths for relatively complicated
models.
Method:
1. Construct a graph like the one above, with the velocity depth model on the
left, and a distance - depth model on the right. The depth and distance scales
must be the same.
2. Given a velocity depth function with linear gradients, draw the function on the
left side.
example: v(0)=2 km/s, v(10)=5 km/s, v(15)=8 km/s
3. Determine the depths at which the velocity would be zero in each layer, and
extend light horizontal lines across the page to the right at those depths.
4. label each region where the velocity gradient is constant with a letter, and
label each of the lines above with the appropriate letter.
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5. Decide where you want to start your ray, and what slope it should have. (It
may be easier to start at a depth where the ray is traveling horizontally, rather
than at the surface.)
6. Placing the point of the compass on the zero velocity line for the appropriate
layer, move the point and change the radius until the slope of the ray is the
desired slope.
7. Draw the ray arc in the region where that particular velocity gradient is valid.
8. At depths where the gradient changes from one value to another, if the
velocity-depth curve is continuous you can easily extend your ray into the next
layer. At the point on the ray where it enters the next layer, the velocity in
both layers is the same, thus Snell's Law tells us that the slope is the same.
This implies that the circular segments in each layer are tangent to each other
at the point where the ray travels from one layer to the other. To find the
center of the circle in the new layer, draw a line through the center of the circle
in the old layer through the point of tangency across the zero velocity line in
the new layer. The point where the line you draw crosses the new zero
velocity line is the center of the new ray circle.
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9. Continue the ray through the model until you hit the surface again or run off
the page.
•
depth, km
0
A
•
velocit y, km/ s
4
8
0
•
5
B
A
10
B
•
15
What is the ray parameter of the above ray?
The depth at which the ray is horizontal has a velocity of about 7.3 km/s, thus the ray
parameter is 0.137.
What if the velocity-depth function has a first-order discontinuity in it?
The circles for those layers will be tangent at the depth where the velocities are equal
- even if the layers don't extend to those depths.
Exercises. See the following pages. For each case, draw the rays requested. Assume
horizontal layers. If you want to use a software package, great...
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