Seismic Rays and The Interior of the Earth

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Seismic Rays and The Interior of the Earth

• Dusty Wilson
• Tina Ostrander
• Eric Baer
• With lots of great help from
• Logan Wallace and Tim Minalia

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The Problem

• How do we know what is beneath our feet?

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How can we find out about the Interior of the
Earth?

• The deepest a human has ever gone
• The deepest well ever drilled
• The deepest a rock has ever been retrieved from
• The center of the Earth

3 km
10 km
250 km
6370 km
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Seismic Waves

• Formed when earthquakes occur

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Seismic compression wave velocity

• In air 0.344 km/sec
• In water 1.5 km/sec
• In Jello 4 km/sec
• In glass 4.5 km/sec
• In rocks 7-15 km/sec

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We can measure the arrival times
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So, lets use this information

• Could the Earth be entirely made of blue cheese?
• Seismic velocity through blue cheese is 5 km/sec

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An Earth made of cheese
0
Geocentric Angle ? (degrees) raypath length (km) Travel Time (s) Travel Time (min)
0
40       
 80      
 120      
 150      
180       
40
80
12760 km
120
150
180
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An Earth made of cheese
0
Geocentric Angle ? (degrees) raypath length (km) Travel Time (s) Travel Time (min)
0 0 0 0
40  4357  871 14.5
 80  8189  1638  27.29
 120 11050   2210 36.8
 150 12305   2461 41
180  12760  2552 42.5
40
80
120
150
180
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Remember, it has to match what we measure!
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Does it work?
Geocentric Angle ? (degrees) raypath length (km) Travel Time (s) Travel Time (min)
0 0 0 0
40   4357  871 14.5 
 80  8189  1638  27.29
 120 11050   2210 36.8 
 150 12305   2461 41
180   12760  2552 42.5 
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Can any work?

• No single velocity
• Upper Earth is slower than deep Earth

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15
12
10
8
12
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So what about an Earth with 2 layers?
??
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Refraction

• When a wave changes speed it bends
• The amount of bending is given by Snells law

Sin (A)
V1

V1
Sin (B)
V2
A
V2
B
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So what about an Earth with 2 layers?
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Unfortunately, that does not work either..

• We need more layers!
• Call in the mathematician!

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Seismic Ray

• a mathematical approach
• Dusty Wilson

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Mathematical Overview

• A description of the problem
• The mathematics of the solution
• Examples of two models
• Lessons learned for next time

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A Description of the Problem

• Create an algorithm to model the path of a
  seismic ray through a planetary body.
• Assume an arbitrary number of layers (or shells)
  in the planetary model.
• Assume rays travel at a constant velocity through
  each layer.
• Assume the trajectory of each ray changes as the
  ray changes layers, subject to Snells law.

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The Mathematics

• The mathematics of this project required topics
  found at pre calculus level.
• The Law of Cosines
• The Quadratic Formula
• Rotation of Axes

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Law of Cosines

• Law of Cosines
• Solve for C by sub-tracting A2 from both sides of
  the equation

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The Quadratic Formula

• This equation (below) is quadratic in C
• Which can be solved using the quadratic formula
• Do I choose or ?

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Rotation of Axes

• After the ray travels through the outer layer,
  all subsequent paths are determined by an angle
  made with a tangent.
• This requires a rotation of axes by the angle ?.

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Definition of a Function

• Does the resultant graph represent a function?
• More importantly What is independent and what is
  dependent when it comes to seismic waves.

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Examples

• Example 1 A model using two layers.
• Example 2 The PREM model which uses 74 layers to
  model the Earth.

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Example 1 A Two Layer Model

• The earthquake takes place at the N. Pole.
• Waves are sent out in all directs at once.
• The model shows individual ray paths but all
  begin at the same time.
• The waves bend subject to Snells Law

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Example 1 Angle vs. Time

• The graph shows angle (around the globe) from the
  rays start to finish versus the time for the way
  to travel through the model.
• The discontinuity correlates to the layer
  change.

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Example 2 The PREM Model

• The Preliminary Reference Earth Model (PREM) is a
  current model used by geologists to understand
  the interior of the Earth
• It uses 74 layers to model the physical results
  of seismographs.

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Example 2 Angle vs. Time

• This output from seismic algorithm models the
  actual output of seismographs around the world.

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Lessons Learned

• Mathematics is difficult when the answer is not
  in the back of the book
• Documentation and prep work is well worth the
  time
• Mathematics doesnt need to be sophisticated to
  pose a serious challenge
• Mathematica beautiful yet aggravating
• Challenges lead to excitement (and )

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Thats all fine and good, but we need something a
student can use!
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Why We Used Java

• Graphical User Interface (GUI)
• Web based Applets
• Runs in a browser
• Platform Independent
• Can be accessed from anywhere

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Program Development
Analyze
Document
Debug
Design
Test
Code
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Converter
InputPanel
Line
RealData
Program Design
LineManager
Display
Graph
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Code Translation
Mathematica
Java
public static void nextSymAlpha() double
symTheta0 thetaj-1 double v0
velocityWInLayernLayers-j-1 double v1
velocityWInLayernLayers-j double symAlpha1
Math.asin((v1Math.sin((Math.PI/2)-symTheta0))/v
0) alpha appendTo1D(alphaIndex,
symAlpha1, alpha)
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Before
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and After!
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Some final notes

• This project required all of our expertise
• The only place a project like this could happen
  is at a community college
• Many times we face problems that require
  knowledge and expertise from outside of our field
• The result is an amazing learning opportunity for
  students!

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Questions?

• Afterwards you are welcome to try the Java program

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How to use the program

• Turn on a laptop
• Press cntrl-alt-del
• Log on as FRCuser
• No Password
• Log on to (this computer)
• Start seismicGraph document on the desktop (not
  the folder)