Seismic Rays and The Seismic Rays and The Interior of the EarthInterior of the Earth
Dusty WilsonDusty WilsonTina OstranderTina Ostrander
Eric BaerEric BaerWith lots of great help fromWith lots of great help from
Logan Wallace and Tim MinaliaLogan Wallace and Tim Minalia
The Problem:The Problem:
How do we know what is beneath our feet?
How can we find out about the How can we find out about the Interior of the Earth?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:
10 km
250 km6370 km
3 km
Seismic WavesSeismic Waves
• Formed when earthquakes occur
Seismic compression wave velocitySeismic compression wave velocity
In air 0.344 km/secIn water 1.5 km/secIn Jello 4 km/secIn glass 4.5 km/secIn rocks 7-15 km/sec
We can measure the arrival timesWe can measure the arrival times
So, lets use this information… So, lets use this information…
• Could the Earth be entirely made of blue cheese?
• Seismic velocity through blue cheese is 5 km/sec
An Earth made of cheeseAn Earth made of cheese
40
80
120
150180
0Geocentric
Angle (degrees)
raypath length
(km)
TravelTime(s)
TravelTime(min)
0
40
80
120
150
180
12760 km
An Earth made of cheeseAn Earth made of cheese
40
80
120
150180
0Geocentri
cAngle
(degrees)
raypath length(km)
TravelTime(s)
TravelTime(min)
0 0 0 040 4357 871 14.5
80 8189 1638 27.29 120 11050 2210 36.8 150 12305 2461 41180 12760 2552 42.5
Remember, it has to match what Remember, it has to match what we measure!we measure!
Does it work?Does it work?Geocentri
cAngle
(degrees)
raypath length(km)
TravelTime(s)
TravelTime(min)
0 0 0 040 4357 871 14.5 80 8189 1638 27.29
120 11050 2210 36.8 150 12305 2461 41180 12760 2552 42.5
Can any work? Can any work?
• No single velocity• Upper Earth is
slower than deep Earth
15 1212 10 8
5
So what about an Earth with 2 So what about an Earth with 2 layers?layers?
??
RefractionRefraction• When a wave
changes speed it bends
• The amount of bending is given by Snell’s law
A V1
V2B
Sin (A) V1
V2Sin (B)
=
So what about an Earth with 2 So what about an Earth with 2 layers?layers?
Unfortunately, that does not Unfortunately, that does not work either…..work either…..
We need more layers!
Call in the mathematician!
Mathematical OverviewMathematical Overview
• A description of the problem• The mathematics of the solution• Examples of two models • Lessons learned for next time
A Description of the ProblemA 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 Snell’s law.
The MathematicsThe Mathematics
• The mathematics of this project required topics found at pre calculus level.– The Law of Cosines– The Quadratic Formula– Rotation of Axes
Law of CosinesLaw of Cosines
• Law of Cosines
• Solve for C by sub-tracting A2 from both sides of the equation
2 2 2 2 cosA B BC C a
2 2 20 2 cosB aC C B A
The Quadratic FormulaThe Quadratic Formula
• This equation (below) is quadratic in C:
• Which can be solved using the quadratic formula:
• Do I choose “+” or “–”?
2 2 2 21 2 cos 4 cos 42
B a B a B AC
2 2 20 2 cosB aC C B A
Rotation of AxesRotation 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 Ө., cos sin , sin cosx y x y x y
ExamplesExamples
• Example 1: A model using two layers.
• Example 2: The PREM model which uses 74 layers to model the Earth.
Example 1: A Two Layer ModelExample 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 Snell’s Law
Example 1: Angle vs. TimeExample 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.
Example 2: The PREM ModelExample 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.
Example 2: Angle vs. TimeExample 2: Angle vs. Time
• This output from seismic algorithm models the actual output of seismographs around the world.
Lessons LearnedLessons 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 doesn’t need to be sophisticated to pose a serious challenge
• Mathematica – beautiful yet aggravating• Challenges lead to excitement (and …)
That’s all fine and good, but That’s all fine and good, but we need something a student we need something a student
can use!can use!
Why We Used JavaWhy We Used Java
• Graphical User Interface (GUI)• Web based Applets
– Runs in a browser– Platform Independent– Can be accessed from anywhere
MyProgram.jaMyProgram.javava
MyProgram.clMyProgram.classass
compile
Program DevelopmentProgram Development
AnalyzeDocument
Debug
Test Code
Design
RealData
ConverterLine
Graph
InputPanel
LineManager
Display
ProgramDesign
Code TranslationCode TranslationNextj_,n_: Module0,1,v0,v1,
0 THETAj;Known angle within layer nearer the surfacev0 VelocityWInLayernj;Layer neared the surfacev1 VelocityWInLayernj1;Layer nearer the center1 ArcSinv1Sin
2 0
v0;Angle found with Snell's Law near the center
AppendToALPHA,1public static void nextSymAlpha() {
double symTheta0 = theta[j-1];
double v0 = velocityWInLayer[nLayers-j-1];
double v1 = velocityWInLayer[nLayers-j];
double symAlpha1 = Math.asin((v1*Math.sin((Math.PI/2)-symTheta0))/v0);
alpha = appendTo1D(++alphaIndex, symAlpha1, alpha); }
Java
Mathematica
Some final notes…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!
Questions?Questions?
• Afterwards you are welcome to try the Java program
How to use the programHow 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)
