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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. The Problem:. How do we know what is beneath our feet?. How can we find out about the Interior of the Earth?. The deepest a human has ever gone:
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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
The Problem: How do we know what is beneath our feet?
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
Seismic Waves • Formed when earthquakes occur
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
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 cheese 0 40 12760 km 80 120 150 180
An Earth made of cheese 0 40 80 120 150 180
Can any work? • No single velocity • Upper Earth is slower than deep Earth 5 15 12 10 8 12
Refraction • When a wave changes speed it bends • The amount of bending is given by Snell’s law Sin (A) V1 = V1 Sin (B) V2 A V2 B
Unfortunately, that does not work either….. We need more layers! Call in the mathematician!
Mathematical Overview • A description of the problem • The mathematics of the solution • Examples of two models • Lessons learned for next time
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 Snell’s law.
The 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 Cosines • Law of Cosines • Solve for C by sub-tracting A2 from both sides of the equation
The Quadratic Formula • This equation (below) is quadratic in C: • Which can be solved using the quadratic formula: • Do I choose “+” or “–”?
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 Ө.
Examples • 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 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. 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 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. Time • This output from seismic algorithm models the actual output of seismographs around the world.
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 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 we need something a student can use!
MyProgram.java MyProgram.class compile Why We Used Java • Graphical User Interface (GUI) • Web based Applets • Runs in a browser • Platform Independent • Can be accessed from anywhere
Program Development Document Analyze Debug Design Test Code
InputPanel Converter Line RealData Program Design LineManager Display Graph
Code Translation Mathematica Java public 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); }
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? • Afterwards you are welcome to try the Java program
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)