Seismic Rays and The Interior of the Earth

1. 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

2. The Problem: How do we know what is beneath our feet?

3. 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

4. Seismic Waves • Formed when earthquakes occur

5. 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

6. We can measure the arrival times

7. So, lets use this information… • Could the Earth be entirely made of blue cheese? • Seismic velocity through blue cheese is 5 km/sec

8. An Earth made of cheese 0 40 12760 km 80 120 150 180

9. An Earth made of cheese 0 40 80 120 150 180

10. Remember, it has to match what we measure!

11. Does it work?

12. Can any work? • No single velocity • Upper Earth is slower than deep Earth 5 15 12 10 8 12

13. So what about an Earth with 2 layers? ??

14. 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

15. So what about an Earth with 2 layers?

16. Unfortunately, that does not work either….. We need more layers! Call in the mathematician!

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

18. 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.

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

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

21. The Quadratic Formula • This equation (below) is quadratic in C: • Which can be solved using the quadratic formula: • Do I choose “+” or “–”?

22. 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.

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

24. 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

25. 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.

26. 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.

27. Example 2: Angle vs. Time • This output from seismic algorithm models the actual output of seismographs around the world.

28. 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 …)

29. That’s all fine and good, but we need something a student can use!

30. 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

31. Program Development Document Analyze Debug Design Test Code

32. InputPanel Converter Line RealData Program Design LineManager Display Graph

33. 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); }

34. Before…

35. …and After!

36. 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!

37. Questions? • Afterwards you are welcome to try the Java program

38. 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)