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Martian Soil Analysis With Linear Algebra

Martian Soil Analysis With Linear Algebra. By Gary Newsom and Jessalyn Timson. Purpose:. Was Water necessary to form Mars’ soil? What Geological processes contributed to Mars’ soil composition?. Method:. Soil is a combination of rocky material and mobile elements

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Martian Soil Analysis With Linear Algebra

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  1. Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson

  2. Purpose: • Was Water necessary to form Mars’ soil? • What Geological processes contributed to Mars’ soil composition?

  3. Method: • Soil is a combination of rocky material and mobile elements • We know Mar’s soil composition from the Mars pathfinder mission

  4. We Know Mars’ Basalt (rock) composition from Martian meteorites primarily the Shergotty meteorite. Objective: Find the mobile elements that make up Martian soil.

  5. We know the geological transformations that can act on the basalt to form the mobile elements through lab tests and observations on earth. Some of these require water or hydrothermal environments

  6. Water Is Important because water leads to life

  7. Our problem: • Basalt + mobile elements = soil • Because the transformations are limited the end result will be • X% basalt + y% mobile elements = soil (x+y=100%) • We can change this into matrix form • [Basalt Mobile elements] [x,y] = [soil]

  8. The Math = X +Y = 1

  9. We can’t solve this using traditional methods. We need to approximate an answer. • We need…

  10. Least-Square Fitting

  11. Least Squares Approximation!!! • The Least Squares method is used to find the line of best fit for a certain number of data points in a given plane • It cannot solve the linear system A*x = b; however, it can solve the system A^T*A*x = A^T*b

  12. The Pseudoinverse • If A is a matrix with linearly independent columns, then the pseudoinverse of A is the matrix A^+ defined by: A^+ = (A^T*A)^-1*A^T • The least squares solution to A*x = b is: x = A^+*b

  13. The Math: Part 2 • A= A^T =

  14. More math!!!! • A^T*A = (A^T*A)^ -1= (A^T*A)^ -1*A^T =

  15. Even more math • x = (A^T*A)^ -1*A^T * mars soil = =

  16. Least Square Fitting with two variables

  17. But what if more than one event altered the rock? • Just add another column to A to represent that transformation.

  18. Least square Fitting with three variables

  19. To find the best fit the data was compared to many different alterations and the one with the least error and the sum closest to 100% was selected. That was the previous slide.

  20. Least square Fitting with three variables

  21. The End!!!

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