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 Mars soil 48.60337 1.160506 9.169147 22.87773 0.7009 9.984948 7.503074 0 Shergotty 1592H AGSII SiO2 49.503341.39015 TiO2 0.8701421.106066 Al2O3 7.5915.90406 FeO 19.8005518.98941 MnO 0.1890.0815 MgO 8.9517.75527 CaO 9.634.529049 Na2O 00 = X +Y = 1 x y

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

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= Shergotty 1592H AGSII SiO2 49.503341.39015 TiO2 0.8701421.106066 Al2O3 7.5915.90406 FeO 19.8005518.98941 MnO 0.1890.0815 MgO 8.9517.75527 CaO 9.634.529049 Na2O 00 A^T = 49.50330.8701427.5919.800550.1898.959.630 41.390151.10606615.9040618.989410.081517.755274.5290490

14. More math!!!! A^T*A = 3073.8782749.164 2663.673 (A^T*A)^ -1= 0.004229-0.00436 0.00488 (A^T*A)^ -1*A^T = 0.02869 - 0.0011 5-0.03732 0.00085 3 0.00044 4-0.03964 0.02095 60 - 0.014070.0016 0.04448 4 0.00624 9 - 0.00043 0.04758 2 - 0.019930

15. Even more math x = (A^T*A)^ -1*A^T * mars soil 0.02869 - 0.00115-0.037320.0008530.000444-0.039640.0209560 -0.014070.00160.0444840.006249-0.000430.047582-0.019930 48.60337 1.160506 9.169147 22.87773 0.7009 9.984948 7.503074 0 = 0.832171 0.194017 x y =

16. Least Square Fitting with two variables Shergott y 1592H AGSI I ModeMars soilCalc Residu al SiO2 49.503341.39015 SNC =0.832171 48.60337 49.23-0.622 TiO2 0.8701421.106066 clay=0.194017 1.160506 0.940.222 Al2O3 7.5915.90406 9.169147 9.40-0.233 FeO 19.8005518.98941 22.87773 20.162.716 MnO 0.1890.0815 0.7009 0.170.528 MgO 8.9517.75527 9.984948 10.89-0.908 CaO 9.634.529049 7.503074 8.89-1.389 Na2O 00 0 0.000.000 Mode Sum1.026 sum r 2 10.900

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

18. Shergotty 11191H AGSI Pantelleri a Mode mars soilCalcResidual SiO2 49.50329 528 49.58864 119 46.14432 168 0.649711 80248.60348.812-0.208 TiO2 0.870141 61 2.614118 896 0.052508 545 0.076748 571.1610.7810.380 Al2O3 7.589999 976 21.15180 467 8.181009 268 0.278319 189.1698.8320.338 FeO 19.80055 118 18.13959 66 31.19874 999 22.87822.940-0.062 K2O 0.189000 035 0.252123 142 1.923169 754 0.7010.6770.023 MgO 8.950000 009 2.985668 79 11.63196 899 9.9859.2810.703 CaO 9.629999 99 4.989384 289 2.226170 614 7.5037.2590.244 Na2O000 0.000 0 Mode Sum 1.004779 553 sum r20.860 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. Shergotty 11191H AGSI Pantelleri a Mode mars soilCalcResidual SiO2 49.50329 528 49.58864 119 46.14432 168 0.649711 80248.60348.812-0.208 TiO2 0.870141 61 2.614118 896 0.052508 545 0.076748 571.1610.7810.380 Al2O3 7.589999 976 21.15180 467 8.181009 268 0.278319 189.1698.8320.338 FeO 19.80055 118 18.13959 66 31.19874 999 22.87822.940-0.062 K2O 0.189000 035 0.252123 142 1.923169 754 0.7010.6770.023 MgO 8.950000 009 2.985668 79 11.63196 899 9.9859.2810.703 CaO 9.629999 99 4.989384 289 2.226170 614 7.5037.2590.244 Na2O000 0.000 0 Mode Sum 1.004779 553 sum r20.860 Least square Fitting with three variables