Martian Soil Analysis With Linear Algebra

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

• Was Water necessary to form Mars soil?
• What Geological processes contributed to Mars
  soil composition?

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Method

• Soil is a combination of rocky material and
  mobile elements
• We know Mars soil composition from the Mars
  pathfinder mission

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• We Know Mars Basalt (rock) composition from
  Martian meteorites primarily the Shergotty
  meteorite.

Objective Find the mobile elements that make
up Martian soil.
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• 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
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Water Is Important because water leads to life
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• Our problem
• Basalt mobile elements soil
• Because the transformations are limited the end
  result will be
• X basalt y mobile elements soil (xy100)
• We can change this into matrix form
• Basalt Mobile elements x,y soil

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The Math
  Shergotty 1592H AGSII
SiO2 49.5033 41.39015
TiO2 0.870142 1.106066
Al2O3 7.59 15.90406
FeO 19.80055 18.98941
MnO 0.189 0.0815
MgO 8.95 17.75527
CaO 9.63 4.529049
Na2O 0 0
Mars soil
48.60337
1.160506
9.169147
22.87773
0.7009
9.984948
7.503074
0
x
y

X Y 1
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• We cant solve this using traditional methods. We
  need to approximate an answer.
• We need

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• Least-Square Fitting

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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 Ax b
  however, it can solve the system ATAx ATb

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The Pseudoinverse

• If A is a matrix with linearly independent
  columns, then the pseudoinverse of A is the
  matrix A defined by
• A (ATA)-1AT
• The least squares solution to Ax b is
• x Ab

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The Math Part 2
  Shergotty 1592H AGSII
SiO2 49.5033 41.39015
TiO2 0.870142 1.106066
Al2O3 7.59 15.90406
FeO 19.80055 18.98941
MnO 0.189 0.0815
MgO 8.95 17.75527
CaO 9.63 4.529049
Na2O 0 0

• A

AT
49.5033 0.870142 7.59 19.80055 0.189 8.95 9.63 0
41.39015 1.106066 15.90406 18.98941 0.0815 17.75527 4.529049 0
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More math!!!!

• ATA

3073.878 2749.164
2749.164 2663.673
(ATA) -1
0.004229 -0.00436
-0.00436 0.00488
(ATA) -1AT
0.02869 -0.00115 -0.03732 0.000853 0.000444 -0.03964 0.020956 0
-0.01407 0.0016 0.044484 0.006249 -0.00043 0.047582 -0.01993 0
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Even more math

• x (ATA) -1AT mars soil

0.02869 -0.00115 -0.03732 0.000853 0.000444 -0.03964 0.020956 0
-0.01407 0.0016 0.044484 0.006249 -0.00043 0.047582 -0.01993 0
48.60337
1.160506
9.169147
22.87773
0.7009
9.984948
7.503074
0

0.832171
0.194017
x
y

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Least Square Fitting with two variables
  Shergotty 1592H AGSII     Mode Mars soil Calc Residual
SiO2 49.5033 41.39015   SNC 0.832171 48.60337 49.23 -0.622
TiO2 0.870142 1.106066   clay 0.194017 1.160506 0.94 0.222
Al2O3 7.59 15.90406       9.169147 9.40 -0.233
FeO 19.80055 18.98941       22.87773 20.16 2.716
MnO 0.189 0.0815       0.7009 0.17 0.528
MgO 8.95 17.75527       9.984948 10.89 -0.908
CaO 9.63 4.529049       7.503074 8.89 -1.389
Na2O 0 0       0 0.00 0.000
                 
        Mode Sum 1.026   sum r2 10.900
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• But what if more than one event altered the rock?
• Just add another column to A to represent that
  transformation.

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Least square Fitting with three variables
  Shergotty 11191H AGSI Pantelleria   Mode mars soil Calc Residual
SiO2 49.50329528 49.58864119 46.14432168   0.649711802 48.603 48.812 -0.208
TiO2 0.87014161 2.614118896 0.052508545   0.07674857 1.161 0.781 0.380
Al2O3 7.589999976 21.15180467 8.181009268   0.27831918 9.169 8.832 0.338
FeO 19.80055118 18.1395966 31.19874999     22.878 22.940 -0.062
K2O 0.189000035 0.252123142 1.923169754     0.701 0.677 0.023
MgO 8.950000009 2.98566879 11.63196899     9.985 9.281 0.703
CaO 9.62999999 4.989384289 2.226170614     7.503 7.259 0.244
Na2O 0 0 0     0.000 0.000 0.000
  0              
        Mode Sum 1.004779553   sum r2 0.860
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• 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.

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Least square Fitting with three variables
  Shergotty 11191H AGSI Pantelleria   Mode mars soil Calc Residual
SiO2 49.50329528 49.58864119 46.14432168   0.649711802 48.603 48.812 -0.208
TiO2 0.87014161 2.614118896 0.052508545   0.07674857 1.161 0.781 0.380
Al2O3 7.589999976 21.15180467 8.181009268   0.27831918 9.169 8.832 0.338
FeO 19.80055118 18.1395966 31.19874999     22.878 22.940 -0.062
K2O 0.189000035 0.252123142 1.923169754     0.701 0.677 0.023
MgO 8.950000009 2.98566879 11.63196899     9.985 9.281 0.703
CaO 9.62999999 4.989384289 2.226170614     7.503 7.259 0.244
Na2O 0 0 0     0.000 0.000 0.000
  0              
        Mode Sum 1.004779553   sum r2 0.860
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The End!!!