Sharon L. Mitchell Althea Maduramente Jessalyn Klein University at Buffalo.
Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.
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Transcript of Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.
![Page 1: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/1.jpg)
Martian Soil Analysis With Linear Algebra
By Gary Newsom and Jessalyn Timson
![Page 2: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/2.jpg)
Purpose:
• Was Water necessary to form Mars’ soil?
• What Geological processes contributed to Mars’ soil composition?
![Page 3: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/3.jpg)
Method:
• Soil is a combination of rocky material and mobile elements
• We know Mar’s soil composition from the Mars pathfinder mission
![Page 4: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/4.jpg)
• We Know Mars’ Basalt (rock) composition from Martian meteorites primarily the Shergotty meteorite.
Objective:
Find the mobile elements that make up Martian soil.
![Page 5: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/5.jpg)
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
![Page 6: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/6.jpg)
Water Is Important because water leads to life
![Page 7: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/7.jpg)
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]
![Page 8: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/8.jpg)
The MathMars soil
48.60337
1.160506
9.169147
22.87773
0.7009
9.984948
7.503074
0
Shergotty1592H
AGSII
SiO249.5033 41.39015
TiO20.870142 1.106066
Al2O37.59 15.90406
FeO19.80055 18.98941
MnO0.189 0.0815
MgO8.95 17.75527
CaO9.63 4.529049
Na2O0 0
=
X +Y = 1
x
y
![Page 9: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/9.jpg)
• We can’t solve this using traditional methods. We need to approximate an answer.
• We need…
![Page 10: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/10.jpg)
![Page 11: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/11.jpg)
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
![Page 12: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/12.jpg)
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
![Page 13: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/13.jpg)
The Math: Part 2
• A= 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^T =
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
![Page 14: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/14.jpg)
More math!!!!
• A^T*A = 3073.878 2749.164
2749.164 2663.673
(A^T*A)^ -1= 0.004229 -0.00436
-0.00436 0.00488
(A^T*A)^ -1*A^T =
0.02869
-0.0011
5 -0.037320.00085
30.00044
4 -0.039640.02095
6 0
-0.01407 0.0016
0.044484
0.006249
-0.00043
0.047582
-0.01993 0
![Page 15: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/15.jpg)
Even more math
• x = (A^T*A)^ -1*A^T * mars soil
0.02869-
0.00115 -0.03732 0.000853 0.000444 -0.039640.02095
6 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
=
![Page 16: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/16.jpg)
Least Square Fitting with two variables
Shergott
y
1592H AGSII
Mode Mars soil CalcResidu
al
SiO249.5033 41.39015
SNC = 0.83217148.60337
49.23 -0.622
TiO2 0.870142 1.106066 clay= 0.194017 1.160506 0.94 0.222
Al2O37.59 15.90406
9.169147
9.40 -0.233
FeO19.80055 18.98941
22.87773
20.16 2.716
MnO 0.189 0.0815 0.7009 0.17 0.528
MgO8.95 17.75527
9.984948
10.89 -0.908
CaO9.63 4.529049
7.503074
8.89 -1.389
Na2O 0 0 0 0.00 0.000
Mode Sum 1.026 sum r2 10.900
![Page 17: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/17.jpg)
• But what if more than one event altered the rock?• Just add another column to A to represent that
transformation.
![Page 18: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/18.jpg)
Shergotty11191H
AGSIPantelleri
a Modemars
soil Calc Residual
SiO249.50329
52849.58864
11946.14432
168 0.649711
802 48.603 48.812 -0.208
TiO20.870141
612.614118
8960.052508
545 0.076748
57 1.161 0.781 0.380
Al2O37.589999
97621.15180
4678.181009
268 0.278319
18 9.169 8.832 0.338
FeO19.80055
11818.13959
6631.19874
999 22.878 22.940 -0.062
K2O0.189000
0350.252123
1421.923169
754 0.701 0.677 0.023
MgO8.950000
0092.985668
7911.63196
899 9.985 9.281 0.703
CaO9.629999
994.989384
2892.226170
614 7.503 7.259 0.244
Na2O 0 0 0 0.000 0.000 0.000
0
Mode
Sum1.004779
553 sum r2 0.860
Least square Fitting with three variables
![Page 19: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/19.jpg)
• 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.
![Page 20: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/20.jpg)
Shergotty11191H
AGSIPantelleri
a Modemars
soil Calc Residual
SiO249.50329
52849.58864
11946.14432
168 0.649711
802 48.603 48.812 -0.208
TiO20.870141
612.614118
8960.052508
545 0.076748
57 1.161 0.781 0.380
Al2O37.589999
97621.15180
4678.181009
268 0.278319
18 9.169 8.832 0.338
FeO19.80055
11818.13959
6631.19874
999 22.878 22.940 -0.062
K2O0.189000
0350.252123
1421.923169
754 0.701 0.677 0.023
MgO8.950000
0092.985668
7911.63196
899 9.985 9.281 0.703
CaO9.629999
994.989384
2892.226170
614 7.503 7.259 0.244
Na2O 0 0 0 0.000 0.000 0.000
0
Mode
Sum1.004779
553 sum r2 0.860
Least square Fitting with three variables
![Page 21: Martian Soil Analysis With Linear Algebra By Gary Newsom and Jessalyn Timson.](https://reader036.fdocuments.us/reader036/viewer/2022062423/56649edd5503460f94bee1d0/html5/thumbnails/21.jpg)