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

• We know Mar’s soil composition from the Mars pathfinder mission

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

Objective:

Find the mobile elements that make up Martian soil.

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

Water Is Important because water leads to life

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]

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

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

• We need…

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

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

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

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

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

=

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

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

transformation.

  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

• 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.

  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