PhD seminar 2018, INRIA Sophia Antipolis

The secrets of the Moon rocks, a basic idea ofremanent magnetization and how to approximate

it with rational analysis

Konstantinos Mavreas

Advisors: Sylvain Chevillard and Juliette LeblondINRIA Sophia Antipolis, FACTAS team

Nov 12, 2018

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PhD seminar 2018, INRIA Sophia Antipolis

Overview

1 Introduction

2 Magnetization - moment estimation

3 Magnetic dipole localizationPole estimationGeometrical method

4 Simulations (with synthetic data)

5 Conclusions & further work

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PhD seminar 2018, INRIA Sophia Antipolis

Introduction

Motivation of the study

Figure 1: Magnetic anomalies on Moon surfaceAn Impactor Origin for Lunar Magnetic Anomalies, Wieczorek et al, Science 335, 2012

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PhD seminar 2018, INRIA Sophia Antipolis

Introduction

Motivation of the study

Natural remanent magnetization (NRM)• Primary components

- Thermoremanent magnetization (TRM, acquired duringcooling)

- Chemical remanent magnetization (CRM, chemical action orgrowth of crystals)

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PhD seminar 2018, INRIA Sophia Antipolis

Introduction

Motivation of the study

Natural remanent magnetization (NRM)• Secondary components

- Isothermal remanent magnetization (IRM, induced through alarge magnetic field)

Figure 2: Gallery of Australian government bureau meteorology

- Viscous remanent magnetization (VRM, exposed in a field fora long period of time)

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PhD seminar 2018, INRIA Sophia Antipolis

Introduction

Motivation of the study

Figure 3: Benjamin P. Weiss, Sonia M. Tikoo:Science 05 Dec 2014, Vol. 346, Issue 6214, 1246753

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PhD seminar 2018, INRIA Sophia Antipolis

Introduction

Motivation of the study

Figure 4: Magnetic field measurements of Apollo samplesThe Lunar dynamo, Weiss et al, Science 346, 2014 7 / 33

PhD seminar 2018, INRIA Sophia Antipolis

Introduction

ANR project MagLune

General purpose

Moon’s past magnetic activityStudying the magnetic field ofMoon rocks (nano Tesla)[MIT, CEREGE-CNRS]

Figure 5: NASA galleries

Measurements in NASA

Portability

Time limitations

Magnetic isolation

Sample’s safety

Goal: samples selection

Magnetic moment

Chronology

Location

Mineral content8 / 33

PhD seminar 2018, INRIA Sophia Antipolis

Introduction

Magnetometer and data acquisition process

Figure 6: Top view Figure 7: Side view

A spinner magnetometer for large Apollo lunar samples, Uehara et al, Review of

Scientific Instruments 88, 2017 9 / 33

PhD seminar 2018, INRIA Sophia Antipolis

Introduction

Magnetometer and data acquisition process

Figure 8: Available data geometry (9 sections)10 / 33

PhD seminar 2018, INRIA Sophia Antipolis

Introduction

Magnetometer and data acquisition process

Figure 9: Illustration of synthetic data for one position (3 sections)

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PhD seminar 2018, INRIA Sophia Antipolis

Introduction

Single dipole assumption model & Inverse problem

Maxwell equations with approximations

Single dipole assumption

Magnetostatic (quasi-static)

Macroscopic

Expression of the magnetic field:

• B is the magnetic field, X are the sensor positions• Xc is the dipole position, Mc is the magnetization moment

−4π

µ0B(X ) =

|X − Xc |2Mc − 3 [Mc · (X − Xc)] (X − Xc)

|X − Xc |5

µ0 = 4π10−7Tm/A: permeability of free space

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetization - moment estimation

Magnetization - moment estimationNumerator analysis

−4π

µ0B(X ) =

|X − Xc |2Mc − 3 [Mc · (X − Xc)] (X − Xc)

|X − Xc |5

Xc = [0, 0, 0] geoscientists approach

Xc estimated, proposed method

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetization - moment estimation

Numerator

Expression of the magnetic field

−4π

µ0B(X )|X − Xc |5 = |X − Xc |2Mc − 3 [Mc · (X − Xc)] (X − Xc)

Note: Mc =

M1

M2

M3

, B =

B1

B2

B3

expanding and simplifying

B: denote the vector formed from the left hand side of theequation, based on the measurements of a B(X ) component

A: denote the matrix which is formed from the right hand side,based on the dipole location estimation and the sensors positions

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetization - moment estimation

Moment recovery

Step 1: System expressed in matrix form

B = AMc

vector B ∈ RN (= number of sensors · points of measurement)matrix A size N × 3vector Mc ∈ R3

Step 2: Least square problem

Arg minM‖B− AM‖2

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetization - moment estimation

Proposed method: recovery of Xc , Mc

B(X ) magnetic field measured at sensors positions

First step: Xc = [xc , yc , zc , ] estimation

• Non linear problem• Initial pole (point) estimation• Final pole estimation (RARL2 a Gradient descent method,

INRIA APICS team )• Rough localization• Geometrical study

Main purpose: Mc = [M1,M2,M3] estimation

• After Xc recovery, problem becomes linear

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetic dipole localization

Magnetic dipole localizationDenominator analysis

−4π

µ0B(X ) =

|X − Xc |2Mc − 3 [Mc · (X − Xc)] (X − Xc)

[|X − Xc |2]52

X = [x , y , h]

ξ = x + iy : sensor’s path onthe measurement plane

h: sensor’s height

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetic dipole localization

Denominator of the magnetic field (1 section study)

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetic dipole localization

Denominator of the magnetic field (1 section study)

Xc = [xc , yc , zc ]

ξc = xc + iyc : projectionon complex plane

zc : projection on 0z axis

h: sensor’s height

Square distance between the sensor and the dipole position:

|X − Xc |2 = |ξ − ξc |2 + (h − zc)2

= (ξ − ξc)(1

ξ− ξc) + (h − zc)2

= −ξcξ

(ξ − ξ−)(ξ − ξ+) = −1

ξp(ξ)

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetic dipole localization

Pole estimation

Pole estimation ξ−

Figure 10: Physicalillustration∗ Figure 11: Mathematical illustration

From circle to diskWe compute the Fourier coefficients and we use• Grid method: minimizing a cost function• Kung Algorithm, Principal Hankel Components method (PHC)• RARL2: gradient descent method

∗Cyril Langlois: Complex LaTeX visualizations (Tikz), Dipolar magnetic field, 201020 / 33

PhD seminar 2018, INRIA Sophia Antipolis

Magnetic dipole localization

Pole estimation

Grid method illustrations (Position1, sensors Br & Bt)

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetic dipole localization

Pole estimation

PHC method illustration (Position1, sensors Br & Bt)

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetic dipole localization

Pole estimation

PHC method illustration (Position1, sensors Br & Bt)

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetic dipole localization

Pole estimation

Information from ξ−

For each section we compute:

Step 1: Rewrite p(ξ) with change of variable ξ = t ξc|ξc | , |ξ| = |t|

|X − Xc |2 = −ξcξ

[t2 − t

|ξc |2 + (h − zc)2 + 1

|ξc |+ 1

]Verify that ∆ > 0 hence two real solutions(solutions have the same argument)

Step 2: Vieta’s formulas (for polynomial αx2 + βx + γ = 0)& λ coefficient

t+t− =γ

α= 1

t+ + t− = −βα

=|ξc |2 + (h − zc)2 + 1

|ξc |=: λ

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetic dipole localization

Geometrical method

Circles method (combination of 3 sections)

Circle equation, with center C =(λ2 , h)

and radius R =√

λ2

4 − 1

|ξc |2 + (h − zc)2 + 1

|ξc |=: λ ⇔

(|ξc | −

λ

2

)2

+ (zc − h)2 =λ2

4− 1

Illustration of the 3 sections combination:

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PhD seminar 2018, INRIA Sophia Antipolis

Magnetic dipole localization

Geometrical method

Circles method (combination of 3 sections)

Circle equation, with center C =(λ2 , h)

and radius R =√

λ2

4 − 1

|ξc |2 + (h − zc)2 + 1

|ξc |=: λ ⇔

(|ξc | −

λ

2

)2

+ (zc − h)2 =λ2

4− 1

Illustration of the 3 sections combination:

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PhD seminar 2018, INRIA Sophia Antipolis

Simulations (with synthetic data)

Simulations (with synthetic data)

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PhD seminar 2018, INRIA Sophia Antipolis

Simulations (with synthetic data)

Simulations with synthetic data: moment recovery

Illustration of 4 different simulations examples

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PhD seminar 2018, INRIA Sophia Antipolis

Simulations (with synthetic data)

Simulations with synthetic data: moment recoverylocationC = (26.5,−37.1,−89) mmsimulation momentMC = (−0.026,−0.079,−0.034) Am2

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PhD seminar 2018, INRIA Sophia Antipolis

Simulations (with synthetic data)

Simulations with synthetic data (without noise): momentrecovery

Position C(26.5,−37.1,−89)mm

MomentComponents

Previousapproach

Proposedmethod

M1 Am2 -0.026 -0.017 -0.026

M2 Am2 -0.079 -0.031 -0.079

M3 Am2 -0.034 0.069 -0.034

Percent Error - 127% ≈0%

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PhD seminar 2018, INRIA Sophia Antipolis

Simulations (with synthetic data)

Simulations with synthetic data (without noise): momentrecovery

Position A(28, 28, 28) mm

MomentComponents

Previousapproach

Proposedmethod

M1 Am2 0.05 0.034 0.05

M2 Am2 0.05 0.034 0.05

M3 Am2 0.05 0.039 0.05

Percent Error - 29% ≈0%

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PhD seminar 2018, INRIA Sophia Antipolis

Simulations (with synthetic data)

Simulations with synthetic data (without noise): momentrecovery

Position B(46, 26, 45) mm

MomentComponents

Previousapproach

Proposedmethod

M1 Am2 0.017 -0.023 0.017

M2 Am2 0.047 0.019 0.047

M3 Am2 0.086 0.054 0.086

Percent Error - 59% ≈0%

Position B(46, 26, 45) mm

MomentComponents

Previousapproach

Proposedmethod

M′1 Am2 0.067 0.017 0.067

M′2 Am2 0.069 0.032 0.069

M′3 Am2 0.017 -0.018 0.017

Percent Error - 73% ≈0%

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PhD seminar 2018, INRIA Sophia Antipolis

Conclusions & further work

Conclusion & further work

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PhD seminar 2018, INRIA Sophia Antipolis

Conclusions & further work

Conclusion & further work

• Dipole location estimations- Numerical illustrations without noise- Geometrical methods

• Moment recovery affected by- Source position- Data analysis/selection- Dipole orientation

More simulations (with noisy synthetic & real data)

Expand the model for more dipoles (2 to 4)

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PhD seminar 2018, INRIA Sophia Antipolis

Thank you for your attention!

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PhD seminar 2018, INRIA Sophia Antipolis

Appendix

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PhD seminar 2018, INRIA Sophia Antipolis

Glossary

INRIA: National Institute for Research in Computer Scienceand Control

FACTAS: Functional Analysis for the ConcepTion andAssessment of Systems

RARL2: Recursive Rational Approximation L2

CEREGE: European Center Research and Teaching inGeosciences of the Environment

CNRS: National Center for Scientific Research

ANR: French National Research Agency

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