The secrets of the Moon rocks, a basic idea of remanent ... · The secrets of the Moon rocks, a...

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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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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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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Introduction


Natural remanent magnetization (NRM)• Primary components

- Thermoremanent magnetization (TRM, acquired duringcooling)

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

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Introduction


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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Introduction


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

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Introduction


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


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


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


Introduction


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


Introduction


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

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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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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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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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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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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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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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Denominator of the magnetic field (1 section study)

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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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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



Pole estimation

Grid method illustrations (Position1, sensors Br & Bt)

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Pole estimation

PHC method illustration (Position1, sensors Br & Bt)

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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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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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Simulations (with synthetic data)


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Simulations with synthetic data: moment recovery

Illustration of 4 different simulations examples

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

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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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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


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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


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


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Conclusions & further work

Conclusion & further work

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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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Thank you for your attention!

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Appendix

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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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The secrets of the Moon rocks, a basic idea of remanent ... · The secrets of the Moon rocks, a...

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