the secrets of the moon rocks, a basic idea of remanent ... · the secrets of the moon rocks, a...
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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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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
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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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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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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Introduction
Motivation of the study
Figure 4: Magnetic field measurements of Apollo samplesThe Lunar dynamo, Weiss et al, Science 346, 2014 7 / 33
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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
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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
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Introduction
Magnetometer and data acquisition process
Figure 8: Available data geometry (9 sections)10 / 33
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Introduction
Magnetometer and data acquisition process
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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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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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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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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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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Magnetic dipole localization
Denominator of the magnetic field (1 section study)
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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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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
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Magnetic dipole localization
Pole estimation
Grid method illustrations (Position1, sensors Br & Bt)
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Magnetic dipole localization
Pole estimation
PHC method illustration (Position1, sensors Br & Bt)
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Magnetic dipole localization
Pole estimation
PHC method illustration (Position1, sensors Br & Bt)
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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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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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Simulations (with synthetic data)
Simulations (with synthetic data)
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Simulations (with synthetic data)
Simulations with synthetic data: moment recovery
Illustration of 4 different simulations examples
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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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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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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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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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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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