Crustal magnetization

Kathy Whaler

University of Edinburgh

Why study the Martian Magnetic Field?

Satellite data - Earth

• Earth– POGO (1960s and 1970s): scalar field – MAGSAT (1979/80): vector– Ørsted (current): vector (high altitude)– CHAMP (current): vector, but I’ve just worked

with scalar data so far

• Need to extract small crustal anomaly field from data dominated by the main field

Scalar data

• The anomaly field is a tiny fraction of the main field generated in the core, Bc

• Thus we can linearize the relationship between the scalar and vector fields:

• Hence any methods developed to treat vector data will work with minor modifications on scalar data

)(c

η

)(

c

BBB

1B

Satellite data - Mars• Mars Global Surveyor

– current– vector– aerobraking phase provided data as low as 120km

above surface– used data in the 120-600km altitude range

• No main field– field is due to remanent magnetization of the crust

and external field– difficult to use scalar data

Martian magnetic field

• The internal magnetic field amplitude is surprisingly high

• The field is much stronger over the heavily cratered region south of the dichotomy

• Greater external field contamination in the horizontal components

Data and Technique• Low-altitude coverage is sparse, suitable for

analysis to degree 50 globally.• Spatially varying magnetization, linear

combination of Green’s functions relating predicted magnetization to observed magnetic field.

• Minimize RMS magnetization within the Martian crust.

• Data by data information matrix is sparse (preconditoned CG).

• Damping controls relative importance of fit and RMS magnetization.

Methodology

• Relate a magnetic field satellite measurement to the magnetic field or magnetization in the crust, e.g.

V jjjj

dVB )(.),(.ˆ)()()( sMsrHlr

r

where (η) denotes the component, rj is the satellite datum position, s positionwithin the magnetized crust, H a known geometrical function, and M magnetization

Methodology

• Express the model as a linear combination of the data kernels

• Find the multipliers that minimize e.g.

so-called minimum norm solutions

• Hence model continuously-varying functions, either downward continued B, or M within the crust

dVV

2M

Numerical considerations

• Minimum norm solutions require solving a data-by-data system of equations - too big

• Reduce by:– expanding in terms of data kernels at a limited

number of points– taking advantage of peaked nature of data

kernels - matrix effectively sparse• Calculation parallelises effectively• Use iterative conjugate gradient algorithm on

resulting sparse system• Improve convergence by Jacobi preconditioning

Green’s function showing how the surface magnetic field contributes to a satellite measurement at 400km altitude. Solid/dashed line: vertical/horizontal component

Its interpretation

A magnetic chronology

Interpretation caveats

• Residuals to our model are strongly non-normal: external fields and unmodelled sources

• Varying misfit by a factor of 3 alters the RMS magnetization by two orders of magnitude: model does not constrain magnetization strength, only direction

Comparison with other magnetization models

•Discrete model of magnetization (Langlais et al., 2003): correlation coefficients are in excess of 0.8.

•Ideal body theory (Parker, 2003) demonstrates that magnetization must be at least 4.76 A/m for a 50 km thick crust.

•Ten isolated bodies described by Arkani-Hamed and Boutin (2003): Five of their ten paleomagnetic poles are within 30º of our poles. The others are at angular distances of 32º, 41º, 59º, 66º and 69º.

Power spectra for downward continued Magsat model (diamonds) and aeromagnetic compilation (crosses) over Africa

This rendition of the North American

compilation has had wavelenths longer

than 500 km replaced with the

Comprehensive model (Sabaka et al., 2002), utilizing a technique

developed by Ravat et al. (2002). The data is

on a 1 km grid projected with a

spherical transverse Mercator. The grid

size is 8901 by 8511.

Inversion for magnetization

directions based on satellite and near-

surface data

Depth-independent magnetization

• Satellite data ‘see’ magnetized crust as a thin sheet

no point in trying to resolve depth dependence• Allowing depth variation seems to cause

problems when integrating aeromagnetic data• Almost finished implementing code to calculate M constant with depth through magnetized layer

Conclusions

• Satellite data have provided a new perspective on the magnetic fields of both Earth and Mars

• The long wavelength crustal magnetization of both planets aids structural and tectonic interpretation