Last Time: Gravity Modeling • Gravitational attraction of an arbitrary body:

• Gravity anomaly due to a sphere: (represents only the vertical component of the total vector!)

• Anomalies for: horizontal cylinder:

vertical cylinder:

fault-bounded basin:

Geology 5660/6660Applied Geophysics

24 Mar 2014

© A.R. Lowry 2014For Wed 26 Mar: Burger 429-449 (§7.1–7.3)

€

Δgz =4πGR3Δρd

3 x 2 + d 2( )

3

2

€

Δgz = 2πGR2Δρd

x 2 + d 2( )

€

Δgz = 2πGΔρ h2 − h1 + R2 + h12 − R2 + h2

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟

€

rg = −4πG ρdV

Vol

∫∫∫

€

Δgz = 2GΔρπt

2+θ2t + x θ2 −

π

2

⎛

⎝ ⎜

⎞

⎠ ⎟sin β cos β + x cos2 β ln

r2

x

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

Magnetics:• Like gravity, a potential field method governed by Laplace’s equation: 2u = f (sources)

• Unlike gravity, source term is a vector rather than a scalar

Gravity:

€

rg = ˆ r

Gm

r 2

Monopole source field is alwaysdirected radially toward a “sink” location

Magnetics:

Dipole source field direction & strengthdepend on one “source”, one “sink”

S

N

Coulomb’s Law describes the force of attraction exerted between two magnetic poles:

€

rF = ˆ r

p1p2

μr 2

where is the distance & direction between two poles p1 & p2 are pole strengths is magnetic permeability, a property of the medium (usually ~1)

p1

p2

€

rr

€

rr

Magnetic Field Strength is the force that would be exerted on a hypothetical “unit monopole” p2 due to p1:

€

rH =

r F

p2

= ˆ r p1

μr 2

(MKS units of Tesla; magnetic anomalies usually given in nT or s)

Definitions:

Magnetic Dipole Moment: Two poles +p and –p separated by a distance l have moment

Intensity of Magnetization:

(where V is volume) is a material property of the magnetic source.

*** If a material that can produce a magnetic field ( > 1) is placed within an external magnetic field H, then intensity of the induced magnetization is

where k = – 1 is magnetic susceptibility of the material.

€

rm = ˆ r lp

€

rI =

r m

V

€

€

rH

€

rI = k

r H

€

rH E

(e.g., magnetite-rich body)

Most Earth materials are diamagnetic: very small negative k ~ –10-5 (e.g., quartz)

Fe-Mg silicates (pyroxene, amphibole, olivine) are paramagnetic (moments can align within small magnetic domains) k ~ 0.02 to 0.2

Hematite (“rust” in aeolian soils) is antiferromagnetic: domains align, equal amounts parallel & antiparallel k ~ 0.05 positive and non-negligible, but still small

Magnetite is ferrimagnetic: domains align parallel & antiparallel but one dominates: k ~ 0.5 to 10! (Also titanomagnetite, ilmenite, pyrrhotite)

Crystalline iron, nickel, cobalt are ferromagnetic: domains align parallel (not common in crust, but common in the core!)

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

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rH E

Gravity Anomaly

Magnetic Anomaly

Magnetic Prospecting: Measure magnitude H of the total field, subtract out magnitude HE of the Earth’s (core dynamo- derived) main field to get a magnetic anomaly

However despite thecomplexities of modeling,magnetic anomalies areheavily used (particularlyby the mining industryand for investigations ofbasement structure)

• Induced magnetization is always in the direction of the ambient field

Must know strength & direction of the Earth’s ambient field to determine location and magnetic susceptibility k of a source body!

• Magnetic field strength falls off proportional to 1/r3

• Total intensity of magnetization

where remanent magnetization IR is in the direction of HE at the time of magnetization…

€

rI = k

r H E

€

rH E

€

rI T = k

r H E +

r I R