last time: gravity modeling gravitational attraction of an arbitrary body: gravity anomaly due to a...
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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)
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Δgz =4πGR3Δρd
3 x 2 + d 2( )
3
2
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Δgz = 2πGR2Δρd
x 2 + d 2( )
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Δgz = 2πGΔρ h2 − h1 + R2 + h12 − R2 + h2
2 ⎛ ⎝ ⎜ ⎞
⎠ ⎟
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rg = −4πG ρdV
Vol
∫∫∫
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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:
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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:
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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
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rr
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rr
Magnetic Field Strength is the force that would be exerted on a hypothetical “unit monopole” p2 due to p1:
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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.
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rm = ˆ r lp
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rI =
r m
V
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rH
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rI = k
r H
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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…
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rI = k
r H E
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rH E
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rI T = k
r H E +
r I R