applied geophysics potential field methods jeannot trampert
TRANSCRIPT
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APPLIED GEOPHYSICS
POTENTIAL FIELD METHODS
JEANNOT TRAMPERT
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GAUSS’ THEOREM
For any vector F
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STOKES’ THEOREM
For any vector F
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POTENTIAL FIELD THEORY
A force F derives from a scalar potential Φ if
The work done by force F (see Stokes)
irrotational conservative field
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POTENTIAL FIELD THEORY
A force field B derives from a vector potential A if
A is not unique (gauge conditions divA=0 or divA=-dφ/dt)
divergence free incompressible solenoidal field
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GRAVITY
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GRAVITY
Gauss
Stokes
PoissonLaplace
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GRAVITY
Gravity measures spatial variations of the gravitational field due to lateral variations in density.
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ELECTROSTATICS (CHARGES AT REST)
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Gauss
Stokes
PoissonLaplace
ε = permittivity
ELECTROSTATICS (CHARGES AT REST)
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MAGNETOSTATICS (MOVING CHARGES)
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MAGNETOSTATICS (MOVING CHARGES)
Lorentz
Ampere
μ = permeability
If no currents (j=0) B derives from a scalar potential
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BOUNDARY VALUE PROBLEMS
Poisson
Laplace
• ρ is a source term• Solutions to the Laplace equation are called harmonic
functions• Poisson and Laplace are elliptic pde • Boundary value problem: Find φ in a volume V given
the source and additional information on the surface:• Dirichlet: φ specified on the surface• Neumann: gradφ specified on the surface
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MAGNETOSTATICS
Geomagnetics measures spatial variations of the intensity of the magnetic field due to lateral variations in magnetic susceptibility.
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ELECTROMAGNETICSMOVING CHARGES IN TIME VARYING FIELDS
Maxwell’s equations
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ELECTRO MAGNETICS
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GRAVITY METHOD
The acceleration of a mass m due to another mass M at a distance r is given by
We can only directly measure g in the vertical direction. In exploration, we usually directly deal with g, in large scale problems it is easier to work with the scalar potential (geoid)
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GRAVITY METHOD
The contributions are summed in the vertical direction.
Unit: 1 m/s2
Earth surface 9.8 m/s2
980 cm/s2
980 Gal980000 mGalanomalies order of mGal
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MEASURING GRAVITY
Falling body measurements
Mass and spring measurements
Pendulum measurements
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PENDULUMThe period T of a pendulum is related to g via K which represents the characteristics of the pendulum
K is difficult to determine accurately Relative measurements
Precision 0.1mGal Precision of T 0.1 ms Long measurements
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MASS ON SPRINGLacoste introduced a zero-length spring (tension proportional to length) first used in the Lacoste-Romberg gravitymeter. Zero length-string is very sensitivity to small changes in g. In the Worden gravitymeter spring and lever are made from quartz minimizes temperature changes 0.01 mGal precision
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ABSOLUTE GRAVITY MEASUREMENTSIf we only survey a small region, relative measurements are enough (assume reference g), but comparing different regions requires the knowledge of absolute gravity. IGSN-71Absolute measurements (z=gt2/2)