applied geophysics potential field methods jeannot trampert
TRANSCRIPT
APPLIED GEOPHYSICS
POTENTIAL FIELD METHODS
JEANNOT TRAMPERT
GAUSS’ THEOREM
For any vector F
STOKES’ THEOREM
For any vector F
POTENTIAL FIELD THEORY
A force F derives from a scalar potential Φ if
The work done by force F (see Stokes)
irrotational conservative field
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
GRAVITY
GRAVITY
Gauss
Stokes
PoissonLaplace
GRAVITY
Gravity measures spatial variations of the gravitational field due to lateral variations in density.
ELECTROSTATICS (CHARGES AT REST)
Gauss
Stokes
PoissonLaplace
ε = permittivity
ELECTROSTATICS (CHARGES AT REST)
MAGNETOSTATICS (MOVING CHARGES)
MAGNETOSTATICS (MOVING CHARGES)
Lorentz
Ampere
μ = permeability
If no currents (j=0) B derives from a scalar potential
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
MAGNETOSTATICS
Geomagnetics measures spatial variations of the intensity of the magnetic field due to lateral variations in magnetic susceptibility.
ELECTROMAGNETICSMOVING CHARGES IN TIME VARYING FIELDS
Maxwell’s equations
ELECTRO MAGNETICS
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)
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
MEASURING GRAVITY
Falling body measurements
Mass and spring measurements
Pendulum measurements
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
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
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)