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

GRAVITYGravity 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

MAGNETOSTATICSGeomagnetics 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)