  We consider a three-dimensional magnetic field produced by an arbitrary
collection of dipoles. Assuming the magnetic vector or its gradient tensor
field is measured above the earth surface, the inverse problem is to use the
measurement data to find the location, strength, orientation and distribution
of the dipoles underneath the surface. We propose a reversible jump Markov
chain Monte Carlo (RJ-MCMC) algorithm for both the magnetic vector and its
gradient tensor to deal with this trans-dimensional inverse problem where the
number of unknowns is one of the unknowns. A special birth-death move strategy
is designed to obtain a reasonable rate of acceptance for the RJ-MCMC sampling.
  Some preliminary results show the strength and challenges of the algorithm in
inverting the magnetic measurement data through dipoles. Starting with an
arbitrary single dipole, the algorithm automatically produces a cloud of
dipoles to reproduce the observed magnetic field, and the true dipole
distribution for a bulky object is better predicted than for a thin object.
Multi-objects located at different depths remain a very challenging inverse
problem.
