Magnetic dipole localization with a gradiometer: obtaining unique solutions

The five independent equations describing the field gradient tensor at a point for a static magnetic dipole source can be inverted to give the bearing vector to the source and the source moment vector divided by the fourth power of the range. The equations have four solutions, two of which are related in a non-trivial way, and two more that are obtained by reflections through the field point. The symmetry of the four solutions in the principal-axis frame of the gradient tensor can be used to express the solutions directly in terms of one another in an arbitrary frame without executing the inversion. This relationship has been exploited to construct explicit proofs that a unique solution for magnetic moment vector and relative position between source and field point can be obtained if either the magnetic field vector, or the rate of change of the gradient tensor and the relative motion of source and field point is known.