A non-standard boundary value problem related to geomagnetism

Consider the following boundary value problem in the exterior space ${\hat S^{d - 1}} = \left \{ {x \in {^d}:\left | x \right | > 1} \right \}$ of a sphere ${S^{d - 1}}$ in two and three dimensions $\left ( d = 2, 3 \right )$: Given a vector field $D:{S^{d - 1}} \to {^d}$ we ask for all harmonic vector fields $B:{\hat S^{d - 1}} \to {^d}$ which decay at least as fast as a dipole field at infinity and are parallel to D on ${S^{d - 1}}$ i.e. there is $f:{S^{d - 1}} \to$ such that $B = fD$. For $d = 3$, this problem is related to the problem of reconstructing the geomagnetic field outside the earth from directional data measured on the earth’s surface. The question for uniqueness or non-uniqueness is of particular interest here.