Analysis of Geomagnetic Directinal Data

It is sometimes desirable to calculate a magnetic field model when only measurements of the direction of the magnetic vector are available. Obviously any such model is indeterminate to an arbitrary constant scalar multiplier; here we investigate the possibility of further ambiguities when the data are perfect and complete. In two dimensions (outside an infinite cylinder with no variation along its axis) we demonstrate the existence of a 2n-parameter family of solutions for the field, where (n + 2) is the number of rotations of the direction of the field vector during one circuit of the bounding circle. In three dimensions (the field outside a sphere) we show by example that many-parameter families of solutions are possible for axisymmetric fields of octupole type (two lines of off-axis dip-poles). These results place the status of the direction problem on the same level as the total intensity problem: in both cases the solutions are known to be non-unique through examples, but no general rule is available to classify the ambiguities. However, the direction problem is homogeneous and we are able to give a method for investigating uniqueness from a local analysis about one particular solution. We argue that field models represented by truncated spherical harmonic expansion with fixed truncation level will be unique (except for some possible special cases), but that severe ambiguities in such models can be investigated by the usual methods of linearized inversion theory to provide a numerical demonstration of qualified ‘uniqueness’. Several ‘realistic’ field models are studied and all are found to be ‘unique’ (in the above sense) to within the arbitrary multiplier; it is most probable that none of the field models published so far suffers from the type of non-uniqueness discovered in our example.