Analyses of substantially different plasma current densities and safety factors reconstructed from magnetic diagnostics data

The problem of plasma current density and safety factor reconstruction using magnetic field measurements is considered. In the traditional formulation, the problem is strongly ill-posed. In particular, substantially different current densities and safety factors can be equally well attributed to the same set of measurements, given their experimental errors. In other words, the problem can be strongly unstable with respect to the input data. Different constraints are used in practice to make the problem more stable. This paper presents an accurate mathematical formulation of the inverse problem and its variants. A numerical algorithm is provided, which permits us to study the stability with respect to variations in the input data, to find all substantially different solutions, or to prove their absence, and to determine the confidence intervals of the reconstructions. The proposed method also allows establishing the maximum error for a given diagnostic (additional constraint), below which the diagnostic efficiently extracts one solution among several substantially different ones. Examples of very different current density and safety factor reconstructions for measurements with finite accuracy are presented for the original formulation of the inverse problem. Cases of MAST, JET and ITER-like plasmas are considered. It is shown that including the motional Stark effect (MSE) measurements as a constraint, provided the accuracy of MSE measurements is sufficient, allows identifying one solution among several very different ones, obtained without such a constraint. The maximum MSE diagnostics error for efficient identification of this solution is estimated for JET. The approach of this paper can be used for a wide range of ill-posed problems in physics and can help in selecting additional conditions, which can identify the most likely solution among several.