Inversion technique to obtain an emissivity profile from tangential line-integrated hard x-ray measurements

A matrix inversion technique has been developed to calculate local x-ray emissivity profiles from line-integrated x-ray measurements from an anisotropic source. A 2D hard x-ray camera has been used to measure tangential line-integrated bremsstrahlung emission from suprathermal electrons generated by lower hybrid current drive in the PBX-M tokamak. Local emissivities, E, on the midplane of the tokamak are useful to localize the lower hybrid-driven current. The anisotropy is assumed to be uniform over the plasma with a form E(R,θ)=E(R)cosn(θ), where E(R) is the forward emissivity at the major radius R, θ is the angle to the camera from tangency, and the exponent n defines the degree of anisotropy. For isotropic emission, n=0. The measured brightness B(R) on the midplane is then related to the forward emissivity E(R) by a matrix M=L(R,φ)cosn[θ(R,φ)], where φ is the toroidal angle, L is a length matrix containing the path lengths through the plasma. By inverting the matrix M, the forward emissivity is obtained from the measured brightness, E=M−1B. The anisotropy exponent n is adjusted to match the emission at the inner and outer radii of the plasma.