Solution to the Bivariate Integral Inversion Problem: The Determination of Emission Measures Differential in Temperature and Density

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Abstract
We present a general solution to the inversion problem of determining the "source function" f(t, n) from integral equations of the form gi = ∫∫ Ki(t, n) f(t, n) dt dn. The function f(t, n) represents the most information that can be extracted from a set of observables {gi} for a source for which the kernel functions Ki(t, n), depending on the two independent variables t and n, can be calculated a priori. Our specific application is to the inversion of the equations for a set of optically thin emission line intensities gi with kernel functions Ki(t, n) which depend on both the electron density n and the temperature t, a problem defined by Jefferies and colleagues in the early 1970s. We determine "regularized" solutions [those for which derivatives of f(t, n) are minimized to constrain the allowed solutions] for f(t, n) from which the usual emission measure differential in temperature ξ(t) = ∫f(t, n) dn can be obtained. Unlike some recent work, our solution is fully two-dimensional and is not restricted to cases where functional dependences are assumed to exist between t and n in f(t, n). We compare our solutions for the source terms (derived from inversions of calculated intensities from input source functions) with input source functions, for typical extreme-ultraviolet and UV lines formed in the solar transition region. Details, refinements, and applications are left to a later paper. This work is likely to be relevant to other areas of astrophysics, and can aid in planning observations with spacecraft such as the Hubble Space Telescope and the upcoming SOHO mission.