Preconditioning the Differential Emission Measure ( T e ) Inverse Problem

Abstract

In an inverse problem of any kind, poor conditioning of the inverse operator decreases the numerical stability of any unregularized solution in the presence of data noise. In this paper we show that the numerical stability of the differential emission measure (DEM) inverse problem can be considerably improved by judicious choice of the integral operator. Specifically, we formulate a combinatorial optimization problem where, in a preconditioning step, a subset of spectral lines is selected in such a way as to minimize explicitly the condition number of the discretized integral operator. We tackle this large combinatorial optimization problem using a genetic algorithm. We apply this preconditioning technique to a synthetic data set comprising of solar UV/EUV emission lines in the SOHO SUMER/CDS wavelength range. Following which we test the same hypothesis on lines observed by the Harvard S-055 EUV spectroheliometer. On performing the inversion we see that the temperature distribution in the emitting region of the solar atmosphere is recovered with considerably better stability and smaller error bars when our preconditioning technique is used, in both synthetic and ``real'' cases, even though this involves the analysis of fewer spectral lines than in the ``All-lines'' approach. The preconditioning step leads to regularized inversions that compare favorably to inversions by singular value decomposition, while providing greater flexibility in the incorporation of physically and/or observationally based constraints in the line selection process.