Resolution Enhancement of Spectra Using Differentiation

Finding the positions and heights of spectral lines becomes an extremely di cult problem when the lines are close. The most popular methods currently available for this task use nonlinear tting. For such methods, a particular challenge is the determination of good starting values for the line positions. An alternative method is based on the reformulation of the discrete nonlinear tting problem as a continuous linear one which does not require any starting values. However, this linear problem turns out to be severely ill-posed, and, in addition, the solution does not have the properties required for the application of the standard convergence theory. The situation is di erent if one only solves the linear problem approximately. While this \approximate problem" is still ill-posed, it can be analysed by applying the standard convergence theory. This approximation procedure is called \enhancement". To a certain extent, enhancement can remedy some of the problems occurring in non-linear tting by preconditioning the data such that the spectral lines are narrower, and thus better separated. It is demonstrated that this enhancement can be done in a stable way using numerical di erentiation. A new technique is proposed which is found to have superior enhancement properties compared to that achieved with the original Taylor-series based method.