Sensitivity of Constrained Linear Inversions to the Selection of the Lagrange Multiplier

Abstract

The sensitivity of constrained linear inversions to the selection of the Lagrange multiplier is demonstrated for the can of inferring columnar aerosol size distributions from spectral aerosol optical depth measurements. Since negative values of the aerosol size distribution constitute an unphysical solution, the Lagrange multiplier is varied within a restricted range until a range of values is reached for which all elements of the solution vector are positive. In addition to the constraint that the solution vector be positive, it is necessary for the final solution to be a smooth function and to satisfy the original integral equation to within the noise level of the measurements. An iterative method is presented whereby an initial estimate of the size distribution is modified until the final solution satisfies both the positivity constraint and the requirement that the regression fit to the data using the inverted size distribution he consistent with the measurement errors. A formula for calculating the variances and covariances in the inversion solution is derived and applied to optical depth measurements obtained at the University of Arizona and at Goddard Space Flight Center. In the former case an estimate of the measurement errors is available and thus the inversion formula and error analysis explicitly includes the magnitudes of the measurement variances. In the latter case the measurement errors are not known and the analysis assumes the errors in the measurements are equal and uncorrelated. Results of the error analysis show that the variances in the solution vector are large for radii where the information content of the measurements is small.