1356 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 39 Sensitivity of Constrained Linear Inversions to the Selection of the Lagrange Multiplier MICHAEL D. KING Laboratory for Atmospheric Sciences, Goddard Space Flight Center, NASA, Greenbelt, MD .Xl77] (Manuscript received 10 June 1981, in final form 28 January 1982) ABSTRACT The sensitivity of constrained linear inversions to the selection of the Lagrange multiplier is demonstrated for the case 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 mul- tiplier 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 be 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. 1. Introduction Inversion methods for solving Fredholm integral equations of the first kind have existed for some 20 years. The earliest method for solving these indirect sensing problems in which due consideration was given to the estimation problem was the linear in- version technique developed by Phillips (1962) and Twomey (1963). This inversion method, which was arrived at independently by Tikhonov ( 1963), is com- monly referred to as constrained linear inversion be- cause it relies on the introduction of an additional condition or constraint, not deriving from the mea- surements, which enables one of the set of possible solution vectors to be selected. Many applications of constrained linear inversion can be found in the lit- erature. These include the inference of atmospheric temperature profiles from satellite-borne radiometers (Wark and Fleming, 1966; Glasko and Timofeyev, 1968a,b), inference of the vertical distribution of ozone from scattered sunlight (Yarger, 1970), and the inference of aerosol size distributions from spec- tral attenuation (Yamamoto and Tanaka, 1969; King et al., 1978; Walters, 1980) or angular light scattering (Dave, 1971; Byrne, 1978; Reagan et al., 1980) measurements. In each of these problems, as in any physics or engineering problem involving Fredholm integral equations of the first kind, the measurements and frequently the kernel functions are known with only finite accuracy. In addition, there is often a high degree of interdependence among some of the kernel functions which leads to highly oscillatory and un- satisfactory solutions in the absence of a suitable constraint. In a Bayesian sense this arises from a vague prior knowledge of the solution vector f(x) in that least-squares assumes only that -cc < f(x) < cc for all values of X. For most problems likely to be encountered in the atmospheric sciences, such as those outlined above, this range for f(x) is unnec- essarily broad since physical necessity dictates that the solution vector must at the very least be positive. In addition to the constraint that the solution vec- tor be positive, it is normal to seek the solution among the set of possible solutions which is the smoothest in some sense. Phillips (1962) introduced a smooth- ing constraint such that the sum of the squares of the second derivatives of the solution points is min- imized. Twomey (1977) discusses many possible con- straints that can be applied in constrained linear in- version problems, but Phillips’ second derivative smoothing constraint remains one of the most pop- ular in the atmospheric sciences. Among mathema- ticians the most popular constraint is that the sum of the squares of the solution points is minimized. This leads to the constraint matrix being the identity matrix and is referred to by Hoer1 and Kennard (1970a,b) as ridge regression, rather than con-
