A general purpose stochastic sequential estimator is described, which allows the inverse problem of remote sounding to be solved in an optimum manner. The use of a sequential estimator allows continuity in the horizontal or in time to be icluded, making the a priori estimate close to the final solution, so that iteration is not required. The scan pattern in which measurements are made is not used and need not be known, so that the problem of spacecraft attitude stability is minimized. The method gives both a solution profile and its error covariance. It is applied to both temperature and composition sounding from a limb scanning instrument, the SAMS on NIMBUS 7.