In this talk we present computational techniques for approximation of Lyapunov exponents based upon smooth matrix factorizations and some potential applications of these techniques to earth system processes. Lyapunov exponents characterize stability properties of time dependent solutions to differential equations. We introduce methods for approximation of Lyapunov exponents, review results on the sensitivity of Lyapuonv exponents to perturbations, describe codes we have developed for their computation, and present results on the approximation of Lyapunov vectors and some possible applications of these results.