The reconstruction of low-order nonlinear dynamics from the time series of a state variable has been an active area of research in the last decade. The 154 year long, biweekly time series of the Great Salt Lake volume has been analyzed by many researchers from this perspective. In this study, we present the application of a powerful state space reconstruction methodology using the method of support vector machines (SVM) to this data set. SVM are machine learning systems that use a hypothesis space of linear functions in a kernel-induced higher-dimensional feature space. SVM are optimized by minimizing a bound on a generalized error (risk) measure rather than just the mean square error over a training set. Under Mercer's conditions on the kernels the corresponding optimization problems are convex; hence global optimal solutions can be readily computed. The SVM-based reconstruction is used to develop time series forecasts for multiple lead times ranging from 2 weeks to several months. Unlike previously reported methodologies, SVM are able to extract the dynamics using only a few past observed data points out of the training examples. The reliability of the algorithm in learning and forecasting the dynamics is tested using split sample sensitivity analysis, with a particular interest in forecasting extreme states. Efforts are also made to assess variations in predictability as a function of initial conditions and as a function of the degree of extrapolation from the state space used for learning the model.