The paper proposes stochastic models for the analysis of ocean surface trajectories obtained from freely drifting satellite-tracked instruments. The time series models proposed are used to summarize large multivariate data sets and to infer important physical parameters of inertial oscillations and other ocean processes. Non-stationary time series methods are employed to account for the spatiotemporal variability of each trajectory. Because the data sets are large, we construct computationally efficient methods through the use of frequency domain modelling and estimation, with the data expressed as complex-valued time series. We detail how practical issues related to sampling and model misspecification may be addressed by using semiparametric techniques for time series, and we demonstrate the effectiveness of our stochastic models through application to both real world data and to numerical model output.