Deep learning methods have surpassed the performance of traditional techniques on a wide range of problems in computer vision, but nearly all of this work has studied consumer photos, where precisely correct output is often not critical. It is less clear how well these techniques may apply on structured prediction problems where fine-grained output with high precision is required, such as in scientific imaging domains. Here we consider the problem of segmenting echogram radar data collected from the polar ice sheets, which is challenging because segmentation boundaries are often very weak and there is a high degree of noise. We propose a multi-task spatiotemporal neural network that combines 3D ConvNets and Recurrent Neural Networks (RNNs) to estimate ice surface boundaries from sequences of tomographic radar images. We show that our model outperforms the state-of-the-art on this problem by (1) avoiding the need for hand-tuned parameters, (2) extracting multiple surfaces (ice-air and ice-bed) simultaneously, (3) requiring less non-visual metadata, and (4) being about 6 times faster.