Ice thickness estimation is an important aspect of ice sheet studies. In this work, we use convolutional neural networks with multiple output nodes to regress and learn the thickness of internal ice layers in Snow Radar images collected in northwest Greenland. We experiment with some state-of-the-art networks and find that with the residual connections of ResNet50, we could achieve a mean absolute error of 1.251 pixels over the test set. Such regression-based networks can further be improved by embedding domain knowledge and radar information in the neural network in order to reduce the requirement of manual annotations.