Accurate mapping of structural faults and stratigraphic sequences is essential to the success of subsurface interpretation, geologic modeling, reservoir characterization, stress history analysis, and resource recovery estimation. In the past decades, manual interpretation assisted by computational tools — i.e., seismic attribute analysis — has been commonly used to deliver the most reliable seismic interpretation. Because of the dramatic increase in seismic data size, the efficiency of this process is challenged. The process has also become overly time-intensive and subject to bias from seismic interpreters. In this study, we implement deep convolutional neural networks (CNNs) for automating the interpretation of faults and stratigraphies on the Opunake-3D seismic data set over the Taranaki Basin of New Zealand. In general, both the fault and stratigraphy interpretation are formulated as problems of image segmentation, and each workflow integrates two deep CNNs. Their specific implementation varies in the following three aspects. First, the fault detection is binary, whereas the stratigraphy interpretation targets multiple classes depending on the sequences of interest to seismic interpreters. Second, while the fault CNN utilizes only the seismic amplitude for its learning, the stratigraphy CNN additionally utilizes the fault probability to serve as a structural constraint on the near-fault zones. Third and more innovatively, for enhancing the lateral consistency and reducing artifacts of machine prediction, the fault workflow incorporates a component of horizontal fault grouping, while the stratigraphy workflow incorporates a component of feature self-learning of a seismic data set. With seven of 765 inlines and 23 of 2233 crosslines manually annotated, which is only about 1% of the available seismic data, the fault and four sequences are well interpreted throughout the entire seismic survey. The results not only match the seismic images, but more importantly they support the graben structure as documented in the Taranaki Basin.