Sentinel-1 is the first of a family of satellites designed to provide a data stream for the European environmental monitoring program known as Copernicus. Sentinel-1 constellation has been specifically designed to perform, over land, advanced Differential Interferometric Synthetic Aperture Radar (DInSAR) analyses for the investigation of Earth's surface displacements. In particular, owning to its 6-day revisit time and its innovative acquisition mode, which is referred to as Terrain Observation by Progressive Scans (TOPS) and is fundamental for guaranteeing a global spatial coverage, Sentinel-1 constellation is contributing to the creation of a framework for the exploitation of "Big Data" for Earth Observation (EO) applications. In this paper, we present an efficient and automatic implementation of the Parallel Small BAseline Subset (P-SBAS) DInSAR algorithm, specifically intended for the processing of Sentinel-1 SAR data. The algorithm is able to run on distributed computing infrastructures by effectively exploiting a large number of resources, and allows the generation of ground displacement time-series. The aim of this paper is to show that it is possible to automatically and continuously process, in a short time frame, very large sequences of Sentinel-1 data, thus allowing us to perform advanced interferometric analyses at an unprecedented large scale. In addition, the proposed Sentinel-1 P-SBAS algorithm has also been tested on commercial public cloud computing platforms, such as those provided by the Amazon Web Services. The presented Sentinel-1 P-SBAS processing chain is well suited to build up operational services for the easy and rapid generation of advanced interferometric products, which can be very useful not only for scientific purposes but also for the risk management and the natural hazard monitoring. (C) 2016 The Authors. Published by Elsevier B.V.