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Parallel stream surface calculation, while highly related to other particle advection-based techniques such as streamlines, has its own unique characteristics that merit independent study. Specifically, stream surfaces require new integral curves to be added continuously during execution to ensure surface quality and accuracy; performance can be improved by specifically accounting for these additional particles. We present an algorithm for generating stream surfaces in a distributed-memory parallel setting. The algorithm incorporates multiple schemes for parallelizing particle advection and we study which schemes work best. Further, we explore speculative calculation and how it can improve overall performance. In total, this study informs the efficient calculation of stream surfaces in parallel for large data sets, based on existing integral curve functionality.