Conventional oil and gas fields are increasingly difficult to explore and image, resulting in the call for more complex wave-equation-based inversion algorithms that require dense long-offset samplings. Consequently, there is an exponential growth in the size of data volumes and prohibitive demands on computational resources. We have developed a method to compress and process seismic data directly in a low-rank tensor format, which drastically reduces the amount of storage required to represent the data. Seismic data exhibit a low-rank structure in a particular transform domain, which can be exploited to compress the dense data in one extremely storage-efficient tensor format when the data are fully sampled or can be interpolated when the data have missing entries. In either case, once our data are represented in the compressed tensor form, we have developed an algorithm to extract source or receiver gathers directly from the compressed parameters. This extraction process can be done on the fly directly on the compressed data, and it does not require scanning through the entire data set to form shot gathers. We apply this shot-extraction technique in the context of stochastic full-waveform inversion as well as forming full subsurface image gathers through probing techniques and reveal the minor differences between using the full and compressed data, while drastically reducing the total memory costs.