S U M M A R Y We propose a theory for rotational and strain ambient noise interferometry, motivated by the recent development of rotational ground motion sensors and distributed acoustic sensing (DAS) technology. In this context, we demonstrate that displacement, strain and rotation interferograms can be generically written in the form of a representation theorem, that is, as a solution to the seismic wave equation that we refer to as the interferometric wavefield. The physical quantity

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... ement, strain or rotation) determines the distributed source of the interferometric wavefield, as well as an observational operator that extracts the correct type of noise correlation function. The proposed interferometric equations are free of assumptions on the distribution of noise sources or the equipartitioning of the ambient field, typically required for Green's function retrieval. In addition to being valid for any kind of heterogeneous source and viscoelastic medium, they allow us to account for measurement details, such as the gauge length in DAS. We illustrate the practical feasibility of our approach with a series of numerical examples, based on regional-scale, spectral-element simulations of the interferometric wavefield. Specifically, we compare displacement and strain interferograms for homogeneous and heterogeneous earth models, and for homogeneous and heterogeneous noise sources. Ultimately, our developments are intended to enable adjoint-based waveform inversion with emerging measurement technologies that provide spatial gradient information in addition to conventional seismic displacement recordings.