Projection-based methods for spectral analysis of data without temporal information

Extracting coherent structures and physically important features from complex systems has become the subject of great interest in fluid dynamics, motivated and enabled by a variety of modal analysis techniques. The availability of spatiotemporally resolved computational and experimental data enables the use of techniques such as dynamic mode decomposition and spectral proper orthogonal decomposition to isolate and analyze structures that correspond to specific temporal frequencies. However, in many scenarios time-resolved data remains challenging to acquire and store. Here, we explore how spectral content may be recovered from spatially-resolved data in the absence of any temporal information. This is achieved by analyzing the spectral content of reduced-order models identified from projecting the governing equations onto a subspace identified from the original data via proper orthogonal decomposition, through either a pseudospectral analysis of the identified linearized system or by considering the spectral proper orthogonal decomposition of time-resolved data generated by evolving nonlinear reduced-order models. We explore the method performance for example problems with various temporal dynamics, ranging from systems with a single dominant frequency to those with broadband frequency content. We find that the method can identify spectral content even in cases where the identified reduced-order models have limited accuracy in predicting the time-evolution of the system dynamics.