Hankel singular vectors are space-time POD modes

We shed light on a connection between space-time proper orthogonal decomposition (POD) and the singular value decomposition (SVD) of a particular block Hankel matrix. The block Hankel matrix formed using successive vector-valued snapshots of the flow state, and the SVD of this matrix, lie at the heart of several popular methods for modeling and analyzing fluid systems. We show that the left singular vectors (and singular values) of this Hankel matrix correspond to a discrete approximation of continuous space-time POD modes (and eigenvalues), which are the solution of an optimization problem defined over a time window of interest. Furthermore, popular variants of POD, namely the standard space-only POD and spectral POD, are recovered in the limits that snapshots used to form each column of the Hankel matrix represent flow evolution over short and long times, respectively. These connections provide insight into the construction of the Hankel matrix, including the weighting in which the SVD modes are optimal, and the impact of the time steps between rows and columns of the matrix on the convergence of the approximation. These insights lead to a modified matrix, depending on the weight and time step, for which the SVD modes are optimal in the desired weighting.