Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition

Dynamic mode decomposition (DMD) provides a practical means of extracting insightful dynamical information from fluids datasets. Like any data processing technique, DMD's usefulness is limited by its ability to extract real and accurate dynamical features from noise-corrupted data. Here, we show analytically that DMD is biased to sensor noise, and quantify how this bias depends on the size and noise level of the data. We present three modifications to DMD that can be used to remove this bias: (1) a direct correction of the identified bias using known noise properties, (2) combining the results of performing DMD forwards and backwards in time, and (3) a total least-squares-inspired algorithm. We discuss the relative merits of each algorithm and demonstrate the performance of these modifications on a range of synthetic, numerical, and experimental datasets. We further compare our modified DMD algorithms with other variants proposed in the recent literature.