  Spectral proper orthogonal decomposition (SPOD) is an increasingly popular
modal analysis method in the field of fluid dynamics due to its specific
properties: a linear system forced with white noise should have SPOD modes
identical to response modes from resolvent analysis. The SPOD, coupled with the
Welch method for spectral estimation, may require long time-resolved data sets.
In this work, a linearised Ginzburg-Landau model is considered in order to
study the method's convergence. SPOD modes of the white-noise forced equation
are computed and compared with corresponding response resolvent modes. The
quantified error is shown to be related to the time length of Welch blocks
(spectral window size) normalised by a convective time. Subsequently, an
algorithm based on a temporal data shift is devised to further improve SPOD
convergence and is applied to the Ginzburg-Landau system. Next, its efficacy is
demonstrated in a numerical database of a boundary layer subject to bypass
transition. The proposed approach achieves substantial improvement in mode
convergence with smaller spectral window sizes with respect to the standard
method. Furthermore, SPOD modes display growing wall-normal and span-wise
velocity components along the stream-wise direction, a feature which had not
yet been observed and is also predicted by a global resolvent calculation. The
shifting algorithm for the SPOD opens the possibility for using the method on
datasets with time series of moderate duration, often produced by large
simulations.
