  Temporal or spatial structures are readily extracted from complex data by
modal decompositions like Proper Orthogonal Decomposition (POD) or Dynamic Mode
Decomposition (DMD). Subspaces of such decompositions serve as reduced order
models and define either spatial structures in time or temporal structures in
space. On the contrary, convecting phenomena pose a major problem to those
decompositions. A structure traveling with a certain group velocity will be
perceived as a plethora of modes in time or space respectively. This manifests
itself for example in poorly decaying singular values when using a POD. The
poor decay is counter-intuitive, since a single structure is expected to be
represented by a few modes. The intuition proves to be correct and we show that
in a properly chosen reference frame along the characteristics defined by the
group velocity, a POD or DMD reduces moving structures to a few modes, as
expected. Beyond serving as a reduced model, the resulting entity can be used
to define a constant or minimally changing structure in turbulent flows. This
can be interpreted as an empirical counterpart to exact coherent structures. We
present the method and its application to a head vortex of a compressible
starting jet.
