Correlation of velocity and density contributions to spectroscopic
  channel maps: Reality check on Kalberla et.al (2022)
  The existence of magnetized turbulence in the interstellar HI is well
accepted. A number of techniques to obtain turbulence spectrum and magnetic
field direction and strength have been developed and successfully applied to HI
spectroscopic data. To better separate the imprints of density and velocity
fluctuations to the channel maps, a new theory-based technique, the Velocity
Decomposition Algorithm (VDA,Yuen et.al 2021), has been created. The technique
demonstrates that the intensity fluctuations are separated into a component pv
that mostly arises from velocity fluctuations and pd that mostly arise from
density fluctuations. The VDA helps to clarify the nature of the filamentary
structure observed in channel maps. A recent publication (Kalberla et.al
2022,K22) claims that the application of VDA to HI4PI data provides negative
correlation of pv and pd,which according to the authors invalidates the
technique since it requires that pv and pd have zero correlation. However, the
quantities pv and pd given by VDA are naturally orthogonal which can be
trivially checked analytically or numerically. That means the correct
application of the VDA to any data must provide zero correlation. This is the
point that we clarify in this paper and search for the cause of the mistake in
the application of the VDA in K22 that resulted in the erroneous conclusion. We
prove analytically that by construction pv and pd are not correlated. We
identify the likely mistake in the VDA expression that K22 used and reproduce
their figures with the incorrect expression. We find that 14 out of 15 figures
in K22 are invalid, and thus their criticism of the VDA is ill-founded and
arises from their use of incorrect expressions. We conclude that the
detrimental mistake that K22 made at their analysis completely invalidate their
scientific claim that Y21 isnt compatible to observation.
