We examine the Raman spectrum of double wall carbon nanotubes with a broad size distribution in agglomerated form and partially filled with $\mathrm{Pb}{\mathrm{I}}_{2}$. Using spectroscopic mapping we find a statistical correlation between the radial breathing mode (RBM) and $D$ band intensity. All the investigated samples show the same correlation using the $514\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ laser excitation. We derive the phonon deformation potential for the $D$ band and extend the dispersion of the $D$ band to the UV. We find that the intensity of the $D$ band decreases with increasing pressure and vanishes at a pressure comparable to what has been observed earlier for the RBM and ${D}^{*}$ band.