Variation in cutoff frequency with height in the umbral models eMaltby (panel a), mMaltby (panel b), lMaltby (panel c), and Avrett2015spot (panel d). The red lines with asterisks show the cutoff values determined from examining phase-difference spectra in numerical simulations with a vertical magnetic field of 3000 G. Color lines indicate the analytical cutoff frequency computed using Eq. (1) (blue line), Eq. (2) (green line), Eq. (3) (violet line), and Eq. (4) (red line). All the plotted heights are in the β <  1 region.