Phase (red) and group (envelope, blue) travel time differences, azimuthally averaged, as functions of radius r of the true (dots) or mid-point-inferred (full curves) middle skip points. Left column: 3 mHz; right column: 4 mHz. Top row: full magnetic sunspot model using ω_c = ω_DG; second row: full magnetic sunspot model using ω_c = ω_I; third row: “thermal spot” with the same thermal and density structure, but with magnetic field artificially suppressed. All points were binned to 1 Mm² squares and averaged both by bin and azimuthally. All data presented here have been pre-filtered to remove any rays with second skip distance outside the range times the first skip distance, or second skip direction more than 20° from the first skip direction. The fraction of points deleted by this pre-filtering for the six panels is (0.023, 0.140, 0.122, 0.143, 0.025, 0.129). The dashed red and blue curves represent, respectively, the equivalent phase and group speed thermal depths of the Wilson depression; see text for details. The grey vertical lines represent the umbral and penumbral boundaries.