Air passing upward through the tropical tropopause is “marked” by an annually varying water vapor mixing ratio much as a tape recorder marks a magnetic tape; as the air ascends in the tropical stratosphere, these marks are effaced by a combination of vertical diffusion within the tropics and dilution of tropical air by sideways (isentropic) mixing-in of midlatitude air. We represent these processes using a one-dimensional advection-diffusion-dilution model, which we inverse-solve for the vertical profiles of three unknowns (vertical advection velocity, vertical diffusion coefficient, and dilution rate coefficient) after prescribing the vertical profiles of time mean methane [CH4] and of amplitude and phase of the annually varying tape recorder signal in 2[CH4]+[H2O]. When tested on synthetic data generated by forward solving the same model, the method for inverse solution proved to be well conditioned and to give accurate results above 18 km. Applying the method to 5 years of smoothed data from the Halogen Occultation Experiment, we find a vertical advection velocity with a minimum of about 0.2 mm s−1 near 20 km, and both dilution rate coefficient and vertical diffusion coefficient with remarkably low minima near 22 km, 1/(6–7 year) and roughly 0.02 m2s−1, respectively. Our derived profile of vertical advection velocity agrees well, between 18 and 24 km, with an independent, radiatively derived, mass-budget-constrained transformed Eulerian mean calculation. Despite the relatively modest values of the diffusion coefficient, vertical diffusion plays a significant role in attenuating the tape recorder signal, according to our model. The minimum value of the dilution rate coefficient corresponds to a relaxation timescale of 6–7 years, much longer than the timescales found in other studies. The long relaxation timescale at 20–24 km is, however, consistent with (1) the minimum in vertical velocity, (2) a reduced attenuation rate in the tape recorder signal, and (3) a decrease, hitherto unremarked, in the tropical vertical gradient of [CH4] there.