= Solution
In <tapping-mode atomic force microscopy>, demodulate the cantilever displacement to obtain its amplitude and compare it with a setpoint below $A_{\rm free}$. A <closed-loop control> system drives a <piezoelectric scanner> in $z$ to restore the setpoint as the raster crosses raised or depressed regions. The interaction remains in its sensitive amplitude range while the scanner follows heights much larger than the oscillation amplitude. \b[The principal height image is the calibrated feedback $z$ displacement at each lateral pixel.] The amplitude-error and phase channels are supplementary signals, rather than interchangeable height measurements. Maximum height is now bounded by scanner travel, clearance and stable tracking. Steep features require adequate feedback bandwidth and sufficiently slow scanning; trace/retrace disagreement tests tracking, while a changed <AFM probe> tests geometric broadening. Surface-dependent dissipation can move the setpoint height even on a geometrically flat specimen.
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