  The relative smoothing rates of various gauge field smoothing algorithms are
investigated on ${\cal O}(a^2)$-improved $\suthree$ Yang--Mills gauge field
configurations. In particular, an ${\cal O}(a^2)$-improved version of APE
smearing is motivated by considerations of smeared link projection and cooling.
The extent to which the established benefits of improved cooling carry over to
improved smearing is critically examined. We consider representative gauge
field configurations generated with an ${\cal O}(a^2)$-improved gauge field
action on $\1$ lattices at $\beta=4.38$ and $\2$ lattices at $\beta=5.00$
having lattice spacings of 0.165(2) fm and 0.077(1) fm respectively. While the
merits of improved algorithms are clearly displayed for the coarse lattice
spacing, the fine lattice results put the various algorithms on a more equal
footing and allow a quantitative calibration of the smoothing rates for the
various algorithms. We find the relative rate of variation in the action may be
succinctly described in terms of simple calibration formulae which accurately
describe the relative smoothness of the gauge field configurations at a
microscopic level.
