  Field-emission resonances (FERs) for two-dimensional Pb(111) islands grown on
\mbox{Si(111)7$\times$7} surfaces were studied by low-temperature scanning
tunneling microscopy and spectroscopy (STM/STS) in a broad range of tunneling
conditions with both active and disabled feedback loop. These FERs exist at
quantized sample-to-tip distances $Z^{\,}_n$ above the sample surface, where
$n$ is the serial number of the FER state. By recording the trajectory of the
STM tip during ramping of the bias voltage $U$ (while keeping the tunneling
current $I$ fixed), we obtain the set of the $Z^{\,}_n$ values corresponding to
local maxima in the derived $dZ/dU(U)$ spectra. This way, the continuous
evolution of $Z^{\,}_n$ as a function of $U$ for all FERs was investigated by
STS experiments with active feedback loop for different $I$. Complementing
these measurements by current-distance spectroscopy at a fixed $U$, we could
construct a 4-dimensional $I-U-Z-dZ/dU$ diagram, that allows us to investigate
the geometric localization of the FERs above the surface. We demonstrate that
(i) the difference $\delta Z^{\,}_n=Z^{\,}_{n+1}-Z^{\,}_n$ between neighboring
FER lines in the $Z-U$ diagram is independent of $n$ for higher resonances,
(ii) the $\delta Z^{\,}_{n}$ value decreases as $U$ increases; (iii) the
quantized FER states lead to the \emph{periodic} variations of $\ln I$ as a
function of $Z$ with periodicity $\delta Z$; (iv) the periodic variations in
the $\ln I - Z$ spectra allows to estimate the absolute height of the tip above
the sample surface. Our findings contribute to a deeper understanding on how
the FER states affect various types of tunneling spectroscopy experiments and
how they lead to a non-exponential decay of the tunneling current as a function
of $Z$ at high bias voltages in the regime of quantized electron emission.
