  The LHC has released precise measurements of elastic proton-proton scattering
that provide a unique constraint on the asymptotic behavior of the scattering
amplitude at high energies. Recent reanalyses of part of these data indicate
that the central values of some forward quantities would be different than
initially observed. We introduce correlation information between the original
and the reanalyzed data sets in a way suitable for a global fitting analysis of
all data. The careful treatment of correlated errors leads to much less
stringent limits on the $\rho$ uncertainty and sets up the stage for describing
the forward data using a scattering amplitude dominated by only crossing-even
terms. In the light of these correlated data we determine the parameters of the
soft Pomeron from the Regge theory. We use Born-level and eikonalized
amplitudes. In the Born-level case we estimate the contribution of the double
Pomeron exchange, while in the latter case we investigate the role of the
eikonalization in both the one- and two-channel models. The role of the
proton-Pomeron vertex form and of the nearest $t$-channel singularity in the
Pomeron trajectory receives particular attention. We discuss the implications
of our results and present predictions for the total cross section and the
$\rho$ parameter in proton-proton collisions at LHC and cosmic ray energies.
