  The late-time nonlinear Lagrangian displacement field is highly correlated
with the initial field, so reconstructing it could enable us to extract
primordial cosmological information. Our previous work [1] carefully studied
the displacement field reconstructed from the late time density field using the
iterative method proposed by Ref. [2] and found that it does not fully converge
to the true, underlying displacement field (e.g., $\sim 8\%$ offset at $k\sim
0.2 \ihMpc$ at $z=0.6$). We also constructed the Lagrangian perturbation theory
model for the reconstructed field, but the model could not explain the
discrepancy between the true and the reconstructed fields in the previous work.
The main sources of the discrepancy were speculated to be a numerical artifact
in the displacement estimator due to the discreteness of the sample. In this
paper, we develop two new estimators of the displacement fields to reduce such
numerical discreteness effect, the normalized momentum estimator~(NME) and the
rescaled resumed estimator~(RRE). We show that the discrepancy Ref. [1]
reported is not due to the numerical artifacts. We conclude that the method
from Ref. [2] cannot fully reconstruct the shape of the nonlinear displacement
field at the redshift we studied, while it is still an efficient BAO
reconstruction method. In parallel, by properly accounting for the UV-sensitive
term in a reconstruction procedure with an effective field theory approach, we
improve the theoretical model for the reconstructed displacement field, by
almost five times, from $\sim 15\%$ to the level of a few \% at $k\sim
0.2\ihMpc$ at the redshift $z=0.6$.
