  We apply the four dimensional variational method to reconstruct the small
scales of three-dimensional turbulent velocity fields with a moderate Reynolds
number, given a time sequence of measurement data on a coarse set of grid
points. The results show that, reconstruction is successful when the resolution
of the measurement data, given in terms of the wavenumber, is at the order of
the threshold value $k_c = 0.2\eta_K^{-1}$ where $\eta_K$ is the Kolmogorov
length scale of the flow. When the data are available over a period of one
large eddy turn-over time scale, the filtered enstrophy and other small scale
quantities are reconstructed with a $30\%$ or smaller normalized point-wise
error, and a $90\%$ point-wise correlation. The spectral correlation between
the reconstructed and target fields is higher than $80\%$ for all wavenumbers.
Minimum volume enclosing ellipsoids (MVEEs) and MVEE trees are introduced to
quantitatively compare the geometry of non-local structures. Results show that,
for the majority samples, errors in the locations and the sizes of the
reconstructed structures are within $15\%$, and those in the orientations is
within $15^\circ$. This investigation demonstrates that satisfactory
reconstruction of the scales two or more octaves smaller is possible if data at
large scales are available for at least one large eddy turn-over time. In
comparison, a direct substitution scheme results in three times bigger
point-wise discrepancy in enstrophy. The spectral difference between the
reconstructed and target velocity fields is more than ten times higher than
what is obtained with the four dimensional variational method.
