  We investigate the reconstruction of a turbulent flow field in the
atmospheric boundary layer from a time series of lidar measurements, using
Large-Eddy Simulations (LES) and a 4D-Var data assimilation algorithm. This
leads to an optimisation problem in which the error between measurements and
simulations is minimised over an observation time horizon. To exploit the
spatial coherence of the turbulence, we use a quadratic regularisation in the
objective function, that is based on the two-point covariance matrix. Moreover,
to improve conditioning, and remove continuity constraints, the problem is
transformed into a Karhunen-Lo\`eve basis. For the optimisation, we use a
quasi-Newton limited-memory BFGS algorithm combined with an adjoint approach
for the gradient. We also consider reconstruction based on a Taylor's frozen
turbulence (TFT) model as point of comparison. To evaluate the approach, we
construct a series of virtual lidar measurements from a fine-grid LES of a
pressure-driven boundary-layer. The reconstruction uses LES on a coarser mesh
and smaller domain, and results are compared to the fine-grid reference. Two
lidar scanning modes are considered: a classical plan-position-indicator mode,
which swipes the lidar beam in a horizontal plane, and a 3D pattern that is
based on a Lissajous curve. We find that normalised errors lie between 15% and
25% (error variance normalised by background variance) in the scanning region,
and increase to $100\%$ over a distance that is comparable to the correlation
length scale outside this scanning region. Moreover, LES outperforms TFT by 30%
to 70% depending on scanning mode and location.
