We reconstruct the turbulent flow field in the neutrally-stable atmospheric boundary layer using large-eddy simulations (LES) from lidar measurements. This reconstruction is approached as a four-dimensional variational data assimilation (4D-VAR) problem with weak constraints. In the strong formulation, the reconstruction is achieved by selecting the optimal initial state, which evolves over time to match the observed measurements [1]. In contrast, the weak formulation introduces greater flexibility by enabling the characterization of spatiotemporal features associated with model errors, providing additional degrees of freedom in the reconstruction process. A good characterization of the these features allows much faster convergence and relatively better reconstruction accuracy compared to the standard strong formulation. The assimilation problem is derived in a Beysian framework combining the likelihood term (mismatch from measurements), the LES model, and the spatiotemporal characteristics of the model error term. To test the assimilation model, we consider a series of virtual lidar measurements in time. We then reconstruct the turbulent field over a time horizon of 200 s, and compare the results of the strong and the waek models using various characterizations of the model error. The initial condition is regularized (spatially) using the HGW model [2, 3]. Figure 1 shows an example of the constructed streamwice velocity field and the corresponding error field.