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Estimation of the initial state of turbulent channel flow from spatially andtemporally resolved wall data is performed using adjoint-variational dataassimilation. The accuracy of the predicted flow deteriorates with distancefrom the wall, most precipitously across the buffer layer beyond which onlylarge-scale structures are reconstructed. To quantify the difficulty of thestate estimation, the Hessian of the associated cost function is evaluated atthe true solution. The forward-adjoint duality is exploited to efficientlycompute the Hessian matrix from the ensemble-averaged cross-correlation of theadjoint fields due to impulses at the sensing locations and times.Characteristics of the Hessian are examined when observations correspond to thestreamwise or spanwise wall shear stress or wall pressure. Most of theeigenmodes decay beyond the buffer layer, thus demonstrating weak sensitivityof wall observations to the turbulence in the bulk. However, when themeasurement time $t_m^+ \gtrsim 20$, some streamwise-elongated Hessianeigenfunctions remain finite in the outer flow, which corresponds to thesensitivity of wall observations to outer large-scale motions. Furthermore, wereport statistics of the adjoint-field kinetic-energy budget at long times,which are distinctly different from those of the forward model. One notabledifference is the high concentration of energy within the buffer layer and itsnarrower support. A large and exponentially amplifying adjoint field in thatregion leads to large gradients of the cost function, smaller step size in thegradient-based optimization, and difficulty to achieve convergence especiallyelsewhere in the domain where the gradients are comparatively small.

From wall observations to turbulence: The difficulty of flow reconstruction