  Estimation of the initial state of turbulent channel flow from spatially and
temporally resolved wall data is performed using adjoint-variational data
assimilation. The accuracy of the predicted flow deteriorates with distance
from the wall, most precipitously across the buffer layer beyond which only
large-scale structures are reconstructed. To quantify the difficulty of the
state estimation, the Hessian of the associated cost function is evaluated at
the true solution. The forward-adjoint duality is exploited to efficiently
compute the Hessian matrix from the ensemble-averaged cross-correlation of the
adjoint fields due to impulses at the sensing locations and times.
Characteristics of the Hessian are examined when observations correspond to the
streamwise or spanwise wall shear stress or wall pressure. Most of the
eigenmodes decay beyond the buffer layer, thus demonstrating weak sensitivity
of wall observations to the turbulence in the bulk. However, when the
measurement time $t_m^+ \gtrsim 20$, some streamwise-elongated Hessian
eigenfunctions remain finite in the outer flow, which corresponds to the
sensitivity of wall observations to outer large-scale motions. Furthermore, we
report statistics of the adjoint-field kinetic-energy budget at long times,
which are distinctly different from those of the forward model. One notable
difference is the high concentration of energy within the buffer layer and its
narrower support. A large and exponentially amplifying adjoint field in that
region leads to large gradients of the cost function, smaller step size in the
gradient-based optimization, and difficulty to achieve convergence especially
elsewhere in the domain where the gradients are comparatively small.
