  We employ a resolvent-based methodology to estimate velocity and pressure
fluctuations within turbulent channel flows at friction Reynolds numbers of
approximately 180, 550 and 1000 using measurements of shear stress and pressure
at the walls, taken from direct numerical simulation (DNS) databases. Martini
et al. (J. Fluid Mech., vol. 900, 2021, A2) showed that the resolvent-based
estimator is optimal when the true space-time forcing statistics are utilized,
thus providing an upper bound for the accuracy of any linear estimator. We use
this framework to determine the flow structures that can be linearly estimated
from wall measurements, and we characterize these structures and the estimation
errors in both physical and wavenumber space. We also compare these results to
those obtained using approximate forcing models - an eddy-viscosity model and
white-noise forcing - and demonstrate the significant benefit of using true
forcing statistics. All models lead to accurate results up to the buffer-layer,
but only using the true forcing statistics allows accurate estimation of
large-scale log-layer structures, with significant correlation between the
estimates and DNS results throughout the channel. The eddy-viscosity model
displays an intermediate behaviour, which may be related to its ability to
partially capture the forcing colour. Our results show that structures that
leave a footprint on the channel walls can be accurately estimated using the
linear resolvent-based methodology, and the presence of large-scale
wall-attached structures enables accurate estimations through the logarithmic
layer.
