Laser-based adaptive optics telemetry contains information about the atmospheric vertical profiles of turbulence strength (C-n(2)), outer scale (L-0) and wind speeds. Various techniques in the literature already process laser-based Shack-Hartmann telemetry and estimate the profiles out of cross-covariance maps from the slopes data. Building local derivative estimates out of the data (i.e. curvatures or even higher-order derivative estimates) is a possible mean to remove the large uncertainties existing on low-order modes in laser-based systems. The present study analyses this filtering strategies in a unified formalism. The modified shapes of the cross-covariance peaks are studied and compared. It explains how the sensitivity to the outer scale is strongly reduced by using such higher-order derivative estimates than slopes. The sharpened covariance peaks also simplify the layer detection problem in atmospheric profiling and may improve vertical resolution of SLODAR-based techniques. As a first application of this analysis, an automated algorithm for turbulence profiling is presented with results on simulated data and also on on-sky registered data from the Adaptive Optics Facility at Paranal Observatory.