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Recent works on wall-bounded flows have corroborated the coexistence of wall-attached eddies, whose statistical features are predicted through Townsend's attached eddy hypothesis (AEH), and very-large-scale motions (VLSMs), which are not encompassed in the AEH. Furthermore, it has been shown that the presence of wall-attached eddies within the logarithmic layer is linked to the appearance of an inverse-power-law region in the streamwise velocity energy spectra, upon significant separation between outer and viscous scales. In this work, a near-neutral atmospheric surface layer (ASL) is probed with a wind LiDAR to investigate the contributions to the streamwise velocity energy associated with wall-attached and VLSMs for a very-high Reynolds-number boundary layer. Energy and linear coherence spectra (LCS) of the streamwise velocity are interrogated to identify the spectral boundaries associated with eddies of different typologies and the maximum height attained by wall-attached eddies. Inspired by the AEH, an analytical model for the LCS associated with wall-attached eddies is formulated. The experimental results show that the identification of the wall-attached-eddy energy contribution through the analysis of the energy spectra leads to an underestimate of the associated spectral range, maximum height attained, and turbulence intensity. This feature is due to the overlap of the energy associated with VLSMs obscuring the inverse-power-law region. On the other hand, the Townsend-Perry constant for the turbulence intensity seems to be properly estimated through the spectral analysis. The LCS analysis estimates wall-attached eddies with a streamwise/wall-normal ratio of about 14.3 attaining a height of about 30% of the outer scale of turbulence.