This work (1) introduces a compensation framework to predict attached-eddy signatures from instantaneous fields and (2) explores the logarithmic decay rate $A_1$ of the streamwise turbulence intensity $\overline{u^{\prime 2}}$, where $u^{\prime }$ is the streamwise velocity fluctuation. While the attached eddy hypothesis implies a constant logarithmic decay rate $A_1$ of $\overline{u^{\prime 2}}$ concerning the wall distance, $A_1$ varies significantly in previous studies. Very recently, energy spectra have been decomposed via spectral linear stochastic estimation (SLSE) and linear coherence function (LCF) to extract the attached-eddy contribution and avoid the contamination of detached eddies. Even after the decomposition, $A_1$ shows an evident increasing trend with Reynolds numbers. This work re-examines the original single-input/single-output (SI/SO) model of SLSE and identifies the near-wall universal signal (noise of the SI/SO system) as one possible reason for lower $A_1\mathrm{s}$ at low and moderate Reynolds numbers. Both rigorous mathematical derivations and intuitive physical understandings are provided to explain the impact of noise on the SLSE and LCF. Then, two theoretically equivalent compensation approaches are proposed and compared, both of which transfer the original SI/SO system to a multi-input/single-output (MI/SO) system by introducing an extra input, near-wall wall-normal velocity fluctuation $v_i^{\prime }$, to eliminate the impact of the near-wall universal signal. The refined attached-eddy spectral subcomponent also complies with the classic model. The decay rate $A_1$ becomes, overall, much larger, reducing the Reynolds number variation. It reaches the asymptotic value 0.98 for ${\text{Re}}_{\tau }\ge 4200$. Although lower-Reynolds-number cases still have smaller $A_1\mathrm{s}$, this work cannot exclude the probability of a constant $A_1$ through qualitative analysis of LCFs between the near-wall $u_i^{\prime }$, $v_i^{\prime }$, and $u_o^{\prime }$ in the logarithmic region using the theory of the MI/SO system. The comprehensive refinement framework for analyzing and eliminating the effect of noise is also applicable to other prediction problems, and promising results in the present work could inspire new frameworks for future investigation to improve the precision of extracting attached-eddy signatures for low-Reynolds-number cases.

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• Structure & turbulence of boundary layers
• Turbulence modeling
• Turbulent boundary layers