A practical limitation to investigating self-similarity in geophysical phenomena from their measured state variables is that measured signals are typically convolved with instrumentation noise at multiple scales. This study develops and tests a multiscale Bayesian model (BEFE) for separating a 1/f-like signal from inherent instrumentation noise and contrasts its performance to the Wiener-type (WAS) and Fourier amplitude (FAS) shrinkage methods. The novel feature in BEFE is that the separation is performed in the wavelet domain and involves the use of a Bayesian inference approach guided by existing theoretical power laws in the filtered signal energy spectrum. We contrast the performance of all three methods for synthetic fractional Brownian motion (fBm) signals and turbulent velocity time series collected in the atmospheric boundary layer. Differences between BEFE and WAS were minor except for the spectral properties at low signal-to-noise ratios and at the finest levels of details in which the filtered signal spectra by BEFE is more consistent with the spectra of the uncontaminated velocity signal.