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 ({\it fBm}) signals and turbulent velocity time series collected in the atmospheric boundary layer. A discussion on the advantages and disadvantages of each method is also presented, particularly when the process is not exactly an {\it fBm}.

A preliminary version of the manuscript is available in postscript and pdf formats