Beyond the Hurst exponent: The role of the spectral pre-factor in fault roughness characterization and its synthetic validation

: Power spectral density analysis, PSD, is a standard method to characterize fault surface roughness. The Hurst exponent, H, often serves as the primary descriptor of self-affine scaling behavior even if surfaces with the same H value show considerable differences in roughness amplitude and apparent morphology. Using synthetic surfaces, we reconcile qualitative observations (e.g., rough and smooth) with quantitative parameters and quantify the sensitivity of the results to three spectral averaging variants and to instrumental noise. For instance, the observation that slip-parallel profiles are smoother despite having lower H values is explained by considering H and the pre-factor, α, together. The recovered parameters are within the expected statistical variability of the periodogram estimator, a characteristic of the method rather than a reflection of data quality. Band-limited instrumental noise leads to a systematic underestimation of H, whose severity is greatest in the direction where the signal decays faster (i.e., higher H), sinking below the noise floor at a lower frequency. The measured anisotropy of H is therefore reduced or even inverted. A multi-scale validation shows that matching α is necessary for spectral alignment between surfaces of different sizes. Informed by the synthetic tests, the analysis is applied to an outcrop-scale patch of the Roccapreturo Fault in the central Apennines, Italy, acquired using SfM-MVS photogrammetry with two camera systems of different resolution and two processing pipelines. The two acquisitions converge at low spatial frequencies but diverge above a threshold, providing an internal criterion for the usable frequency range of an acquisition.