Stochastic Modeling as a Method of Arriving at Higher Frequencies: An Application to κ Estimation

Abstract

AbstractWe present a framework for stochastically modeling the Fourier spectrum of the noisy seismic recording based on the fundamental assumption that the latter constitutes a random rather than a deterministic quantity. First we demonstrate (mathematically and through simulations) that, under the stochastic‐signal assumption, the periodogram ordinates of the noisy recording can be considered independent exponential random variables with a frequency‐dependent mean. Using this finding, the estimation of seismological parameters is translated into a well‐defined maximum likelihood (ML) problem, allowing a fast, accurate, and robust solution. Although the proposed ML methodology constitutes a general estimation framework that we believe can improve any spectral analysis, here we specifically apply it to the high‐frequency attenuation parameter (κ), crucial for understanding rock ground motion. Other seismological parameters could be estimated by appropriately adapting the theoretical model and frequency band used. The greatest advantage of the proposed method is its ability to account for the presence of noise rather than simply try to avoid it, as is the case with all conventional κ approaches and many other spectral analysis approaches. This means that the proposed technique achieves acceptable results even for very low signal‐to‐noise ratios (SNR), thus pushing the boundary of what can be considered usable recording quality. The technique's superior performance in estimation is demonstrated through a series of experiments involving both synthetic and real seismic recordings. Our results also indicate that, compared to this new “noise‐modeling” approach, conventional “noise‐avoiding” approaches are likely to systematically underestimate the “true” kappa value in recordings of moderate‐to‐low SNR.