Robust Empirical Time–Frequency Relations for Seismic Spectral Amplitudes, Part 2: Model Uncertainty and Optimal Parameterization

Abstract

ABSTRACT In a companion article, Safarshahi and Morozov (2020) argued that construction of distance- and frequency-dependent models for seismic-wave amplitudes should include four general elements: (1) a sufficiently detailed (parametric or nonparametric) model of frequency-independent spreading, capturing all essential features of observations; (2) model parameters with well-defined and nonoverlapping physical meanings; (3) joint inversion for multiple parameters, including the geometrical spreading, Q, κ, and source and receiver couplings; and (4) the use of additional dataset-specific criteria of model quality, while fitting the logarithms of seismic amplitudes. Some of these elements are present in existing models, but, taken together, they are poorly understood and require an integrated approach. Such an approach was illustrated by detailed analysis of an S-wave amplitude dataset from southern Iran. The resulting model is based on a frequency-independent Q, and matches the data closer than conventional models and across the entire epicentral-distance range. Here, we complete the analysis of this model by evaluating the uncertainties and trade-offs of its parameters. Two types of trade-offs are differentiated: one caused by a (possibly) limited model parameterization and the second due to statistical data errors. Data bootstrapping shows that with adequate parameterization, attenuation properties Q, κ, and geometrical spreading parameters are resolved well and show moderate trade-offs due to measurement errors. Using the principal component analysis of these trade-offs, an optimal (trade-off free) parameterization of seismic amplitudes is obtained. By contrast, when assuming theoretical values for certain model parameters and using multistep inversion procedures (as commonly done), parameter trade-offs increase dramatically and become difficult to assess. In particular, the frequency-dependent Q correlates with the distribution of the source and receiver-site factors, and also with biases in the resulting median data residuals. In the new model, these trade-offs are removed using an improved parameterization of geometrical spreading, constant Q, and model quality constraints.