Fitting viscoelastic mechanical models to seismic attenuation and velocity dispersion observations and applications to full waveform modelling

The Cole-Cole mechanical model can match a simple seismic attenuation curve over a broad frequency range, but it is not a suitable model for replicating complicated seismic attenuation dispersion curves which exhibit multiple peaks or display pronounced asymmetry. In this case, we use the General Fractional Zener (GFZ) Model, which comprises multiple Cole-Cole elements, to approximate the attenuation observations. The observations here represent the arbitrary (frequency-dependent) dispersion behaviour from actual measurements (phase velocities and/or dissipation factors) or from some physical dissipation mechanism(s) such as local induced fluid flow in effective Biot theory. The key parameters of these viscoelastic models,which include the stress and strain relaxation times and the fractional derivative orders, are determined with a simulated annealingmethod. Instead of searching for the relaxation times directly, we search for the Zener peak attenuation points and corresponding frequencies, each of which corresponds to a pair of relaxation times. We show that just two fractional Zener elements can sometimes provide a satisfactory approximation to the observations over the entire frequency range. A simple deterministic method is developed to extract the parameters of the single element Zener model using phase velocity observations. As a special case of the GFZ model, we found and proved the constancy of the width of the attenuation curves at the half maximum amplitude point (FWHM) for all Zener models which is critical to the design of reasonable observation frequencies. We stress and demonstrate that ignoring the frequency-dependence of Q may result in significant discrepancies of calculated waveforms with observed or predicted values.