Determining Viscoelastic Models from Seismic Attenuation Measurements

Abstract

We have developed two simple deterministic methods to extract the parameters of viscoelastic models from seismic data. One is for the Zener model using phase velocity dispersion observations and the other is for the single fractional Zener model (Cole-Cole model) using attenuation versus frequency observations. The observations here represent either the arbitrary frequency-dependent dispersion behaviour from actual measurements or from some physical dissipation mechanism(s). In this contribution, it is also proved that similar to Zener model, the attenuation factor curve for the Cole-Cole model, on a logarithmic frequency-axis, symmetric about the frequency corresponding to the peak attenuation value, the peak frequency itself is equals to the inverse square root of the product of the two (stress and strain) relaxation times. The Cole-Cole model has a broad dispersion response over an appreciable frequency range, but is not very suitable for replicating complicated seismic attenuation dispersion curves which exhibit multiple peaks. In this case, we use the General Zener (GZ) model comprising multiple Zener elements and the General Fractional Zener (GFZ) model comprising multiple Cole-Cole elements to approximate the attenuation observations. Their parameters, including relaxation times and fractional derivative orders, are determined using a simulated annealing inversion method. Instead of searching for the relaxation times directly, we search for the Zener peak attenuation points (attenuation value and corresponding frequency, each of which corresponds to a pair of relaxation times. There are distinct advantages of such an approach.