A parametric model for seismic wavelets—with estimation and uncertainty quantification

Abstract

Wavelet estimation is an essential step in qualitatively and quantitatively analysing and interpreting seismic data. Applications span from seismic data quality assessment to well ties and seismic inversion. Wavelet estimation methods can be roughly separated into two approaches, data driven inversion methods and analytical definitions. We present a new analytical wavelet definition, which is based on Hermite basis functions. This wavelet model contains four parameters, which correspond to wavelet magnitude, phase, wavelet length and bandwidth. One of our main motivations for this development was to define a compact wavelet representation and an intrinsic parameter uncertainty assessment workflow, which allows us to quantify uncertainties in estimated wavelets, as well as the generation of wavelet realizations to be used, for example, in statistical seismic amplitude inversions. We present a statistical workflow to estimate the model parameters and to explore their posterior uncertainties given well log data and seismic amplitude data. This includes sampling the posterior distribution of the four wavelet parameters using Markov Chain Monte Carlo methods. We then discuss the applicability, limitations and challenges of the approach with the help of synthetic data and a North Sea data set with well logs and processed seismic amplitudes, where we also compare our method to Bayesian least-squares and a commercial wavelet estimation routine. Realizations of wavelets based on the optimized parameters and their uncertainties appear to sample the wavelet space well with reasonable variations in wavelet length, phase and amplitude while not introducing random fluctuations or wavelet lobes. The results indicate that the compact wavelet representation allows for an efficient and rather stable wavelet estimation workflow that achieves useful results in the presence of noisy data.