Bayesian Frequency-domain Mixed-phase Wavelet Estimation and Deconvolution

Enhancing the resolution of reflection data with embedded mixed-phase wavelet requires filtering techniques that omit assumptions about the wavelet phase. We present a blind mixed-phase wavelet estimation and deconvolution algorithm that uses the parameterization of a mixed-phase wavelet as the convolution of the wavelet’s minimum-phase equivalent with a dispersive all-pass filter, includes prior information about the wavelet to estimate in a Bayesian framework, and relies on the assumption of a sparse reflectivity. Solving the normal equations using the data autocorrelation function provides an inverse filter to remove the minimum-phase equivalent leaving traces with a balanced amplitude spectrum but distorted phase. Subsequently, we invert in the frequency domain for an all-pass filter thereby taking advantage of the fact that the all-pass filter action is exclusively contained in its phase spectrum. The ambiguous phase estimation is constraint by seeking the sparsest deconvolution output while honoring prior information about the wavelet. The prior information comprising an anticipation of the wavelet’s onset shape and duration allows resolving the wavelet’s polarity and timing that cannot be found using the sparseness measure alone. Applications to synthetic and field data show that simple phase models are already adequate to describe the all-pass filter.