Air‐gun signatures and the minimum‐phase assumption

Abstract

The air‐gun array signature is close to minimum‐phase as a function of continuous time, in the sense that for processing purposes its phase spectrum can be derived from the Hilbert transform of the logarithm of its amplitude spectrum. This phase spectrum is different, however, from the minimum‐phase spectrum that is estimated by spiking deconvolution for a sampled and time‐windowed version of the signature. As a consequence, there can be large phase errors when spiking deconvolution is applied to an air‐gun signature or to a recording instrument response. The errors can be shown to consist primarily of a time shift and, at least visually over a limited bandwidth, a phase rotation of the output wavelet. The time shift is introduced by time sampling, while the phase rotation is caused by the spectral smoothing generated by time windowing. If the seismic wavelet as a whole, and not just the air‐gun signature, is minimum‐phase, then the total residual phase error after spiking deconvolution, including also the error due to data noise, can also be shown to be close to a time shift and a phase rotation. This may be physical justification for the phase rotation schemes that are often successful in matching seismic data and well‐log synthetics. The minimum‐phase assumption can be used for statistical air‐gun array signature deconvolution, providing that a limited amount of deterministic information (the instrument slopes and the source and receiver depths in the approach used here) is available to guide the process in those areas of the spectrum that are critical to the phase computation. Date examples show that, with care, almost identical results can then be obtained from either purely statistical deconvolution or deterministic deconvolution plus statistical deconvolution of multiples and ghosting.