Appraisal of Instantaneous Phase-Based Functions in Adjoint Waveform Inversion

Complex signal analysis allows separation of instantaneous envelope and phase of seismic waveforms. Seismic attributes have long routinely been used in geological interpretation and signal processing of seismic data as robust tools to highlight relevant characteristics of seismic waveforms. In the context of adjoint waveform inversion (AWI), it is crucial choosing an efficient parameter to describe the seismic data. The most straightforward option is using whole waveforms but the mixing of amplitude and phase parameters increases the nonlinearity inherent to the methodology. Several studies support the good functioning of the instantaneous phase (IP), a more linear parameter to measure the misfit between synthetic and recorded data. The IP is calculated using the inverse of the tangent function, where its principal value can be defined either wrapped in between different limits or also unwrapped. The wrapped phase presents phase jumps that reflect as noise in the inversion results. The conditioning of these discontinuities solves the problem partially and the continuous unwrapped IP is not a good descriptor of the waveform. For this reason, it is worth to explore beyond the traditional description of the IP parameter. Two alternative functions have been studied: 1) a revision of the triangular IP and 2) the first implementation of the normalized signal. The main objective of this paper is therefore, to review the fundamentals of the IP attribute in order to design robust IP-based objective functions which allow mitigating the inherent nonlinearity in the AWI method.