The exponentiated phase measurement, and objective-function hybridization for adjoint waveform tomography

Seismic tomography has arrived at the threshold of the era of big data. However, how to extract information optimally from every available time-series remains a challenge; one that is directly related to the objective function chosen as a distance metric between observed and synthetic data. Time-domain cross-correlation and frequency-dependent multitaper traveltime measurements are generally tied to window selection algorithms in order to balance the amplitude differences between seismic phases. Even then, such measurements naturally favour the dominant signals within the chosen windows. Hence, it is difficult to select all usable portions of seismograms with any sort of optimality. As a consequence, information ends up being lost, in particular from scattered waves. In contrast, measurements based on instantaneous phase allow extracting information uniformly over the seismic records without requiring their segmentation. And yet, measuring instantaneous phase, like any other phase measurement, is impeded by phasewrapping. In this paper,we address this limitation by using a complex-valued phase representation that we call 'exponentiated phase'. We demonstrate that the exponentiated phase is a good substitute for instantaneous-phase measurements. To assimilate as much information as possible from every seismogram while tackling the non-linearity of inversion problems, we discuss a flexible hybrid approach to combine various objective functions in adjoint seismic tomography. We focus on those based on the exponentiated phase, to take into account relatively small-magnitude scattered waves; on multitaper measurements of selected surface waves; and on cross-correlation measurements on specific windows to select distinct body-wave arrivals. Guided by synthetic experiments, we discuss how exponentiated-phase, multitaper and cross-correlation measurements, and their hybridization, affect tomographic results. Despite their use of multiple measurements, the computational cost to evaluate gradient kernels for the objective functions is scarcely affected, allowing for issues with data quality and measurement challenges to be simultaneously addressed efficiently.