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

Yanhua O Yuan, Ebru Bozdağ, Caio Ciardelli, Fuchun Gao, Frederik J Simons

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


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 favor 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 phase wrapping. 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 nonlinearity 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.
Original languageEnglish (US)
JournalGeophysical Journal International
StatePublished - Feb 11 2020
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: This paper grew out of early discussions with Jeannot Trampert on combining different phase measurements in adjoint tomography. We
thank Ryan Modrak and Jeroen Tromp for fruitful discussions. We also thank Ridvan Orsvuran for computing the global kernels. The opensource spectral-element software package SPECFEM and the automated window-selection package FLEXWIN are freely available from the Computational Infrastructure for Geodynamics ( We gratefully acknowledge the computational resources provided by the Princeton Institute for Computational Science & Engineering (PICSciE). CC thanks the Fundac¸ao de Amparo ˜ a Pesquisa do Estado `de Sao Paulo (FAPESP 2018/04918-6) for providing the financial support for his Ph.D. studies, and FJS thanks the U.S. National Science ˜ Foundation (EAR 1736046), the King Abdullah University of Science and Technology (OSR-2016-CRG5-2970-01) and TOTAL E&P for financial support, as well as the Institute for Advanced Study in Princeton for a quiet yet stimulating work environment during 2018–2019. We would like to thank the editor, Dr. Martin Schimmel, and two anonymous reviewers for their careful reading and constructive suggestions, which have helped improve the manuscript.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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