Joint-Domain Full Waveform Inversion Avoids Cycle Skipping in Ultrashallow Water
Authors: Carpenter Chris 
Published: February 2025

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This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 219190, “Joint-Domain Full Waveform Inversion and Its Applications to Ultrashallow Water Oceanbottom-Node Data,” by Yang Yang, Han Yin, and Lian Duan, BGP, et al. The paper has not been peer reviewed.

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Full waveform inversion (FWI) is commonly used for high-resolution velocity model building. Conventional FWI with least-squares objective function often suffers from the problem of cycle skipping. This issue becomes prominent in subsurface structures with high complexity, significant deviations between the initial and true models, or a lack of low-frequency content in the data. To address the cycle-skipping problem, the authors present a novel joint-domain full waveform inversion (JDFWI) framework optimizing travel-time accuracy in both data and model domains.

Introduction
Various strategies have been proposed to address cycle skipping. One set of solutions involves using different objective functions to ameliorate the nonconvex nature inherent in the conventional least-squares objective function. This encompasses using time-lag-based objective functions or quadratic Wasserstein metric-based objective functions. Alternatively, for those adhering to the conventional least-squares objective function, another approach involves manipulating the data to circumvent cycle skipping. Additionally, low-frequency data can be extrapolated from its higher-frequency components through deep-learning techniques. A distinct category of methods, known as domain extension, introduces an additional degree of freedom in the model space to tackle cycle skipping. The inversion process is then reformulated to enforce the transition from a nonphysical model extension to a physical one. This extension can be incorporated either in the model space or in the data domain.
The presented JDFWI framework optimizes travel-time accuracy in both data and model domains. In the model domain, the concept of delay time common image gathers (DTCIGs) is used to achieve precise travel-time measurements, while, in the data domain, the travel-time data misfit is extracted without the amplitude influence. Integrating a time-lag-based misfit function efficiently overcomes the cycle-skipping problem with the proposed JDFWI workflow.
The authors include a subsection of the complete paper in which equations underpinning both conventional and joint-domain FWI are supplied and detailed.

Synthetic Example
The nonlinear state-space response, the curse of dimensionality, and the imperfect low-frequency content in both the model and data often increase the risk for conventional FWI to be trapped in the local minimum when using the objective function. As a simple demonstration, the authors first performed a 2D synthetic test to show the efficiency of the proposed JDFWI to overcome cycle skipping. The exact velocity model used in this synthetic test featured a chalk structure on the left and a large salt body on the right. Both high-velocity structures initiated as shallow as 2 km in depth. In the case of FWI, only when these structures were correctly inverted could the model below them reflect the exact geology. However, in the initial model for FWI, both high-velocity structures were missing and the difference to the exact velocity exceeded 2000 m/s in many places. The acquisition design for the inversion included 1,501 evenly spaced shots with 10-m spacing and 301 evenly spaced receivers with 50-m spacing across the whole model. Field data were simulated with a 1-Hz high-passed flat-spectrum wavelet.