Robust time-domain full waveform inversion with normalized zero-lag cross-correlation objective function

Abstract

S U M M A R Y In full waveform inversion (FWI) with the least-squares (L2) norm, the direct amplitude matching is never perfect and the accurate estimation of the seismic source strength is not always available. In contrast, the normalized zero-lag cross-correlation objective function relaxes on the amplitude constraints and emphasizes the phase information when measuring the closeness between the simulated and observed data. This FWI method becomes insensitive to differences in amplitude. Based on this property, we investigate the effectiveness and robustness of FWI with the normalized zero-lag cross-correlation function (CFWI) against the noise and unpredictable amplitude of the data that cannot be modelled by the wavefield extrapolation operator. The effectiveness is firstly tested by noise-free data and data contaminated by Gaussian white noise. In addition, CFWI can invert the data set with incorrect source strength when compared with the L2 norm. Moreover, the data set with incorrect source signature illustrates that CFWI is slightly more insensitive to the error in source signature than the L2 norm. However, a source inversion is still needed when the source signature is severely erroneous. With non-Gaussian noise data, such as contaminated by strong ground motion noise and even by spike-type noise, CFWI provides a comparable result with that of the robust Huber norm. Numerical experiments with non-Gaussian noise also indicate that CFWI can suppress noise in data to produce clearer images when compared with the Huber norm. Besides, CFWI is free of the threshold criterion that controls the transition between the L2 and L1 norms used with the Huber and Hybrid norms and therefore free from tedious trial-and-error tests. Several numerical examples support that CFWI is an alternative and reliable inversion method. However, a numerical test with a 1-D initial model confirms that CFWI is more sensitive to the cycle-skipping problem caused by less-accurate initial velocity model than the L2 norm, which is due to the wrong matched events contributing to spurious local minima of the objective function of CFWI, but to an increase in the objective function used with the L2 norm.

Full-waveform inversion using a nonlinearly smoothed wavefield

“…Dutta et al (2014) and Zhang et al (2015) demonstrate that the global correlation norm generates more stable and practical results in the least-squares migration on account of relaxing the dependence on amplitude matching often required by the L2-norm. Liu et al (2017) also indicate that FWI using the normalized crosscorrelation function is slightly less sensitive to the noise and the source signature. In this paper, we use the global correlation norm as an alternative to the least-squares misfit to mitigate the dependence on exact amplitude information of the nonlinearly smoothed wavefield.…”

Simulation of x‐ray‐induced acoustic imaging for absolute dosimetry: Accuracy of image reconstruction methods

“…However, when the amplitudes are low, due to attenuation in the tissue, matching the amplitudes directly is challenging. In this case, a cross‐correlation object function that measures the similarities between predicted and observed data or a phase‐only objective function that emphasizes the phase mismatch may be more efficient . Using the optimal objective function requires further investigation.…”

“…In this case, a cross-correlation object function that measures the similarities between predicted and observed data or a phase-only objective function that emphasizes the phase mismatch may be more efficient. 33 Using the optimal objective function requires further investigation. Note that, for this example we assume an advanced acoustic data acquisition technology that uses a custom-built sensor array, capable of acquiring planar acoustic data on a patient.…”

Initial model assessment for intermediate-period full-waveform inversion of the contiguous U.S. and surrounding regions

“…Besides these misfit functions, we also consider the normalized zero-lag cross-correlation coefficient (NZCC), which best reflects both the phase match and the complex similarity of waveform shape, e.g., triplicated waveforms but regardless of the absolute amplitude. It has been used as an objective function for full waveform inversion which robustly recovers seismic structures with strong wave speed contrasts (Liu et al 2016;Tao et al 2017Tao et al , 2018. The NZCC is defined as:…”