Comparison of waveform inversion, part 1: conventional wavefield vs logarithmic wavefield

Abstract

A B S T R A C TThis is the first in a series of three papers focused on using variants of a logarithmic objective function approach to full waveform inversion. In this article, we investigate waveform inversion using full logarithmic principles and compare the results with the conventional least squares approach. We demonstrate theoretically that logarithmic inversion is computational similar to the conventional method in the sense that it uses exactly the same back-propagation technology as used in least-squares inversion. In the sense that it produces better results for each of three numerical examples, we conclude that logarithmic inversion is also more robust. We argue that a major reason for the inherent robustness is the fact that the logarithmic approach produces a natural scaling of the amplitude of the residual wavefield by the amplitude of the modelled wavefield that tends to stabilize the computations and consequently improve the final result. We claim that any superiority of the logarithmic inversion is based on the fact that it tends to be tomographic in the early stage of the inversion and more dependent on amplitude differences in the latter stages. I N T R O D U C T I O NConventional full-waveform inversion estimates the velocity of the subsurface by minimizing the least-squares difference between the modelled wavefield and the observed wavefield. This least-squares approach, as introduced by Lailly (1983) and Tarantola (1984), iteratively computes the optimum solution through a steepest-descent algorithm based on the Frechet derivative of a specific objective function. Practical results using this formulation of the problem have been less than satisfactory. While much is known about why and where the methodology fails, there has been only a nominal amount of effort devoted to alternative full-waveform approaches. A simple adaptation to the conventional least-squares approach is to use a different, more advantageous objective function. Basing

Application of Fullwaveform Tomography to VSP Walkaway Data

“…As a consequence, only the phase was used during the inversion and the logarithmic data residual was adopted. The logarithmic data residual had proven to provide results of comparable resolution to the logarithmic amplitude and phase residual of early arrival waveform inversion Shin et al, 2007;Kamei et al, 2014).…”

Comparison of Frequency-Selection Strategies for 2D Frequency-Domain Acoustic Waveform Inversion

“…The step length is determined using a line search method implemented as an inverse parabolic interpolation (PRESS et al, 1992). For the waveform inversion, we also estimate the source wavelet with the initial velocity model (SONG et al, 1995;PRATT, 1999;MIN, 2006 andSHIN et al, 2007).…”

Ore-body imaging by crosswell seismic waveform inversion: A case study from Kambalda, Western Australia

“…A comparison of inversion schemes using a known source wavelet, an iteratively estimated source wavelet, and normalised amplitude inversion was given by Greenhalgh and Zhou (2004) for surface reflection data. Other recent papers on full waveform acoustic inversion of reflection data in the frequency domain include Operto et al (2004), Shin and Min (2006) and Shin et al (2007). An alternative Laplace-domain inversion is given by Shin and Cha (2008).…”