2. Seismic/Geodetic Imaging Technologies
Comparison of Waveform Inversion: Conventional Wave Field vs Logarithmic Wave Field
Changsoo Shin(1), Sukjoon Pyun(2) and J. Bee Bednar(3)
(1) Division of Civil, Urban and Geosystem Engineering, Seoul National University, Korea. (2) Computational Science & Technology Program, Seoul National University, Korea. (3) Panorama Technologies, USA.
Abstract
Conventional full waveform inversion estimates the velocity of the subsurface by minimizing the least squares difference between the modeled wave field and the observed wave field. This least squares approach, as introduced in the middle 1980's by Lailly and Tarantola, 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 only been 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 but, hopefully, more advantageous objective function. This certainly opens a Pan Dora's Box of optimization possibilities, but basing the inversion on the logarithm of the modeled and observed wave-fields would appear to have some potential. Since the logarithm independently focuses attention on the amplitude and phase of the wave field, an objective function of this type implicitly provides for two independent inversions. One of these is dynamic. The other is temporal. One can choose to jointly or independently invert the amplitude and/or the phase. Because the logarithm tends to flatten the wave field's spectrum, it is not unreasonable to expect that the inversion will be less sensitive to missing or erroneous spectral content. With these concepts in mind, we chose to investigate waveform inversion using logarithmic principles and compare the results with the conventional norm. The results and comments are to be presented in a series of three papers. This first paper demonstrate that like its conventional sibling, logarithmic inversion can proceed using a back-propagation algorithm that is virtually identical to that used in a more conventional approach. In the second paper, we compare a modified version of the logarithmic methodology constructed by assuming that the amplitude of the modeled wavefield is identical to that of the observed wave field. Here, by example, we show that the logarithmic methodology is significantly more robust than conventional optimization. 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 wave field by the amplitude of the modeled wave field that tends to stabilizes the computations and consequently improves the final result. Similar comments apply to the corresponding delay associated with the modeled phase.
Last modified: Mon May 15 13:06:57 2006