Full waveform inversion is taking an increasing important role in exploration seismology. As this role continues to grow, we must develop our understanding of the basic nature of full waveform inversion. We develop an analytic example of anacoustic full waveform inversion for a simple attenuating Earth model. The first gradient of full waveform inversion is calculated for analytic, one-dimensional zero-offset data of the attenuating model. Analysis of the results yields that the gradient predicted an imaginary step function located at the depth of the attenuative target. This imaginary step function seems to be an intuitively correct result as introducing absorption into the wave equation usually involves allowing the wavespeed to have an imaginary component. The imaginary part appears to be moving well towards the right answer, whereas the dispersion and its effect on the real part has left much remaining work to be done to reconstruct the correct result.
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