Image-domain waveform inversion with the adjoint-state method

Waveform inversion is a velocity model building technique based on full seismograms as the input and seismic wavefields as the information carrier. Conventional waveform inversion is implemented in the data-domain. Similar techniques can be formulated in the image domain, with seismic image as the input and seismic wavefields as the information carrier. The objective function for image-domain waveform inversion is designed to optimize the coherency of reflections in extended common-image gathers. The function applies a penalty operator to the gathers, thus highlighting inaccuracies due to the velocity model error. Minimizing the objective function optimizes the model and improves the image quality. The gradient of the objective function is computed using the adjoint-state method in a similar way to the data-domain implementation. We present a parallel derivation for waveform inversion in data-domain and imagedomain, and show that these two approaches are only different in the choice of the objective function. The image-domain formulation is implemented with various types of extended image gathers with almost identical workflows. The results obtained from these approaches applied to the same synthetic model are quite different. The methods implemented with space-lag gathers and common-image-point gathers recover the model and improve the image quality. The approach with time-lag gathers is less effective to reconstruct the model compared to other formulations because it is less sensitive in capturing the image incoherency due to the model error.