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Full waveform inversion (FWI) makes full use of the seismic waveforms to find high-resolution velocity and density models by minimising residuals between the calculated and recorded data.  Q  attenuation widely exists in the subsurface media, leading to weak amplitude and misplacement of reflectors. However, the commonly used  Q -compensated FWI ( Q FWI) based on the second-order wave equation has difficulties in simultaneously inverting velocity and density fields. A  Q FWI method based on new first-order viscoacoustic quasi-differential equations is proposed to simultaneously produce velocity and density fields. Based on the adjoint state inversion theory,  Q -compensated forward-propagated operators, adjoint operators, and gradient equations are derived using the newly derived first-order viscoacoustic quasi-differential wave equations. The time-domain multi-scale decomposition method is introduced to update the velocity and density models from a low to a high wavenumber. Numerical examples on an actual work area model and a modified attenuating Marmousi model show that the proposed  Q FWI method produces higher-accuracy velocity and density models with iterations by correcting the  Q  attenuation than the conventional acoustic FWI. Even when the  Q  model is extremely inaccurate, the proposed QFWI obtains acceptable inversion results. Compared to the conventional  Q FWI, our  Q FWI better inverts velocity field in the case of an inaccurate density model. Finally, we verify the adaptability of our  Q FWI to field data.

Q -compensated full waveform inversion for velocity and density