Attenuation compensation least-square reverse time migration (Q-LSRTM) linearly inverts for the subsurface reflectivity model from lossy data. It can compensate for the amplitude loss and phase distortion due to the strong subsurface attenuation compared to the conventional migration methods. However, the inverted images from Q-LSRTM with a certain number of iterations are often observed to have lower resolution when compared with the benchmark acoustic LSRTM from acoustic data. This because the adjoint Q propagators used for backpropagating the residual are also attenuative. To increase the resolution and accelerate the convergence of Q-LSRTM, we used viscoacoustic deblurring filters as a preconditioner for Q-LSRTM. Numerical tests on synthetic and field data demonstrate that the Q-LSRTM combined with viscoacoustic deblurring filters can produce images with higher resolution and more balanced amplitudes when there is strong atteunation in the background medium. The proposed preconditioning method is also shown to significantly increase the convergence rate of Q-LSRTM.