Some remarks on Q ‐compensated sparse deconvolution without knowing the quality factor Q

The subsurface media are not perfectly elastic, thus anelastic absorption, attenuation and dispersion (aka Q filtering) effects occur during wave propagation, diminishing seismic resolution. Compensating for anelastic effects is imperative for resolution enhancement. Q values are required for most of conventional Q ‐compensation methods, and the source wavelet is additionally required for some of them. Based on the previous work of non‐stationary sparse reflectivity inversion, we evaluate a series of methods for Q ‐compensation with/without knowing Q and with/without knowing wavelet. We demonstrate that if Q ‐compensation takes the wavelet into account, it generates better results for the severely attenuated components, benefiting from the sparsity promotion. We then evaluate a two‐phase Q ‐compensation method in the frequency domain to eliminate Q requirement. In phase 1, the observed seismogram is disintegrated into the least number of Q ‐filtered wavelets chosen from a dictionary by optimizing a basis pursuit denoising problem, where the dictionary is composed of the known wavelet with different propagation times, each filtered with a range of possible Q − 1values. The elements of the dictionary are weighted by the infinity norm of the corresponding column and further preconditioned to provide wavelets of different Q − 1values and different propagation times equal probability to entry into the solution space. In phase 2, we derive analytic solutions for estimates of reflectivity and Q and solve an over‐determined equation to obtain the final reflectivity series and Q values, where both the amplitude and phase information are utilized to estimate the Q values. The evaluated inversion‐based Q estimation method handles the wave‐interference effects better than conventional spectral‐ratio‐based methods. For Q ‐compensation, we investigate why sparsity promoting does matter. Numerical and field data experiments indicate the feasibility of the evaluated method of Q ‐compensation without knowing Q but with wavelet given.