Conventional reflectivity inversion methods are based on a stationary convolution model and theoretically require stationary seismic traces as input (i.e., those free of attenuation and dispersion effects). Reflectivity inversion for nonstationary data, which is typical for field surveys, requires us to first compensate for the earth’s Q-filtering effects by inverse Q filtering. However, the attenuation compensation for inverse Q filtering is inherently unstable, and offers no perfect solution. Thus, we presented a sparse reflectivity inversion method for nonstationary seismic data. We referred to this method as nonstationary sparse reflectivity inversion (NSRI); it makes the novel contribution of avoiding intrinsic instability associated with inverse Q filtering by integrating the earth’s Q-filtering operator into the stationary convolution model. NSRI also avoids time-variant wavelets that are typically required in time-variant deconvolution. Although NSRI is initially designed for nonstationary signals, it is suitable for stationary signals (i.e., using an infinite Q). The equations for NSRI only use reliable frequencies within the seismic bandwidth, and the basis pursuit optimizes a cost function of mixed l2l1 norms to derive a stable and sparse solution. Synthetic examples show that NSRI can directly retrieve reflectivity from nonstationary data without advance inverse Q filtering. NSRI is satisfactorily stable in the presence of severe noise, and a slight error in the Q value does not greatly disturb the sensitivity of NSRI. A field data example confirmed the effectiveness of NSRI.

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