Nonstationary seismograms can be mathematically expressed by the nonstationary convolution model (NSCM), which integrates the equivalent Q-value Qe⁠, source wavelet, and acoustic impedance AI into a single formula. Performing nonstationary AI inversion based on NSCM requires subsurface Qe field distribution and a source wavelet, which are all challenging to perform. The constant Qe field and the extracted wavelet obtained from the early part of the seismograms are commonly used as alternatives; consequently, inaccurate AI inversion results will be easily yielded. To deal with this problem, we decompose nonstationary seismic inversion into three subproblems. First, Qe curves are estimated at well locations using a novel scheme. Because the Gabor transform can induce approximate factorization of NSCM, the time-varying wavelet amplitude spectrum (TVWAS) can be obtained by dividing the time-frequency amplitude spectrum (TFAS) of the seismic signal by the TFAS of the reflection coefficient. Stable Qe curves at well locations then can be estimated by the compensation-based Q-analysis method, whose input data are TVWAS. Second, having an estimation of Qe and AI logging information at well locations, the source-wavelet inversion objective function with wavelet transform-domain sparse constraint is proposed based on the time-domain NSCM. This inverse problem is solved efficiently using projected fast iterative soft-thresholding algorithm. We have developed the AI inversion objective function with the low-frequency constraint and isotropic total variation transform-domain sparse constraint based on frequency-domain NSCM and the split Bregman method for minimizing this objective function. Benchmark model synthetic tests and field-data application indicate the effectiveness of our nonstationary AI inversion algorithm by demonstrating the improved accuracy and resolution of the AI inversion results.

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