In reflection seismology, the inversion of subsurface reflectivity from the observed seismic traces (super-resolution inversion) plays a crucial role in target detection. Since the seismic wavelet in reflection seismic data varies with the travel time, the reflection seismic trace is non-stationary. In this case, a relative amplitude-preserving super-resolution inversion has been a challenging problem. In this paper, we propose a super-resolution inversion method for the non-stationary reflection seismic traces. We assume that the amplitude spectrum of seismic wavelet is a smooth and unimodal function, and the reflection coefficient is an arbitrary random sequence with sparsity. The proposed method can obtain not only the relative amplitude-preserving reflectivity but also the seismic wavelet. In addition, as a by-product, a special $Q$ field can be obtained.
The proposed method consists of two steps. The first step devotes to making an approximate stabilization of non-stationary seismic traces. The key points include: firstly, dividing non-stationary seismic traces into several stationary segments, then extracting wavelet amplitude spectrum from each segment and calculating $Q$ value by the wavelet amplitude spectrum between adjacent segments; secondly, using the estimated $Q$ field to compensate for the attenuation of seismic signals in sparse domain to obtain approximate stationary seismic traces. The second step is the super-resolution inversion of stationary seismic traces. The key points include: firstly, constructing the objective function, where the approximation error is measured in $L_2$ space, and adding some constraints into reflectivity and seismic wavelet to solve ill-conditioned problems; secondly, applying a Hadamard product parametrization (HPP) to transform the non-convex problem based on the $L_p (0 < p < 1)$ constraint into a series of convex optimization problems in $L_2$ space, where the convex optimization problems are solved by the singular value decomposition (SVD) method and the regularization parameters are determined by the L-curve method in the case of single-variable inversion. In this paper, the effectiveness of the proposed method is demonstrated by both synthetic data and field data.