Compressive sensing (CS) is an alternative to regular Shannon sampling that captures similar information from reduced measurements. It relies on randomized sampling patterns and a sparse data representation to reconstruct the regularly sampled object. CS is an important ingredient in afford- able seismic acquisition which can lead to improvements in the near surface mapping and in noise suppression for land data. However, the near surface traps the majority of the source-generated energy, resulting in data that are rich in high-wavenumber content and have amplitudes spanning several orders of magnitude. When dealing with such high dynamic range non-stationary data, the Fourier domain is not optimal for providing a sparse representation - a necessary condition for successful application of CS. In contrast, a discrete complex wavelet transform can localize high energy features, has good directional selectivity, and is near-shift invariant. Combined, these properties allow complex wavelets to represent detail-rich wavefields in a compact form. To leverage these features and achieve good CS reconstructions, we develop a scale- and orientation- dependent iterative soft thresholding scheme (IST) for reconstructing high dynamic range wavefields. Our approach requires little parametrization, is easy to implement, and robust to reconstructed wave- field sampling grid and dynamic range. We test IST on different wavefields with randomly missing traces, and compare the data reconstructions to the spectral projected gradient solver and projection onto convex sets. We quantify the reconstructions by a direct comparison of Fourier coefficients between fully sampled and reconstructed wavefields. Taking log10 of Fourier coefficients prior to computing the quality metric de-emphasizes the importance of magnitude match while highlighting Fourier coefficient support accuracy which usually translates into good structural fidelity of reconstructed data. We find that IST performs consistently among all examples, yielding a good phase match while performing gentle denoising.
