Adaptive multiple subtraction with wavelet-based complex unary Wiener filters

Adaptive subtraction is a key element in predictive multiple-suppression methods. It minimizes misalignments and amplitude differences between modeled and actual multiples, and thus reduces multiple contamination in the dataset after subtraction. Due to the high cross-correlation between their waveform, the main challenge resides in attenuating multiples without distorting primaries. As they overlap on a wide frequency range, we split this wide-band problem into a set of more tractable narrow-band filter designs, using a 1D complex wavelet frame. This decomposition enables a single-pass adaptive subtraction via complex, single-sample (unary) Wiener filters, consistently estimated on overlapping windows in a complex wavelet transformed domain. Each unary filter compensates amplitude differences within its frequency support, and can correct small and large misalignment errors through phase and integer delay corrections. This approach greatly simplifies the matching filter estimation and, despite its simplicity, narrows the gap between 1D and standard adaptive 2D methods on field data.

💡 Research Summary

The paper addresses the long‑standing problem of adaptive multiple attenuation in seismic data, where multiples are highly correlated with primaries and therefore difficult to suppress without damaging the primary reflections. Traditional approaches fall into two categories: (i) transform‑domain methods that separate primaries and multiples by mapping data to a domain where their overlap is reduced, and (ii) model‑based adaptive subtraction, where a multiple model (e.g., from SRME) is subtracted after an adaptive matching filter is estimated. The latter typically relies on long 1‑D Wiener filters or 2‑D adaptive filters. Long 1‑D filters are computationally heavy and can still distort primaries; 2‑D filters improve performance but require substantial parameter tuning and computational resources.

The authors propose a fundamentally different strategy: decompose the wide‑band seismic trace into a set of narrow‑band sub‑problems using a complex Morlet wavelet frame (continuous wavelet transform approximated by a redundant discrete frame). Each scale‑voice pair yields a complex coefficient for the data trace and for the multiple model. Within each sub‑band, the adaptive filter is reduced to a single complex scalar – a “unary” Wiener filter – that simultaneously corrects amplitude, phase, and small time‑shift errors. The optimal scalar a_opt is obtained in closed form by minimizing the mean‑square error: a_opt = ⟨d, x⟩ / ‖x‖², where d and x are the complex wavelet coefficients of the data and the model, respectively.

Large misalignments, however, cannot be handled by a simple scalar phase correction. To address this, the authors introduce an integer delay l and search for the delay that either (a) minimizes the residual energy ξ(a,l) = ‖d – a·x_l‖², or (b) maximizes the normalized cross‑correlation (coherence) between the delayed model and the data. The delay search is performed on overlapping windows in the wavelet domain, and once the optimal delay l_opt is found, the corresponding a_opt