The roles of the spatial regularization in seismic deconvolution

In this paper we investigate the roles of the spatial regularization in seismic deconvolution. The spatial regularization is described as a L2 norm of the lateral reflectivity difference imposed on multi-trace data misfit term. In essence, the spatial regularization acts as a band-pass filter along the spatial direction. Therefore, it can suppress the high-wavenumber components of the estimated reflectivity, for example, noisy trails like noodles which usually caused by temporal regularized deconvolution. As well, the spatial regularization can help recovering the reflectivity of discarding traces by repeatedly and linearly weighting its neighboring reflectivity, thereby exploring the spatial continuities among traces. Moreover, the spatial regularization can help stabilizing inversion, just like the temporal regularization. Both synthetic and field data examples are used to demonstrate the three roles of the spatial regularization by comparing spatial regularized deconvolution with conventional temporal deconvolution implemented by minimizing a data misfit and a L2 norm or a L1 norm of reflectivity. Furthermore, the synthetic examples also clearly illustrate that the spatial regularization can help yielding a high resolution and meanwhile high signal-to-noise ratio deconvolution result, which matches best with the reference reflectivity.