Multidimensional Seismic Data Denoising Using Framelet-Based Order-p Tensor Deep Learning

Multidimensional (M-D) seismic data denoising is cast as an underdetermined inverse problem whose solution hinges on effective image priors extracted from machine learning knowledge. However, modeling seismic image priors is challenging due to the M-D nature of seismic images. Among the most promising prevailing image prior techniques is learning prior knowledge of the underlying structure by various 2-D or 3-D deep learning (DL)-based methods. However, for higher dimensional seismic data such as 4-D prestack data, these DL denoising schemes undoubtedly fail to capture the complete image structure in the absence of the flattening operation. To address this challenge, we present a framelet-based order-
$p$
tensor neural network (dubbed the FPTNN) model to implicitly learn the priors reflecting the typical behavior of clear M-D seismic images in a data-driven manner. First, motivated by the supremacy of the framelet transform over the Fourier transform, replacing the Fourier transform with the framelet gives a new definition with respect to the order-
$p$
tensor–tensor product (t-product). Then, through the redefined order-
$p$
t-product, the order-
$p$
tNN framework is a straightforward extension of the tNN with a standard t-product for M-D seismic denoising. By exploiting the fact that the order-
$p$
t-product can be computed through matrix multiplication in the framelet domain, we can readily reach the optimal weighted parameters in FPTNN via DL on a set of transformed matrix frontal slices. The experiments on both the synthetic and real field seismic datasets comprehensively demonstrate the advantages of our method against other state-of-the-art (SOTA) methods.