GNGTS 2022 - Atti del 40° Convegno Nazionale

508 GNGTS 2022 Sessione 3.3 fail to accurately describe complex vector spaces such as those spanned by seismic data. An alternative is to learn the transform directly from the data, which can be performed effectively and efficiently by the non-linear dimensionality reduction obtained through CNNs. This way the inverse problem we wish to solve writes: Here D θ ( z ): k → m is a non-linear mapping implemented through a trained convolutional decoder. Here, since the functional to be minimized is nonlinear, it is appropriate to use non- linear solvers such as L-BFGS to estimate z. Finally, the solution of the inverse problem is found as xˆ = D θ ( zˆ ). The parameters θ of the decoder D θ (•) are found alongside those of an encoder E θ ( z ): m → k while learning the latent representation by training the convolutional autoencoder AE(•) = D θ (E θ (•)). This is accomplished by minimizing the following loss function over a training dataset of n s samples { u 1 , u 2 ,..., u n } that is representative of the data x we wish to invert. Application to deblending . We demonstrate the effectiveness by applying it to the problem of seismic deblending. In a seismic blended acquisition, multiple sources fire in a short time interval according to a random dithering code: the shots of different sources are then blended. Therefore, in this case the linear operator A implements the delays due to the dithering code. The goal of seismic deblending is to recover the unblended seismic data x from the blended measurements y . The example is designed by means of a field dataset from Gulf of Suez comprised of 128 shots, 128 receivers and 512time samples with spatial and temporal sampling intervals of 12.5 m and 4 ms, respectively. Blending is performed by applying a random dithering conde where the random delay lies in the time range [0, 0.8s]. Some blended data are shown in Fig.1. Fig. 1 - Some blended common shot gathers. An original unblended common shot gather is shown in Fig. 2a and its corresponding pseudo-deblended gather is shown in Fig 2e; notice that the blending noise masks the useful signal especially in the deep area of the first few traces. The deblending result of the proposed method is displayed in Fig. 2d and compared with the results obtained via dictionary learning (Fig 2b) and via end-to-end deep learning (2c). The corresponding residuals are shown in Fig 2f, 2g and 2h. The dictionary learning based deblending method (Fig. 2f) results in strong lending noise residual and signal damage. For the deblending CNN (Fig. 2g), although we can barely