Passive seismic interferometry is a vastly generalized blind deconvolution question, where different paths through the Earth correspond to different channels called Green's functions; the sources are completely incoherent and not shared by the channels, and the question is to estimate paths (channels) that are not present in the dataset. SI, turning noise to signal, has numerous applications, from monitoring industrial activities to crustal structure investigation. No standard method of signal processing will solve SI. Instead, domain scientists resort to a simple cross-correlation operation, a.k.a. correlogram, which can retrieve the Green's function directly, but only under restrictive assumptions of ergodicity (energy equipartitioning) of the random process generating the seismic source. However, in practice, correlograms are not equal to the empirical Green's function, because these assumptions are generally far from being satisfied in realistic situations. In the framework of supervised learning, we propose to train deep neural networks (NNs) to overcome two limitations of correlation-based SI: the temporal limitation of passive recordings and the spatial limitation of the random source distribution. Deep NNs are trained to implicitly find the relationship between the empirical Green's functions and the correlograms and then used to extract the correct Green's functions from ambient noise. The input of the network is correlograms (a virtual shot gather), and the desired output is the empirical Green's function (the active shot gather). The NN can often retrieve Green's functions from 5-min passive recordings with acceptable accuracy in our synthetic example. Although an exact estimation of the source locations may not be necessary, a prior knowledge of the source directionality (through a preliminary beamforming step) is helpful when training the NN to mitigate the challenges associated with inhomogeneous source distributions (directional wave fields). In this work, all the numerical examples are based on the retrieval of P-wave reflections in the exploration scale and are conducted on synthetic data. We use a modified ResNet in our numerical experiments.

Full-waveform inversion (FWI) strongly depends on an accurate starting model to succeed. This is particularly true in the elastic regime: The cycle-skipping phenomenon is more severe in elastic FWI compared to acoustic FWI due to the short S-wave wavelength. In this article, we extend our work on extrapolated FWI (EFWI) by proposing to synthesize the low frequencies of multicomponent elastic seismic records and use those “artificial” low frequencies to seed the frequency sweep of elastic FWI. Our solution involves deep learning: we can either train the same convolutional neural network (CNN) on two training datasets, one with vertical components and one with horizontal components of particle velocities, or train with two components together, to extrapolate the low frequencies of elastic data for 2-D elastic FWI. The architecture of this CNN is designed with a large receptive field by dilated convolution. Numerical examples on the Marmousi2 model show that the 2–4 Hz low-frequency data extrapolated from band-limited data above 4 Hz provide good starting models for elastic FWI of P- and S-wave velocities. In addition, we study the generalization ability of the proposed neural network from acoustic to elastic data. For elastic test data, collecting the training dataset by elastic simulation shows better extrapolation accuracy than acoustic simulation, i.e., a smaller generalization gap.

We present lift and relax for waveform inversion (LRWI), an approach that mitigates the local minima issue in seismic full waveform inversion (FWI) via a combination of two convexification techniques. The first technique (Lift) extends the set of unknown variables to their products, arranged as a moment matrix. This algebraic idea is a celebrated way to replace a hard polynomial optimization problem by a semidefinite programming approximation. Concretely, both the model and the wavefield are lifted from vectors to rank-2 matrices. The second technique (Relax) invites to relax the strict wave-equation constraint-a technique known as wavefield reconstruction inversion (WRI), which introduces wave-equation misfits as a weighted penalty term in the objective function. The relaxed penalty formulation enables balancing the data and wave-equation misfits by tuning a penalty parameter. Together, “Lift” and “Relax” help reformulate the inverse problem as a set of constraints on a rank-2 moment matrix. Such a lifting strategy permits good data and wave equation fits throughout the inversion process while leaving the numerical rank of the rank-2 moment matrix to be minimized down to one. Moreover, LRWI does not require adjoint wavefield to compute the gradient, which mitigates computational burdens. Numerical examples indicate that starting with a poor initial model, LRWI can conduct successful inversions with a starting frequency that is higher than that required by FWI and WRI.

This paper considers recovering L-dimensional vectors w, and x1, x2,⋯, xNfrom their circular convolutions yn= w*xn, n = 1, 2, 3,⋯, N. The vector w is assumed to be S-sparse in a known basis that is spread out in the Fourier domain, and each input xn is a member of a known K-dimensional random subspace. We prove that whenever K+S log2S ≲ L/ log4(LN), the problem can be solved effectively by using only the nuclear-norm minimization as the convex relaxation, as long as the inputs are sufficiently diverse and obey N ≳ log2(LN). By “diverse inputs,” we mean that the xn's belong to different, generic subspaces. To the best of our knowledge, this is the first theoretical result on blind deconvolution where the subspace to which w belongs is not fixed but needs to be determined. We discuss the result in the context of multipath channel estimation in wireless communications. Both the fading coefficients and the delays in the channel impulse response w are unknown. The encoder codes the K-dimensional message vectors randomly and then transmits coded messages xn's over a fixed channel one after the other. The decoder then discovers all of the messages and the channel response when the number of samples taken for each received message are roughly greater than (K + S log2S) log4(LN), and the number of messages is roughly at least log2(LN).