Seismic waves are attenuated and distorted during propagation because of the conversion of acoustic energy to heat energy. We focus on intrinsic attenuation, which is caused by [Formula: see text], which is the portion of energy lost during each cycle or wavelength. Amplitude attenuation can decrease the energy of the wavefields, and dispersion effects distort the phase of seismic waves. Attenuation and dispersion effects can reduce the resolution of image, and they can especially distort the real position of interfaces. On the basis of the viscoacoustic wave equation consisting of a single standard linear solid, we have derived a new viscoacoustic wave equation with decoupled amplitude attenuation and phase dispersion. Subsequently, we adopt a theoretical framework of viscoacoustic reverse time migration that can compensate the amplitude loss and the phase dispersion. Compared with the other variable fractional Laplacian viscoacoustic wave equations with decoupled amplitude attenuation and phase dispersion terms, the order of the Laplacian operator in our equation is a constant. The amplitude attenuation term is solved by pseudospectral method, and only one fast Fourier transform is required in each time step. The phase dispersion term can be computed using a finite-difference method. Numerical examples prove that our equation can accurately simulate the attenuation effects very well. Simulation of the new viscoacoustic equation indicates high efficiency because only one constant fractional Laplacian operator exists in this new viscoacoustic wave equation, which can reduce the number of inverse Fourier transforms to improve the computation efficiency of forward modeling and [Formula: see text]-compensated reverse time migration ([Formula: see text]-RTM). We tested the [Formula: see text]-RTM by using Marmousi and BP gas models and compared the [Formula: see text]-RTM images with those without compensation and attenuation (the reference image). [Formula: see text]-RTM results match well with the reference images. We also compared the field data migration images with and without compensation. Results demonstrate the accuracy and efficiency of the presented new viscoacoustic wave equation.
Deep-learning seismic simulations have become a leading-edge field that could provide an effective alternative to traditional numerical solvers. We present a small-data-driven time-domain method for fast seismic simulations in complex media based on the physics-informed Fourier neural operator (FNO). Unlike most deep learning-based modeling schemes that either solve wave equations by embedding physical constraints into the cost function or conduct physics-informed learning by incorporating wave functions into convolutional neural networks (CNN), the FNO employs a learning architecture similar to the structure of split-step Fourier wave propagators, which is composed of two CNNs formulated in the space and wavenumber domains, respectively. The space-domain CNN acts as a local trainable phase-screen compensation. The wavenumber-domain CNN represents a non-local spatial convolutional operator acting as a trainable wavenumber filter for the phase-shift process. The FNO method approximates the mathematical-physical behavior of wave equations through learning the mapping between seismic wavefields at different times/locations from training seismic data. That is, the learning process parameterizes the integral kernel directly in the Fourier space, so that we can establish an expressive and efficient architecture for a better balance between accuracy and performance than the traditional spatial CNNs. Applications to gradient, layered, and Marmousi velocity models demonstrate its performance in accuracy and efficiency. The FNO seismic simulation is a data-driven method that needs a small amount of training data, especially when using blended source training data. It is a discretization-independent method that is not subject to the limitation of spatial sampling and time steps imposed on traditional numerical solvers, implying that training data can be discretized arbitrarily. It is also a model-independent method that can include absorption attenuation into seismic modeling without the need of viscoelastic wave equations.
Focal beam analysis has built a bridge between the acquisition parameters on the surface and the image quality of underground targets. However, as a practical matter, it is still difficult to answer how to choose a proper acquisition geometry according to the complexity of medium, especially considering the contradictory effects of multiple reflections on spatial resolution as they can be considered to be either potential signal or additional noise, depending on the envisioned imaging technology. We introduce an order-controlled, closed-loop focal beam method in which the migration operator and the resolution function can be analysed in the process of the closed-loop migration with full control over the order of the surface and internal multiples considered. This method highlights the effects of primary and different-order multiple wavefields on the imaging resolution for different acquisition geometries and various overburden strata. We apply the method to analyse the predicted resolution of seismic acquisition geometries considering multiples as either noise or signal. Results show, in the acquisition geometry design, that when the primaries cannot provide a complete spatial illumination for the subsurface target, e.g. because of the limited-aperture acquisition geometries or the complicated overburden, we should use the closed-loop focal beam analysis to assess the contradictory effects of multiples as both signal and noise, in which the maximum order of multiples ought to be chosen according to the core aim of the acquisition analysis. We can apply the second-order closed-loop focal beam analysis to quantify the effects of acquisition geometries on multiple-wave suppression and can also perform the high-order closed-loop focal beam analysis to quantify the effects of acquisition geometries on high-resolution imaging (migration). This method can also be used to choose the optimal order of multiples in the closedloop migration.
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