Preface to special issue on recent advances in computational seismology and its applications

Computational seismology is a relatively new interdisciplinary field spanning computational techniques in theoretical and observational seismology. It studies numerical methods and their implementation in various theoretical and applied problems in seismology.

Multi-axial unsplit frequency-shifted perfectly matched layer for displacement-based anisotropic wave simulation in infinite domain

Multi-axial perfectly matched layer(M-PML),known to have lost the perfect-matching property owing to multi-axial coordinate stretching,has been numerically validated to be long-time stable and it is thus used extensively in linear anisotropic wave simulation and in isotropic cases where the PML becomes unstable.We are concerned with the construction of the M-PML for anisotropic wave simulation based on a second order wave equation implemented with the displacement-based numerical method.We address the benefit of the incorrect chain rule,which is implicitly adopted in the previous derivation of the M-PML.We show that using the frequency-shifted stretching function improves the absorbing efficiency of the M-PML for near-grazing incident waves.Then,through multi-axial complex-coordinate stretching the second order anisotropic wave equation in a weak form,we derive a time-domain multi-axial unsplit frequency-shifted PML(M-UFSPML)using the frequency-shifted stretching function and the incorrect chain rule.A new approach is provided to reduce the number of memory variables needed for computing convolution terms in the M-UFSPML.The obtained M-UFSPML is well suited for implementation with a finite element or the spectral element method.By providing several typical examples,we numerically verify the accuracy and long-time stability of the implementation of our M-UFSPML by utilizing the Legendre spectral element method.

A nearly analytic exponential time difference method for solving 2D seismic wave equations

In this paper, we propose a nearly analytic exponential time difference （NETD） method for solving the 2D acoustic and elastic wave equations. In this method, we use the nearly analytic discrete operator to approximate the high-order spatial differential operators and transform the seismic wave equations into semi-discrete ordinary differential equations （ODEs）. Then, the converted ODE system is solved by the exponential time difference （ETD） method. We investigate the properties of NETD in detail, including the stability condition for 1-D and 2-D cases, the theoretical and relative errors, the numerical dispersion relation for the 2-D acoustic case, and the computational efficiency. In order to further validate the method, we apply it to simulating acoustic/elastic wave propagation in mul- tilayer models which have strong contrasts and complex heterogeneous media, e.g., the SEG model and the Mar- mousi model. From our theoretical analyses and numerical results, the NETD can suppress numerical dispersion effectively by using the displacement and gradient to approximate the high-order spatial derivatives. In addition, because NETD is based on the structure of the Lie group method which preserves the quantitative properties of differential equations, it can achieve more accurate results than the classical methods.

MLReal: Bridging the gap between training on synthetic data and real data applications in machine learning

Among the biggest challenges we face in utilizing neural networks trained on waveform(i.e.,seismic,electromagnetic,or ultrasound)data is its application to real data.The requirement for accurate labels often forces us to train our networks using synthetic data,where labels are readily available.However,synthetic data often fail to capture the reality of the field/real experiment,and we end up with poor performance of the trained neural networks(NNs)at the inference stage.This is because synthetic data lack many of the realistic features embedded in real data,including an accurate waveform source signature,realistic noise,and accurate reflectivity.In other words,the real data set is far from being a sample from the distribution of the synthetic training set.Thus,we describe a novel approach to enhance our supervised neural network(NN)training on synthetic data with real data features(domain adaptation).Specifically,for tasks in which the absolute values of the vertical axis(time or depth)of the input section are not crucial to the prediction,like classification,or can be corrected after the prediction,like velocity model building using a well,we suggest a series of linear operations on the input to the network data so that the training and application data have similar distributions.This is accomplished by applying two operations on the input data to the NN,whether the input is from the synthetic or real data subset domain:(1)The crosscorrelation of the input data section(i.e.,shot gather,seismic image,etc.)with a fixed-location reference trace from the input data section.(2)The convolution of the resulting data with the mean(or a random sample)of the autocorrelated sections from the other subset domain.In the training stage,the input data are from the synthetic subset domain and the auto-corrected(we crosscorrelate each trace with itself)sections are from the real subset domain,and the random selection of sections from the real data is implemented at every epoch of the training.In the inference/application stage,the input data are from the real subset domain and the mean of the autocorrelated sections are from the synthetic data subset domain.Example applications on passive seismic data for microseismic event source location determination and on active seismic data for predicting low frequencies are used to demonstrate the power of this approach in improving the applicability of our trained NNs to real data.

SH-wavefield simulation for a laterally heterogeneous whole-Earth model using the pseudospectral method

We present a scheme to simulate SH-wave propagation in a whole-Earth model with arbitrary lateral heterogeneities employing the Fourier pseudospectral method. Wave equations are defined in two-dimensional cylindrical coordinates and the model is taken through a great circle of the Earth. Spatial derivatives in the wave equations are calculated in the wavenumber domain by multiplication, and the transformation between spatial and wavenumber domains is performed via fast Fourier transformation. Because of the high accuracy and high speed of the Fourier pseudospectral method, the scheme enables us to calculate a short-wavelength global SH-wavefield with accurate waveforms and arrival times for models with heterogeneities that can be approximated as azimuthally symmetric. Comparing with two-dimensional simulation methods based on an axisymmetric model, implementing the seismic source in the present scheme is more convenient. We calculated the global SH-wavefield for the preliminary reference Earth model to identify the generation, reflection and refraction of various seismic phases propagating in the Earth. Applications to a heterogeneous global model with low-velocity perturbation above the core-mantle boundary were conducted to analyze the effect of lateral heterogeneity on global SH-wave propagation.

A New Earthquake Location Method Based on the Waveform Inversion

In this paper,a new earthquake location method based on the waveform inversion is proposed.As is known to all,the waveform misfit function under the L2 measure is suffering from the cycle skipping problem.This leads to a very small convergence domain of the conventional waveform based earthquake location methods.In present study,by introducing and solving two simple sub-optimization problems,we greatly expand the convergence domain of the waveform based earthquake location method.According to a large number of numerical experiments,the new method expands the range of convergence by several tens of times.This allows us to locate the earthquake accurately even from some relatively bad initial values.

An Adaptive Modal Discontinuous Galerkin Finite Element Parallel Method Using Unsplit Multi-Axial Perfectly Matched Layer for Seismic Wave Modeling

The discontinuous Galerkin finite element method(DG-FEM)is a highprecision numerical simulation method widely used in various disciplines.In this paper,we derive the auxiliary ordinary differential equation complex frequency-shifted multi-axial perfectlymatched layer(AODE CFS-MPML)in an unsplit format and combine it with any high-order adaptive DG-FEMbased on an unstructuredmesh to simulate seismicwave propagation.To improve the computational efficiency,we implement Message Passing Interface(MPI)parallelization for the simulation.Comparisons of the numerical simulation results with the analytical solutions verify the accuracy and effectiveness of our method.The results of numerical experiments also confirm the stability and effectiveness of the AODE CFS-MPML.