Half-precision wave simulation

The numerlcal sUmulation of the elastic wave equation has been become one of the fundamental seismic modeling methods. In recent years, because high speed and large storage computers were avaiable, the numerical stimulation of the Blot’s coupled equation was developed. Biot’s equation takes account of energy dissipation due to the relative motion between a viscous pore fluid and the porous solid matrix. But for the samc size model the numerical s!hnulaUon of Biot’s equation takes much more computer storage and hours than that of elastic wave equation. The realistic oilfields are composed of non-porous rocks (elastic media) and fluid-saturated porous rocks (Blot’s media). In this paper, we pmaent a hybrid finite difference scheme to stimulate wave propagation in reallstic oilfields, which describes the wave propagation in an elastic medium by elastic wave equation, and in a fluid-saturated porous medium by Biot’s coupled wave equation. Numerical example is presented to demonstrate the scheme. By comparing the result with that computed by acoustic wave equation, the effects of parameters uch as porosity and fluid content can be investigated.

This study investigates the scientific development of Artificial Intelligence (AI) and Machine Learning (ML) applications in Human Resource Management (HRM) through bibliometric analysis. To this end, 522 academic publications indexed in the Scopus database between 2020 and 2024 were analyzed using the R-based bibliometrix package and VOSviewer software. Descriptive analysis, scientific productivity metrics, and content analysis techniques were employed. The findings revealed three main patterns. First, research on AI and ML applications in HRM has grown significantly—particularly between 2022 and 2024—driven by post-pandemic digital transformation. Second, India, China, and the USA lead in research output, while the UK and France demonstrate strong citation impact, indicating a globally expanding research ecosystem. Third, the thematic focus of research is shifting from technical infrastructure toward more human-centered and ethical dimensions. Additionally, keyword co-occurrence network analysis identified three major thematic clusters: HRM functions, AI applications, and machine learning analytics, highlighting the field’s interdisciplinary nature. Compared to the previous studies, this research provides a more comprehensive bibliometric analysis of AI and ML applications in HRM. It is the first extensive study to map the intellectual evolution of the field from a multidisciplinary perspective. Furthermore, it charts research trends and collaboration networks, revealing a shift from technical implementations to strategic integration. In conclusion, this analysis offers new insights to the literature by illustrating the technological evolution in HRM and highlighting the growing significance of cutting-edge approaches such as AI and ML, reaffirming the field as a timely and impactful area of research.

Floating-point arithmetic precision is limited in length the IEEE single (respectively double) precision format is 32-bit (respectively 64-bit) long. Extended precision formats can be up to 128-bit long. However some problems require a longer floating-point format, because of round-off errors. Such problems are usually solved in arbitrary precision, but round-off errors still occur and must be controlled. Interval arithmetic has been implemented in arbitrary precision, for instance in the MPFI library. Interval arithmetic provides guaranteed results, but it is not well suited for the validation of huge applications. The CADNA library estimates round-off error propagation using stochastic arithmetic. CADNA has enabled the numerical validation of real-life applications, but it can be used in single precision or in double precision only. In this paper, we present a library called SAM (Stochastic Arithmetic in Multiprecision). It is a multiprecision extension of the classic CADNA library. In SAM (as in CADNA), the arithmetic and relational operators are overloaded in order to be able to deal with stochastic numbers. As a consequence, the use of SAM in a scientific code needs only few modifications. This new library SAM makes it possible to dynamically control the numerical methods used and more particularly to determine the optimal number of iterations in an iterative process. We present some applications of SAM in the numerical validation of chaotic systems modeled by the logistic map.

We present a quantum algorithmic framework for simulating linear, anti-Hermitian (lossless) wave equations in heterogeneous, anisotropic, and time-independent media. This framework encompasses a broad class of wave equations, including the acoustic wave equation, Maxwell’s equations, and the elastic wave equation. Our formulation is compatible with standard numerical discretization schemes and allows for the efficient implementation of multiple practically relevant time- and space-dependent sources. Furthermore, we demonstrate that subspace energies can be extracted and wave fields compared through an l2 loss function, achieving optimal precision scaling with the number of samples taken. Additionally, we introduce techniques for incorporating boundary conditions and linear constraints that preserve the anti-Hermitian nature of the equations. Leveraging the Hamiltonian simulation algorithm, our framework achieves a quartic speedup over classical solvers in three-dimensional simulations, under conditions of sufficiently global measurements and compactly supported sources and initial conditions. This quartic speedup is optimal for time-domain solutions, as the Hamiltonian of the discretized wave equations has local couplings. In summary, our framework provides a versatile approach for simulating wave equations on quantum computers, offering substantial speedups over state-of-the-art classical methods.

Heterogeneous wave equations are more complicated numerically than homogeneous wave equations, but are necessary for physical validity. A wide variety of numerical solutions of seismic wave equations is available, but most produce strong numerical artefacts and local instabilities where model parameters change rapidly. Accuracy and stability of heterogeneous equations is achieved through staggered‐grid formulations. A new pseudospectral staggered‐grid algorithm is developed for the poroelastic (Biot) equations. The algorithm may be reduced to handle the elastic and acoustic limits of the Biot equations. Comparisons of results from poroelastic, elastic, acoustic and scalar computations for a 2D model show that porous medium parameters may affect amplitudes significantly. The use of homogeneous wave equations for modelling of a heterogeneous medium, or of a centred rather than a staggered grid, or of simplified (e.g. acoustic) wave equations when elastic or poroelastic media are synthesized, may produce erroneous or ambiguous interpretations.

SUMMARY We first transform the seismic wave equations in 2-D inhomogeneous anisotropic media into a system of first-order partial differential equations with respect to time t. And then we develop a new explicit split-step algorithm (SSA) to solve the transformed equations through using the implicit Adams method and high-order interpolation approximation. Our algorithm enables wave propagation to be simulated in two dimensions through generally isotropic and anisotropic models. The high-order space derivatives in this SSA are determined by using the wave displacement, particle-velocity and its gradients simultaneously, while the time advancing of modelling seismic propagation uses the slit-operator algorithm based on the third-order implicit Adams method. On the basis of such a structure, the so-called SSA can suppress effectively the numerical dispersion and source-generated noises caused by discretizing the wave equations when too-coarse grids are used, and is third-order accuracy in time and fourth-order accuracy in space. We explore the theoretical properties of the SSA including the stability criteria of the SSA for solving 1-D and 2-D scalar wave equations, numerical dispersion, theoretical error and numerical error and computational efficiency when using the SSA to model the acoustic wave fields. Meanwhile, we also present the synthetic seismograms generated by the SSA in the three-layer isotropic medium and the 2-D elastic two-layer isotropic and anisotropic wavefield snapshots computed by the fourth-order Lax–Wendroff correction (LWC) scheme and the SSA, respectively. Numerical calculations of the relative errors show that the numerical error of the SSA is less than those of the conventional finite difference (FD) method and the fourth-order LWC scheme. Promising numerical results further illustrate that the SSA has less numerical dispersions and can suppress effectively the source-generated noises, further resulting in both increasing the computational efficiency and saving the space storage.

The par-axial wave equation is one of the most accurate prestack imaging tools for seismic data. It has been shown that pre-stack par-axial wave equation migration correctly images seismic data even for very complex velocity models (e.g., such as those containing caustics [4]). Furthermore, since the paraxial approximation gives rise to a one-way wave equation it is ideal for migration and imaging since, in principle, it does not generate reflections as do the acoustic and elastic wave equations. Nevertheless, the formulation and implementation of the paraxial wave equation give rise to undesirable numerical artifacts [ 1], [3]. Some thought has already been given to the filtering of these artifacts by using complex coefficients in the rational approximation to the paraxial wave equation [5]. This choice, however, only partially eliminates the artifacts and degrades the performance of the opening and amplitude responses of the par-axial operator. In this paper an efficient, post-extrapolation filtering scheme is described which completely eliminates the numerical artifacts while preserving the opening and amplitude characteristics of the paraxial operator.

In marine seismic exploration, ocean-bottom cable techniques accurately record the multicomponent seismic wavefield; however, the seismic wave propagation in fluid–solid media cannot be simulated by a single wave equation. In addition, when the seabed interface is irregular, traditional finite-difference schemes cannot simulate the seismic wave propagation across the irregular seabed interface. Therefore, an acoustic–elastic forward modeling and vector-based P- and S-wave separation method is proposed. In this method, we divide the fluid–solid elastic media with irregular interface into orthogonal grids and map the irregular interface in the Cartesian coordinates system into a horizontal interface in the curvilinear coordinates system of the computational domain using coordinates transformation. The acoustic and elastic wave equations in the curvilinear coordinates system are applied to the fluid and solid medium, respectively. At the irregular interface, the two equations are combined into an acoustic–elastic equation in the curvilinear coordinates system. We next introduce a full staggered-grid scheme to improve the stability of the numerical simulation. Thus, separate P- and S-wave equations in the curvilinear coordinates system are derived to realize the P- and S-wave separation method.

Full-waveform inversion (FWI) is a non-destructive health monitoring technique that can be used to identify and quantify the embedded anomalies. The forward modeling of the FWI consists of a simulation of elastic wave equation to generate synthetic data. Thus the accuracy of the FWI method highly depends on the simulation method used in the forward modeling. Simulation of a 3-D seismic survey with small-scale heterogeneities is impossible with the classic finite difference approach even on modern super computers. In this work, we adopted a mesh refinement approach for simulation of the wave equation in the presence of small-scale heterogeneities. This approach uses cubic smoothing spline interpolation for spatial mesh refinement step in solving the wave equation. The simulation results for the 2-D elastic wave equation are presented and compared with the classic finite difference approach.

This paper discusses the applicability of a single precision computation with a graphics processing unit for the finite elements method simulation of surface acoustic wave (SAW) devices using the hierarchical cascading technique (HCT). The HCT simulation is performed in both single and double precision, and the required computational time, memory size and accuracy are compared. Three SAW device structures are discussed; (a) a synchronous resonator on 42°YX-LiTaO3 (42-LT), (b) a double mode SAW (DMS) filter on 42-LT, and (c) a synchronous resonator on 42-LT/Si bonded wafer. It is shown that the single precision enables us to double the calculation speed and halve the memory size with negligible accuracy deterioration.

We have extended a new highly accurate numerical scheme for unstructured 2D and 3D meshes based on the discontinuous Galerkin approach to simulate seismic wave propagation in heterogeneous media containing fluid-solid interfaces. Because of the formulation of the wave equations as a unified first-order hyperbolic system in velocity stress, the fluid can be in movement along the interface. The governing equations within the moving fluid are derived from well-known first principles in fluid mechanics. The discontinuous Galerkin approach allows for jumps of the material parameters and the solution across element interfaces, which are handled by Riemann solvers or numerical fluxes. The use of Riemann solvers at the element interfaces makesthe treatment of the fluid particularly simple bysetting the shearmodulus in the fluid region to zero. No additional compatibility relations, such as vanishing shear stress or continuity of normal stresses, are necessary to couple the solid and fluid along an interface. The Riemann solver automatically recognizes the jump of the material coefficients at the interface and provides the correct numerical fluxes for fluid-solid contacts. Therefore, wave propagation in the entire computational domain containing heterogeneous media, namely moving fluids and elastic solids, can be described by a uniform set of acoustic and elastic wave equations. The accuracy of the proposed scheme is confirmed by comparing numerical results against analytic solutions. The potential of the new method was demonstrated in a 3D model problem typical for marine seismic exploration with a fluid-solid interface determined by a complicated bathymetry.

This paper addresses the direct and inverse source problems for the stochastic acoustic, biharmonic, electromagnetic, and elastic wave equations in a unified framework. The driven source is assumed to be a centered generalized microlocally isotropic Gaussian random field, whose covariance and relation operators are classical pseudodifferential operators. Given the random source, the direct problems are shown to be well-posed in the sense of distributions and the regularity of the solutions are given. For the inverse problems, we demonstrate by ergodicity that the principal symbols of the covariance and relation operators can be uniquely determined by a single realization of the far-field pattern averaged over the frequency band with probability one.

Forward seismic modeling is a useful tool for simulating the response of the low velocity zones near the ground surface and delicate structures at a deeper depth in northwestern China. The main objective of forward seismic modeling is to solve the wave equation accurately. The finite difference method (FDM) is a commonly used numerical method for solving the equation. It can obtain very high accuracy with high‐order difference scheme if the grid configuration is appropriate. However it shows numerical dispersion and its stability criteria is also very strict. In order to deal with these problems, a new precise integration method is advanced in this paper, using an integration scheme in the temporal domain rather than a traditional low‐order difference scheme. The precise integration method employs a difference approximation in the spatial domain, converting the acoustic or elastic wave equation into a set of differential equations with respect to time and then solves this equation system using an integration method. The final solution is attributed to an integration of a matrix of exponential functions. Theoretically this solution is accurate in the temporal domain, hence increasing the total accuracy. The whole spatial domain can be divided into several sub‐domains, considering computer capacity and speed and then the solutions put together. This revised sub‐domain precise integration method (SPIM) is more stable than the FDM and more accurate as is tested in the text. Besides it can greatly reduce the numerical dispersion with the same grid configuration as the FDM, though it may take a little more CPU time. A numerical test is presented to justify this method and then an application is shown to indicate that the SPIM is of potential value in the geophysical exploration field.

We extended the pseudo-Laplacian to staggered grids based on the concept of normalized pseudo-Laplacian and applied it to constructing the pseudoanalytical formulations for the variable-density acoustic wave equation and the elastic wave equation. Acoustic wavefields only contain P-waves and therefore only P-wave pseudo-Laplacians are required for acoustic wave propagation. In comparison, two sets of staggered grid pseudo-Laplacians are needed in the elastic case in order to properly compensate for time stepping errors for both P-waves and S-waves. We gave a thorough derivation of the pseudoanalytical method for the elastic wave equation, based on normalized pseudo-Laplacians implemented on staggered grids, and presented the resulting complete discretized formulas. We proved that the staggered grid pseudo-Laplacian reduces to the pseudo-Laplacian for the scalar wave equation on standard grids. When using zero compensation velocities for normalized pseudo-Laplacians, the pseudoanalytical formulas simply reduce to the pseudospectral equations. We demonstrated with numerical examples that staggered grid pseudo-Laplacians effectively compensate for second-order time stepping errors and help generate highly accurate acoustic and elastic wave solutions in variable-density media.

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term