Extrapolation Scheme Research Articles

Efficient simultaneous migration of primary and free-surface related multiples using reformulated two-way wave-equation depth extrapolation scheme

The migration of free-surface related multiples can enhance subsurface illumination and improve overall imaging quality. However, this process encounters two main challenges: crosstalk artefacts resulting from the cross-correlation of non-reflection-related wavefields, and the increased computational burden of imaging different orders of multiples. We propose a novel method that simultaneously and efficiently migrates both primary and multiple reflections while mitigating crosstalk artefacts. The method employs a reformulated two-way wave-equation depth extrapolation scheme that simplifies up/down wavefield separation through straightforward summation and subtraction operations at each depth step. Two innovative algorithms are integrated into this scheme: a generalized up/down separation algorithm, and a simultaneous migration algorithm of primary and free-surface-related multiples. The up/down separation algorithm efficiently separates the up- and down-going wavefields into primary wavefield and multiple reflections of various orders at the measurement surface. The simultaneous migration algorithm then pairs these components as two-way quantities, allowing for efficient depth extrapolation using a unified propagator, followed by effective decomposition into corresponding one-way components for imaging. Numerical experiments conducted on synthetic models, including a two-dimensional two-layer model and the Sigsbee 2B model, as well as on real seismic data from a gas hydrates bearing zone, demonstrate that the proposed method simultaneously migrate both primary and multiple reflections with reduced crosstalk artefacts and limited computational overhead.

A joint time–space extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems

A joint time–space extrapolation approach within the Wiener path integral (WPI) technique is developed for determining, efficiently and accurately, the non-stationary stochastic response of diverse nonlinear dynamical systems. The approach can be construed as an extension of a recently developed space-domain extrapolation scheme to account also for the temporal dimension. Specifically, based on a variational principle, the WPI technique yields a boundary value problem (BVP) to be solved for determining a most probable path corresponding to specific final boundary conditions. Further, the most probable path is used for evaluating, approximately, a point of the system response joint probability density function (PDF) corresponding to a specific time instant. Remarkably, the BVP exhibits two unique features that are exploited in this paper for developing an efficient joint time–space extrapolation approach. First, the BVPs corresponding to two neighboring grid points in the spatial domain of the response PDF not only share the same equations, but also the boundary conditions differ only slightly. Second, information inherent in the time-history of an already determined most probable path can be used for evaluating points of the response PDF corresponding to arbitrary time instants, without the need for solving additional BVPs. In a nutshell, relying on the aforementioned unique and advantageous features of the WPI-based BVP, the complete non-stationary response joint PDF is determined, first, by calculating numerically a relatively small number of PDF points, and second, by extrapolating in the joint time–space domain at practically zero additional computational cost. Compared to a standard brute-force implementation of the WPI technique, the developed extrapolation approach reduces the associated computational cost by several orders of magnitude. Two numerical examples relating to an oscillator with asymmetric nonlinearities and fractional derivative elements, and to a nonlinear structure under combined stochastic and deterministic periodic loading are considered for demonstrating the reliability of the extrapolation approach. Juxtapositions with pertinent Monte Carlo simulation data are included as well.

Ocean windspeeds forecast by Gaidai multivariate risk assessment method, utilizing deconvolution scheme

The current work presents spatiotemporal analysis of areal windspeeds, using state-of-the-art Gaidai multivariate hazards evaluation approach. Environmental dynamic systems being challenging to forecast using existing reliability methods due to their high dimensionality and intricate nonlinear cross-correlations between different environmental system essential dimensions or components. Developing novel multivariate reliability methods for dynamic environmental systems may assist contemporary research on the global climate change effects. Multivariate Gaidai risks evaluation approach is especially well-suited for multi-dimensional structural and environmental dynamic systems, that have been monitored physically or numerically simulated across a representative time-lapse. The current study made use of raw in-situ windspeed data, collected by NOAA (i.e., National Oceanic and Atmospheric Administration) buoys near the Hawaiian Islands in the North Pacific. Resilient island communities/infrastructure can be designed, using advocated multivariate risk/hazard assessment methodology, particularly those that are close to the ocean and hence subject to extreme weather.As was demonstrated in this study, even given limited underlying dataset, it is feasible to conservatively estimate environmental system’s hazard risks. Novel non-parametric deconvolution extrapolation scheme has been utilized to accurately assess hazard risks.

MP2-Based Composite Extrapolation Schemes Can Predict Core-Ionization Energies for First-Row Elements with Coupled-Cluster Level Accuracy.

X-ray photoelectron spectroscopy (XPS) measures core-electron binding energies (CEBEs) to reveal element-specific insights into the chemical environment and bonding. Accurate theoretical CEBE prediction aids XPS interpretation but requires proper modeling of orbital relaxation and electron correlation upon core-ionization. This work systematically investigates basis set selection for extrapolation to the complete basis set limit of CEBEs from ΔMP2 and ΔCC energies across 94 K-edges in diverse organic molecules. We demonstrate that an alternative composite scheme using ΔMP2 in a large basis corrected by ΔCC-ΔMP2 difference in a small basis can quantitatively recover optimally extrapolated ΔCC CEBEs within 0.02 eV. Unlike ΔCC, MP2 calculations do not suffer from convergence issues and are computationally cheaper, and thus, the composite ΔMP2/ΔCC scheme balances accuracy and cost, overcoming limitations of solely using either method. We conclude by providing a comprehensive analysis of the choice of small and large basis sets for the composite schemes and provide practical recommendations for highly accurate (within 0.10-0.15 eV MAE) ab initio prediction of XPS data.

A Two-Stage 2D Channel Extrapolation Scheme for TDD 5G NR Systems

A Two-Stage 2D Channel Extrapolation Scheme for TDD 5G NR Systems

Lucas decomposition and extrapolation methods for the evaluation of infinite integrals involving the product of three Bessel functions of arbitrary order

A method to efficiently evaluate integrals containing the product of three Bessel functions of the first kind and of any non negative real order is presented. The numerical problem is particularly challenging since standard integration techniques are completely unsuccessful in integrating such anomalously oscillating (and possibly slow decaying) functions. The proposed method is based on a decomposition of such a product into a sum of functions which asymptotically approach sinusoidal functions, for which integration schemes based on integration then summation procedures followed by extrapolation methods can be applied. Different extrapolation procedures are compared in order to identify the most efficient extrapolation strategy. Several numerical examples and comparisons with known analytic results are provided to show the robustness and the accuracy of the proposed approach and to help in identifying the most efficient and reliable extrapolation scheme. Particular attention is given to a class of integrals emerging in many fields of physics, in particular in quantum mechanics and particle physics, showing that the proposed method can be even more efficient (sometimes hundred of times faster) than the available analytical formulas.

Efficient Guided Wave Modelling for Tomographic Corrosion Mapping via One-Way Wavefield Extrapolation

Mapping corrosion depths along pipeline sections using guided-wave-based tomographic methods is a challenging task. Accurate defect sizing depends heavily on the precision of the forward model in guided wave tomography. This model is fitted to measured data using inversion techniques. This study evaluates the effectiveness of a recursive extrapolation scheme for tomography applications and full waveform inversion. It employs a table-driven approach, with precomputed extrapolation operators stored across a spectrum of wavenumbers. This enables fast modelling for extensive pipe sections, approaching the speed of ray tracing while accurately handling complex velocity models within the full frequency band. This ensures an accurate representation of diffraction phenomena. The study examines the assumptions underlying the extrapolation approach, namely, the negligible reflection and conversion of modes at defects. In our tomography approach, we intend to use multiple wave modes—A0, S0, and SH1—and helical paths. The acoustic extrapolation method is validated through numerical studies for different wave modes, solving the 3D elastodynamic wave equation. Comparison with an experimentally measured single-mode wavefield from an aluminium plate with an artificial defect reveals good agreement.

Building a New Platform for Significantly Improving Performance of Hartree-Fock and CCSD(T) Correlation Energy Based on Two-Point Complete Basis Set Extrapolation Schemes.

The leading cause of high expense in gold standard coupled cluster theory is that calculations of electronic energies converge exceedingly slowly with an increased basis set size. Extrapolation principally allows for achieving higher-quality outcomes at reduced costs. Numerous extrapolation formulas have been developed, with attempts to predict energies up to the complete basis set limit. Unfortunately, since the intricate shape of the function hinges on the molecular properties with the highest angular momentum of the basis set, the accuracy of the extrapolated energies highly depends on the fitted empirical parameters, which rely on the quality of the data sets for fitting. In this work, to overcome the extrapolation deficiency caused by the very limited data sets and smaller basis sets in the early stages, we constructed a new benchmark platform that includes a broader data set of 183 species (containing open-shell, closed-shell, ionic, and neutral species) and a larger basis set up to aug-cc-pV6Z. The newly optimized parameters can significantly improve the energy-predictive abilities of ten published formulas. Notably, all ten formulas perform quite similarly under the new platform with the reoptimized parameters. Finally, we built an online calculator for researchers to use for these extrapolation schemes. Our work would reignite the interest and applications of the underestimated formulas.

A Crank–Nicolson leap-frog scheme for the unsteady incompressible magnetohydrodynamics equations

This paper presents a Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style’s scheme, the second step employs the Crank–Nicolson extrapolation scheme, and others apply the CNLF scheme. We establish that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. Firstly, we show the stability of the scheme by means of the mathematical induction method. Next, we focus on analyzing error estimates of the CNLF method, where the convergence order of the velocity and magnetic field reach second-order accuracy, and the pressure is the first-order convergence accuracy. Finally, the numerical examples demonstrate the optimal error estimates of the proposed algorithm.

Improved CPS and CBS Extrapolation of PNO-CCSD(T) Energies: The MOBH35 and ISOL24 Data Sets.

Computation of heats of reaction of large molecules is now feasible using the domain-based pair natural orbital (PNO)-coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] theory. However, to obtain agreement within 1 kcal/mol of experiment, it is necessary to eliminate basis set incompleteness error, which comprises both the AO basis set error and the PNO truncation error. Our investigation into the convergence to the canonical limit of PNO-CCSD(T) energies with the PNO truncation threshold shows that errors follow the model . Therefore, PNO truncation errors can be eliminated using a simple two-point CPS extrapolation to the canonical limit so that subsequent CBS extrapolation is not limited by the residual PNO truncation error. Using the ISOL24 and MOBH35 data sets, we find that PNO truncation errors are larger for molecules with significant static correlation and that it is necessary to use very tight thresholds of to ensure that errors do not exceed 1 kcal/mol. We present a lower-cost extrapolation scheme that uses information from small basis sets to estimate the PNO truncation errors for larger basis sets. In this way, the canonical limit of CCSD(T) calculations on sizable molecules with large basis sets can be reliably estimated in a practical way. Using this approach, we report near complete basis set (CBS)-CCSD(T) reaction energies for the full ISOL24 and MOBH35 data sets.

Higher-order galerkin finite element method for nonlinear coupled reaction-diffusion models

In this article, generalized nonlinear coupled reaction-diffusion (NCRD) models are analyzed using the higher-order Galerkin finite element method (FEM). The authors present finite element analysis for one- and two-dimensional NCRD models by applying quadratic Lagrange basis functions. Optimal order convergence in space is proved by applying quadratic FEM in spatial discretization and utilizing the Crank-Nicolson (CN) scheme for time discretization. To address the nonlinearity of NCRD models, CN extrapolation and predictorcorrector schemes are employed. The stability analysis for generalized NCRD models and the applied CN scheme are discussed by utilizing the eigenvalues method. Finally, to assess the accuracy and efficiency of the proposed scheme and to capture different patterns of NCRD models, various one- and two-dimensional models are considered. The obtained results are numerically and graphically compared with existing literature. It is observed that quadratic basis functions in FEM improve accuracy and efficiency of the results.

Adiabatic ionization energies of RuC, RhC, OsC, IrC, and PtC.

The ionization energies (IEs) of RuC, RhC, OsC, IrC, and PtC are assigned by the measurement of their two-photon ionization thresholds. Although late transition metal-carbon bonds are of major importance in organometallic chemistry and catalysis, accurate and precise fundamental thermochemical data on these chemical bonds are mainly lacking in the literature. Based on their two-photon ionization thresholds, in this work, we assign IE(RuC) = 7.439(40) eV, IE(RhC) = 7.458(32) eV, IE(OsC) = 8.647(25) eV, IE(IrC) = 8.933(74) eV, and IE(PtC) = 9.397(32) eV. These experimentally derived IEs are further confirmed through quantum chemical calculations using coupled-cluster single double perturbative triple methods that are extrapolated to the complete basis set limit using a three-parameter mixed Gaussian/exponential extrapolation scheme and corrected for spin-orbit effects using a semiempirical method. The electronic structure and chemical bonding of these MC species are discussed in the context of these ionization energy measurements. The IEs of RuC, RhC, OsC, and IrC closely mirror the IEs of the corresponding transition metal atoms, suggesting that for these species, the (n + 1)s electrons of the transition metals are not significantly involved in chemical bonding.

Massively parallel implementation of gradients within the random phase approximation: Application to the polymorphs of benzene.

The Random-Phase approximation (RPA) provides an appealing framework for semi-local density functional theory. In its Resolution-of-the-Identity (RI) approach, it is a very accurate and more cost-effective method than most other wavefunction-based correlation methods. For widespread applications, efficient implementations of nuclear gradients for structure optimizations and data sampling of machine learning approaches are required. We report a well scaling implementation of RI-RPA nuclear gradients on massively parallel computers. The approach is applied to two polymorphs of the benzene crystal obtaining very good cohesive and relative energies. Different correction and extrapolation schemes are investigated for further improvement of the results and estimations of error bars.

Developing correlation-consistent numeric atom-centered orbital basis sets for krypton: Applications in RPA-based correlated calculations.

Localized atomic orbitals are the preferred basis set choice for large-scale explicit correlated calculations, and high-quality hierarchical correlation-consistent basis sets are a prerequisite for correlated methods to deliver numerically reliable results. At present, numeric atom-centered orbital (NAO) basis sets with valence correlation consistency (VCC), designated as NAO-VCC-nZ, are only available for light elements from hydrogen (H) to argon (Ar) [Zhang et al., New J. Phys. 15, 123033 (2013)]. In this work, we extend this series by developing NAO-VCC-nZ basis sets for krypton (Kr), a prototypical element in the fourth row of the periodic table. We demonstrate that NAO-VCC-nZ basis sets facilitate the convergence of electronic total-energy calculations using the Random Phase Approximation (RPA), which can be used together with a two-point extrapolation scheme to approach the complete basis set limit. Notably, the Basis Set Superposition Error (BSSE) associated with the newly generated NAO basis sets is minimal, making them suitable for applications where BSSE correction is either cumbersome or impractical to do. After confirming the reliability of NAO basis sets for Kr, we proceed to calculate the Helmholtz free energy for Kr crystal at the theoretical level of RPA plus renormalized single excitation correction. From this, we derive the pressure-volume (P-V) diagram, which shows excellent agreement with the latest experimental data. Our work demonstrates the capability of correlation-consistent NAO basis sets for heavy elements, paving the way toward numerically reliable correlated calculations for bulk materials.

Deep Learning–Based Channel Extrapolation and Multiuser Beamforming for RIS-aided Terahertz Massive MIMO Systems over Hybrid-Field Channels

The reconfigurable intelligent surface (RIS) is a promising technology for terahertz (THz) massive multiple-input multiple-output (MIMO) communication systems. However, acquiring high-dimensional channel state information (CSI) and realizing efficient active/passive beamforming for RIS are challenging owing to its cascaded channel structure and lack of signal processing units. To overcome these challenges, this study proposes a deep learning (DL)-based physical signal processing scheme for RIS-aided THz massive MIMO systems over hybrid far-near field channels wherein channel estimation with low pilot overhead and robust beamforming are implemented. Specifically, first, an end-to-end DL-based channel estimation framework that consists of pilot design, CSI feedback, subchannel estimation, and channel extrapolation is introduced. In this framework, only some RIS elements are first activated, a subsampling RIS channel is then estimated, and a DL-based extrapolation network is finally used to reconstruct the full-dimensional CSI. Next, to maximize the sum rate under imperfect CSI, a DL-based scheme is developed to simultaneously design hybrid active beamforming at the base station and passive beamforming at the RIS. Simulation results show that the proposed channel extrapolation scheme achieves better CSI reconstruction performance than conventional schemes while greatly reducing pilot overhead. Moreover, the proposed beamforming scheme outperforms conventional schemes in terms of robustness to imperfect CSI.

Q-compensated wavefield depth extrapolation-based migration using a viscoacoustic wave equation

In a viscoacoustic medium, intrinsic attenuation causes seismic wavefields to attenuate in amplitude and become dispersed in phase, leading to distorted structural imaging and inaccurate migrated amplitudes. To address this problem, viscoacoustic reverse time migration corrects for dispersion and amplitude attenuation effects in wavefield propagation in forward and backward directions. To provide a parallel alternative approach, the time fractional derivative viscoacoustic wave equation in the depth domain is solved and a Q-compensated wavefield depth extrapolation scheme is established to compensate for viscoacoustic effects during recursive wavefield depth extrapolation. This approach decouples the amplitude attenuation and phase dispersion effects from the viscoacoustic vertical wavenumber. To suppress high wavenumber components and address the problem of exponential amplitude growth during wavefield depth extrapolation, we limit the imaginary part of the vertical wavenumber in the frequency wavenumber domain and use an adaptive stabilization solution. Numerical experiments of impulse responses in a viscoacoustic isotropic medium demonstrate that our proposed scheme can observe calculated wavefields exhibiting amplitude attenuation and phase dispersion effects in comparison to wavefields of the acoustic medium, indicating good agreement with theoretical expectations. We also demonstrate the capability of our proposed scheme to recover imaging amplitudes through a variety of numerical experiments, such as imaging of three-layer, Marmousi, and BP gas reservoir models. The results indicate that our proposed scheme can recover amplitude attenuation and phase dispersion effects more accurately than acoustic migration. We also use marine seismic data featuring natural gas hydrates to indicate that our proposed Q-compensated scheme can generate an enhanced imaging result, especially for the bottom simulating reflectors, compared with the conventional imaging algorithm without Q-compensation.

Plane Waves Versus Correlation-Consistent Basis Sets: A Comparison of MP2 Non-Covalent Interaction Energies in the Complete Basis Set Limit.

Second-order Møller-Plesset perturbation theory (MP2) is the most expedient wave function-based method for considering electron correlation in quantum chemical calculations and, as such, provides a cost-effective framework to assess the effects of basis sets on correlation energies, for which the complete basis set (CBS) limit can commonly only be obtained via extrapolation techniques. Software packages providing MP2 energies are commonly based on atom-centered bases with innate issues related to possible basis set superposition errors (BSSE), especially in the case of weakly bonded systems. Here, we present noncovalent interaction energies in the CBS limit, free of BSSE, for 20 dimer systems of the S22 data set obtained via a highly parallelized MP2 implementation in the plane-wave pseudopotential molecular dynamics package CPMD. The specificities related to plane waves for accurate and efficient calculations of gas-phase energies are discussed, and results are compared to the localized (aug-)cc-pV[D,T,Q,5]Z correlation-consistent bases as well as their extrapolated CBS estimates. We find that the BSSE-corrected aug-cc-pV5Z basis can provide MP2 energies highly consistent with the CBS plane wave values with a minimum mean absolute deviation of ∼0.05 kcal/mol without the application of any extrapolation scheme. In addition, we tested the performance of 13 different extrapolation schemes and found that the X-3 expression applied to the (aug-)cc-pVXZ bases provides the smallest deviations against CBS plane wave values if the extrapolation sequence is composed of points D and T, while performs slightly better for TQ and Q5 extrapolations. Also, we propose as a reliable alternative to extrapolate total energies from the DTQ, TQ5, or DTQ5 data points. In spite of the general good agreement between the values obtained from the two types of basis sets, it is noticed that differences between plane waves and (aug-)cc-pVXZ basis sets, extrapolated or not, tend to increase with the number of electrons, thus raising the question of whether these discrepancies could indeed limit the attainable accuracy for localized bases in the limit of large systems.

ROUGH-HESTON LOCAL-VOLATILITY MODEL

In industrial applications it is quite common to use stochastic-volatility models driven by semi-martingale Markov volatility processes. However, in order to fit exactly market volatilities, these models are usually extended by adding a local-volatility term. Here, we consider the case of singular Volterra processes, and we extend them by adding a local-volatility term to their Markov lift by preserving the stylized results implied by these models on plain-vanilla options. In particular, we focus on the rough-Heston model, and we analyze the small-time asymptotics of its implied local-volatility function in order to provide a proper extrapolation scheme to be used in calibration.

Optical Properties of Neutral F Centers in Bulk MgO with Density Matrix Embedding.

The optical spectra of neutral oxygen vacancies (F0 centers) in the bulk MgO lattice are investigated using density matrix embedding theory. The impurity Hamiltonian is solved with the complete active space self-consistent field and second-order n-electron valence state perturbation theory (NEVPT2-DMET) multireference methods. To estimate defect-localized vertical excitation energies at the nonembedding and thermodynamic limits, a double extrapolation scheme is employed. The extrapolated NEVPT2-DMET vertical excitation energy value of 5.24 eV agrees well with the experimental absorption maxima at 5.03 eV, whereas the excitation energy value of 2.89 eV at the relaxed triplet defect-localized state geometry overestimates the experimental emission at 2.4 eV by only nearly 0.5 eV, indicating the involvement of the triplet-singlet decay pathway.

Basis Set Effects on the Stabilities and Interaction Energies of Small Amide Molecules Adsorbed on Kaolinite Surface

Adsorptions of small amide molecules, acetamide (AA) and N-methyl-acetamide (NMA) on the surface of kaolinite are investigated in this study. The focus is on the basis set effects towards the stabilities and the interaction energies of the molecules on the Al–O surface. With a fixed B3LYP functional, we increased the size of the basis sets for the single-point calculations, to find the converged interaction energies and obtain the relative stabilities. We found that, under the direct usage of Pople-type and Dunning’s correlation consistent basis sets, it is not possible to achieve the pattern of convergence for the interaction energies and the relative stabilities. Compared to the complete basis set (CBS) extrapolation scheme, the double zeta basis sets deviated the most, in the range of 21 to 27%, while it is from 1 to 7% for the triple zeta basis sets. Based on the results, we suggest using 6-311++G(2df,2pd) or cc-pVQZ for energy-related quantities. Compared to AA, NMA attached more strongly by 0.5 eV on the surface of Al–O.