Accurate simulations of pure-viscoacoustic wave propagation in tilted transversely isotropic media

Forward modeling of seismic wave propagation is crucial for the realization of reverse time migration(RTM) and full waveform inversion(FWI) in attenuating transversely isotropic media. To describe the attenuation and anisotropy properties of subsurface media, the pure-viscoacoustic anisotropic wave equations are established for wavefield simulations, because they can provide clear and stable wavefields. However, due to the use of several approximations in deriving the wave equation and the introduction of a fractional Laplacian approximation in solving the derived equation, the wavefields simulated by the previous pure-viscoacoustic tilted transversely isotropic(TTI) wave equations has low accuracy. To accurately simulate wavefields in media with velocity anisotropy and attenuation anisotropy, we first derive a new pure-viscoacoustic TTI wave equation from the exact complex-valued dispersion formula in viscoelastic vertical transversely isotropic(VTI) media. Then, we present the hybrid finite-difference and low-rank decomposition(HFDLRD) method to accurately solve our proposed pure-viscoacoustic TTI wave equation. Theoretical analysis and numerical examples suggest that our pure-viscoacoustic TTI wave equation has higher accuracy than previous pure-viscoacoustic TTI wave equations in describing q P-wave kinematic and attenuation characteristics. Additionally, the numerical experiment in a simple two-layer model shows that the HFDLRD technique outperforms the hybrid finite-difference and pseudo-spectral(HFDPS) method in terms of accuracy of wavefield modeling.

Surrogate modeling for unsaturated infiltration via the physics and equality-constrained artificial neural networks

Machine learning(ML)provides a new surrogate method for investigating groundwater flow dynamics in unsaturated soils.Traditional pure data-driven methods(e.g.deep neural network,DNN)can provide rapid predictions,but they do require sufficient on-site data for accurate training,and lack interpretability to the physical processes within the data.In this paper,we provide a physics and equalityconstrained artificial neural network(PECANN),to derive unsaturated infiltration solutions with a small amount of initial and boundary data.PECANN takes the physics-informed neural network(PINN)as a foundation,encodes the unsaturated infiltration physical laws(i.e.Richards equation,RE)into the loss function,and uses the augmented Lagrangian method to constrain the learning process of the solutions of RE by adding stronger penalty for the initial and boundary conditions.Four unsaturated infiltration cases are designed to test the training performance of PECANN,i.e.one-dimensional(1D)steady-state unsaturated infiltration,1D transient-state infiltration,two-dimensional(2D)transient-state infiltration,and 1D coupled unsaturated infiltration and deformation.The predicted results of PECANN are compared with the finite difference solutions or analytical solutions.The results indicate that PECANN can accurately capture the variations of pressure head during the unsaturated infiltration,and present higher precision and robustness than DNN and PINN.It is also revealed that PECANN can achieve the same accuracy as the finite difference method with fewer initial and boundary training data.Additionally,we investigate the effect of the hyperparameters of PECANN on solving RE problem.PECANN provides an effective tool for simulating unsaturated infiltration.

Saturation Estimation with Complex Electrical Conductivity for Hydrate-Bearing Clayey Sediments:An Experimental Study

Clays have considerable influence on the electrical properties of hydrate-bearing sediments.It is desirable to understand the electrical properties of hydrate-bearing clayey sediments and to build hydrate saturation(S_(h))models for reservoir evaluation and monitoring.The electrical properties of tetrahydrofuran-hydrate-bearing sediments with montmorillonite are characterized by complex conductivity at frequencies from 0.01 Hz to 1 kHz.The effects of clay and Sh on the complex conductivity were analyzed.A decrease and increase in electrical conductance result from the clay-swelling-induced blockage and ion migration in the electrical double layer(EDL),respectively.The quadrature conductivity increases with the clay content up to 10%because of the increased surface site density of counterions in EDL.Both the in-phase conductivity and quadrature conductivity decrease consistently with increasing Sh from 0.50 to 0.90.Three sets of models for Sh evaluation were developed.The model based on the Simandoux equation outperforms Archie’s formula,with a root-mean-square error(E_(RMS))of 1.8%and 3.9%,respectively,highlighting the clay effects on the in-phase conductivity.The fre-quency effect correlations based on in-phase and quadrature conductivities exhibit inferior performance(E_(RMS)=11.6%and 13.2%,re-spectively)due to the challenge of choosing an appropriate pair of frequencies and intrinsic uncertainties from two measurements.The second-order Cole-Cole formula can be used to fit the complex-conductivity spectra.One pair of inverted Cole-Cole parameters,i.e.,characteristic time and chargeability,is employed to predict S_(h) with an E_(RMS) of 5.05%and 9.05%,respectively.

STABILITY OF THE RAREFACTION WAVE IN THE SINGULAR LIMIT OF A SHARP INTERFACE PROBLEM FOR THE COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM

This paper is concerned with the global well-posedness of the solution to the compressible Navier-Stokes/Allen-Cahn system and its sharp interface limit in one-dimensional space.For the perturbations with small energy but possibly large oscillations of rarefaction wave solutions near phase separation,and where the strength of the initial phase field could be arbitrarily large,we prove that the solution of the Cauchy problem exists for all time,and converges to the centered rarefaction wave solution of the corresponding standard two-phase Euler equation as the viscosity and the thickness of the interface tend to zero.The proof is mainly based on a scaling argument and a basic energy method.

Some Modified Equations of the Sine-Hilbert Type

Three modified sine-Hilbert(sH)-type equations, i.e., the modified sH equation, the modified damped sH equation, and the modified nonlinear dissipative system, are proposed, and their bilinear forms are provided.Based on these bilinear equations, some exact solutions to the three modified equations are derived.

Viscoacoustic generalized screen propagator in constant-Q model

When seismic waves propagate through the geological formation,there is a significant loss of energy and a decrease in imaging resolution,because of the viscoacoustic properties of subsurface medium.This profoundly impacts seismic wavefield propagation,imaging and interpretation.To accurately image the true structure of subsurface medium,the consensus among geophysicists is to no longer treat subsurface medium as ideal homogeneous medium,but rather to incorporate the viscoacoustic properties of subsurface medium.Based on the generalized screen propagator using conventional acoustic wave equation(acoustic GSP),our developed method introduces viscoacoustic compensation strategy,and derives a one-way wave generalized screen propagator based on time-fractional viscoacoustic wave equation(viscoacoustic GSP).In numerical experiments,we conducted tests on two-dimensional multi-layer model and the Marmousi model.When comparing with the acoustic GSP using the acoustic data,we found that the imaging results of the viscoacoustic GSP using the viscoacoustic data showed a significant attenuation compensation effect,and achieved imaging results for both algorithms were essentially consistent.However,the imaging results of acoustic GSP using viscoacoustic data showed significant attenuation effects,especially for deep subsurface imaging.This indicates that we have proposed an effective method to compensate the attenuated seismic wavefield.Our application on a set of real seismic data demonstrated that the imaging performance of our proposed method in local areas surpassed that of the conventional acoustic GSP.This suggests that our proposed method holds practical value and can more accurately image real subsurface structures while enhancing imaging resolution compared with the conventional acoustic GSP.Finally,with respect to computational efficiency,we gathered statistics on running time to compare our proposed method with conventional Q-RTM,and it is evident that our method exhibits higher computational efficiency.In summary,our proposed viscoacoustic GSP method takes into account the true properties of the medium,still achieves migration results comparable to conventional acoustic GSP.

Analytical solutions fractional order partial differential equations arising in fluid dynamics

This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB.

Travelling wave solutions of nonlinear conformable analytical approaches

The presented study deals with the investigation of nonlinear Bogoyavlenskii equations with conformable time-derivative which has great importance in plasma physics and non-inspectoral scattering problems.Travelling wave solutions of this nonlinear conformable model are constructed by utilizing two powerful analytical approaches,namely,the modified auxiliary equation method and the Sardar sub-equation method.Many novel soliton solutions are extracted using these methods.Furthermore,3D surface graphs,contour plots and parametric graphs are drawn to show dynamical behavior of some obtained solutions with the aid of symbolic software such as Mathematica.The constructed solutions will help to understand the dynamical framework of nonlinear Bogoyavlenskii equations in the related physical phenomena.

THE STABILITY OF BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION AROUND THE HYDROSTATIC BALANCE

This paper is devoted to understanding the stability of perturbations around the hydrostatic equilibrium of the Boussinesq system in order to gain insight into certain atmospheric and oceanographic phenomena.The Boussinesq system focused on here is anisotropic,and involves only horizontal dissipation and thermal damping.In the 2D case R^(2),due to the lack of vertical dissipation,the stability and large-time behavior problems have remained open in a Sobolev setting.For the spatial domain T×R,this paper solves the stability problem and gives the precise large-time behavior of the perturbation.By decomposing the velocity u and temperatureθinto the horizontal average(ū,θ)and the corresponding oscillation(ū,θ),we can derive the global stability in H~2 and the exponential decay of(ū,θ)to zero in H^(1).Moreover,we also obtain that(ū_(2),θ)decays exponentially to zero in H^(1),and thatū_(1)decays exponentially toū_(1)(∞)in H^(1)as well;this reflects a strongly stratified phenomenon of buoyancy-driven fluids.In addition,we establish the global stability in H^(3)for the 3D case R^(3).

INTERFACE BEHAVIOR AND DECAY RATES OF COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND A VACUUM

In this paper,we study the one-dimensional motion of viscous gas near a vacuum,with the gas connecting to a vacuum state with a jump in density.The interface behavior,the pointwise decay rates of the density function and the expanding rates of the interface are obtained with the viscosity coefficientμ(ρ)=ρ^(α)for any 0<α<1;this includes the timeweighted boundedness from below and above.The smoothness of the solution is discussed.Moreover,we construct a class of self-similar classical solutions which exhibit some interesting properties,such as optimal estimates.The present paper extends the results in[Luo T,Xin Z P,Yang T.SIAM J Math Anal,2000,31(6):1175-1191]to the jump boundary conditions case with density-dependent viscosity.

Finite element model updating for structural damage detection using transmissibility data

This paper presents a new finite element model updating method for estimating structural parameters and detecting structural damage location and severity based on the structural responses(output-only data).The method uses the sensitivity relation of transmissibility data through a least-squares algorithm and appropriate normalization of the extracted equations.The proposed transmissibility-based sensitivity equation produces a more significant number of equations than the sensitivity equations based on the frequency response function(FRF),which can estimate the structural parameters with higher accuracy.The abilities of the proposed method are assessed by using numerical data of a two-story two-bay frame model and a plate structure model.In evaluating different damage cases,the number,location,and stiffness reduction of the damaged elements and the severity of the simulated damage have been accurately identified.The reliability and stability of the presented method against measurement and modeling errors are examined using error-contaminated data.The parameter estimation results prove the method’s capabilities as an accurate model updating algorithm.

Data-Driven Ai-and Bi-Soliton of the Cylindrical Korteweg-de Vries Equation via Prior-Information Physics-Informed Neural Networks

By the modifying loss function MSE and training area of physics-informed neural networks(PINNs),we propose a neural networks model,namely prior-information PINNs(PIPINNs).We demonstrate the advantages of PIPINNs by simulating Ai-and Bi-soliton solutions of the cylindrical Korteweg-de Vries(cKdV)equation.

THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS

This paper deals with the problem of limit cycles for the whirling pendulum equation x=y,y=sin x(cosx-r)under piecewise smooth perturbations of polynomials of cos x,sin x and y of degree n with the switching line x=0.The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations,which the generating functions of the associated first order Melnikov functions satisfy.Furthermore,the exact bound of a special case is given using the Chebyshev system.At the end,some numerical simulations are given to illustrate the existence of limit cycles.

THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARDPOTENTIAL

In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary L^(2) weighted estimates.

Rebuilding the theory of isotope fractionation for evaporation of silicate melts under vacuum condition

Isotope eff ects are pivotal in understanding silicate melt evaporation and planetary accretion processes.Based on the Hertz-Knudsen equation,the current theory often fails to predict observed isotope fractionations of laboratory experiments due to its oversimplified assumptions.Here,we point out that the Hertz-Knudsen-equation-based theory is incomplete for silicate melt evaporation cases and can only be used for situations where the vaporized species is identical to the one in the melt.We propose a new model designed for silicate melt evaporation under vacuum.Our model considers multiple steps including mass transfer,chemical reaction,and nucleation.Our derivations reveal a kinetic isotopic fractionation factor(KIFF orα)αour model=[m(^(1)species)/m(^(2)species)]^(0.5),where m(species)is the mass of the reactant of reaction/nucleation-limiting step or species of diffusion-limiting step and superscript 1 and 2 represent light and heavy isotopes,respectively.This model can eff ectively reproduce most reported KIFFs of laboratory experiments for various elements,i.e.,Mg,Si,K,Rb,Fe,Ca,and Ti.And,the KIFF-mixing model referring that an overall rate of evaporation can be determined by two steps jointly can account for the eff ects of low P_(H2)pressure,composition,and temperature.In addition,we find that chemical reactions,diffusion,and nucleation can control the overall rate of evaporation of silicate melts by using the fitting slope in ln(−ln f)versus ln(t).Notably,our model allows for the theoretical calculations of parameters like activation energy(E_(a)),providing a novel approach to studying compositional and environmental eff ects on evaporation processes,and shedding light on the formation and evolution of the proto-solar and Earth-Moon systems.

On entire solutions of some Fermat type differential-difference equations

On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].

Dynamical distribution of continuous service time model involving non-Maxwellian collision kernel and value functions

The distribution of continuous service time in call centers is investigated.A non-Maxwellian collision kernel combining two different value functions in the interaction rule are used to describe the evolution of continuous service time,respectively.Using the statistical mechanical and asymptotic limit methods,Fokker–Planck equations are derived from the corresponding Boltzmann-type equations with non-Maxwellian collision kernels.The steady-state solutions of the Fokker–Planck equation are obtained in exact form.Numerical experiments are provided to support our results under different parameters.

Multi-soliton solutions, breather-like and bound-state solitons for complex modified Korteweg–de Vries equation in optical fibers

Under investigation in this paper is a complex modified Korteweg–de Vries(KdV) equation, which describes the propagation of short pulses in optical fibers. Bilinear forms and multi-soliton solutions are obtained through the Hirota method and symbolic computation. Breather-like and bound-state solitons are constructed in which the signs of the imaginary parts of the complex wave numbers and the initial separations of the two parallel solitons are important factors for the interaction patterns. The periodic structures and position-induced phase shift of some solutions are introduced.

Wave interaction for a generalized higher-dimensional Boussinesq equation describing the nonlinear Rossby waves

Based on an algebraically Rossby solitary waves evolution model,namely an extended(2+1)-dimensional Boussinesq equation,we firstly introduced a special transformation and utilized the Hirota method,which enable us to obtain multi-complexiton solutions and explore the interaction among the solutions.These wave functions are then employed to infer the influence of background flow on the propagation of Rossby waves,as well as the characteristics of propagation in multi-wave running processes.Additionally,we generated stereogram drawings and projection figures to visually represent these solutions.The dynamical behavior of these solutions is thoroughly examined through analytical and graphical analyses.Furthermore,we investigated the influence of the generalized beta effect and the Coriolis parameter on the evolution of Rossby waves.

A new method for deriving broad-band polar motion geodetic excitations

While the geodetic excitationχ(t)of polar motion p(t)is essential to improve our understanding of global mass redistributions and relative motions with respect to the terrestrial frame,the widely adopted method to deriveχ(t)from p(t)has biases in both amplitude and phase responses.This study has developed a new simple but more accurate method based on the combination of the frequency-and time-domain Liouville's equation(FTLE).The FTLE method has been validated not only with 6-h sampled synthetic excitation series but also with daily and 6-h sampled polar motion measurements as well asχ(t)produced by the interactive webpage tool of the International Earth Rotation and Reference Systems Service(IERS).Numerical comparisons demonstrate thatχ(t)derived from the FTLE method has superior performances in both the time and frequency domains with respect to that obtained from the widely adopted method or the IERS webpage tool,provided that the input p(t)series has a length around or more than 25 years,which presents no practical limitations since the necessary polar motion data are readily available.The FTLE code is provided in the form of Mat Lab function.