Irregular Free Surface Research Articles

Model parameter design for modeling surface topography in VTI elastic finite-difference near-surface simulations

The near-surface anisotropy is one of the factors leading to an inaccurate velocity estimation in the shallow area, which has a strong impact on the static correction of land data and seismic imaging of the subsurface in depth. To investigate the anisotropy influence, the consideration of medium anisotropy in near-surface seismic modeling becomes necessary. In the oil and gas industry, finite-difference (FD) methods are commonly used for seismic wave modeling, thanks to their simplicity, accuracy, and high efficiency. However, for near-surface modeling, the implementation of free-surface boundary conditions and nonflat topography representation are the two main obstacles preventing the effectiveness of FD methods. To address these two issues in the vertical transversely isotropic anisotropic scenarios, we have developed a simple and efficient method for the discrete model design which can be easily applied to the conventional Cartesian-grid FD modeling. Our method involves: (1) the parameter-averaging method for implicitly implementing the stress-free condition by a modification of model anisotropy parameters near the (non) flat free-surface boundary and (2) an independent wavefield superposition with modeling results of different parameter configurations to accurately represent the rugged topography and significantly reduce the staircase diffractions caused by a staircase approximation of continuous surface in the Cartesian-grid discretization. For validation of this method, we conduct several numerical tests in 2D and 3D spaces. The accuracy is demonstrated by a comparison of the spectral-element solutions of SPECFEM in modeling the seismic wave propagation in the presence of an irregular free surface. From the aspect of computational efficiency, it is more promising in practical applications due to the use of wavefield superposition strategy in this method which does not require finer spatial sampling to eliminate the staircase diffractions.

Waveform inversion of seismic first arrivals acquired on irregular surface

The irregular topography of the earth’s surface and low signal-to-noise ratios of land seismic data bring challenges to full-waveform inversion (FWI). We propose a robust method for waveform inversion (WI) to invert such land seismic data. The inversion uses finite-difference methods with rectangular meshes to simulate seismic wavefields efficiently. To accurately model irregular free surface topography, we use an improved immersed boundary method with an iterative symmetric interpolation. First-arrival signals including direct waves and refraction waves are used to estimate the P-wave velocity. To overcome the cycle skipping and dynamic inconsistency issues between the modeled data and the observed data, we create an intermediate data set by shifting the first arrivals of the predicted data toward that of the observed data within half a cycle. The intermediate data instead of the observed data are then inverted. Thus, the inversion essentially matches the traveltime information of first arrivals, which is the most reliable information contained in seismic data. Applications on the synthetic and field data sets demonstrate that the proposed WI algorithm is robust for recovering P-wave velocity from land seismic data. The resulting models have higher resolution and deeper support than that of ray-based traveltime tomography.

Topography-dependent eikonal tomography based on the fast-sweeping scheme and the adjoint-state technique

First-arrival traveltime tomography has been widely used for upper crustal velocity modeling, but it usually suffers from the problem of complex surface topography. To overcome this problem, we have developed a new topography-dependent eikonal tomography scheme that combines a developed accurate and efficient traveltime modeling method and introduces a flexible and robust adjoint inversion scheme in the presence of irregular topography. A surface-flattening scheme is used to handle the irregular surface, where the real model is discretized by curvilinear grids and the irregular free surface is mathematically flattened through the transformation from Cartesian to curvilinear coordinates. Based on this parameterization, the forward traveltime modeling is conducted by a monotone fast-sweeping method that discretizes the factored topography-dependent eikonal equation with a point-source condition. This algorithm can circumvent the source-singularity problem and decrease the numerical error in the vicinity of a point source in the curvilinear system. Then, the gradient-based inversion is used to minimize the misfit function, which is achieved by a matrix-free adjoint-state method without cumbersome ray tracing and explicit estimation of the Fréchet derivative matrix in the curvilinear coordinate system. The new tomographic scheme is evaluated through numerical examples with different seismic structures with complex topography, and then applied to a wide-angle profile acquired in the northeastern Tibetan Plateau. The results validate the effectiveness and efficiency of our tomography scheme in constructing shallow crustal velocity models with irregular topography.

Numerical model of energy dissipation and shallow-water sloshing motions in tank under different coupled excitations

Shallow-water sloshing motions in a three-dimensional rectangular tank are investigated. The Boussinesq-type equations in terms of velocity potential and the finite-difference scheme are applied for the solutions of numerical model. Through linking the rate of decay of the wave amplitudes to the energy dissipation due to the friction at the tank walls, a linear damping term is proposed and added into the free surface boundary condition. Taking the tank under excited frequencies near the lowest natural frequency, the maximum transient wave amplitudes and steady-state wave amplitudes of sloshing motions at the tank wall are presented and verified by the experimental results given in the literature. The characteristics of sloshing motions in tank under different coupled excitations are studied. The results indicate that coupled surge-sway excitations lead to the weaker nonlinear sloshing motions in tank than the single degree of freedom excitations. The intersection of sloshing wave crest lines finally tend to the diagonal line of the tank under the coupled surge-sway excitations with different amplitudes. And the irregular free surface oscillations appear at the corners of the tank excited by the coupled surge-sway-roll-pitch-yaw harmonic motions.

A new generalized stiffness reduction method for 2D/2.5D frequency-domain seismic wave modeling in viscoelastic anisotropic media

In frequency-domain seismic wave modeling, absorbing artificial reflections is crucial to obtain accurate numerical solutions. We have determined that, in viscoelastic anisotropic media (VEAM), the most popular absorbing boundary techniques, such as the perfectly matched layer and the generalized stiffness reduction method (GSRM), fail. Then, we develop a new version of the GSRM and incorporate it into a 2D/2.5D spectral element method. We find with extensive nontrivial numerical experiments that the new GSRM exhibits excellent features of simple and efficient implementation, while handling free-surface and subsurface interface topography. Furthermore, we find that sampling the positive wavenumber range is an efficient strategy to compute the 3D wavefield in arbitrary 2D VEAM, and the new version takes full advantage of the symmetry/antisymmetry of the wavefield. The new GSRM removes artificial reflections by damping the real and imaginary viscoelastic moduli in different ways. The wavefields in two vertically transverse isotropic and one orthorhombic viscoelastic homogeneous models are compared with the corresponding analytical solutions to show the high accuracy performance of the new GSRM. Finally, a complex 2D geologic model with irregular free-surface and subinterface is considered to present the modeling technique and its adaptation capacity for complex 2D VEAM.

Frequency-domain elastic-wave modeling for polygonal topography using rotated average-derivative difference operators

Modeling of seismic wave propagation in areas with irregular topography is an important topic in the field of seismic exploration. As a popular numerical method for seismic modeling, the finite difference method is nontrivial to consider the irregular free surface. There have been extensive studies on the time-domain finite difference simulations with irregular topography; however, the frequency-domain finite difference simulation considering irregular topography is relatively less studied. The average-derivative approach is an optimal numerical simulation scheme in the frequency domain, which can produce accurate modeling results at a relatively low computational cost. Nevertheless, this approach can only deal with the modeling problems with a flat free surface. To address this issue, we design a new frequency-domain finite difference scheme by introducing the polygonal representation of topography into the average-derivative method. The irregular topography is represented by line segments with various slopes. An extension of the conventional average-derivative difference operator in the local rotated coordinate system is used for formulating the spatial derivatives aligned with the topographic line segments. As a result, new average-derivative difference schemes are obtained for irregular topography. In this way, the average-derivative optimal method is generalized to the model with irregular topography. Numerical examples show the effectiveness of the presented method.

Propagation of Torsional Surface Waves in Heterogeneous Half-Space with Irregular Free Surface

This study is an attempt to show the impacts of free surface irregularity on the torsional surface waves propagating in heterogeneous, elastic half-space. The surface irregularity is taken in the parabolic form at the surface of the half-space. The governing equation and corresponding closed form solutions are derived. Then, the phase velocity of torsional surface waves is obtained analytically and the influences of surface irregularity are studied in detail. Numerical results analyzing the torsional surface waves propagation are discussed and presented graphically. The analytical solutions and numerical results reveal that, the surface irregularity and heterogeneity have notable effects on the torsional surface waves propagation in the elastic half-space. Since the Earth crust is heterogeneous medium with irregular surface, thus it is important to consider the effects of heterogeneity and surface irregularity on velocity of torsional surface waves propagating in the Earth medium.

An immersed boundary method with iterative symmetric interpolation for irregular surface topography in seismic wavefield modelling

AbstractThe irregular free surface topography has a significant impact on simulations of seismic wave propagation. Therefore, an accurate representation of the irregular free surface is required for an accurate wavefield simulation. We propose an immersed boundary method used in fluid dynamics calculation to simulate acoustic waves with finite-difference in media with irregular surfaces. First, we set the number of ghost layers to half the length of the finite-difference stencil. Then, we define mirror points by orthogonally projecting the ghost points to fractional points below the free surface. We calculate the wavefield at these mirror points using an iterative symmetric interpolation method. Finally, we set the wavefield at the ghost points to the negative value of the wavefield of their corresponding mirror points. The proposed iterative symmetric interpolation method allows computing the wavefield at the mirror points more accurately and stably than the conventional immersed boundary methods. Numerical examples validate the accuracy and stability of this method in seismic forward modelling with strongly varying topography.

3D wave-equation dispersion inversion of surface waves recorded on irregular topography

Irregular topography can cause strong scattering and defocusing of propagating surface waves, so it is important to account for such effects when inverting surface waves for shallow S-wave velocity structures. We have developed a 3D surface-wave dispersion inversion method that takes into account the topographic effects modeled by a 3D spectral element solver. The objective function is the frequency summation of the squared wavenumber differences [Formula: see text] along each azimuthal angle of the fundamental mode or higher-order modes of Rayleigh waves in each shot gather. The wavenumbers [Formula: see text] associated with the dispersion curves are calculated using the data recorded along the irregular free surface. Numerical tests on synthetic and field data demonstrate that 3D topographic wave equation dispersion inversion (TWD) can accurately invert for the S-wave velocity model from surface-wave data recorded on irregular topography. Field data tests for data recorded across an Arizona fault demonstrate that, for this example, the 2D TWD model can be as accurate as the 3D tomographic model. This suggests that in some cases, the 2D TWD inversion is preferred over 3D TWD because of its significant reduction in computational costs. Compared to the 3D P-wave velocity tomogram, the 3D S-wave tomogram agrees much more closely with the geologic model taken from the trench log. The agreement with the trench log is even better when the [Formula: see text] tomogram is computed, which reveals a sharp change in velocity across the fault. The localized velocity anomaly in the [Formula: see text] tomogram is in very good agreement with the well log. Our results suggest that integrating the [Formula: see text] and [Formula: see text] tomograms can sometimes give the most accurate estimates of the subsurface geology across normal faults.

Frequency-Domain Finite-Difference Elastic Wave Modeling in the Presence of Surface Topography

Seismic numerical modeling in the presence of surface topography has become a valuable tool to characterize seismic wave propagation in basin or mountain areas. Regarding advantages of frequency-domain seismic wavefield simulations (e.g., easy implementation of multiple sources and straightforward extension of adding attenuation factors), we propose a frequency-domain finite-difference seismic wavefield simulation in 2D elastic media with an irregular free surface. In the frequency domain, we first transform second-order elastic wave equations and first-order free surface boundary conditions from the Cartesian coordinate system to the curvilinear coordinate system. Then we apply complex frequency-shifted perfectly matched layer (CFS-PML) absorbing boundary condition to second-order elastic wave equations in the curvilinear coordinate. To better couple free surface boundary conditions and CFS-PML absorbing boundary condition, we also apply the complex coordinate stretching method used in CFS-PML to free surface boundary conditions in the curvilinear coordinate. In the first numerical test, the comparison of the seismograms calculated by our algorithm with an analytical solution indicates that our algorithm can accurately simulate seismic wavefield in the frequency domain. Finally, we choose three more elastic models with different types of surface topographies to further characterize seismic wave propagation.

Reflection and transmission of high-frequency acoustic, electromagnetic and elastic waves at a distinguished class of irregular, curved boundaries

AbstractReflection and transmission phenomena associated with high-frequency linear wave incidence on irregular boundaries between adjacent acoustic or electromagnetic media, or upon the irregular free surface of a semi-infinite elastic solid, are studied in two dimensions. Here, an ‘irregular’ boundary is one for which small-scale undulations of an arbitrary profile are superimposed upon an underlying, smooth curve (which also has an arbitrary profile), with the length scale of the perturbation being prescribed in terms of a certain inverse power of the large wave-number of the incoming wave field. Whether or not the incident field has planar or cylindrical wave-fronts, the associated phase in both cases is linear in the wave-number, but the presence of the boundary irregularity implies the necessity of extra terms, involving fractional powers of the wave-number in the phase of the reflected and transmitted fields. It turns out that there is a unique perturbation scaling for which precisely one extra term in the phase is needed and hence for which a description in terms of a Friedlander–Keller ray expansion in the form as originally presented is appropriate, and these define a ‘distinguished’ class of perturbed boundaries and are the subject of the current paper.

Q-Compensated Pre-stack Reverse Time Migration in Viscoelastic Media with an Irregular Free Surface

Pre-stack reverse time migration (RTM) is a powerful tool that provides better imaging of complex geological structures than traditional migration methods. An irregular free surface has significant impacts on the characteristics of seismic wave propagation and seismic imaging in mountain or basin areas. In addition, viscoelastic media produce a better representation of the earth than acoustic or elastic media. A main feature of RTM in viscoelastic media is that it can consider compensation for P- and S-wave attenuation effects. In this study, we develop an approach for Q-compensated pre-stack RTM in 2D viscoelastic isotropic media with an irregular free surface. Based on the boundary-conforming method, the viscoelastic model can be discretized to better match the irregular surface and interfaces. The 2D second-order viscoelastic equations and free surface boundary conditions are converted from the Cartesian coordinate system to the curvilinear coordinate system. The finite-difference operators are applied to approximate the viscoelastic wave equations and the irregular free surface in the curvilinear coordinate system. In addition, the convolutional perfectly matched layer (CPML) absorbing boundary condition is selected to effectively suppress artificial reflections from the model boundaries. In the numerical examples, the snapshots, shot gathers and Q-compensated pre-stack RTM results show that our algorithm can obtain an accurate structure image in viscoelastic media in the presence of surface topography.

Characteristics of breaking irregular wave forces on a monopile

The substructures of offshore wind turbines are subjected to extreme breaking irregular wave forces. The present study is focused on investigating breaking irregular wave forces on a monopile using a computational fluid dynamics (CFD) based numerical model. The breaking irregular wave forces on a monopile mounted on a slope are investigated with a numerical wave tank. The experimental and numerical irregular free surface elevations are compared in the frequency-domain for the different locations in the vicinity of the cylinder. A numerical analysis is performed for different wave steepness cases to understand the influence of wave steepness on the breaking irregular wave loads. The wave height transformation and energy level evolution during the wave shoaling and wave breaking processes is investigated. The higher-frequency components generated during the wave breaking process are observed to play a significant role in initiating the secondary force peaks. The free surface elevation skewness and spectral bandwidth during the wave transformation process are analysed and an investigation is performed to establish a correlation of these parameters with the breaking irregular wave forces. The role of the horizontal wave-induced water particle velocity at the free surface and free surface pressure in determining the breaking wave loads is highlighted. The higher-frequency components in the velocity and pressure spectrum are observed to be significant in influencing the secondary peaks in the breaking wave force spectrum.

2-D poroelastic wave modelling with a topographic free surface by the curvilinear grid finite-difference method

The curvilinear grid finite-difference method, which is suitable for accurate modelling of seismic waves in the presence of topographic free surfaces, is applied in this study to model poroelastic waves. Poroelastic wave equations of Biot’s theory are expressed in curvilinear coordinates and solved on a collocated grid by utilizing the fourth-order Runge–Kutta time integral scheme with alternative applications of forward and backward MacCormack finite differences. Discretization of the irregular free surface with the curvilinear coordinate grid enables a good fit of the topographic surface, and this good fit results in higher numerical accuracy in comparison with stair-case approximation. The formulas of the free-surface boundary conditions for poroelastic media are also derived for the curvilinear coordinates and subsequently implemented with the traction-imaging method, which is a generalization of the stress-imaging method and can maintain high accuracy in curvilinear coordinates. To verify the performance of the proposed algorithm and determine its accuracy, we conduct numerical simulations on several different models, and compare our numerical solutions with other solutions that are calculated by analytical expressions, generalized reflection/tranmission method and spectral-element method. Besides, the fluid–solid dynamic compatibility of porous media is demonstrated using a three-layered half-space model, and effects of the irregular free-surface on poroelastic waves are investigated for a four-layered model with strong irregular free-surface topography. The good performance of the proposed algorithm suggests that the curvilinear grid finite-differece method is a useful tool for the study of poroelastic wave propagation.

Free Surface Reconstruction for Phase Accurate Irregular Wave Generation

The experimental wave paddle signal is unknown to the numerical modellers in many cases. This makes it quite challenging to numerically reproduce the time history of free surface elevation for irregular waves. In the present work, a numerical investigation is performed using a computational fluid dynamics (CFD) based model to validate and investigate a non-iterative free surface reconstruction technique for irregular waves. In the current approach, the free surface is reconstructed by spectrally composing the irregular wave train as a summation of the harmonic components coupled with the Dirichlet inlet boundary condition. The verification is performed by comparing the numerically reconstructed free surface elevation with theoretical input waves. The applicability of the present approach to generate irregular waves by reconstructing the free surface is investigated for different coastal and marine engineering problems. A numerical analysis is performed to validate the free surface reconstruction approach to generate breaking irregular waves over a submerged bar. The wave amplitudes, wave frequencies and wave phases are modelled with good accuracy in the time-domain during the higher-order energy transfers and complex processes like wave shoaling, wave breaking and wave decomposition. The present approach to generate irregular waves is also employed to model steep irregular waves in deep water. The free surface reconstruction method is able to simulate the irregular free surface profiles in deep water with low root mean square errors and high correlation coefficients. Furthermore, the irregular wave forces on a monopile are investigated in the time-domain. The amplitudes and phases of the force signal under irregular waves generated by using the current technique are modelled accurately in the time-domain. The proposed approach to numerically reproduce the free surface elevation in the time-domain provides promising and accurate results for all the benchmark cases.

A hybrid method for calculating seismic wave first-arrival traveltimes in two-dimensional models with an irregular surface

A hybrid method is developed for the calculation of seismic wave first-arrival traveltimes in two-dimensional models with an irregular free surface. The method is based on a combination of the classical eikonal equation (CEE) and the topography-dependent eikonal equation (TDEE); the latter applies only to the part of the explored domain that includes the non-flat surface, while the first equation applies anywhere outside this domain. CEE is solved using a fast-sweeping method, and TDEE is solved after discretizing the numerical Hamiltonian using the Legendre transform and applying the upwind fast-sweeping method. For the implementation of the hybrid method and the appropriate application of CEE and TDEE, we sort the contact mode between spatial sub-domains into three types and give an explanation about the working routine adopted for each one. Several numerical experiments considering different models, mesh sizes, seismic source positions and partition styles demonstrate that the proposed hybrid method can effectively address seismic modeling problems involving undulated free surfaces, keeping the same precision as TDEE but with a fairly rapid convergence. This proves that the hybrid method is a powerful tool for practical applications aimed at obtaining high-quality images of earth structures.

Simulating Seismic Wave Propagation in Viscoelastic Media with an Irregular Free Surface

In seismic numerical simulations of wave propagation, it is very important for us to consider surface topography and attenuation, which both have large effects (e.g., wave diffractions, conversion, amplitude/phase change) on seismic imaging and inversion. An irregular free surface provides significant information for interpreting the characteristics of seismic wave propagation in areas with rugged or rapidly varying topography, and viscoelastic media are a better representation of the earth’s properties than acoustic/elastic media. In this study, we develop an approach for seismic wavefield simulation in 2D viscoelastic isotropic media with an irregular free surface. Based on the boundary-conforming grid method, the 2D time-domain second-order viscoelastic isotropic equations and irregular free surface boundary conditions are transferred from a Cartesian coordinate system to a curvilinear coordinate system. Finite difference operators with second-order accuracy are applied to discretize the viscoelastic wave equations and the irregular free surface in the curvilinear coordinate system. In addition, we select the convolutional perfectly matched layer boundary condition in order to effectively suppress artificial reflections from the edges of the model. The snapshot and seismogram results from numerical tests show that our algorithm successfully simulates seismic wavefields (e.g., P-wave, Rayleigh wave and converted waves) in viscoelastic isotropic media with an irregular free surface.

Joint tomographic inversion of first-arrival and reflection traveltimes for recovering 2-D seismic velocity structure with an irregular free surface

Irregular surface flattening, which is based on a boundary conforming grid and the transformation between curvilinear and Cartesian coordinate systems, is a mathematical method that can elegantly handle irregular surfaces, but has been limited to obtaining first arrivals only. By combining a multistage scheme with the fast-sweeping method (FSM, the method to obtain first-arrival traveltime in curvilinear coordinates), the reflected waves from a crustal interface can be traced in a topographic model, in which the reflected wave-front is obtained by reinitializing traveltimes in the interface for upwind branches. A local triangulation is applied to make a connection between velocity and interface nodes. Then a joint inversion of first-arrival and reflection traveltimes for imaging seismic velocity structures in complex terrains is presented. Numerical examples all perform well with different seismic velocity models. The increasing topographic complexity and even use of a high curvature reflector in these models demonstrate the reliability, accuracy and robustness of the new working scheme; checkerboard testing illustrates the method’s high resolution. Noise tolerance testing indicates the method’s ability to yield practical traveltime tomography. Further development of the multistage scheme will allow other later arrivals to be traced and used in the traveltime inversion.

Dynamic response of normal moving load on a transversely isotropic piezoelectric half-space with parabolic irregularity

Due to unique electro-mechanical behavior, piezoelectric devices are being used in important sensors (mostly based on surface acoustic waves) and other devices of vital importance and therefore exposed to dynamic loading. These devices often work in geological conditions possessing irregular free surface and subject to external moving load. Present study contributes a theoretical framework to study the stresses produced on such devices due to moving load exerted by operating environment or other situational scenario. In present manuscript, authors have investigated the dynamic response of normal moving load in a piezoelectric transversely isotropic half-space with parabolic irregularity. Closed form solutions for the normal stress, shear stress, and dielectric displacement for both the cases of electrically open and electrically short circuit have been found analytically. This study examines in detail the effect of frictional coefficients, depth of the parabolic irregularity, irregularity parameter, and ground depth on dielectric displacement and magnitude of resultant stresses. Many particular cases have been investigated and compared with numerical simulation along with graphical demonstration of observed results.

An upwind fast sweeping scheme for calculating seismic wave first‐arrival travel times for models with an irregular free surface

ABSTRACTThe topography‐dependent eikonal equation formulated in a curvilinear coordinate system has recently been established and revealed as being effective in calculating first‐arrival travel times of seismic waves in an Earth model with an irregular free surface. The Lax–Friedrichs sweeping scheme, widely used in previous studies as for approximating the topography‐dependent eikonal equation viscosity solutions, is more dissipative and needs a much higher number of iterations to converge. Furthermore, the required number of iterations grows with the grid refinement and results in heavy computation in dense grids, which hampers the application of the Lax–Friedrichs sweeping scheme to seismic wave travel‐time calculation and high‐resolution imaging. In this paper, we introduce a new upwind fast sweeping solver by discretising the Legendre transform of the numerical Hamiltonian of the topography‐dependent eikonal equation using an explicit formula. The minimisation related to the Legendre transform in the sweeping scheme is solved analytically, which proved to be much more efficient than the Lax–Friedrichs algorithm in solving the topography‐dependent eikonal equation. Several numerical experiments demonstrate that the new upwind fast sweeping method converges and achieves much better accuracy after a finite number of iterations, independently of the mesh size, which makes it an efficient and robust tool for calculating travel times in the presence of a non‐flat free surface.