Perfectly matched layer boundary conditions for the second‐order acoustic wave equation solved by the rapid expansion method

Araujo, Edvaldo S.; Pestana, Reynam C.

doi:10.1111/1365-2478.12868

Cited by 5 publications

(5 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To introduce the SPML absorbing boundary conditions on to the equations given by (20), we have to split the wavefield P(t ) into two components: x-component (P x ) and z-component (P z ). Since P(t ) = P x (t ) + P z (t ), the system of equations 20is rewritten as (Gao et al, 2015;Araujo and Pestana, 2019…”

Section: Split Perfectly Matched Layer Boundary Conditionmentioning

confidence: 99%

“…To introduce the SPML absorbing boundary conditions on to the equations given by (), we have to split the wavefield

P (t)

into two components: x ‐component (

P_{x}

) and z ‐component (

P_{z}

). Since

P false(t false) = P_{x} false(t false) + P_{z} false(t false)

, the system of equations () is rewritten as (Gao et al ., 2015; Araujo and Pestana, 2019)

({, \begin{matrix} V_{x} false(t + normalΔ t^{'} false) = V_{x} false(t false) + Δ t^{'} \frac{\partial P (t)}{\partial x} \\ V_{z} false(t + normalΔ t^{'} false) = V_{z} false(t false) + Δ t^{'} \frac{\partial P (t)}{\partial z} \\ P_{x} false(t + normalΔ t^{'} false) = P_{x} false(t false) + Δ t^{'} c^{2} \frac{\partial V_{x} false(t false)}{\partial x} \\ P_{z} false(t + normalΔ t^{'} false) = P_{z} false(t false) + Δ t^{'} c^{2} \frac{\partial V_{z} false(t false)}{\partial z} \\ P false(t + normalΔ t^{'} false) = P_{x} false(t + normalΔ t^{'} false) + P_{z} false(t + normalΔ t^{'} false) . \end{matrix})

…”

Section: Numerical Scheme Derivationmentioning

confidence: 99%

“…(2015). We then apply the same damping functions of the SPML in equations (), giving

({, \begin{matrix} V_{x} false(t + normalΔ t^{'} false) = α V_{x} false(t false) + β Δ t^{'} \frac{\partial P (t)}{\partial x} \\ V_{z} false(t + normalΔ t^{'} false) = α V_{z} false(t false) + β Δ t^{'} \frac{\partial P (t)}{\partial z} \\ P_{x} false(t + normalΔ t^{'} false) = α P_{x} false(t false) + β Δ t^{'} c^{2} \frac{\partial V_{x} false(t false)}{\partial x} \\ P_{z} false(t + normalΔ t^{'} false) = α P_{z} false(t false) + β Δ t^{'} c^{2} \frac{\partial V_{z} false(t false)}{\partial z} \\ P false(t + normalΔ t^{'} false) = P_{x} false(t + normalΔ t^{'} false) + P_{z} false(t + normalΔ t^{'} false), \end{matrix})

where the terms

α

and

β

(Araujo and Pestana, 2019) are

α false(r false) = \frac{1 - 0.5 Δ t^{'} σ false(r false)}{1 + 0.5 Δ t^{'} σ false(r false)},

…”

Section: Numerical Scheme Derivationmentioning

confidence: 99%

See 2 more Smart Citations

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

Araujo

Pestana

2020

Geophysical Prospecting

Self Cite

View full text Add to dashboard Cite

We propose a new numerical solution to the first-order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time-steps than traditional finite-difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the firstorder linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post-stack migration results are also shown using the proposed method.

show abstract

Section: Split Perfectly Matched Layer Boundary Conditionmentioning

confidence: 99%

“…To introduce the SPML absorbing boundary conditions on to the equations given by (), we have to split the wavefield

P (t)

into two components: x ‐component (

P_{x}

) and z ‐component (

P_{z}

). Since

P false(t false) = P_{x} false(t false) + P_{z} false(t false)

, the system of equations () is rewritten as (Gao et al ., 2015; Araujo and Pestana, 2019)

({, \begin{matrix} V_{x} false(t + normalΔ t^{'} false) = V_{x} false(t false) + Δ t^{'} \frac{\partial P (t)}{\partial x} \\ V_{z} false(t + normalΔ t^{'} false) = V_{z} false(t false) + Δ t^{'} \frac{\partial P (t)}{\partial z} \\ P_{x} false(t + normalΔ t^{'} false) = P_{x} false(t false) + Δ t^{'} c^{2} \frac{\partial V_{x} false(t false)}{\partial x} \\ P_{z} false(t + normalΔ t^{'} false) = P_{z} false(t false) + Δ t^{'} c^{2} \frac{\partial V_{z} false(t false)}{\partial z} \\ P false(t + normalΔ t^{'} false) = P_{x} false(t + normalΔ t^{'} false) + P_{z} false(t + normalΔ t^{'} false) . \end{matrix})

…”

Section: Numerical Scheme Derivationmentioning

confidence: 99%

“…(2015). We then apply the same damping functions of the SPML in equations (), giving

({, \begin{matrix} V_{x} false(t + normalΔ t^{'} false) = α V_{x} false(t false) + β Δ t^{'} \frac{\partial P (t)}{\partial x} \\ V_{z} false(t + normalΔ t^{'} false) = α V_{z} false(t false) + β Δ t^{'} \frac{\partial P (t)}{\partial z} \\ P_{x} false(t + normalΔ t^{'} false) = α P_{x} false(t false) + β Δ t^{'} c^{2} \frac{\partial V_{x} false(t false)}{\partial x} \\ P_{z} false(t + normalΔ t^{'} false) = α P_{z} false(t false) + β Δ t^{'} c^{2} \frac{\partial V_{z} false(t false)}{\partial z} \\ P false(t + normalΔ t^{'} false) = P_{x} false(t + normalΔ t^{'} false) + P_{z} false(t + normalΔ t^{'} false), \end{matrix})

where the terms

α

and

β

(Araujo and Pestana, 2019) are

α false(r false) = \frac{1 - 0.5 Δ t^{'} σ false(r false)}{1 + 0.5 Δ t^{'} σ false(r false)},

…”

Section: Numerical Scheme Derivationmentioning

confidence: 99%

See 1 more Smart Citation

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

Araujo

Pestana

2020

Geophysical Prospecting

Self Cite

View full text Add to dashboard Cite

show abstract

“…On this basis, Kosloff et al proposed the rapid-expansion method [2] , which was later developed by Pestana et al applied to reverse time migration method [3] . Revelo and Pestana extended the RE method to the separation of up and down going waves of seismic wavefield [4] , reverse time migration imaging of anisotropic media, and noise suppression [5] , and realized the application of the method in the field of anisotropic media imaging [6] . Araujo and Pestana introduced perfectly matched layer (PML) absorption boundary conditions into the rapidexpansion method, which significantly reduced boundary reflection interference [7] .…”

Section: Introductionmentioning

confidence: 99%

Wavefield separation in VTI media with multiple-angle expansion

Wang

2024

Fourth International Conference on Geology, Mapping, and Remote Sensing (ICGMRS 2023)

View full text Add to dashboard Cite

With the improvement of imaging accuracy required by oil and gas exploration, the anisotropy of underground media has attracted more and more attention. In order to improve the accuracy and stability of the forward modelling of seismic wavefield in VTI (Transversely isotropic media with a vertical symmetry axis) media. In this paper, a solution based on multiple-angle formula expansion in VTI medium is proposed. Combined with the multiple-angle formula and Taylor's expansion, the harmonic function operator including the pseudo-differential operator is approximated by spectral method, and then a VTI medium based on the multiple-angle formula is solved. Numerical tests based on two-dimensional uniform and complex models show that the proposed method can maintain the stability of the extrapolated wavefield at large time steps. On this basis, the wavefield separation is realized, and the simulation test is carried out by the layered model. The results show that the proposed method is effective for up-down going separation.

show abstract

“…We restrict ourselves to the last strategy and numerically solve the TFSE by applying the perfectly matched layer (PML) technique, which was originally proposed by Berenger for electromagnetism [16] and has been successfully applied to solve a variety of partial differential equations [17] [18] [19], to study the behavior of quantum mechanical systems without having spurious reflections from waves traveling out of the interested domain. The key idea of PML approach is to surround the physical domain by an artificial unphysical layer to damp the waves entering the layer region without any reflections.…”

Section: Introductionmentioning

confidence: 99%

Stability of Perfectly Matched Layers for Time Fractional Schrödinger Equation

Zhang¹,

Li²

2023

ENG

View full text Add to dashboard Cite

It is an important issue to numerically solve the time fractional Schrödinger equation on unbounded domains, which models the dynamics of optical solitons propagating via optical fibers. The perfectly matched layer approach is applied to truncate the unbounded physical domain, and obtain an initial boundary value problem on a bounded computational domain, which can be efficiently solved by the finite difference method. The stability of the reduced initial boundary value problem is rigorously analyzed. Some numerical results are presented to illustrate the accuracy and feasibility of the perfectly matched layer approach. According to these examples, the absorption parameters and the width of the absorption layer will affect the absorption effect. The larger the absorption width, the better the absorption effect. There is an optimal absorption parameter, the absorption effect is the best.

show abstract

Perfectly matched layer boundary conditions for the second‐order acoustic wave equation solved by the rapid expansion method

Cited by 5 publications

References 36 publications

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

Wavefield separation in VTI media with multiple-angle expansion

Stability of Perfectly Matched Layers for Time Fractional Schrödinger Equation

Contact Info

Product

Resources

About

Perfectly matched layer boundary conditions for the second‐order acoustic wave equation solved by the rapid expansion method

Cited by 5 publications

References 36 publications

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

Wavefield separation in VTI media with multiple-angle expansion

Stability of Perfectly Matched Layers for Time Fractional Schr&#246;dinger Equation

Contact Info

Product

Resources

About

Stability of Perfectly Matched Layers for Time Fractional Schrödinger Equation