This paper develops a calibration methodology of the artificial absorbing techniques typically used by Fourier pseudo-spectral time-domain (PSTD) methods for geoacoustic wave simulations. Specifically, we consider the damped wave equation (DWE) that results from adding a dissipation term to the original wave equation, the sponge boundary layer (SBL) that applies an exponentially decaying factor directly to the wavefields, and finally, a classical split formulation of the perfectly matched layer (PML). These three techniques belong to the same family of absorbing boundary layers (ABLs), where outgoing waves and edge reflections are progressively damped across a grid zone of

The Fourier pseudo-spectral time-domain (PSTD) method has been applied to wave propagation problems in, for example, electromagnetism

It is worth considering that ABLs are not the only techniques used to absorb waves in numerical simulations. For example, Reynolds- or Higdon-type absorbing boundary conditions (ABCs)

The first ABL technique that we will consider in the present work is the damped wave equation (DWE)

The second ABL technique that will be analyzed is the sponge boundary layer (SBL) proposed in

The third and last ABL analyzed is the perfectly matched layer (PML). The PML was introduced in electromagnetism by

In this work, we compare the characteristics of all three ABL methods mentioned above combined with PSTD schemes. In Sect.

The Fourier PSTD method can be considered a particular case of finite differences (FD) on Cartesian grids where spatial derivatives are substituted with differentiation in the spectral (Fourier) domain. This means that any spatial derivative requires a forward and inverse Fourier transform for the direction differentiated. By multiplying the variable in the spectral domain by

The Euler formulation tends to be less memory efficient than the second-order formulation, because it requires more spatial variables to be stored and differentiated, but is well suited to some numerical applications where first derivatives are relevant. This is the case, for example, of the classical split PML formulation that depends on directional derivatives of both the pressure and velocity fields. Other ABLs, such as DWE and SBL, do not require additional differentiation and thus can be solved directly using the second-order formulation.

In the following we will use Cartesian regular grids, where all spatial differential operators employ forward and inverse 1D fast Fourier transform (FFT) along each Cartesian direction. We will consider constant time and spatial sampling,

All ABLs considered in the following will be presented using a unified representation of the grid. We will assume that the computational domain includes both grid points of the physical domain and grid nodes of the absorbing layers. The grid of the physical domain has size

A vertical cross section of the computational mesh, along a

Figure

There remains a last issue in order to solve the wave equation in the computational domain from parameters of the physical domain: the velocity

The DWE is derived from the linear wave Eq. (

We use the standard second-order and central FD approximations for both temporal derivatives in Eq. (

Our second ABL under study is the SBL technique presented by

The PML's formulation

Finally, we mention that the velocity–pressure scheme Eqs. (

Before our application exercises, we briefly comment on the stability of PSTD and its dispersion properties. At uniform grids and using second-order explicit time integration, a von Neumann analysis of PSTD in unbounded acoustic media yields the following bound for conditional stability:

The coupling of the ABL techniques presented above to a PSTD method does not alter the stability and dispersion properties of PSTD in lossless unbound acoustic media. The physical attenuation experienced by acoustic waves at any frequency along the ABL regions only reinforces the boundedness of the numerical solution and thus favors the damping of short period oscillations induced by dispersion.

In the previous section we have written the formulations of all three ABL and remarked that two main parameters control absorption in each of them: the size of the absorbing layer

Our characterization effort involves (1) finding appropriate tests for which a reference exists, (2) finding suitable metrics that measure the absorption performance of the methods against the reference, (3) establishing absorption thresholds that classify the absorption, and (4) for each classification and ABL, finding the parameter tuple that requires the least absorption nodes or

The first step to create an absorption measure is quantifying the total energy in the physical domain (not including the ABL) at any given time sample. Thus, we define the following quantity which is proportional to the (discrete)

On the other side, for problems where the domain has a constant propagation velocity,

Moreover, it is important to highlight that the methodology for calibration of ABLs presented in this work is based on three main components: using representative models, establishing suitable metrics for absorption, and reducing the calibration to two parameters. We are not adding any assumptions regarding the underlaying PDEs used (linear acoustic waves, in our case). Similarly, there are no assumptions tied to the numerical method (pseudo-spectral time domain, in our case). Nevertheless, two modifications are foreseen for broadening the applicability of the method. On one hand, in the case of using other physical models, we would need to modify Eq. (

Finally, we mention that for all scenarios in the following, the main grid remains identical and we modify only the size of the ABL zone and its associated absorption parameter. We will always exploit the spatial discretization characteristics of PSTD, thus using the coarsest grid possible at 2 points per minimum wavelength (ppw). All sources used will be point sources in space and Ricker wavelets in time. All numerical experiments that follow use our bespoke PSTD-ABL implementations using the g++ C compiler version 4.5.3.1-1, under -lm and -O3 optimization flags, and linking the FFTW3 library version 3.3.4-2. All simulations have been performed by an Intel Core i7-6820HK processor running at 2.70GHz under the Linux operating system.

We first consider a cube

We take a grid step of

Next, we perform a numerical exploration of the

Sampling of the absorption parameters and absorbing layer size used for parameter exploration for each ABL.

The gray scale depicts

Figure

Optimal pairs of

The time evolution of the energy proxy

Table

Finally, Fig.

In addition to sheer energy absorption, it is important to analyze the impact of our different ABLs in practical imaging applications. As a simple yet representative test, we analyze a reverse time migration (RTM) case in a homogeneous model with a single source and receiver. In reverse-time migration (e.g., see

Panel

We use the same model, grid steps

Envelope misfits (EM) and phase misfits (PM) obtained when using the three ABL techniques under different

Table

As an additional comparison, we compute the same impulse response exercise using an algorithm popular in geophysical imaging: finite differences with eighth order in space and second order in time, using

The earth's subsurface is largely heterogeneous across many scales. In such environments wavefields become more complex, involving scattering, reflections, and refractions, among other phenomena. As a consequence, a generalized calibration of ABLs is not possible, as all models are fundamentally different from each other. Our goal when studying ABLs in heterogeneous media is assessing whether their fundamental behavior remains, i.e., absorption increases steadily with

First, we consider a 3D cuboid physical domain involving three flat layers of wave speeds

The time evolution of the energy proxy

In Fig.

As a final and more realistic scenario, we use the 3D SEG/EAGE salt model (e.g., see

A vertical cross section, along the

In order to quantify the absorption for such a complex model, we need to run several configurations of ABLs and compare them with a reference. To construct such a reference solution, we use the PSTD simulation that employs PML using the parameters associated with maximum absorption in Table

Snapshots of pressure at

We run simulations using all three ABLs using all absorption parameter pairs reported in Table

Errors

In this section, we discuss the computational times obtained for our different ABLs coupled with PSTD acoustic wave simulations. Of course, observations in terms of computational time are less objective measures, because times are affected by the algorithm design, compilation optimization, coding skills, and libraries employed; hence, we do not suggest that our findings are universal. Nevertheless, we start our analysis with two theoretical aspects or considerations. Finally, we remark that for all methods, we solve the complete absorbing equation for each grid node, only using non-zero values for the absorption parameters inside the absorbing layers.

First, we consider the memory footprint of PSTD using the three ABLs. As formulated in Sect.

Second, we consider the amount of operations required per time update. Both DWE and SBL compute a single 1D spectral derivative of

Computational time of all ABLs at different grids, characterized by their total number of nodes.

Relative computing time

Following the theoretical discussion, we have measured computational times for our PSTD code using the three ABLs for different grid sizes. In Fig.

To further expand our cost analysis we present results for our experiments in Sect.

In this work, we have reviewed and compared the three main ABL methodologies available in the context of PSTD simulations for acoustic wave propagation. Specifically, the damped wave equation, the sponge boundary layer proposed in

Comparing the three ABLs with each other is a complex issue. On one hand, DWE and SBL have very similar formulations and behave similarly in terms of

To assess absorption performance, we have introduced a dimensionless measure proportional to the total acoustic energy in the seismic volume and use its magnitude in the calibration of ABL parameters. This energy proxy is consistent with the reflected energy that we qualitatively observe in all test scenarios, and therefore we recommend it for similar studies of absorption methods. The methodology to calibrate ABLs in this work could be applied to other wave equations such as the elastic wave equation or anisotropic wave equation. We do not expect the same calibration values to hold across all the equations, but the methodology should reveal the optimal values for each case. This will be the subject of future work.

We remark that computational times increase with grid size, but not in a steady or monotonic way, as a result of using modern FFT libraries. Therefore, varying the absorption of ABLs by means of larger

Computer codes to run all three test cases are readily available at the Zenodo site

CS and JdlP implemented the computer codes and carried out the simulations. CS, JdlP, and OR developed the mathematical formulation and designed the test cases. CS and OR worked on manuscript editing.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 777778 MATHROCKS. In addition, the research leading to these results has received funding from the QUSTom project with proposal no. 101046475 under the call HORIZON-EIC-2021-PATHFINDEROPEN-01.

This research has been supported by the Horizon 2020 (grant no. 777778) and the HORIZON EUROPE European Innovation Council (grant no. 101046475).

This paper was edited by James Kelly and reviewed by one anonymous referee.