The forward modeling of a scalar wave equation plays an important role in the numerical geophysical computations. The finite-difference algorithm in the form of a second-order wave equation is one of the commonly used forward numerical algorithms. This algorithm is simple and is easy to implement based on the conventional grid. In order to ensure the accuracy of the calculation, absorption layers should be introduced around the computational area to suppress the wave reflection caused by the artificial boundary. For boundary absorption conditions, a perfectly matched layer is one of the most effective algorithms. However, the traditional perfectly matched layer algorithm is calculated using a staggered grid based on the first-order wave equation, which is difficult to directly integrate into a conventional-grid finite-difference algorithm based on the second-order wave equation. Although a perfectly matched layer algorithm based on the second-order equation can be derived, the formula is rather complex and intermediate variables need to be introduced, which makes it hard to implement. In this paper, we present a simple and efficient algorithm to match the variables at the boundaries between the computational area and the absorbing boundary area. This new boundary-matched method can integrate the traditional staggered-grid perfectly matched layer algorithm and the conventional-grid finite-difference algorithm without formula transformations, and it can ensure the accuracy of finite-difference forward modeling in the computational area. In order to verify the validity of our method, we used several models to carry out numerical simulation experiments. The comparison between the simulation results of our new boundary-matched algorithm and other boundary absorption algorithms shows that our proposed method suppresses the reflection of the artificial boundaries better and has a higher computational efficiency.

Modeling of a seismic wave field is accomplished by simulating the pattern of the seismic waves as they propagate through various geologic media and computing the simulated measurements at observation points on the Earth's surface or underground, given that the underground medium's structure and the relevant physical parameters are known. Numerical modeling of a seismic wave field is an important tool for seismic data processing and interpretation and for geodynamic studies of the Earth's interior. In recent years, many full waveform inversion methods have been widely proposed and applied to seismic exploration. In the waveform inversion process, wave field modeling is one of the key algorithms because it must be performed first to obtain the predicted wave field that is used to compute the residual errors between the predicted and the actual wave field records. In addition, the information provided by the residual errors, which is required for refinement of the initial model, is actually calculated by a modeling algorithm that uses the residual errors as virtual sources. After many iterations of the above processes, an optimized approximate model of the underground medium can be acquired. Numerical modeling of a wave field will be executed thousands of times throughout the waveform inversion process, so a wave field modeling algorithm is crucial in many ways when performing a waveform inversion algorithm, such as for computational precision, speed, and storage requirements.

The main numerical techniques for seismic wave field modeling include the finite-element method (Marfurt, 1984; Yang et al., 2008), the pseudo-spectral method (Kreiss and Oliger, 1972; Dan and Baysal, 1982), and the finite-difference method (Kelly et al., 2012; Virieux, 1984; Yang et al., 2002; Moczo et al., 2007; Zhang et al., 2013). Due to its easy implementation and the satisfactory compromise between accuracy and efficiency, the finite-difference method is the preferred method. For a comprehensive overview of applications of the finite-difference methods, see Moczo et al. (2014). Over the last several decades, many studies have focused on determining the coefficients of the finite-difference method and designing computational templates (Li et al., 2017).

Schematic of our method:

Comparison of the analytical solution (red solid line) with the
proposed (second-order conventional grid (CG) scheme) and classic staggered-grid (SG) perfectly matched layer (PML) methods (second-order
SG scheme) (blue dotted line) at different receivers,

According to the formulation of the wave equations, the finite-difference methods can be implemented based on the first-order velocity–stress equations or the second-order displacement equations, which lead to different computational templates. A staggered grid (SG) is usually set up for the first-order wave equations and has been widely used with the acoustic and elastic wave equations (Virieux, 1984, 1986; Moczo et al., 2014; Madariaga, 1976; Gold et al., 1997; Saenger et al., 2000; O'Brien, 2010). Many methods of optimizing the differential coefficients, based on a SG, have been proposed to increase the accuracy of the numerical solution, such as the time–space domain dispersion-relation-based method (Liu and Sen, 2011), the simulated annealing algorithm (Zhang and Yao, 2013), and the least-squares method (Yang et al., 2015). However, a conventional grid (CG) is often directly obtained from the second-order wave equation. These methods include the central scheme (Alford et al., 1974; Igel et al., 1995), the high-order compact finite-difference method (Fornberg, 1990), the Lax–Wendroff correction (LWC) scheme (Lax and Wendroff, 1964; Dablain, 1986; Blanch and Robertsson, 2010), the nearly analytical discrete method (Yang et al., 2003), and the nearly analytical central difference method (Yang et al., 2012).

Comparison of the relative errors between the analytical solutions
and the proposed method (second-order CG scheme) (red solid line) or the
classic SG PML method (second-order SG scheme) (blue dotted line) at
different receivers and different grid spacings.

The algorithm design of the CG scheme is easier to use than that of the SG
scheme because the variable definition is uniform throughout the grid.
However, it is hard to determine which of the two schemes is more accurate
and efficient. Although the SG scheme has sometimes been regarded as more
precise than the CG scheme (Huang and Dong, 2009), there is also some
theoretical and experimental proof in the literature that does not support
this proposition. Moczo et al. (2011) compared the accuracy of the different
finite-difference schemes with respect to the

Comparison of the analytical solution (red solid line) with the
proposed (fourth-order CG scheme) and classic SG PML methods (fourth-order
SG scheme) (blue dotted line) at different receivers and

Reflection from the artificial boundaries introduced by the limited computational area is another numerical source of error. Over the past 30 years, many techniques have been developed for boundary processing: paraxial conditions (Clayton and Engquist, 1977; Reynolds, 1978; Higdon, 2012), the sponge boundary (Cerjan et al., 1985; Sochacki et al., 1987), the perfectly matched layer (PML) (Berenger, 1994), and the hybrid absorbing boundary conditions (hybrid ABC) (Ren and Liu, 2012). Among these, the PML is one of the most efficient and most commonly used methods. The PML was first introduced for boundary processing of electromagnetic wave equation modeling, after which, it was applied to the elastic–dynamic problem (Chew and Liu, 1996) and acoustic simulations (Liu and Tao, 1998). Many modified versions of the PML, such as the convolutional PML (Komatitsch and Martin, 2007), were subsequently proposed. Gao et al. (2017) compared most of the typical artificial absorbing boundary processing approaches for use with acoustic wave equations and came to the conclusion that a 20-layer PML is ideal for most practical applications using general size models, even in the presence of strong nearly grazing waves, which demonstrates the high performance and efficiency of the PML approach.

Comparison of the relative errors between the analytical solutions
and the proposed method (fourth-order CG scheme) (red solid line) or the
classic SG PML method (fourth-order SG scheme) (blue dotted line) at
different receivers and different grid spacings.

Comparison of the analytical solutions of the proposed (10th-order
CG scheme) (red solid line) and the classic SG PML methods (10th-order SG
scheme) (blue dotted line) at different receivers,

Comparison of the relative errors between the analytical
solutions and the proposed method (10th-order CG scheme) (red solid line)
or the classic SG PML method (10th-order SG scheme) (blue dotted line) at
different receivers and different grid spacings (

In the field of real wave field simulation, most researchers are devoted to unifying the format of the boundary processing algorithm and the wave equation within the computational region. The classic PML is naturally formulated based on the first-order wave equations for velocity and stress (Collino and Tsogka, 1998), which has proven to be very efficient. It is easy to integrate PML boundary processing into a SG finite-difference algorithm. So, some scholars use the SG scheme in the computational region to match the PML equations, while for many CG-based schemes, they need to adopt other boundary processing methods, such as the hybrid ABC method. However, in recent years, some scholars have also made efforts to formulate a PML for a second-order system to match the second-order wave equation. Komatitsch and Tromp (2003) reformulated the classic PML conditions in order to use it with numerical schemes that are based on the elastic wave equation written as a second-order system with displacement. Grote and Sim (2010) proposed a PML formulation for the acoustic wave equation in its standard second-order form, while Pasalic and McGarry (2010) extended the convolutional PML to accommodate the second-order acoustic wave equation. Nevertheless, all of these second-order PML formulations require the derivation of complicated formulas, the introduction of auxiliary variables, and the modification of existing second-order numerical codes in order to handle the first-order equations describing the auxiliary variables, which increases the computational cost and complexity.

Receiver records of the four methods for a different number of
absorbing layers:

Values of the absorption coefficient

In order to preserve the original efficiency of the PML boundary processing method as well as the accuracy and efficiency of the CG scheme, it is worth trying to integrate the classic first-order PML algorithm into the CG finite-difference scheme in a second-order system and make it easy to implement. In this paper, we propose a new boundary-matched algorithm that uses a CG finite-difference scheme within a limited computational area and an SG finite-difference scheme in a PML area. Our approach enables the inner area and the PML condition to be independent during computation, while preserving the individual advantages of the two methods. The algorithm matches the computational area and the absorbing boundary layers simply by point updating along the boundaries of the computational area and avoids complex formula conversion. Thus, none of the original formulas of the CG scheme or the PML equations are modified and no unnecessary variables are added. The assessment of the proposed algorithm is composed of two parts. First, we compared the accuracy and efficiency of the proposed algorithm with those of the classic SG PML method (SG scheme both in computational area and PML area), which demonstrated the rationality of our decision to use the CG scheme in the computational area. To simulate the actual underground medium, a medium with a linearly increasing velocity gradient was selected for the experiment. The experimental results indicated that the accuracy of the two methods for equal grid sizes is almost equal, but the efficiency of our method is approximately 30 %–50 % higher than that of the classic SG PML method. Next, the proposed algorithm was evaluated by comparing its absorption efficiency and computational cost with those of the classic SG PML method, the second-order PML method (CG scheme both in computational area and PML area) introduced by Pasalic and McGarry (2010), and the hybrid ABC method (CG scheme in computational area and hybrid ABC scheme in boundary area) introduced by Ren and Liu (2012). The numerical experimental results indicated that our algorithm provides an excellent absorption effect and was easier to implement.

Receiver records for the four methods for a different number of
absorbing layers:

Although the elastic wave equation can describe the propagation of seismic
waves more comprehensively, modeling an elastic wave field is complex and
computationally expensive. In practice, the acoustic wave equation is also
popularly used to approximate seismic wave propagation. For the convenient
error analysis of these methods, we consider a scalar wave field

The discretization of the acoustic wave (Eq. 1) with a 2

Values of the absorption coefficient

Marmousi velocity model.

Numerical analyses show that grid dispersion increases with increasing grid size, but decreasing the grid size increases the computational cost. High-order finite-difference schemes are able to control this numerical dispersion using a larger grid spacing compared with low-order schemes (Tan and Huang, 2014).

Because the subscripts

Due to limitations in the capacity and speed of computer facilities, the
numerical simulation of a wave field can only be implemented for a limited
area. The computational area is surrounded by artificial boundaries, except
for the free surface. As described above, the PML boundary condition can
effectively absorb the wave field reflections from the artificial boundaries
in order to simulate wave field propagation in an open space. In a PML
medium, the wave field

Using the SG finite-difference scheme to discretize Eq. (3), the results are as
follows:

A finite-difference scheme based on a CG requires no computation of intermediate variables, and thus the computational cost is lower than that of an SG scheme. We will show in the next section that the accuracy of the CG scheme can reach the same level as that of the SG scheme but with lower computational costs. However, it is difficult to incorporate a naturally formulated PML boundary processing algorithm based on an SG scheme into a CG finite-difference scheme. In this paper, we propose a new boundary-matched algorithm that can bridge the gap between an SG-based PML algorithm and a CG-based numerical simulation of a seismic wave field with neither introduction of intermediary variables nor reformulation of the PML equations. The core idea of the scheme is to interface the wave field reasonably along the boundaries between the CG area and the SG absorbing layers. A detailed description of the method is given below.

As shown in Fig. 1, the entire domain consists of two parts: the computational area and the boundary absorbing area. The computational area is located in the center and is surrounded by the absorbing layers. The algorithm uses a CG finite-difference scheme within the computational area and an SG finite-difference scheme within the boundary absorbing area. If we can reasonably interface the computed values of the wave field between the computational area and the boundary absorbing area, then the scheme can perform satisfactorily. For a clearer explanation, we start with a two-dimensional model.

Receiver records for the four methods for a different number of
absorbing layers:

We let the computational area and the PML area overlap each other for one
layer. As shown in Fig. 1a, the bold red boundary line is both the
outermost boundary of the computational area and the innermost boundary of
the PML area. On this overlapped layer, both the particle velocity

Wave field snapshots with different PML at different time; the
three coordinate axes represent the number of grids in three directions:

At the beginning of iteration, take

Calculate the wave field

Calculate the particle velocity and wave field in the PML area layer by
layer. Calculate the values of all particle velocity

It is important to note that the particle velocities in the area between the
blue and red lines are calculated from the wave field

Update the value of

Repeat steps 2–4 until

The two-dimensional algorithm described above can easily be generalized to
three-dimensional. In the three-dimensional model, we need to add a particle
velocity component

As described in the introduction, the errors in the wave field numerical model are mainly caused by differential dispersion and reflected waves that are not fully absorbed by the boundary processing algorithm. In order to verify the validity of our algorithm, we used a variety of models to compare the computational accuracy, the efficiency of the absorption of the reflected waves, and the computational efficiency of the proposed algorithm to the other methods.

In order to obtain a more convincing result when comparing the computational
accuracy, we used a constant-gradient velocity model, the velocity of which
increases linearly with depth. This model is closer to the actual velocity
distribution of an underground medium than a homogeneous model. We
calculated the relative error between our method and the classic SG PML
method using the analytical solutions for different grid spacings and the
order of difference, and then we performed a comparative analysis of the
two methods. The relative error between the two methods and the analytical
solution is defined by the following time function:

For the two-dimensional scalar (Eq. 1), the wave field analytic
solution for a constant-gradient medium can be obtained from the integral
form of the three-dimensional solution using the dimension reduction method
(Cerveny, 2001). The velocity distribution is

When comparing the absorption efficiency, we used three different geological
models to determine the reflected wave absorption effect of our algorithm:
the homogeneous, constant-gradient velocity and the Marmousi models. We
compared the absorption effects of our algorithm with the classic SG PML
method, the second-order PML method, and the hybrid ABC method using the
same conditions to prove whether our algorithm can effectively combine the
CG scheme with the SG scheme PML boundary condition and achieve the same or
better effect as other methods do. In the computational area, the reflection
coefficient

In the comparison of the computational accuracy and efficiency of the absorption of the reflected waves, we determined the computation time of the three methods separately, which can reflect the advantages and disadvantages of all of the methods in terms of the computational efficiency.

Based on the discussion of the performance analysis, in this section, we
present the results of the numerical experiments. All of the numerical
experiments were run on a desktop personal computer with a 3.40 GHz Intel
Core i5-3570 processor, 32 GB of DDR3 memory, on a 64 bit Windows 7 operating
system, using algorithmic software written in C

As shown in Fig. 2, the constant-gradient velocity model has a size of
6000 m

From Figs. 3 and 4, we can see that both of the methods have obvious errors during the first 2 s. In particular, when the grid spacing is 12 m, the error is the largest, and there is significant numerical dispersion. Reducing the grid spacing can reduce the error and the dispersion. When the grid spacing is 10 m, the result improves. In addition, the results for a longer simulation time also prove the numerical stability of our method. Further comparison of the relative error curves shown in Fig. 5 indicates that although neither method is particularly good; the amplitude of relative errors with their analytical solutions are almost the same.

In theory, the error of the numerical solution can be reduced by using a higher-order difference. We compared the experimental results of the proposed method (fourth-order CG scheme in the computational area) and classic SG PML methods (fourth-order SG scheme in the computational area) with the analytic solution, as shown in Figs. 6–8.

From Figs. 6 and 7, we can see that when the fourth-order difference is used, the relative errors between the analytical solution and both methods are significantly reduced compared with when the second-order difference is used. In addition, as with the second-order result above, the relative error also decreases as the grid spacing decreases. Figure 8 illustrates the fact that the relative error curves of our algorithm and the classic SG PML method are also very similar for the fourth-order difference, In addition, it is difficult to distinguish the advantages and disadvantages of the two algorithms. Although the results of the two methods still exhibit a small error at this time, we can continue to use the higher difference order or we can reduce the grid spacing to reduce the error. The laws of the two methods are the same.

In Figs. 9 and 10, we adopt a 10th-order difference scheme and

Table 1 presents the computation times of the two methods at different grid spacings and difference orders. The efficiency percentage is the total computation time of our method divided by the total computation time of the SG method. Under the same circumstances, the total computation time of our method is only 57 %–70 % that of the classic SG PML method. It is noteworthy that the result of our method for the 10th-order difference and a grid spacing of 10 m is much better than that of the classic SG PML method for the fourth-order difference and a 10 m grid spacing, while the computation time is nearly the same. Also, the result of our method for the fourth-order difference and a grid spacing of 12 m is much better than that of the classic SG PML method for the second-order difference and a 10 m grid spacing, while the former computation time is only 53.3 % of the latter. Therefore, for the same computation time as the classic SG PML method, our method always achieves a higher accuracy for a smaller grid spacing and a higher-order difference. We obtained these conclusions in a constant-gradient velocity medium. Therefore, the algorithm we propose works well when the CG scheme is used in the computational area. Next, we discuss the absorption efficiency of the reflected waves of our method in a series of simple and complex models.

Computation time for our and the classic SG PML methods.

Computation times for the four methods.

First, we used a two-dimensional homogeneous model to verify the reflected
wave absorption efficiency of our new boundary-matched algorithm. As shown
in Fig. 11, the model size is 2000 m

As can be seen in Figs. 12 and 13, all of the four methods can absorb the
reflected waves to a certain degree. For the same number of absorbing
layers, the absorption performance of our method and that of the classic SG
PML method are almost the same and both methods are superior to the other
two methods, while the hybrid ABC method is the worst. Specifically, when
the number of absorbing layers equals 10, the absorption coefficients of our
method and classic SG PML method are both

Taking into account the fact that the homogeneous model is relatively simple
and quite different from the actual distribution of an underground
medium, the second model that we use is the constant-gradient velocity
model, as shown in Fig. 14. The velocity is 1500 m s

We next compared the absorption efficiency of the four methods for a complex
Marmousi model. The Marmousi model has a size of 9200 m

Based on the above numerical experiments, although the hybrid ABC method is often used as the boundary condition of the CG-based method because it is easy to deduce its second-order form, its absorption performance is obviously worse than that of the other three PML methods since it is based on a one-dimensional wave equation. Among the three PML methods, the 10-layer classic SG PML method (first-order PML is used inside) for the first-order wave equation is enough to suppress the edge reflections, while the 20-layer second-order PML method is sufficient for the second-order wave equation. However, our first-order PML method only requires a thickness of 10 grid spacings to absorb the outgoing wave entirely. It may have a significant advantage over the second-order PML method. Table 2 shows the computation times of the four methods for different numbers of absorbing layers. Among them, the computation time of our method is the shortest and that of the classic SG PML method is the longest. Given that our method uses the CG scheme in the computational area, it requires much less computation time than the classic SG PML method does. In addition, the second-order PML method requires the transformation of the original first-order PML equation into a second-order form. The required complex formulas and extra variables without physical meaning increase the computation time. In addition, our method naturally implements high-order temporal discretization if necessary, while the second-order PML method does not. Therefore, our method is ideal for seismic wave forward modeling.

In order to facilitate the experiments and comparative analyses, we used the
two-dimensional models described in the above numerical experiments. To
further illustrate the effectiveness of our method, Fig. 19 shows the
experimental results of this method for a three-dimensional homogeneous
velocity model. The model size is 1000 m

We propose a new boundary-matched algorithm that effectively combines the CG scheme in the computational area and the SG scheme in the PML boundary conditions, while preserving the high computational efficiency of the CG scheme and the good absorption effect of PML boundary conditions. Our proposed method is easy to implement, and we only perform appropriate wave field matching at the grid points, which avoids complicated modifications to the PML formulas and the introduction of unnecessary variables. The numerical experiments of the different models indicate that our method is applicable to a variety of simple and complex two-dimensional and three-dimensional geological models. For the same conditions, our method can achieve similar or better accuracy and reflected wave absorption efficiency compared to other boundary absorption methods, while it requires less computation time. Because our method keeps the independence of the computational area and the boundary absorption area, it can also be combined with other CG-based seismic wave numerical algorithms, such as the nearly analytical center difference method with PML boundary conditions, to achieve better numerical simulation accuracy.

Our work is based on the numerical simulation of a scalar equation. Because the elastic wave equation includes more wave field information, it is also widely used in the numerical simulation of seismic waves. The simulation of the elastic wave equation requires more computations and greater storage capacity, while our proposed method can reduce the computational cost. The next step of our work will be the numerical simulation of the elastic wave equation and is expected to significantly improve its computational efficiency.

The processed data required to reproduce these
findings are available to download from

XZ and QC contributed to the conception of the study. XZ, DZ, and YY contributed significantly to analysis and manuscript preparation; XZ, DZ, and QC performed the data analyses and wrote the manuscript; XZ and DZ helped perform the analysis with constructive discussions.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Advances in seismic imaging across the scales”. It does not belong to a conference.

This research work was financially supported by the National Science and Technology Major Project of China under grant 2011zx05003-003. Edited by: Michal Malinowski Reviewed by: two anonymous referees