# A Method for Modeling Acoustic Waves in Moving Subdomains

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## Abstract

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## 1. Introduction

## 2. Methodology

^{r}. The elements in the reduced vector at a given time step comprise only the pressure at those grid points through which a wave is actively propagating. We refer to these grid points as “relevant” grid points. As waves propagate through the modeling domain, the set of relevant grid points changes. To find the reduced vector at each time step, RDM determines the set of relevant grid points without actually evaluating the pressure at all the grid points. The following paragraphs and Figure 1 below explain how RDM achieves this goal.

^{s}. The subinterval length T

^{s}needs to be small in order to keep the set of relevant grid points within subintervals small. On the other hand, decreasing the subinterval length T

^{s}will increase the number of subintervals in the simulation and the time spent finding the set of relevant grid points for all subintervals could start having a large effect on computation time. The best value for T

^{s}depends on the velocity model and the source, and there does not exist an ideal T

^{s}value that gives the best results in every scenario. However, you can still run consistently fast and accurate simulations while always using the same value of T

^{s}. We set T

^{s}to be equal to the period of the dominant frequency of the source, and as can be seen in the following section, RDM drastically reduced computation time and maintained accuracy in all of the tested models.

^{c}is the pressure vector computed in the coarse-grid simulation, N is the number of snapshots in a subinterval, and t

^{snp}is the time between two subsequent snapshots. The magnitude of the sum vector for a given subinterval, at a given grid point, is large if the corresponding pressure on that grid point was large during the subinterval.

^{snp}depend on the length of the subinterval T

^{s}and the time step of the finite difference scheme. Although N and t

^{snp}need not have specific values, typically we set t

^{snp}such that 40 ≥ N ≥ 20. The goal is to use enough snapshots to accurately describe the wavefield during the interval, while also not using so many snapshots as to affect the computation time. It should be noted that changing N and t

^{snp}has very little effect on the performance of the simulation. Thus, we do not think that optimizing those parameters can lead to noticeable improvements.

^{sum}are used to estimate the map of relevant grid points, an equal weight averaging filter is applied to v

^{sum}. Filtering is performed to smooth the results stored in the sum vector, which reduces the length of the bounding curve between the relevant and irrelevant grid points, as can be seen in Figure 2. A big part of the error caused by using RDM is produced at the boundary between relevant and irrelevant grid points. Having wavefield drop from near-zero values to zero can create a source of error in the wavefield. By reducing the length of the boundary between relevant and irrelevant grid points, we reduce the error. This allows us to reduce the computation time more aggressively, while still maintaining a small error. The averaging filter is two dimensional and 32 grid points wide and long. This is because its length is defined as four times the shortest wavelength in the model, i.e., four times the period of the dominant frequency multiplied by the velocity from the slowest area in the model. The processing time of the averaging filter is proportional to the length of the filter and the number of grid points used in the simulation. As RDM was tested on multiple models, the computation time of the averaging filter varied between 3% and 7% of the entire simulation time when using RDM. To further reduce the computation time of the averaging filter, we could use better picks for the window size that are based on the velocity average rather than minimum velocity, and we could also apply the filter to a different vector that has fewer elements than v

^{sum}, for example, a vector containing elements of v

^{sum}for grid points with spacing four times as large as that of the fine grid. Once the sum vector v

^{sum}is determined, the set of relevant grid points is constructed.

^{sum}representing those grid points is greater than or equal to some pre-determined threshold fraction $(1-{e}^{-\delta})$ of the sum of all elements in the sum vector:

^{sum}) is the number of elements in v

^{sum}, and the threshold $(1-{e}^{-\delta})$ is defined by the parameter δ. The threshold is defined in this way so that an increase in δ causes the threshold to increase, converging closer to the value of 1. An increase in the threshold results in more grid points being included in the set of relevant grid points. Therefore, increasing the parameter δ increases the accuracy of RDM but also increases the computation time.

^{s}, t

^{snp}and N should remain the same in all simulations, the parameter δ can be adjusted to best support the FDM we are applying our method to, which is the purpose of the simulation. For example, if we are doing reverse time migration (RTM) for the purpose of locating a seismic event, we can ignore a lot of weak waves that we know will not contribute to the convergence at the source location. In this case we could set up δ to a small value that will result in fewer relevant grid points and faster simulation. Alternatively, if the user is interested in simulating weak reflection, the parameter δ would be set up to a larger value to make sure the weak reflections are represented.

^{o}) and velocity (c) on the fine grid. Specifically, the set of relevant grid points tells us which elements in pressure (or velocity) vectors represent the pressure (or velocity) on the relevant grid points. To generate reduced pressure and velocity vectors, an algorithm goes through all the elements of the pressure vectors p, p

^{o}, and velocity vector c and the values describing the pressure or velocity at relevant grid points are recorded in reduced pressure and velocity vectors p

^{r}, p

^{or}, and c

^{r}. The set of vectors converted to the reduced model may vary between different models and different simulation methods. For example, in cases with heterogeneous density, we also must apply the set of relevant grid points to the density vector (ρ) in order to generate the reduced density vector ρ

^{r}.

^{r}and p

^{or}rather than on p and p

^{o}, creating a significant reduction in computation time. Once the fine-grid simulation reaches the end of the subinterval, the values of the pressure on the standard fine grid p and p

^{o}are updated using the reduced vectors p

^{r}and p

^{or}and the set of relevant grid points U. The process is repeated until the simulation reaches the end of the final subinterval.

## 3. Results

## 4. Discussion

^{−4}, in the fourth test it reaches 0.3. The large relative error in the fourth test occurs because an aggressive criterion is used to select relevant grid points, so that many of the weaker waves are not modeled. The goal in the fourth test is to efficiently yet accurately model the high-amplitude waves. This goal is achieved as the relative error in areas enclosing the strong waves (presented in Figure 5) is much smaller, as shown in Table 3.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) The sum vector v

^{sum}without the averaging filter and (

**b**) the resulting map of relevant grid points. (

**c**) The sum vector v

^{sum}with the averaging filter and (

**d**) the resulting map of relevant grid points.

**Figure 3.**(

**a**) Results for the first limestone model. (

**b**) Results for the second sediment model. From left to right we see: the map of relevant grid points in the first subinterval, nineteenth subinterval, and thirty-seventh subinterval, and finally, the wavefield corresponding to the end of the thirty-seventh subinterval. The figures were acquired during the simulation, with δ set to 12. The relevant grid points are in the yellow region. The brown line represents the surface of the ocean floor, and the blue line represents the steel object.

**Figure 4.**(

**a**) The velocity model from [22]. (

**b**) The map of relevant grid points in the final subinterval. (

**c**) The wavefield at the final subinterval of the simulation. The parameter δ was set to 12.

**Figure 5.**(

**a**) The density model from [22]. (

**b**) The map of relevant grid points at the final subinterval of the simulation with parameter θ set to 0.7. (

**c**) The wavefield at the end of the standard FDM simulation with the strong waves marked with red squares.

First Model | Second Model | |||
---|---|---|---|---|

Parameter δ | Comp. Time Reduction (%) | Relative Error | Comp. Time Reduction (%) | Relative Error |

10 | 56.5 | 0.017 | 63.0 | 0.062 |

12 | 54.8 | 0.016 | 61.8 | 0.043 |

14 | 48.0 | 0.014 | 56.9 | 0.026 |

16 | 40.5 | 0.015 | 49.1 | 0.033 |

18 | 40.1 | 0.005 | 50.2 | 0.020 |

20 | 40.1 | 0.001 | 47.8 | 0.005 |

Parameter δ | Comp. Time Reduction (%) | Relative Error |
---|---|---|

10 | 71.3 | 0.026 |

12 | 67.9 | 0.011 |

14 | 66.9 | 0.003 |

16 | 66.8 | 6.0 × 10^{−4} |

18 | 65.9 | 2.7 × 10^{−4} |

20 | 65.8 | 6.5 × 10^{−5} |

Parameter θ | Comp. Time Reduction (%) | Relative Error | Relative Error in the Area of Interest |
---|---|---|---|

0.5 | 72.5 | 0.310 | 0.112 |

0.6 | 71.7 | 0.229 | 0.073 |

0.7 | 71.5 | 0.136 | 0.025 |

0.8 | 71.0 | 0.066 | 0.002 |

0.9 | 66.7 | 0.005 | 5.7 × 10^{−7} |

1.0 | 66.7 | 5.4 × 10^{−5} | 5.7 × 10^{−7} |

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**MDPI and ACS Style**

Brankovic, M.; Everett, M.E. A Method for Modeling Acoustic Waves in Moving Subdomains. *Acoustics* **2022**, *4*, 394-405.
https://doi.org/10.3390/acoustics4020024

**AMA Style**

Brankovic M, Everett ME. A Method for Modeling Acoustic Waves in Moving Subdomains. *Acoustics*. 2022; 4(2):394-405.
https://doi.org/10.3390/acoustics4020024

**Chicago/Turabian Style**

Brankovic, Milan, and Mark E. Everett. 2022. "A Method for Modeling Acoustic Waves in Moving Subdomains" *Acoustics* 4, no. 2: 394-405.
https://doi.org/10.3390/acoustics4020024