{"title":"Theoretical and quantitative evaluation of hybrid PML-ABCs for seismic wave simulation","authors":"Yuhang Wang , Wei Zhang","doi":"10.1016/j.eqs.2022.05.002","DOIUrl":null,"url":null,"abstract":"<div><p>A good artificial boundary treatment in a seismic wave grid-based numerical simulation can reduce the size of the computational region and increase the computational efficiency, which is becoming increasingly important for seismic migration and waveform inversion tasks requiring hundreds or thousands of simulations. Two artificial boundary techniques are commonly used: perfectly matched layers (PMLs), which exhibit the excellent absorption performance but impose a greater computational burden by using finite layers to gradually reduce wave amplitudes; and absorbing boundary conditions (ABCs), which have the high computational efficiency but are less effective in absorption because they employ the one-way wave equation at the exterior boundary. Naturally, PMLs have been combined with ABCs to reduce the number of PMLs, thus improving the computational efficiency; many studies have proposed such hybrid PMLs. Depending on the equations from which the ABCs are derived, there are two hybrid PML variants: the PML+unstretched ABC (UABC), in which the ABC is derived from a physical equation; or the PML+stretched ABC (SABC), in which the ABC is derived from the PML equation. Even though all the previous studies concluded that hybrid PMLs can improve the absorption performance, none of them quantified how many PMLs can be removed by combining the PML with the ABC compared with the pure PML. In this paper, we systematically study the absorption performance of the two hybrid PML variants. We develop a method to distinguish the artificial reflections from the PML-interior interface and those caused by the PML exterior boundary to accurately approximate the additional absorption achieved by using the UABC and the SABC. The reflection coefficients based on a theoretical derivation and numerical tests both show that the UABC amplifies most reflections and is not recommended in any situation; conversely, the SABC can always diminish reflections, but the additional absorption achieved by the SABC is relatively poor and cannot effectively reduce the number of PMLs. In contrast, we find that simply increasing the damping parameter improves absorption better than the PML+SABC. Our results show that the improvement in absorption achieved by combining the PML with either the SABC or the UABC is not better than that obtained by simply adjusting the damping profile of the PML; thus, combining the PML with the ABC is not recommended in practice.</p></div>","PeriodicalId":46333,"journal":{"name":"Earthquake Science","volume":"35 2","pages":"Pages 105-121"},"PeriodicalIF":1.2000,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1674451922000325/pdfft?md5=9d173f97d0e4bd37f253c651471cb682&pid=1-s2.0-S1674451922000325-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Earthquake Science","FirstCategoryId":"89","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1674451922000325","RegionNum":4,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Earth and Planetary Sciences","Score":null,"Total":0}
引用次数: 0
Abstract
A good artificial boundary treatment in a seismic wave grid-based numerical simulation can reduce the size of the computational region and increase the computational efficiency, which is becoming increasingly important for seismic migration and waveform inversion tasks requiring hundreds or thousands of simulations. Two artificial boundary techniques are commonly used: perfectly matched layers (PMLs), which exhibit the excellent absorption performance but impose a greater computational burden by using finite layers to gradually reduce wave amplitudes; and absorbing boundary conditions (ABCs), which have the high computational efficiency but are less effective in absorption because they employ the one-way wave equation at the exterior boundary. Naturally, PMLs have been combined with ABCs to reduce the number of PMLs, thus improving the computational efficiency; many studies have proposed such hybrid PMLs. Depending on the equations from which the ABCs are derived, there are two hybrid PML variants: the PML+unstretched ABC (UABC), in which the ABC is derived from a physical equation; or the PML+stretched ABC (SABC), in which the ABC is derived from the PML equation. Even though all the previous studies concluded that hybrid PMLs can improve the absorption performance, none of them quantified how many PMLs can be removed by combining the PML with the ABC compared with the pure PML. In this paper, we systematically study the absorption performance of the two hybrid PML variants. We develop a method to distinguish the artificial reflections from the PML-interior interface and those caused by the PML exterior boundary to accurately approximate the additional absorption achieved by using the UABC and the SABC. The reflection coefficients based on a theoretical derivation and numerical tests both show that the UABC amplifies most reflections and is not recommended in any situation; conversely, the SABC can always diminish reflections, but the additional absorption achieved by the SABC is relatively poor and cannot effectively reduce the number of PMLs. In contrast, we find that simply increasing the damping parameter improves absorption better than the PML+SABC. Our results show that the improvement in absorption achieved by combining the PML with either the SABC or the UABC is not better than that obtained by simply adjusting the damping profile of the PML; thus, combining the PML with the ABC is not recommended in practice.
期刊介绍:
Earthquake Science (EQS) aims to publish high-quality, original, peer-reviewed articles on earthquake-related research subjects. It is an English international journal sponsored by the Seismological Society of China and the Institute of Geophysics, China Earthquake Administration.
The topics include, but not limited to, the following
● Seismic sources of all kinds.
● Earth structure at all scales.
● Seismotectonics.
● New methods and theoretical seismology.
● Strong ground motion.
● Seismic phenomena of all kinds.
● Seismic hazards, earthquake forecasting and prediction.
● Seismic instrumentation.
● Significant recent or past seismic events.
● Documentation of recent seismic events or important observations.
● Descriptions of field deployments, new methods, and available software tools.
The types of manuscripts include the following. There is no length requirement, except for the Short Notes.
【Articles】 Original contributions that have not been published elsewhere.
【Short Notes】 Short papers of recent events or topics that warrant rapid peer reviews and publications. Limited to 4 publication pages.
【Rapid Communications】 Significant contributions that warrant rapid peer reviews and publications.
【Review Articles】Review articles are by invitation only. Please contact the editorial office and editors for possible proposals.
【Toolboxes】 Descriptions of novel numerical methods and associated computer codes.
【Data Products】 Documentation of datasets of various kinds that are interested to the community and available for open access (field data, processed data, synthetic data, or models).
【Opinions】Views on important topics and future directions in earthquake science.
【Comments and Replies】Commentaries on a recently published EQS paper is welcome. The authors of the paper commented will be invited to reply. Both the Comment and the Reply are subject to peer review.