A Numerical Framework for Modeling Folds in Structural Geology
Tóm tắt
A numerical framework for modeling folds in structural geology is presented. This framework is based on a novel and recently published Hamilton–Jacobi formulation by which a continuum of layer boundaries of a fold is modeled as a propagating front. All the fold classes from the classical literature (parallel folds, similar folds, and other fold types with convergent and divergent dip isogons) are modeled in two and three dimensions as continua defined on a finite difference grid. The propagating front describing the fold geometry is governed by a static Hamilton–Jacobi equation, which is discretized by upwind finite differences and a dynamic stencil construction. This forms the basis of numerical solution by finite difference solvers such as fast marching and fast sweeping methods. A new robust and accurate scheme for initialization of finite difference solvers for the static Hamilton–Jacobi equation is also derived. The framework has been integrated in simulation software, and a numerical example is presented based on seismic data collected from the Karama Block in the North Makassar Strait outside Sulawesi.
Tài liệu tham khảo
citation_journal_title=Geophys J Int; citation_title=Multiparameter inversion in anisotropic media; citation_author=R Burridge, MV Hoop, D Miller, C Spencer; citation_volume=134; citation_issue=3; citation_publication_date=1998; citation_pages=757-777; citation_id=CR1
citation_journal_title=Trans Am Math Soc; citation_title=Viscosity solutions of Hamilton–Jacobi equations; citation_author=MG Crandall, PL Lions; citation_volume=277; citation_issue=1; citation_publication_date=1983; citation_pages=1-43; citation_doi=10.1090/S0002-9947-1983-0690039-8; citation_id=CR2
citation_journal_title=Trans Am Math Soc; citation_title=Some properties of viscosity solutions of Hamilton–Jacobi equations; citation_author=MG Crandall, LC Evans, PL Lions; citation_volume=282; citation_issue=2; citation_publication_date=1984; citation_pages=487-502; citation_doi=10.1090/S0002-9947-1984-0732102-X; citation_id=CR3
citation_title=Curves and surfaces for CAGD: a practical guide; citation_publication_date=2002; citation_id=CR4; citation_author=G Farin; citation_publisher=Academic Press
Gillberg T (2011) A semi-ordered fast iterative method (SOFI) for monotone front propagation in simulations of geological folding. In: Chan F, Marinova D, Anderssen B (eds) MODSIM2011: 19th international congress on modelling and simulation. Modelling and Simulation Society of Australia and New Zealand, pp 631–647
citation_journal_title=Phys Fluids; citation_title=Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface; citation_author=F Harlow, J Welch; citation_volume=8; citation_issue=12; citation_publication_date=1965; citation_pages=2182-2189; citation_doi=10.1063/1.1761178; citation_id=CR6
citation_journal_title=Math Geosci; citation_title=A Hamilton–Jacobi framework for modeling folds in structural geology; citation_author=Ø Hjelle, SA Petersen; citation_volume=43; citation_issue=7; citation_publication_date=2011; citation_pages=741-761; citation_doi=10.1007/s11004-011-9357-2; citation_id=CR7
citation_journal_title=IEEE Trans Vis Comput Graph; citation_title=Interactive visualization of volumetric white matter connectivity in DT-MRI using a parallel-hardware Hamilton–Jacobi solver; citation_author=WK Jeong, PT Fletcher, R Tao, R Whitaker; citation_volume=13; citation_issue=6; citation_publication_date=2007; citation_pages=1480-1487; citation_doi=10.1109/TVCG.2007.70571; citation_id=CR8
citation_journal_title=SIAM J Numer Anal; citation_title=Fast sweeping methods for static Hamilton–Jacobi equations; citation_author=CY Kao, S Osher, YH Tsai; citation_volume=42; citation_issue=6; citation_publication_date=2005; citation_pages=2612-2632; citation_doi=10.1137/S0036142902419600; citation_id=CR9
citation_journal_title=J Acoust Soc Am; citation_title=Ray theory for moving fluids; citation_author=E Kornhauser; citation_volume=25; citation_issue=5; citation_publication_date=1953; citation_pages=945-949; citation_doi=10.1121/1.1907223; citation_id=CR10
citation_title=Numerical methods for conservation laws; citation_publication_date=1992; citation_id=CR11; citation_author=RJ LeVeque; citation_publisher=Birkhäuser
Lin Q (2003) Enhancement, extraction, and visualization of 3d volume data. PhD thesis, Linköping University
citation_journal_title=J Comput Appl Math; citation_title=Stable marker-particle method for the Voronoi diagram in a flow field; citation_author=T Nishida, K Sugihara, M Kimura; citation_volume=202; citation_issue=2; citation_publication_date=2007; citation_pages=377-391; citation_doi=10.1016/j.cam.2006.01.035; citation_id=CR13
Petersen SA, Hjelle Ø(2008) Earth recursion, an important component in sheared earth model builders. In: EAGE 70th conference & exhibition. Extended abstracts
Petersen SA, Hjelle Ø, Jensen SL (2007) Earth modelling using distance fields derived by fast marching. In: EAGE 69th conference & exhibition. Extended abstracts
Petersen SA, Hjelle Ø, Hustoft S, Haubiers M, Rasmussen R (2012) Process based data-restoration and model-reconstruction workflow for seismic interpretation and model building. In: EAGE 74th conference & exhibition. Extended Abstracts
citation_journal_title=Comput Graph Forum; citation_title=A Lagrangian approach to dynamic interfaces through kinetic triangulation of the ambient space; citation_author=JP Pons, JD Boissonnat; citation_volume=26; citation_issue=2; citation_publication_date=2007; citation_pages=227-239; citation_doi=10.1111/j.1467-8659.2007.01029.x; citation_id=CR17
citation_journal_title=J Sci Comput; citation_title=A fast sweeping method for static convex Hamilton–Jacobi equations; citation_author=J Qian, YT Zhang, HK Zhao; citation_volume=31; citation_issue=1/2; citation_publication_date=2007; citation_pages=237-271; citation_doi=10.1007/s10915-006-9124-6; citation_id=CR18
citation_title=Folding and fracturing of rocks; citation_publication_date=1967; citation_id=CR19; citation_author=JG Ramsay; citation_publisher=McGraw-Hill
citation_title=Algorithms; citation_publication_date=2010; citation_id=CR20; citation_author=R Sedgewick; citation_author=K Wayne; citation_publisher=Addison-Wesley
citation_journal_title=Commun Math Phys; citation_title=Curvature and the evolution of fronts; citation_author=JA Sethian; citation_volume=101; citation_publication_date=1985; citation_pages=487-499; citation_doi=10.1007/BF01210742; citation_id=CR21
citation_journal_title=Proc Natl Acad Sci USA; citation_title=A fast marching level set method for monotonically advancing fronts; citation_author=JA Sethian; citation_volume=93; citation_publication_date=1996; citation_pages=1591-1595; citation_doi=10.1073/pnas.93.4.1591; citation_id=CR22
citation_title=Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science; citation_publication_date=1999; citation_id=CR23; citation_author=JA Sethian; citation_publisher=Cambridge University Press
citation_journal_title=SIAM J Numer Anal; citation_title=Ordered upwind methods for static Hamilton–Jacobi equations: theory and algorithms; citation_author=JA Sethian, A Vladimirsky; citation_volume=41; citation_issue=1; citation_publication_date=2003; citation_pages=325-363; citation_doi=10.1137/S0036142901392742; citation_id=CR24
citation_title=On ray tracing parametric surfaces; citation_inbook_title=SIGGRAPH; citation_publication_date=1985; citation_pages=171-179; citation_id=CR25; citation_author=DL Toth
Vladimirsky A (2001) Fast methods for static Hamilton–Jacobi partial differential equations. PhD thesis, Department of Mathematics, University of California, Berkeley
citation_journal_title=Math Comput; citation_title=A fast sweeping method for eikonal equations; citation_author=H Zhao; citation_volume=74; citation_issue=250; citation_publication_date=2005; citation_pages=603-627; citation_id=CR27