Comparison between L2- and L1-norm and among optimization algorithms for a nonhyperbolic multiparametric approach for converted-wave and OBN data
Abstract
As velocity analysis is an important step in seismic processing, several nonhyperbolic travel-time approximations have been proposed during the last decades, and each nonhyperbolic approximation was developed for different conditions and with different proposals. However, none of them was proposed to consider the combined effect of the nonhyperbolicity coming from layered media with large offsets, wave conversion and difference of datum between source and receiver. For this, a nonhyperbolic multiparametric travel-time approximation, which is capable of describing this combination of effects, was recently proposed. As this approximation was developed to characterize ultra-deep reservoirs, the understanding of its behavior is necessary for an offshore reservoir concerning the objective function topology complexity, as it is important for a better understanding of its behavior during the inversion procedure, and also important to determine the kind of optimization algorithm to be used. Performing the inversion procedure with different optimization algorithms and norms is proposed. The complexity analysis of the objective function is also proposed. Then, a comparison between each norm and among each algorithm concerning their accuracy and efficiency is proposed, to find which combination is the most effective to recover RMS velocity information for this kind of scenario.
Keywords
Full Text:
PDFReferences
Aarts, E. H. L., and P. J. M. van Laarhoven, 1985, Statistical cooling: A general approach to combinatorial optimization problems: Phillips Journal Research, 40, 193–226.
Abedi, M. M., and A. Stovas, 2019a, Extended generalized non-hyperbolic moveout approximation: Geophysical Journal International, 216, 2, 1428–1440, doi: 10.1093/gji/ggy504.
Abedi, M. M., and A. Stovas, 2019b, A new parameterization for generalized moveout approximation, based on three rays: Geophysical Prospecting, 67, 5, 1243–1255, doi: 10.1111/1365-2478.12770.
Abramson, M. A., 2002, Pattern search algorithm for mixed variable general constrained optimization problems: Ph.D. thesis, Rice University, Houston, Texas, 180 pp.
Abramson, M. A., C. Audet, J. E. Dennis Jr., and S. Le Digabel, 2009, A deterministic MADS instance with orthogonal directions: SIAM Journal on Optimization, 20, 948–966, doi: 10.1137/080716980.
Aleardi, M., F. Ciabarri, and A. Mazzotti, 2017. Probabilistic estimation of reservoir properties by means of wide-angle AVA inversion and a petrophysical reformulation of the Zoeppritz equations: Journal of Applied Geophysics, 147, 28–41, doi: 10.1016/j.jappgeo.2017.10.002.
Aleixo, R., and J. Schleicher, 2010, Traveltime approximations for q-P waves in vertical transversely isotropic media: Geophysical Prospecting, 58, 2, 191–201, doi: 10.1111/j.1365-2478.2009.00815.x.
Alkhalifah, T., and I. Tsvankin, 1995, Velocity analysis for transversely isotropic Media: Geophysics, 60, 1550–1566, doi: 10.1190/1.1443888.
Audet, C., and J. E. Dennis Jr., 2006, Mesh adaptive direct search algorithms for constrained optimization: SIAM Journal on Optimization, 17, 188–217, doi: 10.1137/040603371.
Audet, C., and J. E. Dennis Jr., 2009. A progressive barrier for derivative-free nonlinear programming: SIAM Journal on Optimization, 20, 445–472, doi: 10.1137/070692662.
Barros, P. A., M. R. Kirby, and D. N. Mavris, 2004, Impact of sampling techniques selection on the creation of response surface models: World Aviation Congress & Exposition, SAE International, Transaction, 113, 1682–1693, Technical Paper 2004-01-3134, doi: 10.4271/2004-01-3134.
Barton, R. R., 1994, Metamodeling: A state of the art review: Proceedings of the 1994 Winter Simulation Conference, Lake Buena Vista, Florida, 237–244.
Bélisle, C. J. P., H. E. Romeijn, and R. L. Smith, 1993, Hit-and-run algorithms for generating multivariate distributions: Mathematics of Operations Research, 18, 255–266, doi: 10.1287/moor.18.2.255.
Blias, E., 2009, Long-offset NMO approximations for a layered VTI model: Model study: 79th Annual International Meeting, Society of Exploration Geophysics, Expanded Abstract, Houston, p. 3745–3749, doi: 10.1190/1.3255647.
Bokhonok, O., 2011, Sísmica de Reflexão Rasa Multicomponente: Aquisição e inversão de tempos de trânsito e amplitudes: Ph.D. thesis, Universidade de São Paulo, SP, Brazil, 162 pp.
Bourbaki, N., 1987, Topological vector spaces: Springer-Verlag Berlin and Heidelberg, 362 pp, doi: 10.1007/978-3-642-61715-7.
Costa, F. T., M. A. C. Santos, and D. M. Soares Filho, 2020, Wavenumbers illuminated by time-domain acoustic FWI using the L1 and L2 norms: Journal of Applied Geophysics, 174, 103935, doi: 10.1016/j.jappgeo.2019.103935.
Coxeter, H. S. M., 1948, Regular Polytopes: Methuen, London, 321 pp.
Custódio, A. L., and L. N. Vicente, 2007, Using sample and simplex derivatives in pattern search methods: SIAM Journal on Optimization, 18, 2, 537–555, doi: 10.1137/050646706.
Custódio, A. L., and L. N. Vicente, 2008, SID-PSM: A pattern search method guided by simplex derivatives for use in derivative-free optimization: Technical Report of the Department of Mathematics of the University of Coimbra, Portugal, p. 21.
Custódio, A. L., H. Rocha, and L. N. Vicente, 2010, Incorporating minimum Frobenius norm models in direct search: Computational Optimizing and Applications, 46, 265–278, doi: 10.1007/s10589-009-9283-0.
Dix, C. H., 1955, Seismic velocities from surface measurements: Geophysics, 20, 68–86, doi: 10.1190/1.1438126.
Du, Q., and H. Yan, 2013, PP and PS joint AVO inversion and fluid prediction: Journal of Applied Geophysics, 90, 110–118, doi: 10.1016/j.jappgeo.2013.01.005.
Farra, V., and I. Pšen?ík, 2018, Reflection moveout approximation for a P-SV wave in a moderately anisotropic homogeneous vertical transverse isotropic layer: Geophysics, 84, 2, C75–C83, doi: 10.1190/GEO2018-0474.1.
Gilmore, P., and C. T. Kelley, 1995, An implicit filtering algorithm for optimization of functions with many local minima: SIAM Journal on Optimization, 5, 269–285, doi: 10.1137/0805015.
Golikov, P., and A. Stovas, 2012, Accuracy comparison of nonhyperbolic moveout approximations for qP-waves in VTI media: Journal of Geophysics and Engineering, 9, 428–432, doi: 10.1088/1742-2132/9/4/428.
Hansen, N., 2006, The CMA evolution strategy: A comparing review, in Lozano, J. A., P. Larrañaga, I. Inza, and E. Bengoetxea, eds., Towards a New Evolutionary Computation: Advances on Estimation of Distribution Algorithms: Springer-Verlag, Berlin, Heidelberg, Studies in Fuzziness and Soft Computing, STUDFUZZ, 192, 75–102, doi: 10.1007/3-540-32494-1_4.
Hao, Q., and A. Stovas, 2015, Generalized moveout approximation for P-SV converted waves in vertically inhomogeneous transversely isotropic media with a vertical symmetry axis: Geophysical Prospecting, 64, 6, 1469–1482, doi: 10.1111/1365-2478.12353.
Holmström, K., Quttineh, N.-H., Edvall, M. M., 2008, An adaptive radial basis algorithm (ARBF) for expensive black-box mixed-integer constrained global optimization: Optimization and Engineering 9, 311–339, doi: 10.1007/s11081-008-9037-3.
Horst, R., P. M. Pardalos, and N. V. Thoai, 2000, Introduction to global optimization: 2nd ed., Springer, NY, 354 pp, doi: 10.1007/978-1-4615-0015-5.
Huyer, W., and A. Neumaier, 1999, Global optimization by multilevel coordinate search: Journal of Global Optimization, 14, 331–355, doi: 10.1023/A:1008382309369.
Huyer, W., and A. Neumaier, 2008, SNOBFIT – Stable Noisy optimization by branch and fit: ACM Transactions on Mathematical Software, 35, 1–25, doi: 10.1145/1377612.1377613.
Jones, D. R., 2001, A taxonomy of global optimization methods based on response surface: Journal of Global Optimization, 21, 345–383, doi: 10.1023/A:1012771025575.
Jones, D. R., M. Schonlau, and W. J. Welch, 1998, Efficient global optimization of expensive black-box functions: Journal of Global Optimization, 13, 455–492, doi: 10.1023/A:1008306431147.
Kan, A. H. G., and G. T., Timmer, 1987a, Stochastic global optimization methods part I: Clustering methods: Mathematical Programming, 39, 27–56, doi: 10.1007/BF02592070.
Kan, A. H. G., and G. T. Timmer, 1987b, Stochastic global optimization methods part II: Multi level methods: Mathematical Programming, 39, 57–78, doi: 10.1007/BF02592071.
Kelley, C. T., 2011, Implicit Filtering, in Software, Environments, and Tools Series: SIAM, Philadelphia, 158 pp, doi: 10.1137/1.9781611971903.
Khaleelulla, S. M., 1982, Counterexamples in topological vector spaces: 2nd ed., Lecture Notes in Mathematics Series, Springer-Verlag Berlin and Heidelberg, 184 pp, doi: 10.1007/BFb0097678.
Kirkpatrick, S., C. D. Gelatt Jr., and M. P. Vecchi, 1983, Optimization by simulated annealing: Science, 220, 4598, 671–680, doi: 10.1126/science.220.4598.671.
Kurt, H., 2007, Joint inversion of AVA data for elastic parameters by bootstrapping: Computers & Geosciences, 33, 3, 367–382, doi: 10.1016/j.cageo.2006.08.012.
Lagarias, J. C., J. A. Reeds, M. H. Wright, and P. E. Wright, 1998. Convergence properties of the Nelder-Mead simplex method in low dimensions: SIAM Journal on Optimization, 9, 112–147, doi: 10.1137/S1052623496303470.
Larsen, J. A., 1999, AVO Inversion by Simultaneous P-P and P-S Inversion: M.S. thesis, University of Calgary, Department of Geology and Geophysics, Calgary, 124 pp.
Lewis, R. M., V. Torczon, and M. W. Trosset, 2000, Direct search methods: Then and now: Journal of Computational and Applied Mathematics, 124, 1–2, 191–207, doi: 10.1016/S0377-0427(00)00423-4.
Li, X.-Y., 2003, Converted-wave moveout analysis revisited: The search for a standard approach: 73rd Annual International Meeting, Society of Exploration Geophysics, Expanded Abstract, p. 805–808, doi: 10.1190/1.1818059.
Li, X.-Y., and J. Yuan, 1999, Converted-waves moveout and parameter estimation for transverse isotropy: 61st EAGE Conference, Helsinki, Finland, Expanded Abstract, 1, p. 4–35.
Li, X.-Y., and J. Yuan, 2001, Converted wave imaging in inhomogeneous, anisotropic media - Part I - Parameter estimation: 63rd EAGE Conference & Exhibition, Amsterdam, Netherlands, Expanded Abstract, 1, p. 109, cp-15-00103, doi: 10.3997/2214-4609-pdb.15.P109.
Li, X.-Y., and J. Yuan, 2003, Converted-wave moveout and conversion-point equations in layered VTI media: theory and applications: Journal of Applied Geophysics, 54, 3–4, 297–318, doi: 10.1016/j.jappgeo.2003.02.001.
Lu, J., Z. Yang, Y. Wang, and Y. Shi, 2015, Joint PP and PS AVA seismic inversion using exact Zoeppritz equations: Geophysics, 80, 5, R239–R250, doi: 10.1190/geo2014-0490.1.
Lu, J., Y. Wang, and J. Chen, 2018, Joint velocity updating for anisotropic PP and PS prestack time migration based on hyperbolic correction of nonhyperbolic moveout: Journal of Geophysics and Engineering, 15, 4, 1171–1186, doi: 10.1088/1742-2140/aaacae.
Malovichko, A. A., 1978, A new representation of the traveltime curve of reflected waves in horizontally layered media: Applied Geophysics (in Russian), 91, 1, 47–53.
Margrave, G. F., 2000, New seismic modelling facilities in Matlab: CREWES Research Report, 12, 45 pp.
Margrave, G. F., 2003, Numerical methods of exploration seismology with algorithms in MATLAB: CREWES Research Report, 219 pp.
Matheron, G., 1967, Principles of geostatistics: Economic Geology, 58, 1246–1266, doi: 10.2113/gsecongeo.58.8.1246.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E., 1953. Equation of state calculations by fast computing machines: The Journal of Chemical Physics, 21, 1087–1092, doi: 10.1063/1.1699114.
Muir, F., and J. Dellinger, 1985, A practical anisotropic system: SEP-44. Stanford Exploration Project, p. 55–58.
Nelder, J. A., and R. Mead, 1965, A simplex method for function minimization: The Computer Journal, 7, 308–313, doi: 10.1093/comjnl/7.4.308.
Neumaier, A., O. Shcherbina, W. Huyer, and T. Vinkó, 2005, A comparison of complete global optimization solvers: Mathematical Programming, 103, 335–356, doi: 10.1007/s10107-005-0585-4.
Pintér, J. D., 1995, Global optimization in action: Continuous and Lipschitz optimization: Algorithms, Implementations and Applications, Kluwer Academic Publisher, 422 pp.
Pintér, J. D, K. Holmström, A. O. Göran, and M. M. Edvall, 2006, User’s Guide for TOMLAB/LGO: TOMLAB Optimization.
Powell, M. J. D., 2006, The NEWUOA software for unconstrained optimization without derivatives, in Di Pillo, G., and M. Roma, eds., Large-Scale Nonlinear Optimization: Nonconvex Optimization and Its Applications, Springer, Boston, MA, 83, p. 255–297, doi: 10.1007/0-387-30065-1_16.
Powell, M. J. D., 2008, Developments of NEWUOA for minimization without derivatives: IMA Journal of Numerical Analysis, 28, 649–664, doi: 10.1093/imanum/drm047.
Rios, L. M., and N. V. Sahinidis, 2013, Derivative-free optimization: A review of algorithms and comparison of software implementations: Journal of Global Optimization, 56, 1247–1293, doi: 10.1007/s10898-012-9951-y.
Romeo, F., and A. Sangiovanni-Vincentelli, 1991, A theoretical framework for simulated annealing: Algorithmica, 6, 302–345, doi: 10.1007/BF01759049.
Schonlau, M., 1997, Compute Experiments and Global Optimization: Ph.D. thesis, University of Waterloo, 131 pp.
Slotboom, R. T., 1990, Converted wave moveout estimation: 60th Annual International Meeting, Society of Exploration Geophysics, Expanded Abstract, p. 1104–1106.
Sotirov, R., and T. Terlaky, 2013, Multi-start approach for an integer determinant maximization problem: Optimization, 62, 101–114, doi: 10.1080/02331934.2011.568617.
Spendley, W., G. R. Hext, and F. R. Himsworth, 1962, Sequential Application of Simplex Designs in Optimisation and Evolutionary Operation: Technometrics, 4, 4, 441–461, doi: 10.1080/00401706.1962.10490033.
Terlaky, T., and R. Sotirov, 2010, Multi-start approach to global conic optimization: ISE ?Technical Reports, Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, Report: 10T-001, 16 pp.
Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966, doi: 10.1190/1.1442051.
Thorbecke, J. W., and D. Draganov, 2011, Finite-difference modeling experiments for seismic interferometry: Geophysics, 76, H1–H18, doi: 10.1190/geo2010-0039.1.
Tseng, P.-Y., Y.-F. Chang, C.-H. Chang, and R.-C. Shih, 2016, Traveltimes and conversion-point positions for P-SV converted wave propagation in a transversely isotropic medium: Numerical calculations and physical model studies: Exploration Geophysics, 49, 1, 30–41, doi: 10.1071/EG15123.
Ursin, B., and A. Stovas, 2006, Traveltime approximations for a layered transversely isotropic medium: Geophysics, 71, 2, 23–33, doi: 10.1190/1.2187716.
Vaz, A. I. F., and L. N. Vicente, 2007, A particle swarm pattern search method for bound constrained global optimization: Journal of Global Optimization, 39, 197–219, doi: 10.1007/s10898-007-9133-5.
Wang, W., and L. D. Pham, 2001, Converted-wave prestack imaging and velocity analysis by pseudo-offset migration: 63rd EAGE Conference, Amsterdam, Netherlands, Expanded Abstract, L-12, cp-15-00295, doi: 10.3997/2214-4609-pdb.15.L-12.
Wang, P., J. Liu, Y. Chen, and T. Hu, 2014, Converted-wave imaging technology and application for complex structures: Exploration Geophysics, 45, 2, 105–115, doi: 10.1071/EG13052.
?Winslow, T. A., R. J. Trew, P. Gilmore?,? and C. T. Kelley, 1991, Simulated performance optimization of GaAs MESFET amplifiers: IEEE/Cornell Conference on Advanced Concepts in High Speed Semiconductor Devices and Circuits, Ithaca, NY, USA, p. 393–402, 10.1109/CORNEL.1991.170009.
Xu, S., and A. Stovas, 2018, Estimation of the conversion point position in elastic orthorhombic media: Geophysics, 84, 1, C15–C25, doi: 10.1190/geo2018-0375.1.
Xu, S., and A. Stovas, 2019, Traveltime approximation for converted waves in elastic orthorhombic media: Geophysics, 84, 5, C229–C237, doi: 10.1190/geo2018-0828.1.
Zuniga, N. R. C. F., 2017, Análise comparativa de aproximações não-hiperbólicas dos tempos de trânsito de dados sísmicos multicomponente utilizando tecnologia OBN: M.S. dissertation, Universidade de São Paulo, SP, Brazil, 86 pp.
Zuniga, N. R. C. F., 2021, Nonhyperbolic multiparametric travel-time approximation for converted-wave and OBN data: Ph.D. thesis, Universidade de São Paulo, SP, Brazil, 277 pp.
Zuniga, N. R. C. F., E. C. Molina, and R. L. Prado, 2017, Comparison of travel-time approximations for unconventional reservoirs from Santos Basin, Brazil: Revista Brasileira de Geofísica, 35, 273–286, doi: 10.22564/rbgf.v35i4.906.
Zuniga, N. R. C. F., F. B. Ribeiro, and V. I. Priimenko, 2018, Relation between the model and the topography of the objective function in a velocity analysis using a nonhyperbolic multicomponent travel-time approximation: Revista Brasileira de Geofísica, 36, 375–384, doi: 10.22564/rbgf.v36i4.988.
Zuniga, N. R. C. F., E. C. Molina, and R. L. Prado, 2019a, Comparative analysis of nonhyperbolic traveltime approximations of multicomponent seismic data using OBN Technology: Brazilian Journal of Geophysics, 37, 397–407, doi: 10.22564/rbgf.v37i4.2017.
Zuniga, N. R. C. F., F. B. Ribeiro, and V. I. Priimenko, 2019b, Comparison of L2- and L1-norm to perform the inversion of travel-time curves using nonhyperbolic multiparametric approximations with unimodal and multimodal behavior: Brazilian Journal of Geophysics, 37, 397–407, doi: 10.22564/rbgf.v37i3.2008.
Zuniga, N. R. C. F., F. B. Ribeiro, and V. I. Priimenko, 2019c, L2- and L1-norm applied to inversion of nonhyperbolic travel-time: Brazilian Journal of Geophysics, 37, 155–161, doi: 10.22564/rbgf.v37i2.1998.
DOI: http://dx.doi.org/10.22564/brjg.v40i3.2168
This work is licensed under a Creative Commons Attribution 4.0 International License.
a partir do v.37n.4 (2019) até o presente
v.15n.1 (1997) até v.37n.3 (2019)
Brazilian Journal of Geophysics - BrJG
Sociedade Brasileira de Geofísica - SBGf
Av. Rio Branco 156 sala 2509
Rio de Janeiro, RJ, Brazil
Phone/Fax: +55 21 2533-0064
E-mail: editor@sbgf.org.br
Since 2022, the BrJG publishes all content under Creative Commons CC BY license. All copyrights are reserved to authors.
