1887
Volume 16 Number 1
  • ISSN: 1569-4445
  • E-ISSN: 1873-0604

Abstract

ABSTRACT

We propose a new method for including sparsity constraints into potential field data inversion using a Laplacian kernel. The method obtains three‐dimensional density distributions, which are at best a proxy for geological structures. Compressive sensing has been used in many domains to recover the original data from the acquisition data. Compressive sensing is an inversion philosophy where sparsity constraints are applied, and thus, we use this principle into geophysical inversion. Here, we extend this algorithm to potential field data inversion. We introduced the Laplacian smoothed L0 norm into the stabiliser function as a sparse constraint, and the density constraint function has been introduced into our inversion method to guarantee the inverted density to be in the geological and physical meaning range. Compared to the traditional smooth inverse algorithm, our proposed method can obtain accurate geological structures with sharp boundary and sub‐surface “block” structures. This method permits reconstruction of (non‐smooth) density functions that represent a blocky geological structure. Our results using synthetic gravity data show that the Laplacian smoothed L0 norm inversion method with sparsity constraints predicts more focused and accurate depth and density anomalies than smooth inversion method. Application of this Laplacian smoothed L0 norm sparsity constraints method to the inversion of gravity data collected over the Humble salt dome, Harris County, near Houston, United States, leads to improved interpretation of geological structures. These results confirm the validity of the proposed method and its potential application for other potential field data inversions to explore geological structures.

Loading

Article metrics loading...

/content/journals/10.3997/1873-0604.2017038
2017-07-01
2024-03-29
Loading full text...

Full text loading...

References

  1. AbdelrahmanE.M., BayoumiA.I. and El‐ArabyH.M.1991. A least‐squares minimization approach to invert gravity data. Geophysics56(1), 115–118.
    [Google Scholar]
  2. AbdelrahmanE.M., El‐ArabyH.M., El‐ArabyT.M. and Abo‐EzzE.R.2001. Three least‐squares minimization approaches to depth, shape, and amplitude coefficient determination from gravity data. Geophysics66(4), 1105–1109.
    [Google Scholar]
  3. BarbosaV.C. and SilvaJ.B.2006. Interactive 2D magnetic inversion: a tool for aiding forward modeling and testing geologic hypotheses. Geophysics71(5), L43–L50.
    [Google Scholar]
  4. BassreiA.1993. Regularization and inversion of 2‐D gravity data. In: 1993 SEG annual meeting Society of Exploration Geophysicists.
    [Google Scholar]
  5. Bertete‐AguirreH., CherkaevE. and OristaglioM.2002. Non‐smooth gravity problem with total variation penalization functional. Geophysical Journal International149(2), 499–507.
    [Google Scholar]
  6. BhattacharyyaB.K.1980. A generalized multi‐body model for inversion of magnetic anomaliesGeophysics45(2), 255–270.
    [Google Scholar]
  7. BlakeA. and ZissermanA.1987. Visual Reconstruction, Vol. 2. Cambridge: MIT Press.
    [Google Scholar]
  8. BoulangerO. and ChouteauM.2001. Constraints in 3D gravity inversionGeophysical Prospecting49(2), 265–280.
    [Google Scholar]
  9. CandesE.J. and TaoT.2005. Decoding by linear programming. IEEE Transactions on Information Theory51(12), 4203–4215.
    [Google Scholar]
  10. CandesE. and RombergJ.2005. l1‐magic: recovery of sparse signals via convex programming. www.acm.caltech.edu/l1magic/downloads/l1magic.
    [Google Scholar]
  11. CaoJ., WangY. and WangB.2014. Accelerating seismic interpolation with a gradient projection method based on tight frame property of curvelet. Exploration Geophysics.
    [Google Scholar]
  12. Caratori TontiniF., RondeC.E.J, YoergerD., KinseyJ. and TiveyM.2012. 3‐D focused inversion of near‐seafloor magnetic data with application to the Brothers volcano hydrothermal system, Southern Pacific Ocean, New Zealand. Journal of Geophysical Research: Solid Earth (1978–2012)117(B10).
    [Google Scholar]
  13. CardarelliE. and FischangerF.2006. 2D data modelling by electrical resistivity tomography for complex subsurface geology. Geophysical Prospecting54(2), 121–133.
    [Google Scholar]
  14. ChenS., DonohoD.L. and SaundersM.A.1996. Atomic Decomposition by Basis Pursuit (Technical Report). Stanford, CA: Department of Statistics, Stanford University.
    [Google Scholar]
  15. ChenS.S., DonohoD.L. and SaundersM.A.1999. Atomic decomposition by basis pursuit. SIAM Journal on Scientific Computing20(1), 33–61.
    [Google Scholar]
  16. ConstableS.C., ParkerR.L. and ConstableC.G.1987. Occam’s inversion: a practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics52(3), 289–300.
    [Google Scholar]
  17. DaubechiesI., DefriseM. and De MolC.2003. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. arXiv preprint math/0307152.
    [Google Scholar]
  18. DonohoD.L. and EladM.2002. Maximal sparsity representation via l1 minimization. IEEE Transactions on Information Theory.
    [Google Scholar]
  19. DonohoD.L., EladM. and TemlyakovV.N.2006. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Transactions on Information Theory52(1), 6–18.
    [Google Scholar]
  20. DonohoD.L.2006. For most large underdetermined systems of linear equations the minimal ℓ1‐norm solution is also the sparsest solutionCommunications on Pure and Applied Mathematics59(6), 797–829.
    [Google Scholar]
  21. EssaK.S.2007. Gravity data interpretation using the s‐curves method. Journal of Geophysics and Engineering4(2), 204.
    [Google Scholar]
  22. FarquharsonC.G.2007. Constructing piecewise‐constant models in multidimensional minimum‐structure inversions. Geophysics73(1), K1–K9.
    [Google Scholar]
  23. Field, D.J.1994. What is the goal of sensory coding?Neural Computation6(4), 559–601.
    [Google Scholar]
  24. FigueiredoM.A. and NowakR.D.2003. An EM algorithm for wavelet‐based image restoration. IEEE Transactions on Image Processing12(8), 906–916.
    [Google Scholar]
  25. FigueiredoM.A. and NowakR.D.2005. A bound optimization approach to wavelet‐based image deconvolution. In: IEEE International Conference on Image Processing, ICIP 2005. Vol. 2, pp II–782.
    [Google Scholar]
  26. GabryschR.K.1980. Approximate land‐surface subsidence in the Houston–Galveston region, Texas 1906–1978 1943–1978 and 1973–1978.
    [Google Scholar]
  27. Gallardo‐DelgadoL.A., Pérez‐FloresM.A. and Gómez‐TreviñoE.2003. A versatile algorithm for joint 3D inversion of gravity and magnetic data. Geophysics68(3), 949–959.
    [Google Scholar]
  28. GorodnitskyI.F. and RaoB.D.1997. Sparse signal reconstruction from limited data using FOCUSS: a re‐weighted minimum norm algorithm. IEEE Transactions on Signal Processing45(3) 600–616.
    [Google Scholar]
  29. GuillenA. and MenichettiV.1984. Gravity and magnetic inversion with minimization of a specific functional. Geophysics49(8), 1354–1360.
    [Google Scholar]
  30. HaázI.B.1953. Relations between the potential of the attraction of the mass contained in a finite rectangular prism and its first and second derivatives. Geophysical Transactions II7, 57–66.
    [Google Scholar]
  31. HuangZ.2012. Multidisciplinary Investigation of Surface Deformation above Salt Domes in Houston, Texas (Doctoral dissertation).
    [Google Scholar]
  32. HuettelS.A., SongA.W. and McCarthyG.2004. Functional Magnetic Resonance Imaging, Vol. 1. Sunderland: Sinauer Associates.
    [Google Scholar]
  33. JinS., van DamT. and WdowinskiS.2013. Observing and understanding the Earth system variations from space geodesy. Journal of Geodynamics721‐10.
    [Google Scholar]
  34. KimH.J., SongY. and LeeK.H.1999. Inequality constraint in least‐squares inversion of geophysical data. Earth, Planets and Space51(4), 255–259.
    [Google Scholar]
  35. KolesovaV. and CherkaevaE.1987. A vector model of the magnetic field based on a system of dipole sources. Geomagnetism and Aeronomy27, 151–153.
    [Google Scholar]
  36. LastB.J. and KubikK.1983. Compact gravity inversion. Geophysics48(6), 713–721.
    [Google Scholar]
  37. LelièvreP.G. and OldenburgD.W.2009. A comprehensive study of including structural orientation information in geophysical inversions. Geophysical Journal International178(2), 623–637.
    [Google Scholar]
  38. LewickiM.S. and SejnowskiT.J.2000. Learning overcomplete representations. Neural Computation12(2), 337–365.
    [Google Scholar]
  39. LeW. and ZhdanovM. S.2008. Focusing inversion of marine full‐tensor gradiometry data in offshore geophysical exploration. 2008 SEG annual meeting. Society of Exploration Geophysicists.
    [Google Scholar]
  40. LiX. and ChouteauM.1998. Three‐dimensional gravity modeling in all space. Surveys in Geophysics19(4), 339–368.
    [Google Scholar]
  41. LiY. and OldenburgD.W.1996. 3‐D inversion of magnetic data. Geophysics, 61(2), 394–408.
    [Google Scholar]
  42. LiY. and OldenburgD.W.1998. 3‐D inversion of gravity dataGeophysics63(1), 109–119.
    [Google Scholar]
  43. LiY., CichockiA. and AmariS.I.2003. Sparse component analysis for blind source separation with less sensors than sources. In: ICA2003, pp. 89–94.
    [Google Scholar]
  44. McGaugheyJ. and MilkereitB.2007. Geological models, rock properties and the 3D inversion of geophysical data. In: Proceedings of Exploration, Vol. 7, pp. 473–483.
    [Google Scholar]
  45. MenkeW.1989. Geophysical Data Analysis: Discrete Inverse Theory.Academic Press.
    [Google Scholar]
  46. MohimaniH., Babaie‐ZadehM. and JuttenC.2009. A fast approach for overcomplete sparse decomposition based on smoothed norm. IEEE Transactions on Signal Processing57(1), 289–301.
    [Google Scholar]
  47. MohanN.L., AnandababuL. and RaoS.S.1986. Gravity interpretation using the Mellin transform. Geophysics51(1), 114–122.
    [Google Scholar]
  48. NettletonL.L.1976. Gravity and Magnetics in Oil ProspectingMcGraw‐Hill Companies.
    [Google Scholar]
  49. OlshausenB.A.1996. Emergence of simple‐cell receptive field properties by learning a sparse code for natural images. Nature381(6583), 607–609.
    [Google Scholar]
  50. OruçB.2010. Depth estimation of simple causative sources from gravity gradient tensor invariants and vertical component. Pure and Applied Geophysics167(10), 1259–1272.
    [Google Scholar]
  51. PilkingtonM.2008. 3D magnetic data‐space inversion with sparseness constraints. Geophysics74(1), L7–L15.
    [Google Scholar]
  52. PortniaguineO. and ZhdanovM.S.1999. Focusing geophysical inversion images. Geophysics64(3), 874–887.
    [Google Scholar]
  53. PortniaguineO. and ZhdanovM.S.2002. 3‐D magnetic inversion with data compression and image focusing. Geophysics67(5), 1532–1541.
    [Google Scholar]
  54. RodiW. and MackieR.L.2001. Nonlinear conjugate gradients algorithm for 2‐D magnetotelluric inversion. Geophysics66(1), 174–187.
    [Google Scholar]
  55. SacchiM.D. and UlrychT.J.1996. Estimation of the discrete Fourier transform, a linear inversion approach. Geophysics61(4), 1128–1136.
    [Google Scholar]
  56. SalemA., RavatD., MushayandebvuM.F. and UshijimaK.2004. Linearized least‐squares method for interpretation of potential‐field data from sources of simple geometry. Geophysics69(3), 783–788.
    [Google Scholar]
  57. ShamsipourP., MarcotteD., ChouteauM., RivestM. and BoucheddaA.2013. 3D stochastic gravity inversion using nonstationary covariances. Geophysics78(2), G15–G24.
    [Google Scholar]
  58. ShawR.K. and AgarwalB.N.P.1997. A generalized concept of resultant gradient to interpret potential field maps. Geophysical Prospecting45(6), 1003–1011.
    [Google Scholar]
  59. TarantolaA.2005. Inverse Problem Theory and Methods for Model Parameter Estimation.siam.
    [Google Scholar]
  60. TikhonovA.N. and ArseninV.I.1977. Solutions of Ill‐Posed Problems.Vh Winston.
    [Google Scholar]
  61. VignoliG. and ZanziL.2005. Focusing inversion technique applied to radar tomographic data. In: Near surface 2005‐11th European meeting of environmental and engineering geophysics.
    [Google Scholar]
  62. WangY., LiuP., LiZ., SunT., YangC. and ZhengQ.2013. Data regularization using Gaussian beams decomposition and sparse norms. Journal of Inverse and Ill‐Posed Problems21(1), 1–23.
    [Google Scholar]
  63. ZeyenH. and PousJ.1991. A new 3‐D inversion algorithm for magnetic total field anomalies. Geophysical Journal International104(3), 583–591.
    [Google Scholar]
  64. ZhangC., SongS., WenX., YaoL. and LongZ.2015. Improved sparse decomposition based on a smoothed L0 norm using a Laplacian kernel to select features from fMRI data. Journal of Neuroscience Methods245, 15–24.
    [Google Scholar]
  65. ZhangY., YanJ., LiF., ChenC., MeiB., JinS. and DohmJ.H.2015. A new bound constraints method for 3‐D potential field data inversion using Lagrangian multipliers. Geophysical Journal International201 (1), 267–275.
    [Google Scholar]
  66. ZhdanovM.S.2002. Geophysical Inverse Theory and Regularization Problems, Vol. 36. Elsevier.
    [Google Scholar]
  67. ZhdanovM.S., EllisR. and MukherjeeS.2004. Three‐dimensional regularized focusing inversion of gravity gradient tensor component data. Geophysics69(4), 925–937.
    [Google Scholar]
  68. ZhdanovM.S.2009. New advances in regularized inversion of gravity and electromagnetic data. Geophysical Prospecting57(4), 463–478.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/journals/10.3997/1873-0604.2017038
Loading
/content/journals/10.3997/1873-0604.2017038
Loading

Data & Media loading...

  • Article Type: Research Article

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error