1887
Volume 20, Issue 1
  • ISSN: 1569-4445
  • E-ISSN: 1873-0604

Abstract

ABSTRACT

Joint inversion strategies and physical constraints on model parameters may be used to mitigate equivalence problems caused by solution non‐uniqueness. This strategy is quite a common practice in exploration geophysics, where dedicated rock physical studies are usually carried out, while it is not so frequent in near surface geophysics. We use porosity as a constraint among seismic wave velocities and electrical resistivity in a deterministic joint inversion algorithm for surface wave dispersion, P‐wave traveltimes and apparent resistivity from vertical electrical sounding. These data are often available for near surface characterization. We show that the physical constraint among model parameters leads to internally consistent geophysical models in which solution non‐uniqueness is mitigated. Moreover, an estimate of soil porosity is obtained as a relevant side product of the procedure. In particular, we consider a clean sand deposit and hence the appropriate formulations for the computation of porosity from seismic velocities and resistivity are implemented in the algorithm. We first demonstrate how the non‐uniqueness of the solution is reduced in a synthetic case and then we applied the algorithm to a real‐case study. The algorithm is here developed for one‐dimensional condition and for granular soils to better investigate the physical constraint only, but it can be extended to the two‐dimensional or three‐dimensional case as well as to other materials with the adoption of proper rock physical relationships.

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2022-01-14
2024-03-28
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References

  1. Archie, G.E. (1942) The electrical resistivity log as an aid in determining some reservoir characteristics. Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers, 146, 54–62.
    [Google Scholar]
  2. Aster, R.A., Brochers, B. & Thurber, C.H. (2005) Parameter estimation and inverse problems. 1, Cambridge: Elsevier Academic Press.
    [Google Scholar]
  3. Auken, E. & Christiansen, A.V. (2004) Layered and laterally constrained 2D inversion of resistivity data. Geophysics, 69(3), 752–761. https://doi.org/10.1190/1.1759461
    [Google Scholar]
  4. Bachrach, R., Dvorkin, J. & Nur, A. (1998) High‐resolution shallow‐seismic experiments in sand, Part II: Velocities in shallow unconsolidated sand. Geophysics, 63(4), 1234–1240. https://doi.org/10.1190/1.1444424
    [Google Scholar]
  5. Bachrach, R., Dvorkin, J. & Nur, A.M. (2000) Seismic velocities and Poisson’s ratio of shallow unconsolidated sands. Geophysics, 65(2), 559–564. https://doi.org/10.1190/1.1444751
    [Google Scholar]
  6. Backus, G. & Gilbert, F. (1970) Uniqueness in the inversion of inaccurate gross Earth data. Philosophical Transactions of the Royal Society, 266, 123–192. https://doi.org/10.1098/rsta.1970.0005
    [Google Scholar]
  7. Bergamo, P. & Socco, L.V. (2016) P‐ and S‐wave velocity models of shallow dry sand formations from surface wave multimodal inversion. Geophysics, 81(4), R197–R209. https://doi.org/10.1190/geo2015‐0542.1
    [Google Scholar]
  8. Bellotti, R. & Selleri, G. (1969) Correlazione tra le caratteristiche geotecniche di alcuni terreni di fondazione e confronto tra i risultati ottenibili con l'applicazione di diversi metodi di calcolo del carico ammissibile. Rivista Italiana di Geotecnica, 2, 95–103.
    [Google Scholar]
  9. Berryman, J.G., Berge, P.A. & Bonner, B.P. (2002) Estimating rock properties and fluid saturation using only seismic velocities. Geophysics, 67(2), 391–404. https://doi.org/10.1190/1.1468599
    [Google Scholar]
  10. Biot, M.A. (1956a) Theory of propagation of elastic waves in a fluid‐saturated porous solid. I. Low‐frequency range. Journal of the Acoustical Society of America, 28(2), 168–178.
    [Google Scholar]
  11. Biot, M.A. (1956b) Theory of propagation of elastic waves in a fluid‐saturated porous solid. II. Higher frequency range. Journal of the Acoustical Society of America, 28(2), 179–191.
    [Google Scholar]
  12. Boiero, D. & Socco, L.V. (2014) Joint Inversion of Rayleigh‐wave dispersion and P‐wave refraction data for laterally varying layered models. Geophysics, 79(4), EN49–EN59. https://doi.org/10.1190/geo2013‐0212.1
    [Google Scholar]
  13. Bruggeman, D.A.G. (1935) Berechnung verschiederner physikalischer konstanten von heterogenen substanzen. Annalen Der Physik, 24, 636–679.
    [Google Scholar]
  14. Bussian, A.E. (1983) Electrical conductance in a porous medium. Geophysics, 48(9), 1258–1268. https://doi.org/10.1190/1.1441549
    [Google Scholar]
  15. Carmichael, R.S. (1982) Handbook of physical properties of rock. Boca Raton, FL: CRC Press.
    [Google Scholar]
  16. Colombo, P. (1964) Studio delle caratteristiche dell'argilla limosa degli argini del Po: Istituto di costruzioni marittime. Università di Padova.
    [Google Scholar]
  17. Colombo, D. & Rovetta, D., (2018) Coupling strategies in multiparameter geophysical joint inversion. Geophysical Journal International, 215(2), 1171–1184. https://doi.org/10.1093/gji/ggy341
    [Google Scholar]
  18. Comina, C., Foti, S., Socco, L.V. & Strobbia, C. (2004) Geophysical characterization for seepage potential assessment along the embankments of the Po River. Proceedings ISC‐2 on Geotechnical and Geophysical Site Characterization, 451–458. http://hdl.handle.net/2318/58979
    [Google Scholar]
  19. Comina, C., Cosentini, R.M., Foti, S. & Musso, G. (2010) Electrical Tomography as laboratory monitoring tool. Rivista Italiana di Geotecnica, 44, 15–26.
    [Google Scholar]
  20. Dal Moro, G. (2008) VS and VP vertical profiling via joint inversion of Rayleigh waves and refraction travel times by means of bi‐objective evolutionary algorithm. Journal of Applied Geophysics, 66, 15–24. https://doi.org/10.1016/j.jappgeo.2008.08.002
    [Google Scholar]
  21. de Nardis, R., Cardarelli, E. & Dobroka, M. (2005) Quasi‐2D hybrid joint inversion of seismic and geoelectric data. Geophysical Prospecting, 53, 705–716. https://doi.org/10.1111/j.1365‐2478.2005.00497.x
    [Google Scholar]
  22. de Lima, O.A.L. (1995) Water saturation and permeability from resistivity, dielectric, and porosity logs. Geophysics, 60(6), 1756–1764. https://doi.org/10.1190/1.1443909
    [Google Scholar]
  23. Dell'Aversana, P., Bernasconi, G., Miotti, F. & Rovetta, D. (2011) Joint inversion of rock properties from sonic, resistivity and density well‐log measurements. Geophysical Prospecting, 59, 1144–1154. https://doi.org/10.1111/j.1365‐2478.2011.00996.x
    [Google Scholar]
  24. Dobróka, M., Gyulai, Á., Ormos, T., Csókás, J. & Dresen, L. (1991) Joint inversion of seismic and geoelectric data recorded in an underground coal mine. Geophysical Prospecting, 39(5), 643–665. https://doi.org/10.1111/j.1365‐2478.1991.tb00334.x
    [Google Scholar]
  25. Doetsch, J., Linde, N. & Binley, A. (2010) Structural joint inversion of time‐lapse crosshole ERT and GPR traveltime data. Geophysical Research Letters, 37, L24404. https://doi.org/10.1029/2010GL045482
    [Google Scholar]
  26. Domenico, S.N. (1984) Rock lithology and porosity determination from shear and compressional wave velocity. Geophysics, 49(8), 1188–1195. https://doi.org/10.1190/1.1441748
    [Google Scholar]
  27. Eberhart‐Phillips, D., Han, D.‐H. & Zoback, M.D. (1989) Empirical relationships among seismic velocity, effective pressure, porosity and clay content in sandstone. Geophysics, 54(1), 82–89. https://doi.org/10.1190/1.1442580
    [Google Scholar]
  28. Feng, X., Ren, Q., Liu, C. & Zhang, X. (2017) Joint acoustic full‐waveform inversion of crosshole seismic and ground‐penetrating radar data in the frequency domain. Geophysics, 82(6), H41–H56. https://doi.org/10.1190/geo2016‐0008.1
    [Google Scholar]
  29. Foti, S., Lai, C. & Lancellotta, R. (2002) Porosity of fluid‐saturated porous media from measured seismic wave velocities. Geotechnique, 52(5), 359–373. https://doi.org/10.1680/geot.2002.52.5.359
    [Google Scholar]
  30. Friedman, S.P. (2005) Soil properties influencing apparent electrical conductivity: a review. Computers and Electronics in Agriculture, 46(1‐3), 45–70. https://doi.org/10.1016/j.compag.2004.11.001
    [Google Scholar]
  31. Gallardo, L.A. & Meju, M.A. (2003) Characterization of heterogeneous near‐surface materials by joint 2D inversion of dc resistivity and seismic data. Geophysical Research Letters, 30(13), 1658. https://doi.org/10.1029/2003GL017370
    [Google Scholar]
  32. Gallardo, L.A. & Meju, M.A. (2004) Joint two‐dimensional DC resistivity and seismic travel time inversion with cross‐gradients constraints. Journal of Geophysical Research, 109(B3), B03311. https://doi.org/10.1029/2003JB002716
    [Google Scholar]
  33. Gao, G., Abubakar, A. & Habashy, T. (2012) Joint petrophysical inversion of electromagnetic and full‐waveform seismic data. Geophysics, 77(3), WA3–WA18. https://doi.org/10.1190/geo2011‐0157.1
    [Google Scholar]
  34. Garofalo, F., Sauvin, G., Socco, L.V. & Lecomte, I. (2015) Joint inversion of seismic and electrical data applied to 2D media. Geophysics, 80(4), EN93–EN104. https://doi.org/10.1190/geo2014‐0313.1
    [Google Scholar]
  35. Gase, A.C., Bradford, J.H. & Brand, B.D. (2018) Estimation of porosity and water saturation in dual‐porosity pyroclastic deposits from joint analysis of compression, shear, and electromagnetic velocities. Geophysics, 83(3), ID1–ID11. https://doi.org/10.1190/geo2017‐0234.1
    [Google Scholar]
  36. Gassmann, F. (1951) Uber die Elastizitat poroser Medien. Vierteljahrsschrift der Naturforschenden Gesellschaft, 96, 1–23.
    [Google Scholar]
  37. Haber, E. & Oldenburg, D. (1997) Joint inversion: a structural approach. Inverse Problems, 13(1), 63–77. https://doi.org/10.1088/0266‐5611/13/1/006
    [Google Scholar]
  38. Hadamard, J. (1902) Sur les problèmes aux dériveès partielles et leur signification physique. Princeton University Bulletin, 13, 49–52.
    [Google Scholar]
  39. Haskell, N.A. (1953) The dispersion of surface waves on multilayered media. Bulletin of the Seismological Society of America, 43(1), 17–34. https://doi.org/10.1785/BSSA0430010017
    [Google Scholar]
  40. Hellman, K., Ronczka, M., Günther, T., Wennermark, M., Rücker, C. & Dahlin, T. (2017) Structurally coupled inversion of ERT and refraction seismic data combined with cluster‐based model integration. Journal of Applied Geophysics, 143, 169–181. https://doi.org/10.1016/j.jappgeo.2017.06.008
    [Google Scholar]
  41. Hofmann, B.A., Sego, D.C. & Robertson, P.K. (2000) In situ ground freezing to obtain undisturbed samples of loose sand. Journal of Geotechnical and Geoenvironmental Engineering, 126(11), 979–989. https://doi.org/10.1061/(ASCE)1090‐0241(2000)126:11(979)
    [Google Scholar]
  42. Hu, W., Abubakar, A. & Habashy, T.M. (2009) Joint electromagnetic and seismic inversion using structural constraints. Geophysics, 74(6), R99–R109. https://doi.org/10.1190/1.3246586
    [Google Scholar]
  43. Jackson, P.D., Taylor Smith, D. & Stanford, P.N. (1978) Resistivity‐porosity‐particle shape relationships for marine sands. Geophysics, 43(6), 1250–1268. https://doi.org/10.1190/1.1440891
    [Google Scholar]
  44. Kennedy, W.D. & Herrick, D.C. (2012) Conductivity models for Archie rocks. Geophysics, 77(3), WA109–WA128. https://doi.org/10.1190/geo2011‐0297.1
    [Google Scholar]
  45. Koefoed, O.C. (1979) Geosounding principles, 1 ‐ resistivity sounding measurements. Amsterdam: Elsevier.
    [Google Scholar]
  46. Lesmes, D.P. & Friedman, S.P. (2005) Relationships between the electrical and hydrogeological properties of rocks and soils. Hydrogeophysics, Water Science and Technology Library. 50, 87–128. https://doi.org/10.1007/1‐4020‐3102‐5_4
    [Google Scholar]
  47. Levenberg, K. (1944) A method for the solution of certain nonlinear problems in least squares. Quarterly of Applied Mathematics, 2, 164–168.
    [Google Scholar]
  48. Marquardt, D.W. (1963) An algorithm for least squares estimation of nonlinear parameters. Journal of the Society of Industrial Applied Mathematics, 11(2), 431–441. https://doi.org/10.1137/0111030
    [Google Scholar]
  49. Mavko, G., Mukerji, T. & Dvorkin, J. (2009) The rock physics handbook: tools for seismic analysis of porous media, 2nd edition. 2, Cambridge: Cambridge University Press.
    [Google Scholar]
  50. Moorkamp, M., Heincke, B., Jegen, M., Roberts, A. & Hobbs, R.W. (2011) A framework for 3‐D joint inversion of MT, gravity and seismic refraction data. Geophysical Journal International, 184(1), 477–493. https://doi.org/10.1111/j.1365‐246X.2010.04856.x
    [Google Scholar]
  51. Moorkamp, M., Heincke, B., Jegen, M., Hobbs, R.W. & Roberts, A.W. (2016) Joint inversion in hydrocarbon exploration. In: Moorkamp, M., Lelièvre, P. G., Linde, N. & Khan, A. (Eds.) Integrated imaging of the earth: theory and applications, 1st edition. Hoboken, NJ: John Wiley & Sons, Inc, pp. 167–189.
    [Google Scholar]
  52. Mori, K. & Sakai, K. (2016) The GP sampler: a new innovation in core sampling. Australian Geomechanics Society, 51(4), 131–166.
    [Google Scholar]
  53. Olsen, P.A. (2011) Coarse‐scale resistivity for saturation estimation in heterogeneous reservoirs based on Archie's formula. Geophysics, 76(2), E35–E43. https://doi.org/10.1190/1.3541966
    [Google Scholar]
  54. Piatti, C., Socco, L.V., Boiero, D. & Foti, S. (2013) Constrained 1D joint inversion of seismic surface waves and P‐refraction travel times. Geophysical Prospecting, 61(1), 77–93. https://doi.org/10.1111/j.1365‐2478.2012.01071.x
    [Google Scholar]
  55. Reynolds, J.M. (1997) An Introduction to Applied and Environmental Geophysics. 1, Chichester, England: John Wiley and Sons Ltd, 1–712. 978‐0‐471‐48535‐3
    [Google Scholar]
  56. Ronczka, M., Wisén, R. & Dahlin, T. (2018) Geophysical pre‐investigation for a Stockholm tunnel project: joint inversion and interpretation of geoelectric and seismic refraction data in an urban environment. Near Surface Geophysics, 16(3), 258–268. https://doi.org/10.3997/1873‐0604.2018009
    [Google Scholar]
  57. Salem, H.S. (2000) Poisson's ratio and the porosity of surface soils and shallow sediments, determined from seismic compressional and shear wave velocities. Geotechnique, 50(4), 461–463. https://doi.org/10.1680/geot.2000.50.4.461
    [Google Scholar]
  58. Santamarina, J.C. (2001) Soils and waves: particulate materials behaviour, characterization and process monitoring. Hoboken, NJ: Wiley and Sons. ISBN: 978‐0‐471‐49058‐6
    [Google Scholar]
  59. Senkaya, M., Karsli, H., Socco, L.V. & Foti, S. (2020) Obtaining reliable S‐wave velocity depth profile by joint inversion of geophysical data: the combination of active surface‐wave, seismic refraction and electric sounding data. Near Surface Geophysics, 18, 659–682. https://doi.org/10.1002/nsg.12126
    [Google Scholar]
  60. Singh, S., Seed, H.B. & Chan, C. (1982) Undisturbed sampling of saturated sands by freezing. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 108(2), 247–264. https://doi.org/10.1061/AJGEB6.0001242
    [Google Scholar]
  61. Socco, L.V., Boiero, D., Foti, S. & Wisén, R. (2009) Laterally constrained inversion of ground roll from seismic reflection records. Geophysics, 74, G35–G45. https://doi.org/10.1190/1.3223636
    [Google Scholar]
  62. Tarantola, A. (1987) Inverse problem theory: methods for data fitting and model parameter estimation. Amsterdam: Elsevier Science Publisher.
    [Google Scholar]
  63. Thomson, W.T. (1950) Transmission of elastic waves through a stratified solid medium. Journal of Applied Physics, 21, 89–93. https://doi.org/10.1063/1.1699629
    [Google Scholar]
  64. Toksöz, M.N., Cheng, C.H. & Timur, A. (1976) Velocities of seismic waves in porous rocks. Geophysics, 41(4), 621–645. https://doi.org/10.1190/1.1440639
    [Google Scholar]
  65. Viezzoli, A., Christiansen, A.V., Auken, E. & Sørensen, K. (2008) Quasi‐3D modeling of airborne TEM data by spatially constrained inversion. Geophysics, 73(3), F105–F113. https://doi.org/10.1190/1.2895521
    [Google Scholar]
  66. Waxman, M.H. & Smits, L.J.M. (1968) Electrical conductivities in oil‐bearing shaly sand. Society of Petroleum Engineering Journal, 8(2), 107–122. https://doi.org/10.2118/1863‐A
    [Google Scholar]
  67. Wyllie, M.R.J., Gregory, A.R. & Gardner, G.H.F. (1956) Elastic wave velocities in heterogeneous and porous media. Geophysics, 21(1), 41–70. https://doi.org/10.1190/1.1438217
    [Google Scholar]
  68. Wyllie, M.R.J., Gregory, A.R. & Gardner, G.H.F. (1958) An experimental investigation of factors affecting elastic wave velocities in porous media. Geophysics, 23(3), 459–493. https://doi.org/10.1190/1.1438493
    [Google Scholar]
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  • Article Type: Research Article
Keyword(s): Electrical resistivity; Inversion; Porosity; Refraction; Surface wave

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