A non-parametric resampling method for uncertainty analysis of geophysical inverse problems

Document Type : Research Paper

Authors

Institute of Geophysics, University of Tehran, Tehran, Iran.

10.22059/ijmge.2025.382369.595194

Abstract

Due to non-uniqueness of geophysical inverse problems and measurement errors, the inversion uncertainties within the model parameters are one of the most significant necessities imposed on any modern inverse theory. Uncertainty analysis consists of finding equivalent models which sufficiently fit the observed data within the same error bound and are consistent with the prior information. In this paper, we present a non-parametric block-wise bootstrap resampling method called moving block bootstrapping (MBB) for uncertainty analysis of geophysical inverse solutions. In contrast to conventional bootstrap in which the dependence structure of data is ignored, the block bootstrap considers the dependency and correlation among the observed data by resampling not individual observations, but blocks of observations. The application of the proposed strategy to different synthetic inverse problems as well as to synthetic and real datasets of geo-electrical sounding inversion is presented. The results demonstrated that through the block bootstrap, it is possible to effectively sample the equivalence regions for a given error bound.

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References

[1] Backus, G. E., Gilbert, F., (1967). Numerical Applications of a Formalism for Geophysical Inverse Problems. Geophysical Journal of the Royal Astronomical Society, 13, 1-3, 247–276.
[2] Backus, G.E., Gilbert, F., (1968). The Resolving power of Gross Earth Data, Geophysical Journal of the Royal Astronomical Society, 16, 169–205.
[3] Backus, G.E., Gilbert, F., (1970). Uniqueness in the Inversion of inaccurate Gross Earth Data, Philosophical Transactions of the Royal Society of London A, 266, 123-192.
[4] Mosegaard, K., Tarantola, A., (1995). Monte Carlo sampling of solutions to inverse problems. J Geophys Res Solid Earth 100(B7),12431–12447
[5] Gouveia, W. P., Scales, J. A., (1997). Resolution of seismic waveform inversion: Bayes versus Occam, Inverse Problems 13, 323–349.
[6] Moorkamp M, Jones AG, Eaton DW (2007) Joint inversion of teleseismic receiver functions and magnetotelluric data using a genetic algorithm: are seismic velocities and electrical conductivities compatible? Geophysical Research Letters, 34(16):L16, 311
[7] Akca I, Basokur AT (2010) Extraction of structure-based geoelectric models by hybrid genetic algorithms. Geophysics 75(1):F15–F22
[8] Roy L, Sen MK, Blankenship DD, Stoffa PL, Richter TG (2005) Inversion and uncertainty estimation of gravity data using simulated annealing: an application over Lake Vostok, East Antarctica. Geophysics 70(1):J1–J12
[9] Wang R, Yin C, Wang M, Wang G (2012) Simulated annealing for controlled-source audio-frequency magnetotelluric data inversion. Geophysics 77(2):E127–E133.
[10] Reading, A. M., Cracknell, M. J., and Sambridge, M., (2011). Turning geophysical data into geological information or why a broader range of mathematical strategies is needed to better enable discovery. Preview, 151,24–29. https://doi.org/10.1071/PVv2011n151p24.
[11] Fernández-Muñiz , Z., Khaniani, H., and , J. L. (2019). Data kit inversion and uncertainty analysis. Journal of Applied Geophysics, 161, 228–238.
[12] Efron, B., (1979). Bootstrap methods: another look at the jackknife. Ann. Statist. 7, 1-26.
[13] McLaughlin, K.L., (1988). Maximum-likelihood event magnitude estima- tion with bootstrapping for uncertainty estimation, Bull. seism. Soc. Am., 78(2), 855–862.
[14] Tichelaar, B.W. and Ruff, L.J., (1989). How good are our best models? Jack- knifing, bootstrapping, and earthquake depth, EOS, Trans. Am. geophys. Un., 55(12), 1613–1624.
[15] Shearer, P.M., (1997). Improving local earthquake locations using the l1 norm and waveform cross correlation: application to the Whittier Narrows, California, aftershock sequence, Journal of Geophysical Research, 102(B4), 8269–8283.
[16] Parsekian, A. D., and Grombacher, D. (2015). Uncertainty estimates for surface nuclear magnetic resonance water content and relaxation time profiles from bootstrap statistics. Journal of Applied Geophysics, 119, 61–70. https://doi.org/10.1016/
j.jappgeo.2015.05.005.
[17] Hertrich, M., (2008). Imaging of groundwater with nuclear magnetic resonance: Progress in Nuclear Magnetic Resonance Spectroscopy, 53, 227–248.
[18] Schnaidt, S., and Heinson, G. (2015). Bootstrap resampling as a tool for uncertainty analysis in 2-D magnetotelluric inversion modelling. Geophysical Journal International, 203(1), 92–106. https://doi.org/10.1093/gji/ggv264.
[19] Campanya, J., Ledo, J., Queralt, P., Marcuello, A. and Jones, A.G., (2014). A new methodology to estimate magnetotelluric (MT) tensor relationships: estimation of local transfer-functions by combining interstation transfer- functions (ELICIT), Geophysical Journal International, 198(1), 484–494.
[20] Neukirch, M. and Garcia, X., 2014. Nonstationary magneto-telluric data processing with instantaneous parameter, Journal of Geophysical Research, 119, 1634–1654.
[21] Ebtehaj, M., Moradkhani, H., and Gupta, H. V. (2010). Improving robust-ness of hydrologic parameter estimation by the use of moving block bootstrap resampling, Water Resources Res., 46, W07515,
[22] Kunsch, H. R. (1989). The jackknife and the bootstrap for general stationary observations. Ann.Statist. 17 1217–1261.
[23] Liu, R. Y. and Singh, K. (1992). Moving blocks jackknife and bootstrap capture weak dependence. In Exploring the Limits of Bootstrap (R. Lepage and L. Billard, eds.) 225–248. Wiley, New York.
International Journal of Mining and Geo-Engineering
Volume 59, Issue 1
March 2025
Pages 61-67

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