The Cryosphere Discussions www.the-cryosphere-discuss.net 1994-0432 1994-0440 3 1 2009 10.5194/tcd-3-181-2009 http://www.the-cryosphere-discuss.net/3/181/2009/ http://www.the-cryosphere-discuss.net/3/181/2009/tcd-3-181-2009.html http://www.the-cryosphere-discuss.net/3/181/2009/tcd-3-181-2009.pdf 181 222 2009-02-05 Estimating basal properties of glaciers from surface measurements: a non-linear Bayesian inversion approach M. J. Raymond G. H. Gudmundsson ghg@bas.ac.uk Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie (VAW), ETH Zürich, 8092 Zürich, Switzerland British Antarctic Survey, High Cross, Madingley Rd., Cambridge CB3 0ET, UK We propose a new approach to indirectly estimate basal properties of a glacier, i.e. bedrock topography and basal slipperiness, from observations of surface topography and surface velocities. We demonstrate how a maximum a posteriori estimate of basal conditions can be determined using a Bayesian inference approach in a combination with an analytical linearisation of the forward model. Using synthetic data we show that for non-linear media and non-linear sliding law only a few forward-step model evaluations are needed for convergence. The forward step is solved with a numerical finite-element model using the full Stokes equations. Forward Fréchet derivatives are approximated through analytical small-perturbation solutions. This approximation is a key feature of the method and the effects of this approximation on model performance are analyzed. The number of iterations needed for convergence increases with the amplitude of the basal perturbations, but generally less than ten iterations are needed. Gouveia, W P. and Scales, J A.: Bayesian seismic waveform inversion: Parameter estimation and uncertainty analysis, J. Geophys. Res., 103, 2759–2780, doi:10.1029/97JB02933, 1998. Gudmundsson, G H.: Transmission of basal variability to a glacier surface, J. Geophys. Res., 108(B5), 2253, doi:10.129/2002JB002107, 2003. Gudmundsson, G H.: Glacier Science and Environmental Change, chap.\ Estimating basal properties of glaciers from surface measurements, Blackwell Publishing, Oxford, p. 415, 2006. Gudmundsson, G H. and Raymond, M J.: On the limit to resolution and information on basal properties obtainable from surface data on ice streams, The Cryosphere Discuss., 2, 413–445, 2008. Hutter, K.: Theoretical glaciology; material science of ice and the mechanics of glaciers and ice sheets, D. Reidel Publishing Company/Tokyo, Terra Scientific Publishing Company, 1983. Joughin, I., MacAyeal, D R., and Tulaczyk, S.: Basal shear stress of the Ross ice streams from control method inversions, J. Geophys. Res., 109, B09405, doi:10.1029/2003JB002960, 2004. Kitanidis, A K.: Introduction to geostatistics: applications to hydrogeology, Cambridge University Press, 1997. Leysinger Vieli, G. J.-M C. and Gudmundsson, G H.: On estimating length fluctuations of glaciers caused by changes in climatic forcing, J. Geophys. Res., 109, F01007, doi:10.1029/2003JF000027, 2004. MacAyeal, D R.: The Basal Stress Distribution of Ice Stream E, Antarctica, inferred by control methods, J. Geophys. Res., 97, 596–603, 1992. MacAyeal, D R., Bindschadler, R A., and Scambos, T A.: Basal friction of Ice Stream E, West Antartica, J. Glaciol., 41, 247–262, 1995. Paterson, W. S B.: The physics of glaciers, Butterworth-Heinemann, Oxford, third edn., 1994. Raymond, M J.: Estimating basal properties of glaciers and ice streams from surface measurements, Mitteilung 202, VAW, 152 pp., 2007. Raymond, M J. and Gudmundsson, G H.: On the relationship between surface and basal properties on glaciers, ice sheets, and ice streams., J. Geophys. Res., 110, B08411, doi:10.1029/2005JB003681, 2005. Rodgers, C D.: Inverse Methods for Atmospheric Sounding. Theory and Practice, World Scientific Publishing Co. Pte. Ltd. Singapore, 2000. Smith, G D. and Morland, L W.: Viscous relations for the steady creep of polycrystalline ice, Cold Reg. Sci. Technol., 5, 141–150, 1981. Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation, Society for Industrial and Applied Mathematics, 2005. Thorsteinsson, T., Raymond, C F., Gudmundsson, G H., Bindschadler, R B., Vornberger, P., and Joughin, I.: Bed topography and lubrication inferred from surface measurements on fast flowing ice streams, J. Glaciol., 49, 481–490, 2003. Truffer, M.: The basal speed of valley glaciers: an inverse approach, J. Glaciol., 50, 236–242, 2004. Van~der Veen, C J. and Whillans, I M.: Force Budget: I. Theory and Numerical Methods, J. Glaciol., 35, 53–60, 1989. Vieli, A. and Payne, A J.: Application of control methods for modelling the flow of Pine Island Glacier, West Antarctica, Ann. Glaciol., 36, 197–204, 2003.