The Cryosphere www.the-cryosphere.net 1994-0416 1994-0424 2 2 2008 10.5194/tc-2-167-2008 http://www.the-cryosphere.net/2/167/2008/ http://www.the-cryosphere.net/2/167/2008/tc-2-167-2008.html http://www.the-cryosphere.net/2/167/2008/tc-2-167-2008.pdf 167 178 2008-12-05 On the limit to resolution and information on basal properties obtainable from surface data on ice streams G. H. Gudmundsson M. Raymond British Antarctic Survey High Cross Madingley Rd. Cambridge CB3 0ET, UK Section of Glaciology, VAW-ETHZ, 8092 Zurich, Switzerland An optimal estimation method for simultaneously determining both basal slipperiness and basal topography from variations in surface flow velocity and topography along a flow line on ice streams and ice sheets is presented. We use Bayesian inference to update prior statistical estimates for basal topography and slipperiness using surface measurements along a flow line. Our main focus here is on how errors and spacing of surface data affect estimates of basal quantities and on possibly aliasing/mixing between basal slipperiness and basal topography. We find that the effects of spatial variations in basal topography and basal slipperiness on surface data can be accurately separated from each other, and mixing in retrieval does not pose a serious problem. For realistic surface data errors and density, small-amplitude perturbations in basal slipperiness can only be resolved for wavelengths larger than about 50 times the mean ice thickness. Bedrock topography is well resolved down to horizontal scale equal to about one ice thickness. Estimates of basal slipperiness are not significantly improved by accurate prior estimates of basal topography. However, retrieval of basal slipperiness is found to be highly sensitive to unmodelled errors in basal topography. Arthern, R J. and Hindmarsh, R. C A.: Optimal estimation of changes in the mass of ice sheets, J. Geophys. Res., 108(F1), 6007, \doi10.1029/2003JF000021, 2003. Box, G., Jenkins, G M., and Reinsel, G.: Time Series Analysis: Forecasting and Control, Prentice Hall, 3rd edition, 1994. Dach, R., Beutler, G., and Gudmundsson, G H.: Analysis of GPS data from an Antarctic ice stream, in: Observing our Changing Earth, edited by: Sideris, M G., volume 133 of International Association of Geodesy Symposia, pages 569–579, Springer-Verlag, Berlin Heidelberg, 2009. Gudmundsson, G H.: Transmission of basal variability to a glacier surface, J. Geophys. Res., 108(B5), 2253, doi:10.1029/2002JB002107, 2003. Gudmundsson, G H.: Glaciers and earth's changing environment, chap. Estimating basal properties of glaciers from surface measurements, Blackwell Publishing Ltd, 2004. Gudmundsson, G. H.: Analytical analysis of small-amplitude perturbations in the shallow ice stream approximation, The Cryosphere, 2, 77–93, 2008. Joughin, I., MacAyeal, D R., and Tulaczyk, S.: Basal shear stress of the Ross ice streams from control method inversions, J. Geophys. Res., 109, B0940, \doi10.1029/2003JB002960, 2004. Joughin, I., Bamber, J L., Scambos, T., Tulaczyk, S., Fahnestock, M., and MacAyeal, D R.: Integrating satellite observations with modeling: Basal shear stress of the Filcher-Ronne Ice Stream, Antarctica, Proceedings of the Royal Society of London, Ser. A, 364, 1795–1814, \doi10.1098/rsta.2006.1799, 2006. Joughin, I R.: Estimation of Ice Sheet Topography and Motion Using Interferometric Synthetic Aperture Radar, Ph.D. thesis, University of Washington, 1995. King, M A.: Rigorous GPS data-processing strategies for glaciological applications, J. Glaciol., 50, 601–607, 2004. Larour, E., Rignot, E., Joughin, I., and Aubry, D.: Rheology of the Ronne Ice Shelf, Antarctica, inferred from satellite radar interferometry data using an inverse control method, Geophys. Res. Lett., 32, 1–4, 2005. 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.: A tutorial on the use of control methods in ice-sheet modeling, J. Glaciol., 39, 91–98, 1993. MacAyeal, D R., Bindschadler, R A., and Scambos, T A.: Basal friction of Ice Stream E, West Antartica, J. Glaciol., 41, 247–262, 1995. Menke, W.: Geophysical Data Analysis: Discrete Inverse Theory, vol 45 of Internationla Geophysics Series, Academic Press, New York, 2nd revised edition edn., 1989. Purser, R J. and Huang, H.-L.: Estimating effective data density in a satellite retrieval or an objective analysis, J. Appl. Meterol., 32(6), 1092–1107, 1993. Raymond, M. and Gudmundsson, G H.: On the relationship between surface and basal properties on glaciers, ice sheets, and ice streams, J. Geophys. Res., 110, B08411, \doi10.1029/2005JB003681, 2005. Raymond, M. and Gudmundsson, G.: Estimating basal properties of glaciers from surface measurements: a non-linear Bayesian inversion approach, The Cryosphere Discuss., submitted, 2008. Rodgers, C D.: Inverse Methods for Atmosperic Sounding: Theory and Practice, vol 2 of Series on atmospheric, oceanic and planetary physics, World Scientific, Singapore, 2000. Schoof, C.: Basal perturbations under ice streams: form drag and surface expression, J. Glaciol., 48, 407–416, 2002. Schoof, C.: On the mechanics of ice-stream shear margins, J. Glaciol., 50, 208–218, 2004. Shannon, C E. and Weaver, W.: The mathematical theory of communication, The University of Illinoies Press, IL., 1949. Tarantola, A.: Inverse Problem Theory and Methods for Model Estimation, Society for Industrial and Applied Mathematics, Philadelphia, 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, 2002. Togneri, R. and DeSilva, C. J S.: Fundamentals of Information Theory and Coding Design, Discrete Mathemtics and its Application, Chapman & Hall, 2003. Truffer, M.: The basal speed of valley glaciers: an inverse approach, J. Glaciol., 50, 236–242, 2004. Vaughan, D G., Corr, H. F J., Ferraccioli, F., Frearson, N., O'Hare, A., Mach, D., Holt, J W., Blankenship, D D., Morse, D L., and Young, D A.: New boundary conditions for the West Antarctic ice sheet: Subglacial topography beneath Pine Island Glacier, Geophys. Res. Lett., 33, L09501, \doidoi:10.1029/2005GL025588., 2006. 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.