Journal of Glaciology, Vol. 58, No. 212, 2012 doi: 10.3189/2012JoG11J249 1 Using surface velocities to calculate ice thickness and bed topography: a case study at Columbia Glacier, Alaska, USA R.W. MCNABB, 1 R. HOCK, 1,2 S. O’NEEL, 3 L.A. RASMUSSEN, 4 Y. AHN, 5 M. BRAUN, 6 H. CONWAY, 4 S. HERREID, 1 I. JOUGHIN, 7 W.T. PFEFFER, 8 B.E. SMITH, 7 M. TRUFFER 1 1 Geophysical Institute, University of Alaska Fairbanks, Fairbanks, AK, USA 2 Department of Earth Sciences, Uppsala University, Uppsala, Sweden 3 United States Geological Survey Alaska Science Center, Anchorage, AK, USA 4 Department of Earth and Space Sciences, University of Washington, Seattle, WA, USA 5 School of Technology, Michigan Technological University, Houghton, MI, USA 6 Department of Geography, University of Erlangen-N¨ urnberg, Erlangen, Germany 7 Polar Science Center, Applied Physics Laboratory, University of Washington, Seattle, WA, USA 8 Institute of Arctic and Alpine Research, University of Colorado at Boulder, Boulder, CO, USA ABSTRACT. Information about glacier volume and ice thickness distribution is essential for many glaciological applications, but direct measurements of ice thickness can be difficult and costly. We present a new method that calculates ice thickness via an estimate of ice flux. We solve the familiar continuity equation between adjacent flowlines, which decreases the computational time required compared to a solution on the whole grid. We test the method on Columbia Glacier, a large tidewater glacier in Alaska, USA, and compare calculated and measured ice thicknesses, with favorable results. This shows the potential of this method for estimating ice thickness distribution of glaciers for which only surface data are available. We find that both the mean thickness and volume of Columbia Glacier were approximately halved over the period 1957–2007, from 281 m to 143 m, and from 294 km 3 to 134 km 3 , respectively. Using bedrock slope and considering how waves of thickness change propagate through the glacier, we conduct a brief analysis of the instability of Columbia Glacier, which leads us to conclude that the rapid portion of the retreat may be nearing an end. 1. INTRODUCTION Knowledge of glacier volume and ice thickness distribution are essential for hydrological applications, ice flow mod- eling, assessing the impact of climate change on glaciers, and sea-level rise predictions, among other applications. Direct measurements of ice thickness at a point (e.g. from a borehole) or along a track (e.g. using radio-echo sounding or seismic methods) are time-consuming and expensive. Direct measurements of total ice volume generally require many such points or tracks. In addition, errors in interpolation or extrapolation from along-track measurements of ice thicknesses can introduce large anomalies in calculated ice- flux divergence if the interpolation technique used does not conserve mass or ice flux (Seroussi and others, 2011). A mass-conserving method for interpolating ice thickness has been developed by Morlighem and others (2011). Techniques using radar tomography and interferometry have recently been developed, but they have mostly been applied to portions of the Greenland ice sheet with flat surfaces, and their performance and applicability to outlet and tidewater glaciers is not yet known (Paden and others, 2010). Previous studies have focused on calculating ice volume and ice thickness distribution of glaciers without direct measurement. One of the simplest methods is volume– area scaling (e.g. Bahr and others, 1997; Radi´ c and others, 2008). These methods are easily implemented, requiring information about glacier area and parameterization of the shape of the bed topography. In general, the shape of the bed of individual glaciers is not well known, but the method has proven useful for characterizing regional ice volumes (Radi´ c and Hock, 2010). Other studies have inferred ice thickness distribution through direct evaluation of the mass continuity equation (e.g. Rasmussen, 1988; Morlighem and others, 2011), while others use a simplification of the equation to overcome data gaps (e.g. Fastook and others, 1995; Warner and Budd, 2000; Farinotti and others, 2009a), or through applications of the shallow-ice approximation (e.g. Li and others, 2011). More recently, a method has been proposed which uses neural networks along with simplifications of the mass continuity equation to estimate the bed topography and ice volumes of entire regions, where little more than surface topography might be known (Clarke and others, 2009). Here we propose a new method for calculating ice thickness, by evaluating the mass continuity equation between adjacent flowlines, rather than through a local solution or on a large grid. With velocity fields that cover some portion of a glacier, a digital elevation model (DEM) of the glacier surface, rates of surface mass balance and thinning, and knowledge of the ice thickness at the boundary of the domain of interest, it is possible to calculate the ice thickness distribution of the glacier over the region covered by the velocity field used. Because no assumption is made about the ice flux through the terminus of the glacier, this method is directly applicable to both land and marine/lake- terminating glaciers. We apply this method to Columbia Glacier, a large tidewater glacier in Alaska, USA. We investigate the sensitivity of the calculated ice thicknesses to flowline igs.cls v2.01 September 20, 2012 Article ref t11J249 Typeset by Sukie Proof no 1