Glacier volume estimation as an ill-posed boundary value problem

Abstract. Estimating a glacier's volume by inferring properties at depth from properties observed at the surface creates an instability that grows exponentially with the size of the glacier. Random errors from this instability can overwhelm the volume calculation unless problematic short spatial wavelengths are specifically excluded. Volume-area scaling inherently filters these short wavelengths and automatically eliminates the instability, but typical numerical inversions must filter the correct wavelengths explicitly. The world's total glacier and ice cap (GIC) volume is calculated from the sum of hundreds of thousands of ice masses, dramatically reducing the random error from the "ill-posed" instability. However, the variance of the total GIC volume estimate will be exponentially larger if the instability is not carefully controlled. When all other considerations are equal, greater confidence should be placed in total volume estimates that eliminate the ill-posed instability versus those estimates that do not.