2.1. Balance Thickness
 If ice is treated as an incompressible material, mass conservation requires the velocity vector, v, to be divergence free, ∇ · v = 0. Using the kinematics of the glacier bed and surface, this equation is vertically integrated using the Leibnitz integral rule to obtain the two-dimensional form of the mass conservation equation:
where H is the ice thickness and (x,y) = (x,y) is the depth-averaged horizontal velocity, b is the basal melting rate (m/yr ice equivalent, positive when melting, negative when freezing), and s is the surface mass balance (m/yr ice equivalent, positive for accumulation, negative for ablation). This equation states that the ice flux divergence is balanced by the rate of thickness change and the net surface and basal mass balances.
 Let Ω be the two-dimensional ice domain and ∂Ω its boundary. We define the inflow and outflow boundaries as follows:
with n the outward-pointing unit normal vector. We also define T ∈ Ω as the flight tracks where data are collected within the model domain. The balance ice thickness is calculated by solving
where = s − b − ∂H/∂t, is the apparent mass balance following Farinotti et al.  and Hobs is an observed thickness. This equation requires that ice thickness be constrained once and only once for each flow line. Constraining ice thickness at the inflow boundary is a simple way of achieving this condition. Note that even though we call this solution “balance thickness”, it shall not be confused with the steady-state glacier thickness because it incorporates the rate of thickness change, ∂H/∂t.
 Equation (3) is a steady hyperbolic partial differential equation of first order. Such equations are difficult to solve numerically [e.g., Donea, 1984]. Here we employ a streamline upwinding finite element method to solve it.
2.2. Multi-Parameter Optimization
 Solving for the balance thickness (equation (3)) requires precise knowledge of the apparent mass balance and ice velocity. These data sets are not always available, or include errors, which strongly affect the solution. For example, slight errors in the velocity data yield large errors in glacier thickness [Rasmussen, 1985]. To reduce these deviations, we optimize the apparent mass balance and depth-averaged velocity to minimize the misfit between observed and modeled ice thickness along the flight tracks, T. We define an objective function as
The first term measures the mismatch between modeled and measured thickness and the second term is a regularizing constraint, which penalizes wiggles in ice thickness. The addition of regularization is essential to infer an ice thickness that is smooth enough yet close enough to the original data along flight tracks. γ is a parameter used to adjust the influence of the regularization in the objective function. A large value of γ will result in a smoother thickness map that deviates more significantly from the observations whereas a small value of γ will produce in a good fit with observations but with strong gradients in the proximity of flight tracks.
 The tolerance interval for the apparent mass balance, , is defined here as the sum of errors in surface mass balance, basal melting rate and thickness change. For 79North Glacier, we estimate this error to be on the order of ±1 m/yr. The admissible space for is therefore
 The tolerance interval for ice velocity is more difficult to evaluate a priori. Even though nominal errors in ice velocity are around ±2 m/yr, ice thickness and velocity are usually not measured at the same time and surface velocity is not identical to depth-averaged velocity. In areas where ice is frozen to the bed, the depth-averaged velocity may be 15% lower. In that case, the approach discussed herein needs to be revised to include differences between surface and depth-averaged velocities through 3D modeling of ice sheet flow. Here, on 79North Glacier, differences between surface and depth-averaged velocities are at the 1% level [Seroussi et al., 2011], so it is reasonable to use surface velocities. To allow enough flexibility in the optimization process, however, we need a tolerance on ice velocity that is larger than the nominal error. We obtain satisfactory results with a tolerance of ±50 m/yr, which is less than 5% of the surface velocity of the main flow. The admissible space for becomes
 Given uncertainties in ice velocity, δ, and apparent mass balance, δ, it is possible to estimate the maximum error in ice thickness, δH, as detailed in the supplementary material. Along a flow line, we show that the maximum error is given by
where x is the curvilinear coordinate of the flow line starting from the inflow boundary. The error is therefore inversely proportional to the magnitude of the ice velocity. For a track spacing of 5 km and using values typical for 79North Glacier discussed next, the error in ice thickness is 10 m when ice velocity is about 1000 m/yr and 100 m when ice velocity is about 100 m/yr. This method is therefore most reliable in fast flowing areas.