## 1. Introduction

The projected shrinkage of Earth’s glaciers and ice caps will raise sea level (e.g., Radić and Hock 2011) and affect the water cycle over large areas of Asia, Europe, and the Americas (e.g., Kaser et al. 2010). Improved knowledge of the rate and magnitude of these changes, on a region-by-region basis, is essential and ice flow modeling provides one method to quantify these changes and make projections. Before such models can be used it is necessary to obtain a digital elevation model (DEM) of the underlying subglacial topography. For Earth’s ~200 000 glaciers this is problematic because few have been geophysically mapped and at present no satellite remote sensing instrument can image subglacial topography.

Recent work on ice thickness estimation includes methods that are predominantly geometrical, such as that of Clarke et al. (2009), which is based on artificial neural networks, and those that incorporate assumptions from glacier physics (e.g., Farinotti 2010; Farinotti et al. 2009b; Huss and Farinotti 2012; Linsbauer et al. 2009, 2012; Li et al. 2011, 2012; Marshall et al. 2011; Morlighem et al. 2011; Paul and Linsbauer 2011). The attraction of the former approach is its parsimony, but it is cumbersome to implement and can lead to subglacial topography that diverges from the true topography—a concern when the estimated bed topography is to be used as a boundary condition for ice dynamics modeling. For this reason physics-rooted approaches are favored.

The aims of the present contribution are to develop a physically based method for ice thickness estimation, to validate the method by applying it to artificial datasets generated by a numerical ice dynamics model, and to use the method to estimate the subglacial topography of glaciers in the mountainous regions of British Columbia (BC) and Alberta (AB) in western Canada. The a.d. 2005 ice volume and its sea level equivalent are then calculated by summing calculated ice volume for individual glaciers. A recent inventory of glaciers in the study area indicated that in 2005 the number of glaciers was ~17 600 and the area of glacierized terrain was ~26 700 km^{2} (Bolch et al. 2010). For reference, in the 1970s some 5050 “perennial surface ice bodies” with a combined area of 2909 km^{2} were identified in the European Alps (Haeberli and Hoelzle 1995) and for the Swiss Alps alone there are 1483 glaciers with a total area of ~1063 km^{2} (Farinotti et al. 2009a). Our point in making these comparisons is that the BC–AB dataset is too large to be dissected on a glacier-by-glacier basis, so any procedure for generating ice thickness estimates must heavily rely on unguided automatic computation rather than expert intervention.

Comparisons between scientific knowledge of glaciers in western North America and those in Europe also justify the departure from methods that have been successfully applied to glaciers in the Swiss Alps. The western North America study region is data poor and of the ~17 600 glaciers, few have received scientific attention, yielding only a handful of published ice thickness measurements (e.g., Doell 1963; Kanasewich 1963; Paterson 1970; Raymond 1971a,b; Holdsworth et al. 2006). Most of these measurements were taken decades before our study and, in most cases, the map locations and surface elevations of the sites were not tied to a conventional geodetic reference frame. Furthermore the glaciers have thinned substantially so that the surface elevation is now much lower than at the time of measurement.

Historically, methods of ice thickness estimation have used the idea that glacier ice can be approximated as an ideal plastic material so that bed stress *τ** corresponds to the constant plastic yield stress *τ*_{0} (Orowan 1949). While the implication of a well-defined yield stress is conceptually attractive it suggests that *τ*_{0} is a physical property of ice and invites misleading assertions such as the “yield stress of glacier ice is 1 bar.” If this were truly the case then a single yield stress value could be applied to all glaciers that were sufficiently healthy to maintain the basal stress at this value. Nonetheless several authors have made good use of the plasticity idea. From a 1938 map of the ice surface topography and the assumption that *τ** = 88 kPa, Nye (1952a) produced a first map of the subglacial topography of the Greenland Ice Sheet. Later, Reeh (1982) presented an elegant account of three-dimensional plasticity modeling of ice sheet form and in subsequent publications Reeh (1984) and Fisher et al. (1985) applied this to the contemporary Greenland Ice Sheet, ice caps of the Canadian Arctic islands, and the Laurentide Ice Sheet. Crucially, Reeh noted that the assignment of basal shear stress depended on “accumulation rate, basal temperature, etc.” (Reeh 1984, p. 116).

*τ** for each glacier or, more ambitiously, a spatially varying bed stress

*Z*=

*Z*

_{H}−

*Z*

_{L}with

*Z*

_{H}being the highest elevation of the flowshed,

*Z*

_{L}is the lowest, and

*τ** has units of bars (hPa). Therefore (1) is empirical (Haeberli and Hoelzle 1995, Fig. 1) but clever and is based on data from the European Alps. The elevation span Δ

*Z*is an indirect though readily observed indicator of the mass balance turnover for a particular glacier. However, there is a concern that (1) must be tuned to specific geographical settings. For this reason, we focus on estimating glacier-specific but space-varying bed stress (section 3).

## 2. Thickness estimation as an optimization problem

We assume that surface topography is represented by a matrix of elevation values *S _{i}*

_{,j}expressing the elevation map positions (

*x*,

_{i}*y*) in a Cartesian coordinate system. The cells are assumed to be square with dimensions Δ

_{j}*x*× Δ

*y*which, for our study, are 200 m × 200 m, which matches the resolution of a prognostic ice flow model that we are also developing. Coregistered with this DEM is a second matrix, referred to as the ice mask, which has the properties

*I*

_{i}_{,j}= 1 when the ice cover is greater or equal to 50% and

*I*

_{i}_{,j}= 0 otherwise. Given

*S*

_{i}_{,j}and

*I*

_{i}_{,j}together with information on the mass balance forcing and the rate of surface elevation change, we estimate the ice thickness

*H*

_{i}_{,j}for the

*I*

_{i}_{,j}= 1 cells and, from this, produce a map of the bed topography

*B*

_{i}_{,j}, where

*B*

_{i}_{,j}=

*S*

_{i}_{,j}−

*H*

_{i}_{,j}when

*I*

_{i}_{,j}= 1 and

*B*

_{i}_{,j}=

*S*

_{i}_{,j}when

*I*

_{i}_{,j}= 0.

*θ*, the relationship between bottom stress

*τ** and slab thickness iswhere

*ρ*is the density, and

*g*is the gravity acceleration. In the shallow ice approximation (e.g., Fowler and Larson 1978), (2) is valid everywhere, and thus can be written

*H*

_{i}_{,j}=

*h*

_{i}_{,j}/cos

*θ*

_{i}_{,j}so (2) givesThe obvious implication of (3) is that ice thickness

*H*

_{i}_{,j}can be estimated if the surface slope

*θ*

_{i}_{,j}and bed stress

*θ*

_{i}_{,j}to correspond to the glacier surface slope |

**∇**

_{xy}

*S*

_{i}_{,j}| = tan

*θ*

_{i}_{,j}, where

**∇**

_{xy}

*S*

_{i}_{,j}denotes the two-dimensional gradient of the surface topography at the grid point (

*i*,

*j*). Then (3) can be writtenSolving (4) is equivalent to minimizing a cost function of the formwhere the summation is performed over all ice-covered cells.

*H*

_{i}_{,j}is to use Laplacian interpolation (e.g., Press et al. 2007, p. 151), which is equivalent to solving

*H*

_{i}_{,j}= 0 beyond the glacier margins. Combining the two approaches and introducing a spatially-varying trade-off parameter

*χ*

_{i}_{,j}, the ice thickness estimates are obtained by minimizing the modified cost functionThe factor

*χ*

_{i}_{,j}weightings;

*σ*is an estimate of the standard deviation of the first square-bracketed term and

_{H}*σ*

_{LapH}is that for the second. Hereafter we assume

*σ*= 25 m and

_{H}*σ*

_{LapH}= 0.0025 m

^{−1}, so that

*λ*is fixed at

*λ*= 100 m, and then use

*χ*

_{i}_{,j}to manage the trade-off between the cost terms. For

*χ*

_{i}_{,j}= 1 the thickness estimate is entirely based on the stress relation (4) and for

*χ*

_{i}_{,j}= 0 the estimate is generated by Laplacian interpolation among neighboring cells.

The trade-off parameter *χ _{i}*

_{,j}can be set to vary with spatial position to give the greatest weight to the estimator that has the most authority at a given point. For example, in the central regions of ice caps, where the surface slope is small and the stress-based estimator (3) becomes sensitive to small fluctuations in

*θ*

_{i}_{,j}, the trade-off parameter

*χ*

_{i}_{,j}can be set to a small value so that the thickness estimate is largely or entirely based on Laplacian interpolation.

*H*

_{i}_{,j}and setting the result to zero gives a system of equations that minimizes

*α*:=

*λ*

^{4}/(Δ

*x*)

^{4}. Reorganizing (8) giveswhich represents a set of linear equations having the form

**H**=

**C**, where

**H**is a column vector formed from the unknown ice thickness values

*H*

_{i}_{,j},

**C**is a column vector formed from the known right-hand-side terms of (9). For a large domain the coefficient matrix

*H*

_{i}_{,j}values in roughly 4 h of machine time on a desktop workstation). From test runs it was established that computing time increases linearly with problem size.

The trade-off parameter *χ _{i}*

_{,j}controls the weighting that is assigned to the stress-based estimator relative to that assigned for the Laplacian interpolation estimator. First a default value

*χ*

_{0}must be assigned and this is decided by balancing the conflicting requirements of resolving changes in ice thickness while maintaining a smooth spatial pattern. By applying the inversion method to the output of a numerical ice dynamics model (for which the simulated ice thickness is perfectly known), we found that

*χ*

_{0}= 0.40 yields a satisfactory compromise. Where surface slopes were small we reduced

*χ*

_{i}_{,j}in a smooth fashion, as described in section 4.

## 3. Estimation of basal stress

Our approach to estimating basal stress is similar to that of Farinotti et al. (2009b) and is based on automated delineation of glacier flowsheds and application of the continuity equation. The term “flowshed” has been adopted to describe a glacier flow unit that is defined by its ice catchment. In many situations there is no distinction between a glacier and a glacier flowshed, but for glaciers that share a common catchment region the boundary between individual glaciers emanating from that catchment is defined by ice drainage divides.

_{xy}denotes the two-dimensional divergence,

**q**is the vertically integrated volume flux of ice per unit width (m

^{2}yr

^{−1}),

^{−1}), and

^{−1}). In the glacier accumulation zone the mass balance rate is positive; in the ablation zone it is negative. For notational efficiency and consistency with antecedent work (Farinotti et al. 2009b), we define an apparent balance rate

*α*the limits are

_{α}denote the curve that separates the lower boundary of zone

*α*from the upper boundary of the zone immediately below it, Γ

_{α}traces the lower boundary of a catchment area

*A*for which

_{α}^{3}yr

^{−1}) across Γ

_{α}isDefining

*l*as the length of Γ

_{α}_{α}, the length-averaged ice flux (m

^{2}yr

^{−1}) traversing Γ

_{α}is

_{α}line. The volume flux of ice per unit width of channel is given by(e.g., Cuffey and Paterson 2010, p. 310), where

*q*=

_{s}*υ*

_{s}*h*is the sliding contribution and the second term represents ice flux due to creepThe parameters

*n*are the creep rate factor and exponent of Glen’s flow law for ice. Taking

*τ** =

*ρgh*sin

*θ*, (13) can be rewritten asleading to the expression

*q*

_{α}_{c}and

*q*

_{α}_{s}are the creep and sliding contributions to

*q*

_{α}_{s}, we postulate that it is some fixed fraction of the total ice flux, and therefore take

*q*

_{α}_{c}=

*ξq*and

_{α}*q*

_{α}_{s}= (1 −

*ξ*)

*q*, to obtainwhere

_{α}_{α}. Next we assume that the calculated value for

*α*within a given flowshed. The resulting

*i*,

*j*) grid points that lie within balance band

*α*to obtain

*H*

_{i}_{,j}estimates.

*q*

_{f}and averaged surface slope

*h*to obtainThis is similar to our approach of estimating

*τ** from (16) and then using (3) to calculate the corresponding ice thickness. Their parameter

*C*is a dimensionless correction factor to account for the partitioning between creep and sliding contributions to ice flow. The main points of difference between our method and that of Farinotti et al. (2009b) are that we do not restrict our analysis to flowlines (and hence do not need to delineate them, automatically or otherwise) and that spatial smoothing is applied implicitly as an integral part of the inversion procedure (by means of the Laplacian interpolator), rather than as a distinct and explicit step (e.g., initial smoothing of the surface slope

*θ*).

## 4. Technical matters

This section contains much of the technical detail and justifications that underlie our approach to ice thickness estimation. The casual reader can skip this section and move directly to the section on performance analysis.

### a. Physical constants and glaciological parameters

For the flow law parameters we take *n* = 3 and ^{−24} Pa^{−3} s^{−1}, which match the values recommended in (Cuffey and Paterson 2010, Table 3.4), together with *ρ* = 910 kg m^{−3} for ice density and *g* = 9.81 m s^{−2} for the gravity acceleration. We make the approximation that sliding does not contribute to ice flow [*ξ* = 1 in (16)]. In reality *ξ* = *ξ _{i}*

_{,j}could vary from cell-to-cell in the model but lacking this information we apply a single value to all cells. Although for large active glaciers sliding can be significant, for most cells

*ξ*≈ 1 is probably acceptable. In any case, we will show that

*ξ*is not a sensitive parameter of the inversion scheme.

### b. Calculation of surface slope

*S*

_{i}_{,j}. Our preferred method is to calculate the slope in each of four quadrants (NE, SE, SW, and NW) and take the average. For example, for the NE quadrant,and the quadrant-averaged slope is

*θ*→0. We deal with this problem by applying a limiter to the slope expression (19) to avoid the |

**∇**

*S*

_{i}_{,j}|=0 limit. In the present work we use the slope limiterwith

*δ*

_{0}= 0.01 and

*δ*

_{1}= 0.03. This sets the minimum slope to

*δ*

_{0}= 0.01, which corresponds to an angle of 0.57°. When |

**∇**

*S*| <

*δ*

_{1}and the slope limiter is active we also reduce the trade-off parameter

*χ*

_{i}_{,j}in a systematic manner, tapering it to zero, as follows:Thus the minimum slope

*δ*

_{0}is never actually applied because the trade-off parameter assigns no weight to the stress-based estimator when |

**∇**

*S*| ≤

*δ*

_{0}.

### c. Balance zone delineation and calculation of width and slope at zone boundaries

_{α}can be found algorithmically by searching for cells having

*I*

_{i}_{,j}= 1 (ice-covered) with

*l*, by summing over the length contribution from individual cells using a flux-weighted estimate of the length. Some form of weighting is necessary because the direction of ice flow is not usually aligned with the cell orientation; thus, for an individual cell having dimensions Δ

_{α}*x*× Δ

*y*, its length contribution Δ

*l*

_{i}_{,j}is unlikely to be Δ

*l*

_{i}_{,j}= Δ

*x*or Δ

*y*. For the faces of the cell (

*i*,

*j*), the surface slope components in the cardinal directions can be writtenand the corresponding

*outward*creep flux magnitudes [

*Q*

_{N}]

_{i,j}, [

*Q*

_{E}]

_{i,j}, etc. are proportional to the following:where the negative signs in the north and east terms have been applied so that outward cell fluxes are positive. As the length contribution for a single cell, we take the flux-weighted average for all cell walls through which there is an outward flux of ice, that is,where [

*l*]

_{α}_{g}is the length of the Γ

_{α}balance zone boundary for the

*g*th flowshed.

### d. Ice masks and flowsheds

It is important to distinguish between connected regions of ice that lie within a single ice mask and the individual ice flow units that can subdivide a mask. A high-elevation icefield, for example, might function as the common collection area for many individual glaciers but different flow units can be distinguished, sometimes very subtly, by topographic divides. The analogy with watersheds is obvious.

There is a substantial literature on algorithms for automated delineation of watersheds (e.g., Marks et al. 1984; Fairfield and Leymarie 1991; Meyer 1994; Tarboton 1997) but automated delineation of glacier flowsheds presents special challenges. Although the flows of water and ice are both gravity driven, glaciers have morphological differences that cause problems for conventional watershed algorithms. For example, a single glacier can have a multilobed terminus which, in a watershed algorithm, can be misinterpreted as multiple distinct glaciers, causing the algorithm to assign labels to many more flowsheds than actually exist. Rather than start from scratch we tailored the watershed algorithm included in the TopoToolbox (Schwanghart and Kuhn 2010) software package to deal with this problem (see next subsection for details). Figure 1 shows the results of applying this new flowshed delineation algorithm to a DEM of surface topography *S* (Fig. 1a) and ice mask *I* (Fig. 1b) to generate a flowshed map *G* (Fig. 1c) in a test example.

### e. Orphaned, fissioned, and problematic flowsheds

Accurate delineation of glacier flowsheds is required for accurate estimation of *Q _{α}* from (11) and

Orphaned glacierized cells are readily identified because for these *I _{i}*

_{,j}= 1 and

*G*

_{i}_{,j}is unassigned. If such cells are adjacent to cells that have been assigned a flowshed label they are merged with that flowshed; if they lack an adjacent neighboring flowshed they are assigned a new label and treated as an additional flowshed. This situation is very rare and arises from shortcomings of conventional watershed algorithms.

Fissioned flowsheds are a consequence of applying a watershed algorithm to glaciers that have a multilobed terminus. If the algorithm erroneously designates each lobe as a separate flowshed then this distinction will be preserved farther upstream even though the flowsheds are separated by a flowline rather than a flow divide. Fortunately the situation is easily detected. Cells *p* and *q* that are separated by a flowline boundary will tend to have the same slope direction whereas those that are separated by a flow divide boundary will tend to have opposite slope directions. A simple dot product test **∇**_{xy}*S _{p}* ·

**∇**

_{xy}

*S*is sufficient to discriminate between the two situations.

_{q}We classify a flowshed as “problematic” if *Q _{α}* ≤ 0 everywhere within it. For such flowsheds [

*Q*

_{ela}]

_{g}and [

*l*

_{ela}]

_{g}either vanish or are unacceptably small and the estimate of the ELA ice flux

### f. Last resort estimates of bed stress

*V*∝

*A*that relates glacier volume to glacier area is derived in important papers by Bahr (1997) and Bahr et al. (1997). A lesser known but potentially useful result from the same work is

^{γ}*τ** ∝

*A*, which we write aswhere

^{β}*k*is a proportionality constant having units of stress,

*A*

_{0}is a characteristic glacier area (we arbitrarily take

*A*

_{0}= 1 km

^{2}), and

*γ*=

*γ*= 1.36. Accepting

*γ*=

*β*=

*k*is estimated by evaluating the statisticfor all viable flowsheds and then calculating the mean using a weighted average. In (27)

*k*is the calculated proportionality constant, [

_{g}*τ*

_{ela}]

_{g}is the estimated bed stress at the ELA, and

*A*is the area of the

_{g}*g*th flowshed. We find that root area-weightinggives satisfactory results.

### g. Adjustment of the mass balance field

*A*, the integral form of the expression for ice volume balance is given bywhich expands towhere

**q**=

**v**

*H*,

**v**is the column-averaged ice velocity vector, and

**n**is the outward normal vector (in two dimensions) to the curve

**v**vanishes along

*A*as well as the surface balance rate

*A*

^{(±)},

*b*

_{0}is a constant, and

*A*

^{(±)}is the estimated flowshed area. A nonvanishing result in (32) can lead to systematic over- or underestimates of the ice discharge

*Q*in (11) and thus to errors in estimates of basal stress and ice thickness. If, for example,

_{α}*b*

_{0}is negative then

*Q*will vanish along a line Γ

_{α}_{α}, which is upglacier from the glacier terminus. Thus in the region between the terminus and Γ

_{α}the computed ice discharge will be negative, corresponding to up-slope transport of ice. To deal with this situation we adjust the estimated apparent mass balance field

*b*

_{0}, in effect assuming

### h. Influence of uncertainty in values of flow law coefficient and flow partitioning parameter

*ξ*can be inferred from the relationshipobtained from (3) and (15). Concentrating attention on the parameters

*ξ*we assume that

*H*=

*H*(

*ξ*) and take logarithmic derivatives to obtainWith

*n*= 3 the fractional change in ice thickness is ⅕th the fractional change in

*ξ*and

*ξ*leading to increased ice thickness and increased

*ξ*in the inversion models.

### i. Shape factors and debris cover

Paul and Linsbauer (2011) apply a shape factor correction to their bed stress estimates but assume this to be constant at *f* = 0.80 so its only effect is to systematically increase the estimated ice thickness. In contrast, Farinotti et al. (2009b) do not include a shape factor, although their correction factor *C* in (17) could be adjusted to include a channel shape correction. We do not include an explicit shape factor because we are wary of applying an all-embracing correction factor to our thickness estimates, preferring to view the role of shape factors as a potential source of systematic error that is subject to scrutiny.

In their analysis of Swiss Glaciers, Farinotti et al. (2009b) include the effect of debris cover on mass balance. At present Tiedemann Glacier, BC is one of the few glaciers in our study region for which a debris mask has been generated. For our study region this effort would need to be expanded before we could follow their example.

### j. Regularization

We use *B _{i}*

_{,j}=

*S*

_{i}_{,j}−

*H*

_{i}_{,j}rather than the ice thickness

*H*

_{i}_{,j}. This can lead to

*negative*ice thickness estimates, which must then be dealt with. To avoid this difficulty we use the

**∇**

_{xy}H_{i}_{,j}| regularization, finding that it gives comparable results.

## 5. Performance analysis and error estimates

To assess the performance of the ice thickness estimation scheme, we used a numerical ice dynamics model to generate synthetic ice cover over known deglacierized topography and then tested the skill of the inversion scheme for a range of situations. A potentially serious shortcoming of this stratagem is that the ice dynamics model could have unknown physical or numerical defects that cause it to produce nonphysical representations of glaciers.

^{−24}Pa

^{−3}s

^{−1},

*n*= 3,

*ρ*= 910 kg m

^{−3},

*g*= 9.81 m s

^{−2}, and

**v**

_{s}= 0. We assume the glaciers are isothermal at the melting temperature of ice so there is no need to include an energy equation. For the vast majority of glaciers in the study area this is likely to be a valid assumption but it would become dubious if applied universally to glaciers of the Yukon interior, farther to the north. Huss and Farinotti (2012) face this problem in estimating the glacier contribution to global ice volume and use the mean annual temperature at the ELA as a basis for modifying the creep rate factor

*B*(

*x*,

*y*). The labels and coordinates of the map centers for these regions are BC1 (55.4635°N, 124.8151°W), BC2 (58.9000°N, 125.8714°W), BC3 (59.4545°N, 130.3566°W), and YT1 (61.6014°N, 133.3684°W). Ice cover was then grown on this landscape by applying simplified mass balance forcings of the formwhere

*z*

_{L}and

*z*

_{H}are the lowest and highest elevation of the ELA, and

*T*

_{0}is the assumed periodicity of the climate cycle. We explore a wide range of mass balance forcings for steady-state runs involving the four test regions by varying

*z*

_{ela}(Fig. 2). The resulting ice masks, though not the ice thicknesses, are quite similar for the low- and high-rate models so we only plotted those for the high-rate case.

The numerical ice dynamics model generates time-evolving surface topography *S*(*x*, *y*, *t*) from which we can compute the ice thickness *H*(*x*, *y*, *t*), surface slope **∇***S*(*x*, *y*, *t*), time-evolving ice mask *I*(*x*, *y*, *t*), and flowshed map *G*(*x*, *y*, *t*). We then perform calculations to estimate [*Q _{α}*]

_{g}, [

*l*]

_{α}_{g},

*H*

_{i}_{,j}at any given snapshot time

*t*. Because

*B*

_{i}_{,j}is known a priori, the true ice thickness

*H*

_{i}_{,j}is known and can be compared with the estimated thickness

*z*

_{ela}=

*z*

_{L}=

*z*

_{H}) and the simulations continued until a steady-state was achieved. Model names, such as BC1·1550H, combine information about the geographical site (BC1), ELA (1550 m), and whether the model is strongly forced [large elevation gradients of mass balance rate

*α*

_{I}do not differ much between the H model and the L model but the ice volumes differ by roughly a factor of two. This has possible implications for the effectiveness of the volume–area-scaling approach (e.g., Bahr 1997; Bahr et al. 1997), which would predict that glaciers having similar area would also have similar volume assuming that the scaling constant is universal. To be fair, Bahr has never viewed the scaling constant as universal but users of his theory have occasionally treated it as such.

Input and derived properties of steady-state glacier test models. *α*_{I} = fractional area of ice cover.

### Results of performance test

There are substantial performance differences among the model runs analyzed and among the performance measures for any given run (Table 2). Rather than dwell on individual cases we shall treat the 32 model runs as an ensemble and summarize the ensemble properties by taking the mean and median of the performance indicators for each member of the ensemble. These indicators include *r* (the correlation coefficient between the true ice thickness *H _{i}*

_{,j}and the estimated thickness

Summary of estimation errors in performance tests of steady-state models. Model averages are calculated for ice-covered cells and not the entire map.

We single out model YT1·1700H for additional attention. Relative to other members of the ensemble, the performance indicators are neither bad nor good: of the 32 model runs analyzed, it ranks 24th for standard deviation (23.29 m) and rms ice thickness error (23.29 m), 12th for mean thickness error (2.27 m), and 9th for fractional volume error (4.40%). Nonetheless the overall quality of the ice thickness estimate is encouraging. Plots of the estimated ice thickness versus the ice thickness in the simulation runs (Fig. 3a) show good correspondence except for large values of ice thickness, where the estimated thickness greatly exceeds the true thickness. However, the number of these outliers is small relative to the total number of points. A histogram of the distribution of the ice thickness estimation error (Fig. 3b) shows that the distribution is slightly asymmetrical, consistent with the fact that thickness cannot be negative. The distribution functions for the actual and estimated ice thickness (Fig. 3c) match reasonably well.

A suite of maps illustrates the performance of the estimation model for this particular case (Fig. 4). The assigned bed stress, based on (16) with *ξ* = 1 and mass balance adjustment of each flowshed to enforce (31), is shown in Fig. 4a. Also shown are the estimated ice thickness (Fig. 4b), the thickness error (Fig. 4c), and the resulting estimate of bed surface topography (Fig. 4d). It is reassuring that the estimated bed surface topography actually resembles a deglaciated landscape because our larger aim is to use such estimates as the starting geometry for projection modeling of the climate-forced deglaciation of western Canada.

Similar performance tests have been carried out for models having a cyclic variation in ELA. We accomplish this by setting *z*_{L} = *z*_{0} − Δ*z*_{ela}, *z*_{H} = *z*_{0} + Δ*z*_{ela}, and *t*_{0} = 1000 yr in (38) and running the model for 80 000 simulation years until a periodically repeating state is achieved. Then for each model we select output states from the final cycle of the simulation that correspond to intervals of fastest deglaciation and fastest reglaciation. Owing to system lags and geometrical effects these usually differ from the times at which the ELA is changing most rapidly so the snapshot times can vary among models. Transient model outputs are assigned labels such as BC1·1550 ± 100H↑ or BC1·1550 ± 100L↓ where BC1 denotes the region 1550, the mean ELA in meters, ±100 the amplitude of the sinusoidal elevation excursions, H or L, a high or low mass balance forcing rate, and ↑ a maximally increasing or ↓ maximally decreasing rate of ice area change.

We summarize the results of performance tests carried out when the glacier cover is rapidly decreasing (Table 3) or rapidly increasing (Table 4). Comparing these results with those in Table 2 indicates that the ensemble mean and median of mean thickness error and fractional volume error are larger for the transient inversions than for the steady-state ones. Interestingly, for both the deglaciation and reglaciation datasets there is a strong positive bias to the mean thickness error and fractional volume error, so for both situations there is a tendency to overestimate the ice volume. We have no explanation for this. Of the 64 cases listed in Tables 3 and 4, only BC3·1650 ± 100H↓ yields an underestimate of average ice thickness and volume. Ensemble mean and median thickness error and fractional volume error are greater, by a substantial margin, for the situation of rapid reglaciation than for rapid deglaciation. Worldwide, rapid deglaciation predominates (Lemke et al. 2007) so for this situation the smaller errors should apply.

Summary of estimation errors in performance tests of transient models having temporally decreasing ice cover. Model averages are calculated for ice-covered cells and not the entire map.

Summary of estimation errors in performance tests of transient models having temporally increasing ice cover. Model averages are calculated for ice-covered cells and not the entire map.

## 6. Application to glaciers of western Canada

### a. Comparison with thickness measurements on three glaciers

We have previously commented that actual measurements of ice thickness in the study region are few and their usefulness for testing the ice thickness estimation method is open to question. The main sources of difficulty are that the geographical locations and true elevations of the measurements are not well controlled and that substantial surface lowering has occurred between the time that the measurements were taken and the date of the digital elevation model and ice mask used in the inversion. We have attempted to recover the measurement positions as best we can and, where possible, have applied the 1985–99 measurements of surface elevation change (Schiefer et al. 2007) to correct for thickness changes that have occurred between the measurement dates and the estimation date.

Figure 5 compares measured and estimated ice thicknesses for the three glaciers in the study area that have been geophysically surveyed and for which a published record exists. For Athabasca Glacier (52.19°N, 117.26°W) seismic sounding and drilling data were collected from 1959 to 1961 and are tabulated in Paterson’s doctoral thesis (Paterson 1962). Although measurement locations were precisely determined using optical surveying, the survey coordinate system was not georeferenced and lacks an elevation datum. It was therefore necessary to adjust its orientation and offset to align with the DEM of estimated bed topography. Since 1961 the surface elevation and spatial extent of the glacier have changed substantially. The Southern Rockies region, which contains both Athabasca and Peyto Glaciers, includes parts of British Columbia and Alberta. For BC glaciers, the Schiefer et al. (2007) time-averaged thinning rates were calculated by subtracting two DEMs to obtain cell-by-cell estimates. Both Athabasca and Peyto Glaciers are located in Alberta and one of the two DEMs is not defined beyond the BC boundary. Thus we have applied less accurate and more coarsely-resolved elevation-dependent estimates (Schiefer et al. 2007, Fig. 3) to obtain a thinning correction. The average amount of thinning between 1960 and 2005 was 52.4 m.

Radio echo soundings of Peyto Glacier (51.66°N, 116.56°W), documented in Holdsworth et al. (2006), were taken from 1983 to 1985. The results are tabulated in Table 1 of that work and include Universal Transverse Mercator (UTM) coordinates of all measurement sites. Accepting these at face value (making plausible assumptions concerning the assumed geodetic datum), the data need only be corrected for changes in ice extent and surface thinning. Using the same elevation-dependent relationship as for Athabasca Glacier, we found the average glacier-wide thinning for 1984–2005 to be 14.3 m.

For Salmon Glacier (56.15°N, 130.19°W), in British Columbia, depths were measured using seismic sounding and drilling (Mathews 1959; Doell 1963) but were presented as plotted cross sections and maps. We extracted thickness data from a contour map of surface and bed elevation (Doell 1963, Fig. 5) so the associated uncertainties are considerable. The average thinning for 1956–2005 was 107.7 m and ranged from 145.9 to 32.4 m. Gravity survey results are also available for Athabasca Glacier (Kanasewich 1963) and Salmon Glacier (Russell et al. 1960) but these are expected to be less accurate than direct sounding methods and have not been used.

Figure 5a shows the estimated ice thickness versus measured ice thickness for the three glaciers with no correction for ice thinning. Figure 5b shows the same data but corrected for surface lowering. Although the thinning correction improves the agreement between measurements and estimates there is still a general tendency for our model to underestimate the thickness of the three glaciers. Some of the disagreement can be attributed to uncertainties in the measurement locations and the thinning correction but one plausible explanation for underestimated ice thickness is that the stress system is more complex than that for an inclined slab from (2). The simplest approach to correcting for this situation is to introduce a dimensionless shape factor 0 < *f* ≤ 1 (e.g., Nye 1964; Cuffey and Paterson 2010) and rewrite the bed stress as τ* = *f ρgh* sin*θ* which, reworking (13), leads to a systematic increase in estimated ice thickness by a factor *F* = (1/*f*)^{n/n+2}. Taking *n* = 3 and assuming *f* = 0.70, a reasonable value, gives *F* = 1.24. Multiplying the estimated ice thickness values by this “correction factor” yields an improved fit to the 1:1 line in Fig. 5b but we have substantial misgivings about the trustworthiness of this three-glacier performance test and many concerns about applying this as a global correction to a region that contains more than 17 000 glaciers. Thus for our thickness estimates we take *f* = 1 and flag this assumption as a potential source of error. The shape factor is only relevant to the portions of glaciers that flow through confined channels so, for icefields, *f* = 1 is more likely to apply. Furthermore the shape-corrected bed stress is only meant to apply along the central flow axis of confined channels and not near the channel walls.

### b. Ice volume estimates

Finally we apply the ice thickness estimation method that has been described and validated in previous sections to the problem of estimating the ice volume and subglacial topography of all glaciers in the study region. Our ice mask (Fig. 6) represents glacier extent in Alberta and British Columbia from a.d. 2005 with subregions chosen to match those of Bolch et al. (2010). The DEM for BC and AB is from the Shuttle Radar Topography Mission (SRTM) version 4.1 with 90-m spatial resolution (Farr et al. 2007) and downloaded from http://srtm.csi.cgiar.org/.

For the BC–AB study region we relied mainly on ice masks that were derived from Landsat Enhanced Thematic Mapper (ETM) data and based on scenes captured in 2005 (Bolch et al. 2010). The original masks are in the form of vector graphic polygons but, for present purposes, have been converted to rasterized objects that align with the 200 m × 200 m cells of our computational grid. The St. Elias (SE) and northern coast (NC) subregions required special treatment because the Bolch et al. (2010) masks are only available for British Columbia, whereas several glaciers cross the British Columbia–Alaska boundary. Hence, for these subregions we used the same ice masks as Berthier et al. (2010). These masks were extracted from the GLIMS glacier database (Beedle 2006) and are mainly derived from U.S. Geological Survey (USGS) sources but are heterogeneous in terms of the data sources and dates of acquisition. The most serious consequence of this methodological inconsistency is that our calculated ice areas (hence estimated volumes) for the SE and NC subregions differ slightly from those tabulated in Bolch et al. (2010).

We model the mass balance fields (F. S. Anslow et al. 2013, unpublished manuscript) using climate fields that have been downscaled from the North American Regional Reanalysis (Mesinger et al. 2006) following a methodology that has been described and validated by Jarosch et al. (2012a). As input DEMs for our downscaling methods, we resample the 90-m SRTM dataset at 1 km for precipitation and 200 m for surface topography and temperature. We do not consider knowledge of the glacier mass balance fields to be an onerous requirement because our main motivation for estimating subglacial topography is to perform computer simulations of the climate-forced deglaciation of our study regions. Mass balance fields are essential for any serious modeling effort, so we would require these in any case.

Because few contemporary glaciers are in balance with their climate forcings, we also require estimates of the ice thinning (or thickening) rates. We are obliged to use two different sources for these data. Within British Columbia (BC) the spatial variation of thinning rates is based on the datasets published by Schiefer et al. (2007) and applies to the time interval 1985–99. These were generated by differencing the SRTM DEM for February 2000 and an approximately 1985 DEM based on aerial photography (British Columbia Ministry of Environment 1992). We reprojected these data from the native BC Albers projection to the Lambert conical conformal projection used in the North American Regional Reanalysis (NARR) and then resampled at 200 m to match our computational grid. For Alberta and those parts of Alaska and Yukon that are contiguous to BC, no suitable DEMs existed for the 1980s so spatial representation of the thinning rate was not possible. For these cells, beyond the limits of the BC data, we applied elevation-dependent thinning rates using data from Fig. 3 of Schiefer et al. (2007).

Downscaled NARR climate fields are used to construct annually-averaged glacier mass balance rates for the study region. The degree of time averaging that should be applied to these data is not clear cut and is likely to depend on glacier size as well as other factors. A one-year time-average is too short because it is considerably smaller than representative values of the glacier response time. However, in a warming climate, a century-long time average might assign too much weight to the past state of glaciers. This sensitivity of ice volume estimates depends on how the mass balance field is time averaged (Table 5). We take decadal averages over the time spans 1980–89, 1990–99, and 1999–2008 as well as the 29-yr average 1980–2008 and denote these time-averaged balance rates by

Sensitivity of ice volume estimates to changes in mass balance field. From downscaled mass balance rates fields ^{8} km^{2} with *ρ* = 910 kg m^{−3}, and *ρ*_{w} = 1000 kg m^{−3}.

We foresee that good estimates of the time-averaged thinning rate fields will not necessarily be available for all regions where ice thickness estimates are needed. Various treatments of the thinning rate affect the inversion results (Table 6). The different possibilities that were considered are indicated by

Summary of ice volume estimates for glaciers of western Canada. The ^{8} km^{2} with *ρ* = 910 kg m^{−3}, and *ρ*_{w} = 1000 kg m^{−3}.

We also calculate the total number of ice masses (i.e., unique ice masks with no regard to whether they constitute one or more flowsheds) and the total number of delineated flowsheds (Table 6). For gridded data each cell has neighbors to the north, east, south, and west as well as diagonally situated neighbors to the NE, SE, SW, and NW. Whether one allows diagonal connectivity (C8) or disallows it (C4) influences the results of these calculations. We assume C8 connectivity for ice masses and flowsheds (Table 6). The total ice area for each region is given in column 4 and for most regions closely matches that presented in Bolch et al. (2010); this is not surprising because we used the same ice masks. However our ice areas for the SE and NC regions differ from those of Bolch et al. (2010) because for these regions we use the USGS ice masks that spanned the political boundary between BC and Alaska.

We now compare the estimated sea level equivalent from our inversion method to those derived from volume–area scaling (Bahr 1997; Bahr et al. 1997; Radić and Hock 2010). From *V* = *K*(*A*/*A*_{0})^{γ} with *γ* = 1.375 and *K* = 0.036 544 km^{3} (which with dimensional adjustments corresponds to the *c* = 0.2055 m^{3−2γ} adopted by Chen and Ohmura 1990; Radić and Hock 2010) the estimated volume using the scaling formula is 6.214 mm SLE (ice volume 2470 km^{3}). It is interesting that the volume–area-scaling method yields ice volumes that are not vastly different from those obtained by our estimation technique. Before error analysis, our best estimate of the present day (ca. 2005) ice volume for glaciers of British Columbia and Alberta is 5.83 mm SLE (ice volume 2320 km^{3}).

### c. Error analysis

Table 7 represents an attempt to summarize and quantify the known sources of error. For temperate ice, the flow law coefficient is uncertain and recent studies cited in Cuffey and Paterson (2010) (Hubbard et al. 1998; Gudmundsson 1999; Adalgeirsdóttir et al. 2000; Albrecht et al. 2000; Truffer et al. 2001), in which glacier flow modeling is used to calibrate the flow law, have led to a substantial revision of ^{−24} Pa^{−3} s^{−1} encloses their spread. We assume that all ice in the study region is temperate so there is no need to consider temperature effects on

Summary of sources of error and estimates of error magnitude. Ocean area is taken as 3.62 × 10^{8} km^{2} with *ρ* = 910 kg m^{−3}, and *ρ*_{w} = 1000 kg m^{−3}. Thus 1 km^{3} ice volume corresponds to 2.51 *μ*m of sea level rise and 1 mm SLE corresponds to 398 km^{3} ice volume. For the error estimates the total area of ice cover is taken as 26 586 km^{2} (the 2005 value from Table 6) and the total ice volume as 5.8 mm SLE (from Table 5).

From (34) it is apparent that errors in the flow law coefficient do not strongly affect the thickness estimates. A nonsliding glacier will be thicker than a sliding glacier in the same setting subjected to the same mass balance forcing. Thus our assumption that *ξ* = 1 (no sliding) contributes to an overestimation of ice thickness. In reality *ξ* varies from glacier-to-glacier and from point-to-point in any given glacier. Although for parts of some fast-flowing surging glaciers *ξ* < 0.1, we suspect that *ξ* ≈ 0.8 is typical of the majority of healthy mountain glaciers; for glaciers in retreat, the sliding contribution is likely to be even smaller. Lastly, (34) indicates that ice thickness estimates are comparatively insensitive to uncertainly in *ξ*.

We set the default value of the trade-off parameter to *χ*_{0} = 0.4 to smooth out the estimated bed topography but this also leads to a reduction in estimated thickness. With *χ* → 1 the average estimated thickness is maximized but the exaggeration of bed topography is unacceptable and could pose problems when used as the substrate geometry for ice dynamics modeling. By rerunning the inversion model for a range of *χ*_{0} values we conclude that for *χ*_{0} = 0.4 an overestimate of ice volume is unlikely and that underestimation should not exceed 5% (0.3 mm SLE).

Other potential sources of systematic error are associated with suspected errors in the ice masks for the St. Elias and northern coast subregions and with the physical assumptions of the inversion model. As previously discussed, the ice mask areas for the SE and NC subregions differ slightly from than those calculated by Bolch et al. (2010) and for which his estimated error is small. If one accepts the Bolch et al. values as correct then the ice volume for these subregions, taken together, could be underestimated by as much as 0.1 mm SLE.

The modeling assumptions that warrant scrutiny are (i) that *τ** = *ρgh* sin*θ* provides an acceptable approximation to the bottom stress irrespective of proximity to valley walls, and (ii) that *τ**. We view this as the weakest link of the inversion procedure and one that could lead to underestimates of ice thickness and thus of total ice volume. The magnitude of this underestimate might be as large as 1.5 mm SLE.

For sources of systematic error that would lead to an overestimate of ice volume (Table 7) errors total to 0.6 mm SLE; for sources that would lead to an underestimate the total is 2.2 mm SLE. However, it is highly unlikely that the combined systematic errors would conspire to produce either of these extrema but deciding how best to combine systematic and random error contributions is a subjective task. For each source of systematic error we shall postulate a form for the error distribution function and use the lower and upper range estimates to guide our assignment of the mean value and standard deviation for each distribution. Thus, for the flow law coefficient *σ* = 0.3 mm. The remaining sources of systematic error are either all negative (e.g., *ξ* errors) or all positive (e.g., model physics and shape factor *f*), and we approximate these by exponential distributions. A convenient property of exponential distributions is that the magnitudes of the mean and standard deviation are identical. The range limits in Table 7 are intended to indicate extreme limits of the individual distribution functions so we shall associate the magnitudes of the range limits with the 3*σ* values of the exponential distribution. The sign of the limit determines whether the exponential function is left or right sided. The remaining errors are random and assumed to be Gaussian distributed with zero mean and standard deviations given by the range values. Thus, for example, the standard deviation of the DEM elevation error is *σ* = 0.001 mm SLE. A Monte Carlo procedure (with *N* = 100 000) was then followed to generate a statistical dataset formed by summing the random contributions from each error term and the mean and standard deviation were then calculated for the combined dataset. The mean value of the combined error is 0.54 mm and its standard deviation is 0.55 mm. Thus we conclude that when random and systematic errors are taken into account the estimated ice volume is 6.3 ± 0.6 mm SLE or 2530 ± 220 km^{3}. [It is a simple matter to convert between glacier ice volume and the equivalent sea level rise in Table 7. Taking the ocean area as 3.62 × 10^{8} km^{3} (Lemke et al. 2007) with *ρ* = 910 kg m^{−3} and *ρ*_{w} = 1000 kg m^{−3} the conversion relations are 1 km^{3} ice = 251 *μ*m SLE and 1 mm SLE = 398 km^{3} ice.]

## 7. Discussion and conclusions

Scientific interest in the thickness of glaciers (e.g., Agassiz 1847) preceded, by almost a century, the advent of geophysical instruments capable of measuring this quantity. Recent interest has focused on ice volume and the potential contribution to sea level rise. Volume–area scaling (Chen and Ohmura 1990; Bahr 1997; Bahr et al. 1997; Radić and Hock 2010) was a first response to the problem of estimating the volume of Earth’s mountain glaciers and has the attraction of involving a readily observable quantity (area) as its sole input. Our estimates of ice volume (Table 6) show good agreement with those based on volume–area scaling. We suspect that, in part, this is fortuitous but both methods start from similar physical assumptions so the result is not altogether surprising. Whatever the merits of estimating ice volume using volume–area scaling, the method has limited usefulness for estimating the map of bed topography lying beneath the surface of glaciers—essential information for using computational ice dynamics models to project the future volume and extent of Earth’s mountain glaciers. However, the use of geophysical inversion methods to estimate bed topography has its own pitfalls. As emphasized by Bahr et al. (1994) the problem of calculating the basal stress from boundary conditions imposed at the ice surface yields a boundary value problem that is ill-posed and unstable, causing surface errors to increase exponentially as depth increases.

In our study we have described an approach to estimating ice thickness that is based on simplified glacier physics and on mass balance accounting applied to automatically delineated glacier flowsheds. By framing the question as a geophysical inversion problem, smoothness can be controlled using a space-varying trade-off parameter rather than applied separately at some later stage. The method performs best when glaciers are near equilibrium with a steady climate. Tests on synthetically generated ice cover indicate a tendency for ice thickness to be overestimated when climate is varying, irrespective of whether this leads to glacier growth or to shrinkage. Applying the method to the mountain glaciers of the Canadian cordillera yields a DEM of the subglacial topography and new estimates of ice volume for this region. Our best estimate of the ice volume is 2530 ± 220 km^{3}, equivalent to 6.3 ± 0.6 mm of sea level rise.

We see many areas where future improvements are called for. Accurate DEMs and ice masks are an essential starting point. Development of reliable algorithms for delineating glacier flowsheds should be viewed as a high priority. More challenging will be to remove the reliance on simple stress assumptions and to reframe the question as a nonlinear inverse problem. This will require substantial ingenuity combined with abundant computing resources. Without better knowledge of the mass balance fields this level of complexity is not yet warranted.

## Acknowledgments

We thank the Canadian Foundation for Climate and Atmospheric Sciences (CFCAS), the Natural Sciences and Engineering Research Council of Canada, BC Hydro, the Columbia Basin Trust, and the Universities of British Columbia and Northern British Columbia for financial support. Simon Ommanney provided references for geophysical measurements of ice thickness in western Canada. This paper is a contribution to the Polar Climate Stability Network and to the Western Canadian Cryospheric Network, which were funded by CFCAS and by consortia of Canadian universities. We are grateful to reviewers David Bahr and Daniel Farinotti and to Editor in Chief, Tony Broccoli, for their constructive contributions.

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