The IGM emulator is a CNN that takes as input two-dimensional arrays of glacier thickness h, surface topography s, Arrhenius factor A a number that determines the ice viscosity;, and a sliding coefficient c based on a Weertman sliding law;, and outputs both horizontal velocity components u,v as three-dimensional arrays Fig. 3. All variables are defined over a rectangular domain Ω, and on a structured uniform orthogonal axis-aligned grid with cell width H. Each input array has dimensions Nx×Ny (width times height of the domain Ω in terms of number of cells), and each output array has dimensions Nx×Ny×Nz, where Nz represents the number of vertical layers. The neural network can be represented as follows, where the subscript (⋅)H denotes the discretized variable: 1Nλ:(hH,sH,AH,cH)↦(uH,vH). Hereafter we will drop the subscript (⋅)H for clarity, with the understanding that all variables are discretized within IGM.

As described in , the weights λ of the CNN are trained to minimise the energy functional associated with the Blatter–Pattyn model , thereby replacing the solver that would usually be present in a glacier model, and making this method a form of physics-informed neural network (PINN). PINNs are a method of approximating the solution of a differential equation by way of a neural network. PINNs typically take spatial and/or temporal coordinates as input, and incorporate physical laws into their loss function in the form of the residual of a differential equation and its boundary conditions . IGM's ice flow emulator differs from the original PINN method in two important ways: the inputs to the neural network are fields defined on the entire horizontal grid, rather than spatial coordinates, and the loss function is the energy functional similar to the “variational PINN” or “VPINN” described by, not the residual of the equations and boundary conditions. The formulation of the energy functional of the Blatter–Pattyn model is omitted, but can be found elsewhere in the literature e.g..

We use a pretrained CNN (available from the IGM repository on GitHub), which has the architecture described in : 16 two-dimensional convolutional layers each with a 3×3 kernel and 32 output filters, with padding to conserve the frame size, and leaky rectified linear units as the activation functions, resulting in a CNN with about 140 000 trainable parameters. The CNN has been pretrained on a catalogue of mountain glaciers specifically chosen to be heterogenous and representative of a variety of glacier shapes .

The benefit of the CNN approach over that of traditional ice flow models is that evaluation of the pretrained neural network is fast, especially on a GPU. Additionally, it allows the use of the Blatter–Pattyn model, which for mountain glaciers is more appropriate than the SIA used in the Millan global thickness inversion study.

The encoding of the Blatter–Pattyn model equations as a neural network also makes the inversion framework simple to set up, taking advantage of automatic differentiation, and using tools from the Tensorflow and Keras Python libraries. The aim of the inversion module in IGM is to invert the ice flow mapping (Eq. ). We take known surface velocity data, surface topography data, and thickness measurements (where present), and pick some combination of thickness h, surface topography s, Arrhenius factor A, and sliding coefficient c to be control variables which we can adjust. The aim is to obtain an optimal fit of these control variables across the whole glacier, in the sense that the misfit between the surface velocity output from the emulator (Eq. ) and the observed surface velocity is small .

In this study, we focus on thickness h as the only control variable. The optimized map is obtained by minimising the cost function 2J(h)=Cu+Ch+Rh+Ph. Letting M denote the ice-covered domain, the misfit between modelled and observed surface velocities is 3Cu=12σu21area(M)∫Musobs-Nλ(h,s,A,c)2dm, where usobs=(usobs,vsobs) denotes the vector field of observed horizontal velocity at the surface and |⋅| denotes the Euclidean norm. Letting T⊆M denote the subset of cells for which thickness observations exist, the misfit between modelled and observed ice thickness is 4Ch=12σh21#cells∈T∑cells∈Th-hobs2, where |⋅| denotes absolute value. Details of the computation of the discretized hobs are given in Sect. .

A regularization term is included to minimize the presence of non-physical small-scale spatial variations in the thickness field. Although our control variable is thickness h, we instead regularize the bedrock b=s-h (note that the surface elevation s, which is derived from a DEM, does not change at all in our inversion setup, and that b=s outside M). While the inclusion of this term has the effect of regularizing the ice thickness, it additionally imposes smoothness of the bedrock at the boundary of M. The regularization term for the bedrock elevation is Rh=αh2β1area(Ω)5×∫Ω∇b⋅ũsobs2+β∇b⋅ũsobs,⟂2dm where ũsobs is the observed surface velocity after smoothing with a Gaussian filter with standard deviation 3 and then normalizing, (ũsobs)⟂ is the unit vector field orthogonal to it, and β is a coefficient which allows for anisotropic smoothing (β=1 corresponds to isotropic smoothing, β closer to 0 enforces more smoothing in the along-flow direction). In all of our experiments, we fix the anisotropy factor at β=0.2. The final term is a quadratic penalty term Sect. 17 to enforce non-negative ice thickness and zero thickness outside the ice-covered domain: Ph=1010(1area(Ω∖M)∫Ω∖Mh2dm6+1area(cells where h<0)∫h<0h2dm).

The parameters σu,σh are user-defined weights which can be thought of as “confidence levels” for the observed surface velocity and thickness respectively – a lower σ corresponds to more “confidence” and more weight placed on matching these observations. Note that these σ values do not represent measurement error; they are intended to be tuned by the user . The regularization parameter αh controls how much weight is placed on smoothing the thickness field.

In this study, the thickness field is initialized with the Millan thickness estimate. A sensitivity test was conducted to confirm that results are independent of the starting point: the ice thickness field was initialized at a constant 100 m inside M and zero elsewhere, and the results were indeed very close to the results obtained by initializing with the Millan thickness. We use the Millan thickness as the initial thickness as it is expected to be relatively close to the solution (compared to a constant 100 m thickness), thereby helping the optimization to converge sooner.

The cost J(h) is minimized using the Adam optimizer , which is a popular method for training neural networks, and is a variation on stochastic gradient descent with momentum. The inversion begins with a pretrained neural network Nλ, which goes through one iteration of retraining (also with Adam) after each inversion iteration, using the current glacier geometry as the training data. The aim of retraining is to keep the neural network loss (the energy functional associated with the Blatter–Pattyn model) low, so that the emulator continues to produce output that is close to Blatter–Pattyn solutions. We used one retraining iteration per inversion iteration (the model default), which is expected to provide sufficient fine-tuning, as the thickness field is expected not to deviate much between inversion iterations, especially later on in the optimization procedure. Examples of how the inversion cost (Eq. ) and neural network loss evolve over iterations of the optimization algorithm in practice can be found in Appendix .

We emphasize that it is not necessary for the velocity to be modelled as the output of a neural network: the method of minimizing Eq. () would also be possible with an alternative method of finding the velocity, e.g. by directly minimizing the energy functional of the Blatter–Pattyn model to find the solution. Among the benefits of using a neural network emulator are fast evaluation (especially on GPU), differentiability, and the resulting simplicity of the cost function, particularly the velocity component (Eq. ).

We note also that IGM's inversion module allows for additional control fields and cost terms, as outlined in . We opted not to use the surface elevation as a control variable as this would double the degrees of freedom, possibly admitting non-unique solutions. Furthermore, we decided not to use terms constraining the flux divergence, as the options are to either match to observed flux divergence (this would require reliable surface mass balance observations, which are not available for our glacier of interest), or to match to a linear regression of the modelled flux divergence, which is recomputed at every iteration and can reinforce unrealistically high flux divergence values.

Bhote Kosi glacier (whose RGI v6.0 ID is RGI60-15.03422) was chosen for an in-depth study. It is one of the largest glaciers in the dataset and has good coverage of thickness observations, with several longitudinal (along-flow) and across-flow profiles, and observations across a wide elevation range (see Fig. ). notes that different parts of this glacier are known by different names: Bhote Kosi is the southern main “trunk” of the glacier, with the west tributary and north tributary called Lunag and Nangpa respectively. Throughout this paper we typically use the name Bhote Kosi to refer to the entire glacier, since it is dynamically connected and appears as one glacier in RGI v6.0.

We perform several thickness inversion experiments on Bhote Kosi glacier, starting with a simple configuration, and progressively introducing more complexity. We begin by performing inversions without calibrating to thickness observations, in order to tune the ratio between σu and αh. We then introduce thickness observations into the inversion, aiming to find an appropriate value for σh, and perform cross-validation experiments to explore the value added by calibrating to thickness observations.

Initial experiments with IGM inversion showed that the choice of parameters σu, σh, and αh can have a significant effect on the resulting thickness field (see Fig. ). With our in-depth study on Bhote Kosi, we aim to find a set of parameters that provide an optimal balance between matching observed surface velocities and producing a realistically smooth thickness field.

Following the experiment described in Sect. , Fig. a plots the final values of Cu against Rh/αh, and a clear “L” shape is seen. Heuristically, values of αh close to the “corner” of the L-curve provide a good compromise between minimal velocity mismatch and low spatial variability in the thickness field. Here we find that αh=6,10 lie in this region, and are therefore considered acceptable by this method.

During this experiment, we also measured the MAE and MBE between our final thickness field and the observed thicknesses. We found that the lowest MAE values (between 42–46 m) were achieved for αh=2,3,6,10 (Fig. b), including values which were, incidentally, also found in the corner of the L-curve. The MBE values closest to zero (between -4 and -7 m) are achieved for αh=0.6,1,2; however, the resulting thickness fields are visibly under-regularized compared to those achieved for higher values of αh (e.g. Fig. b). Additionally, the thickness fields for αh=0.6,1 show poorer accuracy in terms of MAE than the thickness fields for αh=6,10. Figure b also demonstrates clearly that MAE and MBE need not be minimized simultaneously: the value of αh producing the lowest MAE is not the same as the αh with the lowest MBE.

Additionally worth noting is that the value of αh appears to have a direct effect on the volume of the modelled glacier, with lower αh corresponding to a higher final volume (Figs. c and ), possible reasons for which are discussed in Sect. .

Based on the results found here, we acknowledge that there is no single optimal value of αh. As the L-curve is a heuristic method, the choice of the “best” αh among 2,3,6,10 is subjective. Note, however, that among these values, as αh increases, the regularization cost decreases, but the magnitude of the MBE increases, and the velocity cost also increases slightly. We use the value αh=6 in all of the following experiments, as it provides a good compromise between a well-regularized thickness field and a low MAE with observed thicknesses.

Figure a–d show the profiles chosen for each experiment described in Sect. . Figure e–h show the MAE against σh, for each of the four cross-validation experiments. In all four experiments, the error on the calibration set decreases as the strength of thickness constraint is increased (by decreasing σh). For all four experiments, the MAE on the calibration set drops below 10 m for any σh≤300, and is very small for any σh≤100. On the other hand, there does not appear to be a clear relationship between the value of σh and the validation set error, which remains between 32–45 m for all four experiments and all values of σh.

Figure  shows the resulting thickness maps when constraining to no profiles (i.e. the velocity-only result), all available profiles, low-elevation profiles only and longitudinal profiles only.

The results of the experiment described in Sect.  are presented in Fig. . In Fig. a, we find that the lowest surface velocity cost is found for A=52 MPa-3yr-1, but that there is only a modest reduction of around 4.5 % in surface velocity cost against the result for A=78 MPa-3yr-1. Figure b shows that the lowest mean absolute error is achieved for A=52 MPa-3yr-1, but that the mean absolute errors for the results for A between 39–91 MPa-3yr-1 are all very close, lying between 39–42 m.

In Fig. c, we see that the mean bias errors are negative for any A value. This is, however, consistent with our results from Fig. b showing that the mean bias error is negative for αh=6. In general, increasing αh has the effect of lowering the volume, whereas decreasing A corresponds to colder, stiffer ice, resulting in a higher volume (since a higher thickness is required to “explain” the same velocity). Low A values in this experiment are likely counteracting the choice of αh which already resulted in a negative MBE (indicative of an underestimation of the glacier volume). The mean bias error closest to zero is achieved for A=13 MPa-3yr-1, but since this value also produces a very high mean absolute error, there is insufficient evidence to recommend this value.

We initially endeavoured to produce the “best possible” thickness field with velocity data only, by making a careful choice of the regularization parameter αh, guided by the L-curve analysis in Sect. .

It is clear from our results that the choice of regularization parameter αh has a strong effect on the final volume of the modelled glacier (Figs. c and ). To understand why this is the case, observe that the regularization term (Eq. ) is non-negative, and reaches zero if and only if ∇b=0 everywhere in Ω, i.e. when the bedrock elevation b is constant. Although the value of b does not change outside the glacier mask M, flattening the bedrock inside M and attempting to match the bedrock elevation at the boundary of M will still reduce the regularization cost. In addition, the penalty term (Eq. ) enforces zero thickness outside M. These two effects together encourage low ice thickness h. This thinning effect is stronger when the regularization term Rh is larger in comparison to the other terms in Eq. (), i.e. when αh is larger. Hence, increasing the value of αh results in a decrease in modelled volume. Although this effect is somewhat undesirable, inclusion of a regularization term is nonetheless necessary to avoid non-physical small-scale spatial variations in the thickness field (as seen in e.g. Fig. a and b).

In future investigations it may be possible to avoid this undesirable effect through a reformulation of the regularization cost function. A potential strategy is to ignore gradients in the direction normal to the glacier boundary, within cells at the boundary. However, implementing such an approach is beyond the scope of our study.

We observe in Fig. b that in the IGM estimate, there is an area of very low thickness present about halfway up the north tributary of the glacier, coinciding with a location where measured velocities are low (Fig. a). Small ice velocities are known to have high relative errors due to limitations associated with feature tracking methods . Since the ice is observed as almost stationary, there is insufficient information to infer the ice thickness from the surface velocity at this location. A similar issue may also be present at the area near the glacier terminus, as we also observe low velocity and low modelled thickness there. Despite using the same surface velocities from which the Millan estimate was derived, these low-thickness features were not present in the Millan estimate (Fig. a). This phenomenon was also observed on preliminary inversions of Ngozumpa and Lumsamba glaciers (not shown); however, although those glaciers are well-surveyed, we do not have sufficient thickness measurements for validation at the affected locations.

Another non-physical artefact visible in the resulting thickness field (Fig. b) is checkerboarding, which is particularly apparent in the tributary halfway up the north tributary of the glacier, immediately to the west of the “low-velocity” area. This artefact is likely the result of the approximation of spatial gradients with simple finite differences in the cost function, which can produce undesirable modes. Further work is required to address this issue within IGM. In the present case, we found that this artefact was triggered most severely in a region showing discontinuity in the input velocity field, and vanishes when using smoothed data (Appendix ).

In our velocity-only inversion, we succeeded in making a reconstruction of the thickness field with errors comparable to the Millan estimate by using surface velocity data only. The observed thickness data (which was not available at the time the Millan estimate was published) was valuable in this experiment to validate the results, as otherwise, we would have had no way of validating the accuracy of our results, or indeed the Millan estimate. While it is not appropriate to consider the difference between our velocity-only thickness estimate and that of as an “error”, as neither study represents the ground truth, it is of interest to compare the two results, as both attempt to estimate glacier thickness based on the same surface velocity data, without using observed thickness.

Interestingly, in Fig. , we observe that IGM produces a more accurate representation of the thickness of the west tributary as compared to the Millan estimate. Measurements show that the west tributary is significantly thinner than the main trunk and north tributary, a detail that is reproduced by IGM even when it is given no direct information about the thickness in this location. Furthermore, Millan generally overestimates thickness in the west tributary, while underestimating thickness in the rest of the glacier. Improvements by IGM over the Millan study may be due in part to higher-order physics, since IGM is based on the Blatter–Pattyn model, while Millan used the SIA, which excludes stress components which are non-negligible in mountain glaciers.

The IGM-inverted thickness exhibits poorer accuracy in the north tributary as compared to Millan, an error which can be attributed in part to the underestimation of the thickness at the low-velocity area discussed in Sect. . We observe also that both Millan and IGM underestimate the thickness of the main trunk of Bhote Kosi (Fig. ).

compared their observed thicknesses with the Millan and consensus estimates, finding that both estimates are biased, with errors of over 100 m in places. In our velocity-only inversions, no value of αh succeeded in bringing the MAE below 42 m (Fig. b), indicating that significant bias is also present in our estimated thickness field. Relative errors (not shown) are over 50 % in several places, and over 100 % in a handful of locations. Such biases undermine the efficacy of these thickness products when predicting the evolution of a glacier over time. In the case of IGM, the systematic underestimation of thickness may result from the implementation of the regularization term (as discussed in Sect. ). Additionally, although IGM is based on the Blatter–Pattyn model, which contains higher-order stress terms that are missing from the SIA as used by Millan, it is still not a full-Stokes model. The use of a neural network as an emulator for the Blatter–Pattyn model, rather than a true solver, may also compromise IGM's accuracy, although high-fidelity results have been achieved previously .

Inter-comparison exercises focus on assessing the quality of thickness estimates by error with unseen measured thickness, but one can also consider a high-quality thickness estimate as one which yields an ice velocity field which is consistent with both data and physics. Although our velocity-only IGM thickness estimate is of comparable accuracy to Millan's thickness estimate when considering MAE or MBE across all measured thicknesses over the entire glacier, it does have the additional benefit of higher-order physical consistency between modelled thickness and satellite-observed velocity data. IGM achieves this by enforcing low misfit between modelled and observed surface velocity (Eq. ) during the thickness optimization.

In Fig. , we observe that increasing the strength of the thickness misfit term (Eq. ) effectively reduces the MAE on the calibration set, but does not reduce the MAE on the validation set. The MAE on the validation set remains between 32–45 m for all four partitions of the data and all values of σh, showing at best a modest reduction from the MAE of more than 42 m achieved when omitting thickness profiles from the inversion (Sect. ). This indicates that the thickness observations have limited interpolative power in this model setup, i.e. constraining to thickness observations has only a weak effect on the accuracy of the modelled thickness at unconstrained locations. This is further confirmed by Fig. , which shows the resulting thickness map when taking σh=300 and constraining to different subsets of the available thickness profiles. We see that the thickness distribution constrained to all profiles (Fig. b) appears largely the same as that obtained without thickness constraints (Fig. a), with only local-scale changes visible in the resulting thickness map, and a small increase in total volume. We observe also that between the thickness maps obtained by constraining to low-elevation profiles (Fig. c) and longitudinal profiles only (Fig. d), the calibration sets in these two cases have no profiles in common and no intersection between profiles in the two sets. Both thickness fields look very similar to the velocity-only thickness field at locations without observations.

While this result indicates that overfitting to observed thicknesses is not an issue, it also indicates that constraining to thickness observations has little effect on the thickness at unconstrained locations with this modelling approach. A potential reason for the low interpolative power of thickness observations is the limited spatial influence of stress perturbations according to the Blatter–Pattyn stress balance. Changes in speed and driving stress are felt over the membrane length scale , typically on the order of a few ice thicknesses. In the present context, consider a location where there are no thickness constraints. If a thickness constraint is introduced close by, this will influence velocities at the location, and the optimization therefore will need to adjust thickness at the location in order to reduce the misfit with observed surface velocities. If the introduced thickness constraints are sufficiently far away, the influence on velocity is minimal, and therefore so is the adjustment. With strong enough regularization, there could be a smoothing length scale that is larger than the membrane length scale, leading to non-physical interpolation of thickness constraints.

The disappearance of the erroneous low-thickness zone in the north tributary when adding thickness constraints (compare Fig. a and b) highlights the importance of constraining to thickness profiles where available, as in this case their inclusion was able to correct for a lack of information from the surface velocity data.

We point out that some methods to estimate thickness involve “flux conservation” or “mass conservation” , in which the depth-integrated glacial mass balance equation essentially propagates information down-glacier or away from thickness profiles, explicitly carrying information to non-measured areas of the glacier. These methods, however, either require accurate surface mass balance information, which is difficult to obtain in our region, or depend on surface mass balance parametrizations at a regional level. Importantly, our approach is distinct from these continuity-based methods in that it provides a thickness that is dynamically consistent with a higher-order stress balance, making it more suitable for subsequent model investigations.

IGM's default value for A is 78 MPa-3yr-1, which heuristically corresponds to temperate glaciers around 0 °C Table 3.4, and may not be optimal for glaciers in HMA. On the other hand, the value used by Millan for glaciers in the Everest area was 13 MPa-3yr-1, which was based on a very small number of thickness observations for glaciers elsewhere in HMA. Their value approximately corresponds to between -5 and -10 °C Table 3.4, colder than measured englacial temperatures on nearby Khumbu glacier . Indeed, the use of different values of A may partially explain why Millan's modelled volume is slightly higher than ours (Fig. ): when directly estimating thickness via the SIA Eq. 2, there is an inverse relationship between A and ice thickness.

We acknowledge that using a different Arrhenius value for our previous experiments may change our results slightly (Fig. ), but we did not identify an ideal value. If we were to switch to a different (spatially constant) value of A, the aspect of our result that would be most affected would be the glacier volume (Fig. c). However, changing A would be unlikely to affect our earlier conclusions: that the regularization parameter αh affects the volume and can be optimized with L-curve analysis, and that tuning the thickness confidence parameter σh reduces the error at locations with thickness observations seen by the model, but seems to have little effect on unseen thickness observation locations. Future work could consider spatially varying A, a scenario which is more realistic since englacial temperatures (and therefore A) may vary throughout a glacier.

As mentioned in Sect. , the acquisition dates of the DEM are from 2010–2015, while the velocity observations are from 2017–2018, and the thickness observations are from 2019. We consider the nominal date of our thickness inversions to follow that of the velocities, which are relatively close in time to thickness observations. However, they are separated from the surface elevations used for the DEM by 3–8 years. This is still a short time relative to glacial evolution time scales, though it potentially introduces uncertainties in our thickness estimates.

Recent surface elevation change rates on Bhote Khosi have been observed at approximately -2 myr-1 , which implies the DEM surface is approximately 6–16 m higher than at the time of velocity acquisition. As the relationship between velocity, surface, and recovered thickness is non-local and complex, it is difficult to quantify the relationship between this surface bias and any resulting bias in estimated thickness. Regardless, the bias in surface elevation is relatively small, and in particular, smaller than the errors in estimated thickness in comparison with the thickness observations (Fig. b). Thus, we do not expect non-contemporaneity of our inputs to be a major source of uncertainty in our study.

On the other hand, even small errors arising from non-contemporaneity can lead to dynamical imbalance which causes spurious behaviour in model projections, as seen in similar settings in ice-sheet models . Therefore, it is important to ensure different data sources are as close in time as possible.

Our results demonstrate that a lack of thickness observations need not be a barrier to producing a reasonable thickness estimate, as we were able to produce results with estimated accuracy comparable to the leading thickness estimates by constraining to surface velocity data alone, by making an informed choice for the regularization parameter αh. Although our results showed that significant bias was present in non-thickness-constrained results, and that thickness constraints introduced only limited changes to the thickness at unconstrained locations, results on other glaciers may vary.

For a glacier with no thickness observations (which is the case for most glaciers), there is no way to perform a validation test against the true thickness, and therefore it would not be possible to create a plot equivalent to Fig. b. Regardless, it would still be possible to create an L-curve plot similar to Fig. a, and, assuming the desired L-shape is present, choose a value in the corner of the L (indeed, for this example we could have chosen the value αh=6 based on this alone). In our example, the αh values in the corner of the L-curve also gave a low MAE. It would also be possible to compare the total volume with the consensus and Millan total volumes, as in Fig. c: while neither of these studies represent the ground truth, this comparison serves as a consistency check to flag up any thickness fields whose volume deviates implausibly from other estimates.

Preliminary inversions were performed on the other 12 glaciers (as outlined in RGI v6.0) with thickness observations from . The parameters found to be optimal for Bhote Kosi were used (αh=6, σh=300), with results (not shown) varying in quality. Very low modelled volume was observed for multiple glaciers, particularly smaller glaciers where a resolution of 100 m may not be fine enough to resolve physical details. This demonstrates that the parameters, particularly αh, may be somewhat glacier-dependent. We note that the term Cu (Eq. ) involves a mean over the glacier mask M, but Rh (Eq. ) involves a mean over the rectangular domain Ω. When switching to a different glacier, the ratio between the areas of M and Ω is likely to be slightly different. This is one factor that may affect the regularization parameter required. In order to produce accurate thickness maps for other glaciers, a full L-curve analysis per glacier would likely be beneficial. A finer spatial resolution may also be required for smaller glaciers.

found that it is advisable not to have all thickness observations concentrated at the lowest elevations of the glacier, and that covering the thickest parts (which often coincide with the flattest parts) is the most effective way to ensure accurate volume estimation.

Our investigation demonstrated that there is a lack of “signal” from low-velocity locations, which often leads to erroneous low modelled thickness, as thin ice can also yield a similar near-stagnant ice flow. This effect can be corrected by the inclusion of a thickness profile for calibration. We therefore propose, in addition to the above considerations from , that fieldworkers should aim to measure the thickness at such low-velocity areas, because this may the best way to ensure accuracy at those locations when estimating thickness.

Conversely, thickness observations are also valuable in fast-flowing locations, since this is where velocity inversions are most effective. In such locations, thickness observations can aid in model-parameter optimization, i.e. the choices of αh and A in our setup. Furthermore, the membrane length scale is longer where there is faster ice flow, thereby increasing the interpolative power of thickness observations.

Following the discussion in Sect. , we reiterate that the inclusion of thickness constraints does not guarantee improved estimates of thickness at distant locations. Therefore, in our setup, the calibration of ice thickness at locations with no nearby measurements relies heavily on making optimal choices for the model parameters αh and A.