IMAU-ICE is a vertically integrated ice-sheet model, which has been specifically designed for large-scale, long-term simulations of ice-sheet evolution (Berends et al., 2022). It solves the depth-integrated viscosity approximation (DIVA; Goldberg, 2011; Lipscomb et al., 2019) to the stress balance, which is similar to the hybrid SIA/SSA but which remains close to the full-Stokes solution at significantly higher aspect ratios (Berends et al., 2022). Proper grounding-line migration is achieved by using a sub-grid friction-scaling scheme, based on the approaches used in PISM (Feldmann et al., 2014) and CISM (Leguy et al., 2021).

For this study, a new sliding law was added to IMAU-ICE, based on the work of Zoet and Iverson (2020). This recent work presents a sliding law based on laboratory experiments, contrasting with previous sliding laws which were based chiefly on theoretical considerations. Here, the basal shear stress τb depends on the basal velocity ub as follows: 1τb=u^bNtan⁡φ|ub||ub|+u01/p. Here, N is the (effective) overburden pressure, which we assume to be identical to the ice overburden pressure (i.e. no subglacial water); u^b is the unit vector parallel to the basal velocity; and φ is the bed roughness, expressed as a till friction angle. By default, the exponent p has a value of p=3, and the transition velocity u0 has a value of u0=200 myr-1. At low sliding velocities, this sliding law behaves like a Weertman-type power law (Weertman, 1957), with the basal shear stress approaching zero as the basal velocity approaches zero. At high sliding velocities, the basal shear stress asymptotes to the Coulomb friction limit (Iverson et al., 1998). This two-regime behaviour agrees with the theoretical considerations underlying previous sliding laws (e.g. Schoof, 2005; Tsai et al., 2015).

For this study, we developed a novel inversion procedure. It is based on the procedure used in CISM (Lipscomb et al., 2021), which in turn is a variation on the geometry-based approach from Pollard and DeConto (2012). In the CISM procedure, as in the Pollard and DeConto approach, the ice-sheet model is run forward in time, and the bed roughness field is adapted based on the difference between the modelled and the target ice sheet. However, whereas the Pollard and DeConto approach only considers the mismatch in ice thickness, a newer, unpublished approach in CISM additionally includes the mismatch in surface velocity, leading to faster convergence (since the velocity responds more quickly to changes in bed roughness than the geometry). We extend this approach by adopting a flowline-averaged rather than a purely local scheme to calculate the mismatch in terms of ice thickness and velocity. The rationale behind this is that changing the bed roughness at any location will affect the ice geometry and velocity not just at that location but also upstream and downstream. Reducing the basal roughness at one location will increase the ice velocity along the entire flowline, causing the ice both locally and upstream to become thinner. By including these effects in the inversion procedure, numerical stability is improved, and artefacts arising from differences in the flotation mask between the modelled and the target state are reduced. The bed roughness produced by the inversion is not affected by these changes, as the inclusion of a regularisation term usually ensures that the bed roughness converges to the same solution. The approach outlined here mainly improves the numerical stability and robustness under changing ice sheet/ice shelf/ocean masks of the inversion. This is shown in Appendix A, where we compare the convergence behaviour of our new inversion procedure to a method currently used in CISM, which also uses both the geometry and velocity mismatch but without the flowline-averaging approach.

Let p=[x,y] be a point on the ice sheet. We divide the flowline passing through p into an upstream part Lu(p,s) and a downstream part Ld(p,s), which can be found by integrating the ice surface velocity field u^=u|u|: 2aLu(p,s+ds)=Lu(p,s)-u^(Lu(p,s))ds,2bLd(p,s+ds)=Ld(p,s)+u^(Ld(p,s))ds,2cLu(p,0)=Ld(p,0)=p. Here, s is the distance along the flowline. In the upstream (downstream) direction, the integral is terminated at su (sd) at the ice divide (ice margin), i.e. when u=0 (H=0), so that 3au(Lu(p,su(p)))=0,3bH(Ld(p,sd(p)))=0. In order to calculate the rate of change dφ/dt of the till friction angle φ, the velocity mismatch (defined as the difference between the modelled absolute surface velocity |um| and the target absolute surface velocity |ut|) is averaged over both the upstream (Eq. 4a) and downstream (Eq. 4b) part of the flowline, whereas the ice thickness mismatch is evaluated only in the upstream direction (Eq. 4c; preliminary experiments showed that including a downstream ice thickness term was detrimental to the results): 4aI1(p)=∫s=0su(p)|um(Lu(p,s))|-|ut(Lu(p,s))|u0wu(s,su(p))ds,4bI2(p)=∫s=0sd(p)|um(Ld(p,s))|-|ut(Ld(p,s))|u0wd(s,sd(p))ds,4cI3(p)=∫s=0su(p)Hm(Lu(p,s))-Ht(Lu(p,s))H0wu(s,su(p))ds. Here, I1 represents the distance-weighted average of the velocity anomaly over the half-flowline upstream of p, I2 represents the distance-weighted average of the velocity anomaly over the half-flowline downstream of p, and I3 represents the distance-weighted average of the geometry anomaly over the half-flowline upstream of p. The default values for the scaling parameters are u0=250 myr-1 and H0=100 m. The linear scaling functions wu and wd serve to assign more weight to anomalies close to p, decreasing to zero at the ends of the flowline, as well as to normalise the integral: 5awu(s,su(p))=2su(p)1-ssu(p),5bwd(s,sd(p))=2sd(p)1-ssd(p). The scaling functions are constructed such that ∫s=0s=su(p)wuds=∫s=0s=sd(p)wdds=1. It is possible that integrating a finite distance from p, rather than over the entire flowline, might improve the rate of convergence; we did not perform any preliminary experiments to test this. The three line integrals from Eq. (4a–c) are then added together and scaled with the local ice thickness H(p) and velocity |u(p)|. This reflects the fact that bed roughness underneath slow-moving and/or thin ice has less effect on the large-scale ice-sheet geometry than the roughness underneath fast-flowing and/or thick ice: 6Itot(p)=(I1(p)+I2(p)+I3(p))R(p),7R(p)=|u(p)|H(p)usHs,0≤R(p)≤1. By default, the scaling parameters are us=3000 myr-1 and Hs=300 m. These values are based on preliminary experiments to attain fast convergence without creating numerical artefacts. Finally, the rate of change dφ/dt of the till friction angle φ can be calculated: 8dφ(p)dt=-φ(p)Itot(p)ts. The default value for the timescale is ts=10 years, again based on preliminary experiments to balance the convergence rate against the numerical stability of the procedure. While the flowline integrals in Eq. (4a–c) are calculated over the entire flowline (including floating ice), dφ/dt is calculated only for grounded ice; it is then extrapolated to fill the entire model domain using a simple Gaussian kernel. This approach helps to prevent artefacts in grid cells that switch over time between grounded and floating, or ice-covered and ice-free states, which typically present as individual or clustered grid cells where the iterative roughness adjustment overshoots, quickly diverging to extreme values.

The routine performing these calculations is run asynchronously from the other components of the ice-sheet model, with a time step of Δtφ=5 years. The till friction angle is updated every time this routine is called: 9φn+1=F2φn+ΔtφF1dφdt. Here, F1 and F2 are Gaussian smoothing filters, with their respective radii defined relative to the grid resolution: σ1=Δx/1.5 and σ2=Δx/4. These filters serve as a regularisation of the bed roughness, to prevent overfitting. Pattyn (2017) uses a similar regularisation approach, with a Savitzky–Golay filter instead of a Gaussian filter. Pollard and DeConto (2012) do not report any regularisation term in their inversion, while in CISM, the inclusion of a dH/dt term likely results in some smoothing. The radii of the two Gaussian filters, which were determined during preliminary experiments, are the lowest values we found that effectively repress small-wavelength terms in the inverted bed roughness, which are most likely a result of overfitting (Habermann et al., 2012). Increasing the radii of the filters does not significantly affect the inverted roughness until it is increased to several grid cells. Roughness variations of a small spatial scale could therefore potentially be obscured by the smoothing in our approach. However, such small variations would quickly approach the ice-dynamical limit of roughness variations that can be resolved by inverting from surface observations (about 50 ice thicknesses; Gudmundsson and Raymond, 2008), so this would likely not pose a serious problem in practical applications. The degree of overfitting in our approach is explored in more detail in Appendix A, where we demonstrate that it does not pose a significant problem.

Our inversion method does not include weighting of the velocity–elevation mismatch based on uncertainty estimations in the observations. However, including these weights in the method would not be difficult and is worth considering when applying this method to the Greenland and/or Antarctic ice sheets.

It might be possible to improve upon the inversion procedure presented here, achieving faster or more robust convergence or better computational performance. For example, our flowline-averaged approach might be difficult to implement in parallel models with a distributed-memory architecture (i.e. where a processor might not have access to all the data on a flowline), which is not the case in IMAU-ICE. However, the aim of this paper is not to find the most efficient way to perform a basal inversion but rather to investigate the uncertainties that remain in the result of that inversion even when the procedure itself works perfectly.

In order to quantify the compensating errors from one particular model component, we use what we call a perfect-model approach. We first use the ice-sheet model to calculate the steady-state ice-sheet geometry for a known bed roughness field in a simulation we call the “target run”. The known bed roughness will be called the target roughness and the resulting ice-sheet the target geometry. If we then apply the inversion routine, with all model parameters set to the same values as were used to create the target geometry, then theoretically the resulting inverted bed roughness (which we call the unperturbed roughness) should be exactly the same as the target roughness. The difference between the unperturbed roughness and the target roughness is the model error of the inversion routine. If the inversion procedure works adequately, this error should be small.

We then perform a “perturbed” inversion, where we change one or more of the model parameters/components (e.g. viscosity, surface mass balance (SMB), subglacial topography) with respect to the target run. As long as the change is small enough that its effect on the steady-state geometry can be compensated for by a change in bed roughness, the inversion will produce an ice sheet that still matches the target geometry and velocity but with a different bed roughness, which we call the perturbed roughness. The difference between the perturbed and unperturbed roughness is the compensating error in the bed roughness caused by the error in the model parameter that was changed in the perturbed run. This procedure is illustrated schematically in Fig. 1.

The first of our two idealised-geometry ice sheets is based on the EISMINT-1 “moving margin” experiment (Huybrechts et al., 1996). It describes an ice sheet on an infinite, non-deformable flat bed, with a radially symmetrical surface mass balance which is independent of the ice-sheet geometry: 10M(r)=min⁡(Mmax,S(E-r)). The values of the parameters are listed in Table 1; the radial distance r from the grid centre is expressed in metres. The ice viscosity is described by a uniform value of Glen's flow law factor A (i.e. no thermomechanical coupling). Lastly, we introduce a non-uniform till friction angle: 11φ(x,y)=φmax-(φmax-φmin)e-12x-xcσx2+y-ycσy2. The values of the parameters are listed in Table 1.

The equation thus describes a strip of reduced bed roughness running along the negative y axis of the domain, which results in the formation of an ice stream with higher ice velocities and a protruding ice lobe, as illustrated in Fig. 2. The ice sheet is initialised to a steady state by integrating the model through time for 50 000 years.

The second idealised-geometry ice sheet is based on the MISMIP+ geometry (Asay-Davis et al., 2016). This describes a laterally symmetric glacial valley, about 800 km long and 80 km wide, with a slightly over-deepening bed, followed by a sill, before dropping sharply into a deep ocean. A uniform accumulation rate of 0.3 myr-1 leads to the formation of a fast-flowing ice stream feeding into a small embayed shelf. The grounding line rests on a retrograde slope, kept in place by buttressing forces. As in experiment I, we introduce a non-uniform bed roughness, which is again described by Eq. (11); the parameters for this experiment are listed in Table 2. Following the MISMIP+ protocol set out by Asay-Davis et al. (2016), the uniform value for Glen's flow law factor A=1.13928×10-17 Pa-3yr-1 is tuned to achieve a steady-state geometry with a mid-stream grounding-line position at x=450 km, in the middle of the retrograde-sloping part of the bed. The resulting ice-sheet geometry is illustrated in Fig. 3. The ice sheet is initialised to a steady state by integrating the model through time for 50 000 years.

In order to verify that the inversion procedure is working properly, we first apply it to both idealised-geometry experiments with all model parameters unchanged. For experiment I, we perform these unperturbed inversions at resolutions of 40, 20, and 10 km; for experiment II we use values of 5 and 2 km. The 50 000-year steady-state initialisation is performed separately at all resolutions. The till friction angle is initialised with a uniform value of φ=5∘, and the model is run forward in time for 100 000 years. With this choice of initial value, the bed roughness typically converges to a stable solution within ∼30000 years (as demonstrated by the additional experiments in Appendix A).

The resulting inverted bed roughness fields for both sets of simulations are shown in Figs. 4 and 5, respectively. The errors in the inverted bed roughness, and the resulting ice-sheet geometry and velocity, are very small at all resolutions and in both experiments (typically <5 % for the bed roughness, <5 m for the surface elevation, and <5 % for the surface velocity), indicating that the inversion procedure works well in the simple geometries of these two experiments.

To quantify the compensating errors in the inverted bed roughness, we perform a number of perturbed inversions, where we introduce errors in several model components. First, we increase (decrease) the uniform value for Glen's flow law factor A by a factor of 1.25. We assume that, in reality, this factor depends on the englacial temperature through an Arrhenius relation. The uncertainty in the annual mean surface temperature during the last glacial cycle is about 1 K for Antarctica (Jouzel et al., 2007) and 4 K for Greenland (Alley, 2000; Kindler et al., 2014). In realistic applications, a flow enhancement factor is often applied to account for anisotropic rheology and damage. Since estimated values of this factor differ significantly (Ma et al., 2010), an uncertainty of an order of magnitude is plausible, but we chose a smaller range to ensure that the inversion procedure was still able to reproduce the target geometry. Second, we increase (decrease) the SMB by a factor of 1.05. This seemingly small range is motivated by the fact that, for simplicity's sake, we alter the SMB over the entire model domain. Whereas estimates of local mass balance contain significant uncertainties, ice-sheet-integrated values are additionally constrained by satellite gravimetry, so that an uncertainty of 5 % seems plausible (Fettweis et al., 2020). Next, we increase (decrease) the transition velocity u0 in the Zoet–Iverson sliding law by a factor of 2, and we increase (decrease) the exponent p in the sliding law by 2. Zoet and Iverson (2020) report a range of transition velocities between 50 and 200 myr-1, whereas in CISM a default value of 200 myr-1 is used. For the exponent, Zoet and Iverson (2020) report a value of 5, CISM uses a value of 3, and a value of 1 yields a linear sliding law, which is still used in some ice-sheet models. We also perform two perturbed inversions where we add an error to the bed topography of ±10 % of the ice thickness, resulting in a bump (depression) of just over 250 m beneath the ice divide. The ice thickness is adjusted accordingly to keep the surface elevation unchanged. While the surface elevation of the Greenland and Antarctic ice sheets is generally known very accurately, estimates of ice thickness and bedrock elevation are based on interpolation of local radar measurements. In the BedMachine Greenland v4 dataset (Morlighem et al., 2017), the reported uncertainty in the bedrock elevation exceeds 10 % of the ice thickness over about 30 % of the ice sheet. Our choice of increasing/decreasing the estimated ice thickness by 10 % everywhere therefore serves as an upper bound, as it is unlikely that all of the data and extrapolations are biased in the same direction.

These five parameters (viscosity, SMB, transition velocity, exponent, topography), each with a high and a low value, result in 10 perturbed inversion simulations. The resulting errors in the inverted bed roughness, steady-state ice geometry, and surface velocity for experiment I are shown in Fig. 6.

The top-leftmost panel in Fig. 6 shows the error in the inverted bed roughness for the high-viscosity perturbed inversion. In this experiment, the overestimated ice viscosity means that the ice flow due to vertical shearing is underestimated, which is compensated for by decreasing the bed roughness, leading to increased basal sliding. The leftmost panels in the third and fifth rows of Fig. 6 show the errors in the resulting steady-state ice geometry and surface velocity, which are negligibly small. For these two quantities, the errors in the viscosity and the bed roughness are indeed compensating errors. This is true for almost all perturbed inversions, except for the low-viscosity and high-topography runs (high-topography means an added depression in the bedrock, such that the target ice thickness is overestimated). In these two experiments, the added perturbations cause the deformational ice flow to be overestimated so much that even preventing all basal sliding cannot entirely compensate for this perturbation. Note that this results from perturbing Glen's flow law factor A by a factor of 1.25, which is rather conservative. In realistic applications, the uncertainty in this quantity is typically an order of magnitude.

The underestimated value of the Zoet–Iverson sliding law exponent p=1 (Fig. 6, fourth column, lower set of rows), which implies a linear sliding law, yields negligible errors in the geometry and velocity but results in the inverted bed roughness being overestimated by a factor of 3 on average. The overestimated value of p=5 yields negligible differences, as do both over- and underestimated values of the transition velocity u0.

In the remaining four perturbed viscosity/mass balance/topography simulations, the errors in the inverted geometry are acceptably small, compared to the errors reported for initialised models in realistic intercomparison projects (e.g. initMIP-Greenland; Goelzer et al., 2018). The errors in the inverted bed roughness, however, are as large or larger than the “signal” of the prescribed bed roughness pattern (i.e. ∼5∘ of till friction angle change in the ice-stream area). These errors show prominent spatial patterns, despite the fact that the perturbations are spatially uniform. This implies that one should be cautious when interpreting the spatial patterns yielded by a basal inversion procedure, as they could reflect errors in some other physical quantity rather than realistic variations in bed roughness.

For experiment II, we perform the same set of perturbed inversions as for experiment I, introducing the same perturbations to the ice viscosity, the surface mass balance, the subglacial topography, and the sliding law parameters. We additionally perturb the sub-shelf melt rate, applying values of ±1 myr-1 (in the target run, no basal melt is applied). The results of the perturbed inversions are shown in Fig. 7. The results of the perturbed Zoet–Iverson sliding law transition velocity u0 are omitted, since that only has a small effect. Similar to experiment I, the relatively small errors introduced in the ice viscosity, mass balance, and subglacial topography lead to large errors in the inverted bed roughness but still produce a steady-state ice geometry that is close to the target geometry. The only exceptions are, again, the low-viscosity and high-topography runs, as well as the low-BMB (basal mass balance) run (i.e. too much sub-shelf melt), where the ice flow is increased more than can be compensated for by increasing the basal friction. However, even here the errors in the inverted geometry are relatively small. The errors in the inverted velocities are mostly small, except for the inversions with the perturbed sub-shelf melt rates. While these inversions produce relatively accurate geometries (about 120 m of ice loss near the grounding line in the increased-melt simulations), they contain large errors in the shelf velocities (about -500 myr-1 in the increased-melt simulation, relative to a target value of about 1000 myr-1).

As in experiment I, the introduced perturbations (which are spatially uniform) lead to prominent spatial patterns in the inverted bed roughness, with the errors being as large as the actual (prescribed) signal. This underlines the conclusion that spatial patterns in inverted bed roughness do not necessarily correspond to spatial patterns in the true bed roughness.

Finally, we perform a perturbed inversion for experiment II where we chose a non-equilibrated target geometry. We achieve this by terminating the initialisation after 10 000 years, instead of the default of 50 000 years, so that the ice has only reached about 90 % of its steady-state thickness. This non-steady-state geometry serves as the target for the inversion. Since the present-day observed geometry of the Antarctic ice sheet likely does not represent a steady state but already displays sustained and accelerating thinning rates (Rignot et al., 2019), this experiment mimics the effects of erroneously assuming that the ice sheet is in equilibrium (a common assumption in modelling studies; Seroussi et al., 2019). The results of this experiment are shown in Fig. 8. Here too, the inversion procedure results in very small errors in the ice geometry and relatively small errors in the velocity (note that the high velocity ratios occur in the slow-moving interior; in the fast-moving part of the ice stream, the errors are around 25 %) but substantial errors in the bed roughness.

To investigate the effect of compensating errors in basal inversions on the dynamic response of the ice sheet, we perform a series of simulations based on experiment II, where we increase the basal melt, forcing the ice sheet to retreat. We use the schematic basal melt parameterisation from the MISMIP+ Ice1r experiment (Asay-Davis et al., 2016) and run the model for 500 years. We initialise our simulations with the perturbed parameters, inverted bed roughness, and steady-state ice geometry from the perturbed inversions presented in Sect. 4.2. For the “non-equilibrated” experiment, note that the ice sheet at the end of the inversion is in a steady state; it has achieved this by lowering the bed roughness far enough to match the target geometry, which was not in a steady state. The resulting ice volume above flotation (relative to the steady state at t=0) and the mid-stream grounding-line position over time for all experiments are shown in Fig. 9.

In the 500-year unperturbed simulation, the grounding line retreats by about 150 km, causing the ice volume above flotation to decrease by about 1.7×1013 m3. As a result of the introduced errors in the perturbed simulations, this mass loss is increased (decreased) by up to 30 % (35 %) relative to the unperturbed simulation. The errors in the subglacial topography have the strongest effect, with the high-perturbed run showing nearly twice as much ice loss as the low-perturbed run. This is followed by the sliding law exponent (-18 % to +3 %) and the ice viscosity (-14 % to +11 %). The effects of the errors in the SMB, the BMB, the sliding law transition velocity, and the non-equilibrated target geometry are small.