Among the most important challenges faced by ice flow models is how to represent basal and rheological conditions, which are challenging to obtain from direct observations. A common practice is to use numerical inversions to calculate estimates for the unknown properties, but there are many possible methods and not one standardised approach. As such, every ice flow model has a unique initialisation procedure. Here we compare the outputs of inversions from three different ice flow models, each employing a variant of adjoint-based optimisation to calculate basal sliding coefficients and flow rate factors using the same observed surface velocities and ice thickness distribution. The region we focus on is the Amundsen Sea Embayment in West Antarctica, the subject of much investigation due to rapid changes in the area over recent decades. We find that our inversions produce similar distributions of basal sliding across all models, despite using different techniques, implying that the methods used are highly robust and represent the physical equations without much influence by individual model behaviours. Transferring the products of inversions between models results in time-dependent simulations displaying variability on the order of or lower than existing model intercomparisons. Focusing on contributions to sea level, the highest variability we find in simulations run in the same model with different inversion products is 32 %, over a 40-year period, a difference of 3.67

Many ice flow models use inverse methods to calculate initial conditions for properties of the ice for which directly observed data do not exist or are of poor quality. Inversion is an iterative process which starts from an initial guess and obtains improved values for the unknown property based on its relationship to a well-observed property, such as surface velocity. This process is generally undertaken for at least one of the following properties: ice rheology (flow rate factor,

However, these inverse problems are not well posed, and a unique solution is never guaranteed, regardless of the method used. In fact, a given inverse problem may have an infinite number of different solutions producing identical outputs of the forward model

Aspects of inversion processes within individual models have been the subject of several recent studies.

The differences between inversion outputs from different modelling platforms have not been given attention under controlled conditions as it is generally thought that the products of inversions are highly model-dependent. In model intercomparison projects

For this study, the focus is on inversions for basal sliding coefficients and ice rheology rate factors using an adjoint method and using the same input datasets. We compare the results of inversions from three ice flow models, identify the factors which cause differences between them and investigate the effect these differences have when transferring the products of inversions between models. We are interested in the extent to which the inversion processes are reflective of the physical ice flow described by the model equations and by how much numerical model behaviour might be influencing the outputs. If the inversion outputs from the models are similar, we can be sure that they represent a solution to the given physical equations without the results being heavily influenced by model-specific differences in the processes used. As part of our investigation, we will assess whether the products of inversions can be used outside their model of origin and whether the fields produced by inversions from different models result in similar behaviour in transient simulations.

Amundsen Sea Embayment shaded with speed measurements from

Our chosen study area is the Amundsen Sea Embayment (ASE) in West Antarctica (Fig.

In this work, we start by giving details of the models used and their respective inversion procedures in Sect.

Three models are used in this study: Úa

Each model performs inversions for two parameters, a rheological parameter and a basal sliding coefficient. To describe ice rheology all models use the constitutive equation

All three models employ the Weertman sliding law

All of the inversion methods involve minimising a cost function of general form

All of the inversion methods contain regularisation parameters which must be chosen.

In Úa, the cost function is

Úa employs Tikhonov regularisation, for which the regularisation term has the form

In ISSM, the cost function is written as

The regularisation term is defined as

In ISSM, the inversions for each parameter are carried out independently of each other. First,

In STREAMICE, the parameters inverted for are

All three model domains used for our inversions, displayed in Fig.

The meshes used by each model for the inversions. All domains cover our area of interest, including Thwaites and Pine Island glaciers and the Dotson and Crosson ice shelves. The main grounding line is shown in red.

The STREAMICE domain is a 528

A Dirichlet boundary condition is used to set all velocities along the grounded parts of Úa's boundary to zero since the boundary generally follows the edges of drainage basins. ISSM also uses Dirichlet boundary conditions, setting the velocities along the grounded parts of the boundary according to the velocity measurements. STREAMICE applies a no-flow boundary condition as its boundaries are sufficiently far from the area of interest for this not to affect the outcome. All models apply the ice-front stress boundary condition along the seaward boundaries. In Úa, this is at the edge of the computational domain, and in the other two models the ice/ocean boundary is set using a mask derived from the BedMachine geometry data.

In the time-dependent simulations, surface mass balance is from a climatological record of RACMO2.1

The first experiments involve a single inversion from each model. For the initial comparison, each model performs an inversion using the same geometry (bedrock and surface elevation) and velocity measurements, detailed in Sect.

The resulting fields of rate factor and basal sliding coefficients are compared directly in order to see whether the models produce similar results. The velocity misfits, defined as the difference between the modelled and observed values, are also compared as an indicator of how well the inversion processes have performed. The results of this comparison are found in Sect.

Following this, further experiments seek to test the sensitivity of inversion outputs to particular details of the inversion procedure, such as the choices of optimisation scheme, algorithm sequence, mesh resolution and priors. An overview of the results of these experiments is found in Sect.

The final stage (Sect.

We first look at the outputs from inversions in the three ice flow models following the procedures previously described. The fields we compare are the speed misfit and the values of

For the purpose of the comparisons in this section, outputs from Úa and ISSM were interpolated linearly onto the rectangular grid of the STREAMICE domain. As can be seen from the shapes of the domains in Fig.

Difference in the calculated speeds after inversion, compared to the measurements, for each model. The grounding line is indicated in black.

Speeds

The speed misfits for each model are displayed in Fig.

A visual comparison reveals that Úa has minimised the difference furthest, with misfit under 50

In general across all three models, the greatest differences are seen on the floating ice downstream of the grounding line and on the fastest-flowing grounded ice. For a clearer picture of the misfit on faster-flowing ice, we can take the mean misfits on regions above a certain measured velocity threshold. The values for a few chosen thresholds are displayed in Table

Mean values for the magnitude of misfit in inversions from the three models on regions with ice over chosen measured speed thresholds.

While we can see similarities in the locations of high-misfit regions, the overall correlation between the distributions of misfit is not high. We calculated the Pearson correlation coefficient

The results from the rate factor inversion (Fig.

The

On floating ice, ISSM and STREAMICE produce similar results, with differences between their outputs (Fig.

To provide some quantification of the differences between the rate factor fields calculated by the models, we use Pearson correlation coefficients as before. The coefficient values can be found in Table

Inverted

However, there are still some notable differences between the

The STREAMICE output occasionally contains “loops” of lower values. These can appear due to the model inverting for

Once again, calculating Pearson correlation coefficients between the outputs gives us a quantitative idea of how alike the distributions are. We find strong positive correlation coefficients in the region of 0.8 for each comparison pair (see Table

The

Pearson correlation coefficients calculated for different inversion outputs between all model pairs.

There are many factors which could cause differences in inversions. We have investigated several of these within Úa in an attempt to identify in particular why the difference in the misfit produced by Úa's inversions is lower than the other two models and what causes patches of lower

The difference in misfit appears to be due to a combination of factors. As noted in Sect.

Major factors affecting the inversion results appear to be the section of the domain over which

In general, the inversions were found to agree on large-scale distributions of

We performed three time-dependent simulations in each of the three models using a pair of inversion products from each model as inputs for the rate factor and basal sliding coefficient. The inversion outputs from Úa and ISSM are those used in the comparisons of Sect.

The models were allowed to evolve for 40 years from the initial state described by our geometry datasets. The only differences between these simulations are the

Before running the full time-dependent simulations, we also looked at diagnostic simulations. However, we found that these were not a good indicator of the quality of inversions or forward model performance due to specific differences in the methods employed by our models at the grounding line. Details of this are given in Appendix

Changes in ice mass and grounded area over 40 years of simulation in Úa using the rate factor and basal sliding coefficient fields resulting from each of the three model inversions.

The changes in volume above flotation, ice mass and grounded area for the domain over 40 years are displayed in Fig.

Thickness changes after 40 years of simulation using the rate factor and basal sliding coefficient fields resulting from each of the three model inversions. The initial grounding line position is indicated in red and the final position in black.

Figure

Some differences between forward runs in different models, even when using the same inputs, are to be expected, as model intercomparisons demonstrate

Our aim was to investigate the transferability of inversions between models, and for this we examine the results in sets of three, comparing the outcomes produced in the same forward model using the three different inversion outputs. In Úa, the range of sea level contribution after 40 years resulting from the three sets of inversion outputs is 1.82

It should be noted that, again using sea level contributions as the metric for comparison, the largest difference between the results from each model under normal usage (i.e. all models using their own inversion outputs) is 2.78

Another important note to emphasise is that the low variability in our time-dependent results demonstrates that the exact magnitude of misfit is not a direct reflection of the quality of an inversion. Higher misfit values were observed in the ISSM inversions (Sect.

After 40 years, the contributions to sea level calculated in our three models using the three sets of model inversions differ by up to 4.8

The variability in ice mass loss found in our experiments is less than that of the control experiment of

Our results display lower variability in terms of sea level contribution than that seen between models in the Antarctic intercomparison of the SeaRISE project

These favourable comparisons demonstrate the value of the standardisation of input datasets in our inversions, which helps to minimise uncertainty when transferring them. As long as the same geometry and densities are used as in the inversion process, no more uncertainty is introduced into a forward problem by choosing to use an inversion product from another model than by other standard modelling choices such as which sliding law to use.

In this work, we have investigated the differences between inversions for flow rate factor and basal sliding coefficients calculated in three different ice flow models. They each use different inversion equations and techniques, but despite this they display a high degree of agreement in patterns of distribution, with strong positive correlations particularly evident between the fields of basal sliding coefficients. The implication of this is that outputs of inversions contain minimal representation of model-specific numerical behaviour and strongly reflect the underlying equation system the models are designed to solve. The results of inversion processes used by our models are shown to be consistent with each other to a higher extent than may have been expected from the ill-posedness of the problem being solved. The minimal model dependence demonstrates that ice flow models are as robust in their inversions as they are in their forward simulations.

Further to this, we have shown that the products of inversions performed in any one of these three models can be used in any of the two other models as an input for transient simulations and that the results obtained this way are similar to those obtained when each model uses its own inversion products. Hence, the inversion products can be described as transferable between models. In our 40-year transient simulations, the variation in sea level contributions produced by a single model did not exceed 32 %, and further efforts to standardise modelling procedures would likely improve this figure. The smallest variation found between simulations using the three different sets of inversion products was 13 %. We found that using inversion products from different models results in similar variability to that which already exists between each of the models operating normally with their own inversion outputs.

Due to our careful control of input datasets, the results of our time-dependent simulations show variability lower than those of other intercomparison experiments. When the process is managed well, the variability introduced by transferring inversion outputs from one model into another is not significantly high and thus is not prohibitive to wider applications. With provision of sufficient details of the models involved, it would be possible to produce fields of basal sliding coefficients and rate factors which could be used by multiple models for the purpose of increasing uniformity in the boundary conditions and ice properties of intercomparison projects or could be used as inputs for models which cannot perform their own inversion calculations.

The choice of regularisation parameters in our inversions is based on

In Úa, there are technically four different regularisation parameters. In Eq. (

In ISSM, since the inversions for

The regularisation parameters for STREAMICE were initially chosen to be the values resulting from previous work

An

After observing the differences between the inversion results of the three ice flow models, we investigated possible causes for them. Each of the models approaches the inversion process in a slightly different way, and further testing would reveal which factors are the most influential in affecting the outcome. We tested different factors by performing independent inversion calculations for each case in Úa and in one case across all three models. We looked at the velocity misfits, rate factors and basal sliding coefficients produced as indicators of inversion performance compared to the original results. We attempted to determine from this how robust our inversion results are with respect to these procedural differences.

One possible source of inconsistency between the models is the optimisation scheme used during the inversion process. ISSM and STREAMICE both make use of a scheme called M1QN3

A comparison of the performance of the interior-point algorithm in MATLAB used by default in Úa and the M1QN3 optimisation scheme used by ISSM and STREAMICE, showing minimisation of the cost function during the inversion process.

The misfit fields resulting from these inversions (Fig.

A comparison of the speed misfit, rate factor and basal sliding coefficients for several cases of inversions in Úa under different conditions.

The Pearson correlation coefficients of various models and tests with Úa's original inversion. The first two columns show the correlation of the original ISSM and STREAMICE inversions, and the remaining columns show the correlation with the cases displayed in Fig.

It is interesting to note that use of the M1QN3 algorithm results in slightly lower values of

The way in which ISSM performs its

The results of this test (Fig.

The correlation of this

A comparison of the outputs of inversions using the original priors from each model in the first row and the two specified sets, Priors1 and Priors2, in the second and third rows.

The models are performing their calculations over different meshes, so experiments to test the mesh dependence of inversions were performed. We first tested the mesh of ISSM within Úa and found that the inversion outputs are not particularly sensitive to the location of mesh points if the resolutions are similar, as is the case here. A comparison produced a strong positive correlation in the

The minimum length of the elements in Úa's original mesh is 500

The Pearson correlation coefficients for

The Pearson correlation coefficients for

Using Mesh3 (Fig.

Within the range of resolutions of our models in their original states, the inversions for

In the original inversion comparison, each model was given the freedom to pick its own default priors for

The results in Fig.

In the

It may well be the case that only the choice of prior for

The strong correlations between each model's original output and the Priors2 experiments lead us to conclude that the choice of priors is not a major factor in the differences between the original inversions as neither STREAMICE nor ISSM was using uniform priors.

While the priors do not appear to affect the inversion outputs greatly, it was found that forward runs in STREAMICE using its original inversion outputs encountered some convergence issues, while this was not the case with the outputs from inversion using Priors2. For this reason, the Priors2 inversion from STREAMICE is used in the forward runs of Sect.

A technical difference between the inversion procedures in our models is the derivation of the adjoint. Úa and ISSM use an exact adjoint, following the terminology of

This factor was not specifically investigated for the inversions in this project, but we note it here in the interest of completeness. We do not believe that it would be a major cause of differences.

A diagnostic model step calculates an instantaneous velocity from the given boundary conditions and geometry without any time evolution. We ran diagnostic calculations in Úa using the fields of

Investigating the phenomenon of large diagnostic velocity discrepancies further, we found that the position and definition of the grounding line is the major cause of the velocity differences. There are two reasons for inconsistencies to appear at the grounding line when transferring inversion products between models, which we shall discuss in this section.

Differences between the speed calculated diagnostically in Úa using the

Firstly, each model carries out inversions on a different mesh, and the outputs must be interpolated for use in other models. This is particularly important when transferring the

Secondly, the models employ different treatments of the grounding line in their equations. Inversions in STREAMICE are calculated using a flotation relationship containing a Heaviside function, which indicates whether ice in a mesh element is floating or grounded. Úa uses a modified version of this as discontinuities in the equations can cause problems in the model's numerical solvers. The Heaviside function is smoothed by use of a parameter named

For diagnostic calculations in Úa, we can change the value of

For the ISSM outputs, lowering the value of

The STREAMICE outputs follow a slightly different pattern. Beyond the tipping point, the difference on the ice shelves continues to follow the same trajectory, becoming negative as

Differences between diagnostic speeds calculated in Úa using the ISSM and STREAMICE inversion outputs and a range of values for

The effect of the grounding line regularisation is different in time-dependent simulations. This is illustrated by the comparisons of speed and grounding line position shown in Fig.

Speeds and grounding lines after 1 and 10 years of simulation in Úa using its own inversion outputs and those of ISSM with two different values of

With both values for

This suggests that diagnostic calculations are not indicative of performance in time-dependent simulations and that large velocity differences in the diagnostic calculations do not necessarily mean that similarly large differences will be present in forward simulations. It also means that tuning grounding line regularisation terms based on the diagnostics is not a method which should be used. Thus, for the time-dependent comparisons in Sect.

Source code for Úa can be downloaded at

Input and output data for the modelling experiments presented in this paper can be accessed at

All authors were involved in the conception of the project and discussions throughout. JMB coordinated the project and carried out the modelling work in Úa. TDdS and DG carried out the modelling work in ISSM and STREAMICE, respectively. JMB led the writing of the manuscript, and all authors provided comments and feedback during the editing process.

The authors declare that they have no conflict of interest.

The authors would like to thank Cyrille Mosbeux and one anonymous reviewer for their insightful feedback, which helped to improve the manuscript, and Olivier Gagliardini for handling the editing of the paper. This work is from the PROPHET project, a component of the International Thwaites Glacier Collaboration (ITGC). This is ITGC contribution no. 019.

This research has been supported by the Natural Environment Research Council (grant nos. NE/S006745/1, NE/S006796/1, and NE/T001607/1) and the National Science Foundation (grant no. 1739031).

This paper was edited by Olivier Gagliardini and reviewed by Cyrille Mosbeux and one anonymous referee.