An important component of any modern computational ice-flow model is the initialisation of the model. This ensures that the initial modelled state of the ice sheet, particularly in relation to hidden fields which are poorly constrained by direct observations, creates an ice-flow which behaves in agreement with measurements. The application of inverse methods in the model initialisation to determine the hidden fields dates back to . Since then an increasing number of computational ice-flow models have been developed that include an inversion step in the model initialisation, for example Úa , BISICLES , ISSM , MALI , STREAMICE , Elmer/Ice and WAVI .

This inversion step typically involves minimising the misfit between the modelled velocities (u,v) and observed surface velocities (uobs,vobs), with respect to two poorly constrained properties of the ice sheet:

The basal slipperiness coefficient, c, which is one of the model parameters entering the sliding law that characterises the basal drag experienced by an ice sheet. Different conventions are used for the definition of this coefficient, but in what follows we always refer to it as basal slipperiness for consistency.

For example, in the Úa ice-sheet model the misfit between the modelled and observed velocities is defined as 1I=12A∫u-uobsuerr2dA+12A∫v-vobsverr2dA where A=∫dA is the total area of the domain, and (uerr,verr) are the data errors in the surface measurements. Note Úa also has the ability to minimise the misfit of the observed surface height change, dh/dt, measurements in addition to the velocity measurements. Regularisation is often applied to this minimisation procedure. For example, the ice sheet model Úa uses Tikhonov regularisation : 2R=∑i=1,212Aγa2∫pi-pi,prior2dA+∑i=1,212Aγs2∫||∇(pi-pi,prior)||2dA where pi are each of the hidden fields to be retrieved, and (γa,γs) are parameters that control the degree of regularisation. The prior estimate for each hidden field, pi,prior, is often taken as a single starting value set at a realistic scale for the field. The cost function minimised in the step-wise inversion procedure is the sum of the data misfit I and regularisation R terms. Typically the inversion calibrates the velocity misfit to a single snapshot of data, but some recent studies have explored improvements gained from calibrating the model to multiple time snapshots of data .

While it is common practise to perform this inversion step in the initialisation of ice-sheet models, there are a number of design choices that vary between models; for example the degree of regularisation in the inversion is typically left to the modeller to decide, and yet recent studies on uncertainty quantification have shown that, of the model parameters chosen by the modeller, this is one of the largest contributors to uncertainty in sea level rise projections from Antarctica . Even more significant is the choice of whether to perform the inversion jointly for the viscosity and basal slipperiness, or whether to restrict to only solving for basal slipperiness over grounded ice and viscosity over ice shelves, and yet there has been little discussion of this in the literature.

In fact, there is a lack of consensus on whether it is possible to uniquely retrieve the viscosity and basal slipperiness from a joint inversion. For example, simulations with the large-scale ice-sheet models Úa , WAVI and MALI , all choose to perform a joint inversion for ice stiffness (a factor related to the viscosity) and basal slipperiness over any model domain that contains grounded ice. The same is true of the ice-sheet model BISICLES , although many of the publications using BISICLES stress that regularisation is essential otherwise the joint inversion problem is under-determined, for example and . Simulations with the STREAMICE ice flow package of MITgcm and the large-scale ice-sheet model ISSM choose to restrict to inverting solely for basal slipperiness over grounded ice, believing that extracting information about the viscosity field at the same time is not possible. Similarly, the deep learning ice-flow model IGM restricts the inversion to a linear combination of the viscosity rate factor and basal slipperiness amid claims that the simultaneous inference of viscosity and basal slipperiness leads to non-unique solutions . Simulations with the Elmer/Ice ice-flow model often only estimate the basal slipperiness through the inversion procedure, for example and , although when considering continent-wide ice-sheet simulations a joint inversion with the ice stiffness parameter is often used, for example .

Despite the disagreement in these modelling approaches, there have been few studies investigating the extent to which ice viscosity and basal slipperiness can be resolved independently when performing a joint inversion procedure. studied the performance of the joint inversion within the ice-flow model Úa for a synthetic ice-stream, and suggested an improved regularisation scheme weighted by strain rates to improve the accuracy of the estimations. A study by performed a Bayesian analysis to highlight the large uncertainties that arise from restricting the inversion to only consider the basal slipperiness, and yet these uncertainties are often ignored in practise leading to overconfident and biased results.

The lack of consensus and wide variety of modelling approaches motivates us to investigate the problem from a theoretical stand point, to understand whether it is possible to uniquely retrieve viscosity and basal slipperiness fields through a joint inversion, and what the limitations are with such an approach. Previous work by developed a theoretical framework using Bayesian inverse methods to understand the limitations of simultaneous retrieval of bed topography and basal slipperiness. The Bayesian inversion maximises the posterior probability of the hidden fields, Φ, given the observed data measurements at the surface, β, to find an estimate for the hidden fields: 3Φ^=argmaxP(Φ|β) The inversion procedure relies on a forward model which relates the hidden properties of the ice-sheet to the observed data measurements at the surface: 4β=f(Φ) The framework developed by uses a linearised set of equations which relate the perturbations in the hidden fields to perturbations in the surface fields through a set of transfer functions : 5β=TΦ where the transfer function matrix T has a component for the transference to the surface of each of the hidden fields in Fourier space. These concepts are elaborated on further in later sections. The relationship between the bed characteristics and surface fields has been the subject of study over a number of decades e.g.. In the analysis by , they derive transfer functions for the bed topography and basal slipperiness, in the context of a uniform slab of ice extending infinitely in all directions and flowing down an inclined place. In this study, we extend their work to include the retrieval of the viscosity field, and derive transfer functions for flow along a laterally confined channel. This more complicated domain, which has horizontal variations in stresses, is an essential part of extending the retrieval to include the ice viscosity.

This paper is organised as follows, in Sect.  we describe the forward model which relates the hidden properties of the ice sheet to the surface fields, and present the relevant transfer functions for a laterally confined channel, with the derivation of these transfer functions given in Appendix ; in Sect.  we describe the essential points relating to the Gaussian maximum posterior probability inversion procedure, with further details presented in Appendix . We present our results in Sect. . This is split into an analysis within the MAP framework in Sect. , which assumes a linearisation of the SSA, followed by a numerical inversion within Úa for our synthetic example in Sect. . We finish with a discussion of the potential impact of the assumptions made in this study in Sect. , and a summary of our findings in Sect. .

In this section we describe the linearised forward model used in this study, which relates the hidden properties of the ice sheet to the surface fields, through a set of transfer functions.

We restrict our analysis to ice-flows that can be described by the shallow ice stream approximation (SSA), where the ice thickness is small compared to the horizontal span , commonly used in large-scale, vertically-integrated ice-sheet models. The SSA momentum-conservation equations for flow down an inclined plane in a tilted coordinate system aligned to the bed topography are 6∂x2hη2∂xu+∂yv+∂y(hη(∂xv+∂yu))-tbx=ρgh∂xscos⁡α-ρghsin⁡α∂y2hη2∂yv+∂xu+∂x(hη(∂xv+∂yu))-tby=ρgh∂yscos⁡α where α is the angle of the coordinate system to the horizontal. The driving stresses due to gravity and any pressure gradients are on the right-hand side of the equations. These are balanced on the left-hand side by the basal drag components (tbx,tby) and the viscous terms resisting internal deformation (proportional to the vertically-integrated effective viscosity η). In this notation (u,v) are the horizontal velocities in the (x,y) directions respectively; s is the location of the upper glacial surface; h is the ice thickness; and g is the acceleration due to gravity. We use the short-hand ∂x≡∂∂x.

The viscosity relates the strain rates, ϵ˙ij, to the deviatoric stresses, τij, in the ice. It is usually modelled by Glen's flow law : 7ϵ˙ij=Aτn-1τij Thus η=A-1/nϵ˙(1-n)/n/2, where A is a temperature-dependent rate factor, and the exponent n varies from n=1 for linear viscous ice to n=4 for dislocation creep. The inversion procedure in numerical ice-flow models either solves directly for the spatial distribution of the rate factor A Úa, or they solve for the spatial distribution of an enhancement factor ϕ which represents how far the rate factor deviates from an Arrhenius temperature law BISICLES. The exponent in Glen's flow law is usually set to a fixed value, typically n=3. For the purposes of a standard inversion, which looks at a single point-in-time snapshot of velocity data, spatial variation in n is subsumed into the inferred spatial variation in A. In our study we assume a linear model for viscosity with n=1, so that the viscosity is independent of the strain rates.

The basal drag can be represented by a number of different sliding laws, a common choice being Weertman's sliding law , with basal slipperiness coefficient, c, and sliding exponent, m: 8tb=c-1/m‖v‖1/m-1v where v≡(u,v). Many studies have looked to improve estimates of the appropriate sliding regime for the Antarctic and Greenland ice-sheets using inversion techniques . found evidence that the optimal Weertman sliding exponent varied between linear viscous (m=1) to close-to-plastic bed rheology (m≫1) beneath Pine Island Glacier in West Antarctica using surface measurement data at multiple time observations. For the purposes of this study, we follow the standard inversion procedure which estimates the spatial distribution of the basal slipperiness coefficient c, while the exponent in the sliding law is held fixed; a common choice is m=3, although for the purposes of finding analytical solutions in this study we choose m=1.

The constitutive equations in Eq. () determine how the viscosity and basal slipperiness impact the surface fields. Notice how the viscosity, η, only ever appears in these equations multiplying the derivatives of the horizontal velocity fields. This means that in the absence of velocity gradients, the viscosity has no visible effect on the surface to first order. This is an important point that should be stressed: horizontal gradients of the velocity field are essential for the viscosity field retrieval in any ice-flow model. This is usually satisfied by ice sheets in the real world.

The linearised forward model in Eq. (), relates the viscosity and basal slipperiness fields to the surface velocity measurements through a set of transfer functions. In principle it can simply be inverted to find the hidden fields for a given set of surface velocity measurements: 18c(k)η(k)=Tuc(k)Tuη(k)Tvc(k)Tvη(k)-1u(k)v(k) In practise, errors in the surface measurements make this an extremely noisy and unreliable estimate. However, it is an important check on the feasibility of the inversion. Inverting this equation is only possible if the transfer matrix T is not singular: det|T|≠0. This is true wherever the transfer functions are non-zero and not linearly dependent. The transfer functions for viscosity are zero in regions of zero velocity gradients (a rare occurrence for ice-streams) and for very short wavelength features. The two sets of transfer functions being linearly dependent is equivalent to saying that there is mixing between the basal slipperiness and viscosity fields, such that the response at the surface is the same for the two fields, as has been suggested in some of the literature . However, from Eqs. () to (), it is clear that the transfer functions (Tuη,Tvη) are not a linear scaling of the transfer functions (Tuc,Tvc) in general. The two hidden fields have distinct frequency profiles. Thus, in principle, it is possible to extract the spatial distribution of the viscosity field uniquely at the same time as the slipperiness distribution.

To understand how this works in practice with errors in the surface data measurements, we use the theoretical framework developed by , which considers a Bayesian maximum a-posteriori (MAP) estimation with Gaussian probabilities. This also allows us to understand the role of regularisation in the inversion procedure, expressed as a prior distribution in this framework. The minimisation of a cost function, comprising a data misfit and regularisation term as in Eqs. () and (), is equivalent to maximising the a-posteriori distribution in the Bayesian MAP framework with uncorrelated Gaussian measurement errors and a prior distribution modelled using the Matérn covariance function e.g.. In general, the relationship between the covariance of the prior distribution and regularisation terms in a cost function is not explicit, although it is possible to reproduce a covariance parameterised with Matérn correlation functions using a diffusion operator, as done in FeniCs .

We provide details of the MAP estimation framework in Appendix . We consider retrieval along the x-direction for a particular flowline y=y0. Prior information about the ice viscosity and basal slipperiness is combined with surface velocity measurements of u(x) and v(x), to give an updated estimate of the viscosity η(x) and basal slipperiness c(x) fields. This estimate maximises the posterior probability according to Bayes' theorem: 19P(c,η|u,v)=P(u,v|c,η)P(c)P(η)P(u,v) The likelihood P(u,v|c,η) of the surface measurements, given a particular instance of the viscosity and basal slipperiness fields, is determined by the measurement errors and the forward model. The prior probabilities P(c) and P(η) for the distribution of the viscosity and basal slipperiness fields, respectively, can be chosen to impose smoothness on the retrieved fields.

Measurements and retrieval are at discrete points along the x-direction, which comprise the elements of vectors denoted in bold. The retrieval estimates of the hidden fields are denoted by Φ^=(c^,η^)T. As derived in Appendix , the estimate that maximises the posterior probability is 20Φ^MAP=CΦΦ^CΦΦ̃-1Φ̃+TβΦHCββ^-1β^ where we have defined 21CΦΦ^≡CΦΦ̃-1+TβΦHCββ^-1TβΦ-1 which can be thought of as the covariance matrix for the error in the retrieval . The data measurements at the surface are denoted by β^=(u^,v^)T, with associated error covariance matrix Cββ^ which is diagonal for uncorrelated errors. The prior distribution for the hidden fields is assumed to follow a multivariate Gaussian distribution with mean Φ̃ and covariance matrix CΦΦ̃. The transfer functions in Eqs. () to () are assumed to perfectly capture the forward model relating the true hidden fields Φ to the true surface fields β in Fourier space, such that 22β=TβΦΦ. where we have defined the combined transfer function matrix 23TβΦ=TucTuηTvcTvη and each transfer function T⋆⋆ is a d×d diagonal matrix, representing the d discrete measurement sites along the flowline, with the transference at each frequency along the diagonal. Note that all the matrices and vectors in these equations have been converted to Fourier space. We can think of the maximum a-posteriori estimate in Eq. () as having one component coming from the priors Φ̃, scaled by the errors in that estimate, and another component which is an update to the prior estimate coming from the observed surface measurements β^, scaled by their measurement errors.

The mean value of the prior is not significant, it quickly gets updated with the data measurements, but the prior covariance matrix CΦΦ̃ is important. It can, for example, determine the relative contribution of different frequencies in the update, or impose a correlation structure on the learnt distribution. In the results that follow, we set the prior mean to zero and choose a covariance matrix which is equivalent to the Tikhonov regularisation in Eq. (), used in many ice sheet models: 24Cηη̃-1=γa2I+γs2D in Fourier space, where D=diag(k2). The derivation of this covariance matrix is given in Appendix . The first regularisation parameter, γa, controls the degree to which large amplitude perturbations are suppressed in the retrieval, while the second regularisation parameter, γs, controls the suppression of higher frequency perturbations. This form of the covariance matrix effectively assigns higher confidence to the prior for larger k. The end result is a smoothing of the retrieved field. This is particularly important for the retrieval of the basal slipperiness and viscosity where higher frequency perturbations are poorly transmitted to the surface fields. Note, when the variations in the hidden fields are defined as fractional perturbations, then γa is dimensionless and γs has dimensions of length. We choose the same form for the slipperiness covariance matrix, Ccc̃-1, and construct a block diagonal covariance matrix for the combined covariance matrix, CΦΦ̃-1.

The results presented in this paper have made a number of necessary simplifying assumptions, so that the inversion procedure could be broken down and understood from a theoretical stand point. In this section we discuss these assumptions and their potential impact.

All experiments have been done with respect to idealised flow down a uniform, laterally-confined channel. Different regions of the flow experience stresses from bed sliding and lateral drag, each to a varying degree, and while the overall flow is simplified, these different regions should be representative of the stresses found within localised regions of a real ice-sheets. The transfer functions derived in this study have distinct frequency profiles, indicating that in general there is not mixing between the viscosity and slipperiness fields. Our expectation is that more complicated stress patterns observed in real ice-sheets should differentiate the transference to the surface of perturbations in slipperiness and viscosity even further.

We have also simplified the flow by assuming linear bed sliding and rheology, when in practice non-linear flow is supported by observations. Higher values of m in the bed sliding law will generally lead to faster velocities, and this will be the dominant impact on the transference to the surface of perturbations in slipperiness. Higher values of n will concentrate higher strain rates into a narrow band close to the lateral margin, and so the gradients in velocity – which are necessary for the transference of viscosity perturbations to the surface – will occur in a narrower region of the domain. Our expectation is that, if anything, values of n>1 and m>1 will likely differentiate the transference to the surface of perturbations in the slipperiness and viscosity even further. Studies by performed a numerical analysis of the transfer functions for higher order n and m, although applied to the full Stokes equations and with a different background state, which showed limited impact on the amplitude and phase of the transfer functions. Nonetheless, an exploration of the effect of non-linear bed friction and rheology on the inversion could be the subject of a further numerical study exploring the wider parameter space.

Another key assumption feeding into these experiments is that the SSA equations are a true representation of the ice-sheet physics. This approximation is appropriate for shallow ice-sheets where the thickness is small compared to the horizontal span, but will break down for thicker tributaries. In this study we have only considered measurement errors in the surface velocities, and assumed perfect knowledge of the surface elevation, while considering only uncorrelated measurement errors. Relaxing these assumptions could lead to more difficulty in separating the effects of slipperiness and viscosity in the inversion.

Finally, we would like to refer the reader to the transfer functions in Fig. . These illustrate the strength of transference to the surface across a range of perturbation frequencies, while the experiments in Figs.  to are for a particular choice of perturbation frequency, and for a particular phase shift between the viscosity and slipperiness perturbations. The space of different frequencies and different phase differences for the perturbations hasn't been explored in the experimental results.

In this study, we have shown that it is feasible to perform a joint inversion procedure to infer the viscosity and basal slipperiness distribution in ice-sheet models. This does not lead to an underdetermined problem where it is impossible to find a unique solution for the two fields, although our formulation does assume independent measurements of the two horizontal velocity fields (u,v) whereas, for example, BISICLES only considers the magnitude of the velocity.

In this paper, we have explored the inversion procedure from an analytical standpoint. Using a linearisation of the SSA equations, we can visualise the transference to the surface of features in the viscosity and basal slipperiness fields under different conditions. This analytical approach helps us to understand the circumstances where the retrieval is more difficult. For example, the ability to retrieve the viscosity field accurately is strongly dependent on the horizontal stresses within the ice-sheet. Close to the margins of an ice-sheet where lateral drag is causing a significant variation in the horizontal velocity field, the viscosity is well resolved; while closer to the centre of a wide ice-sheet where the flow is uniform, the viscosity is poorly resolved. Short wavelength features in both fields are also difficult to resolve. However difficulty retrieving the viscosity field does not necessarily mean the slipperiness field is more difficult to infer.

Poor retrieval can be assisted by the inclusion of a Bayesian prior for the hidden fields. The inclusion of a prior distribution, for certain forms of the covariance matrix, is equivalent to the regularisation employed in the misfit minimisation in many inversion procedures. Strong confidence in the prior distribution is similar to large regularisation coefficients. Some form of regularisation applied to the two fields is essential as the noise in the surface measurement data increases. This suppresses high-frequency noise in the retrieval and helps the inversion procedure extract the signal within the noise. A strong prior is also necessary for regions where features in the the viscosity are poorly transferred to the surface. Ice-sheet models which ignore spatial variation in the viscosity, during the inversion procedure, are effectively prescribing a prior on the viscosity field with 100 % confidence. We recommend making this choice explicit in the modelling by including the viscosity field as an unknown field in the inversion procedure with strong confidence in the prior, or equivalently large regularisation parameters.