Mass loss due to dynamic changes in ice sheets is a significant contributor to sea level rise, and this contribution is expected to increase in the future. Numerical codes simulating the evolution of ice sheets can potentially quantify this future contribution. However, the uncertainty inherent in these models propagates into projections of sea level rise is and hence crucial to understand. Key variables of ice sheet models, such as basal drag or ice stiffness, are typically initialized using inversion methodologies to ensure that models match present observations. Such inversions often involve tens or hundreds of thousands of parameters, with unknown uncertainties and dependencies. The computationally intensive nature of inversions along with their high number of parameters mean traditional methods such as Monte Carlo are expensive for uncertainty quantification. Here we develop a framework to estimate the posterior uncertainty of inversions and project them onto sea level change projections over the decadal timescale. The framework treats parametric uncertainty as multivariate Gaussian and exploits the equivalence between the Hessian of the model and the inverse covariance of the parameter set. The former is computed efficiently via algorithmic differentiation, and the posterior covariance is propagated in time using a time-dependent model adjoint to produce projection error bars. This work represents an important step in quantifying the internal uncertainty of projections of ice sheet models.

The dynamics of ice sheets are strongly controlled by a number of physical properties which are difficult (or intractable) to observe directly, such as basal traction and ice stiffness

Control methods

Uncertainty quantification (UQ) in projections of ice sheet behavior is a crucial challenge in ice sheet modeling. Studies of fast-flowing Antarctic glaciers have shown that uncertainties in the parameters controlling ice flow can lead to large variability in modeled behavior

The uncertainty associated with ice sheet model calibration can be quantified through Bayesian inference, in which prior knowledge is “updated” with observational evidence. Such methods have been applied to continental-scale ice sheet models and models of coupled ice–ocean interactions

Applying such ensemble-based Bayesian methods to glacial flow models and parameter sets of dimension

Thus, there is at present a disconnect between the dual aims of (i) modeling ice sheets as realistically as possible, i.e., through the implementation of higher-order stresses and without making limiting assumptions regarding “hidden” properties of the ice sheet, and (ii) uncertainty quantification (UQ) of models by approximate inference by reducing the dimensionality of the set of parameters.

By augmenting control methods using a Hessian-based Bayesian approach, it is possible to quantify parametric uncertainty without sacrificing parameter dimension or model fidelity. Just as control methods can be interpreted as returning the

Once determined, the Hessian-based parameter covariance can then be used to quantify the variance of a scalar quantity of interest (QoI) of the calibrated model (e.g., ice sheet sea level contribution over a specified period). One approach to this is projecting the parameter covariance on to a linearized model prediction

In this study we introduce a framework for time-dependent ice sheet uncertainty quantification and apply it to an idealized ice sheet flow problem

To facilitate readability of this and subsequent sections we adopt formatting conventions for different mathematical objects. Coefficient vectors corresponding to finite-element functions appear as

An ice sheet flow model can be thought of as a (nonlinear) mapping from a set of input fields, which might be unobservable or poorly known (such as bed friction), to a set of output fields, which might correspond to observable quantities (such as surface velocity). Here, our focus is on the probability distribution function (PDF) of a hidden field

As described in Sect.

The distribution

Models of ice sheet dynamics are in general nonlinear, however, and Eq. (

Equation (

By contrast with Bayesian methods, the control methods generally used in glaciological data assimilation

Our framework effectively uses a control method – but one which allows calculation of the posterior covariance after the MAP point is found. As such we use a fixed set of points, as described above, in our misfit cost term. Thus, the Hessian of the cost function of our control method is equal to the inverse of the posterior covariance given by Eq. (

In the previous section we establish that the posterior covariance is equivalent to the inverse of the Hessian of the (suitably defined) cost function. With a large parameter space, though, calculating the complete Hessian (and its inverse) can become computationally intractable. Still, in many cases, the constraints on parameter space provided by observations can be described by a subspace of lower dimension. In the present study, our idealized examples are small enough that the full Hessian can be calculated, but to provide scalable code we seek an approximation to the posterior covariance that exploits this low-rank structure.

The following low-rank approximation follows from

While

In this study, the problems considered are sufficiently small that we calculate all eigenvalues; i.e., we do not carry out a low-rank approximation. In general, though, a strategy for deciding

Often of interest is how the observational data constrain outputs of a calibrated model as opposed to how they constrain the calibrated parameters themselves. (A simple analogy is an extrapolation using a regression curve, which is generally of more interest than the regression parameters.) Such an output is termed a quantity of interest (QoI)

The distribution of

Note that the assumption of linearity in Eq. (

In this study we use a new numerical code,

The

The

The

In addition to the momentum balance, the continuity equation is solved:

We discretize velocity (

To carry out an inversion, a cost function is minimized using the L-BFGS-B algorithm

In this study, we aim to do the following.

Establish that control method optimizations can be carried out with

Calculate eigendecompositions of the prior-preconditioned model misfit Hessian as described in Sect.

Propagate the posterior uncertainty onto a quantity of interest

Establish, through simple Monte Carlo sampling, that the variance found through Eq. (

Control method optimizations using ice sheet models have been done extensively with parameter sets of very high dimension (e.g.,

Investigating these and similar factors comprehensively, as well as validating the assumption of Gaussian statistics that leads to Eq. (

In our experiments, the momentum balance (Eq.

Our parameter-to-observable map

An inverse solution

ISMIP-C does not prescribe a time-dependent component, but it is straightforward to evolve the thickness

In our error propagation we evolve the ISMIP-C thickness for 30 years and use the time-dependent adjoint capabilities of

An

An

While misfit does not vary greatly in a proportional sense, it suggests

Results of the control method inversion with

Similar to Fig.

The effect of regularization on reduction of uncertainty can be seen from examining the eigenvalues defined by Eq. (

Uncertainty reduction factor

Approximating the posterior covariance of

The impacts of grid resolution on eigenvalue spectra are investigated (Fig.

The low-rank approximation of the posterior covariance of

The trajectory of uncertainties for

Ideally, the assumptions implicit in the calculation of QoI uncertainties shown in Fig.

The assumptions in our propagation of observational and prior uncertainty to the quantity of interest uncertainty are (i) Gaussianity of the distribution of

A randomly sampled vector

Using this method of sampling the posterior, an ensemble of 1000 30-year runs is carried out for both low- and high-regularization experiments (

In all results presented to this point, the imposed locations for observational data

Eigendecompositions of the prior-preconditioned misfit Hessian (

Comparison of eigenspectra relies on the corresponding eigenvectors being the same, or similar, between the experiments. As in the regularization and resolution experiments, the eigenvectors depend on the exact form of

The results described above imply that posterior uncertainty could be made arbitrarily small by increasing the spatial density of observations (although we do not examine observations more dense than 500 m). However, the decreasing uncertainty relies on the observations being statistically

Section

In the context of our idealized experiments we calculate the GNaH (or rather, its action on a vector) to compare against the full Hessian. The

In Fig.

The inversion of surface velocities for basal conditions is ubiquitous in ice sheet modeling – but in most studies in which this is done, the uncertainty of the resulting parameter fields is not considered, and the implications of this parametric uncertainty for projection uncertainty are not quantified. We introduce

We apply our framework to a simple idealized test case, Experiment C of the ISMIP-HOM intercomparison protocol, involving an ice stream sliding across a doubly periodic domain with a varying basal friction parameter. An idealized time-varying QoI is defined, equivalent to the fourth moment of thickness in the domain, as thickness evolves due to mass continuity. The posterior probability density is examined, suggesting mesh independence (provided resolution is high enough). It is shown that the level of uncertainty reduction relative to the prior distribution depends on the amount of information in the prior (or, equivalently, the degree of regularization). Uncertainty of the QoI is found along its trajectory and is found to increase with time and also found to be larger with less-constrained priors. However, the difference in the uncertainty of the QoI is far less than that of the parametric uncertainty due to insensitivity of the QoI to high-frequency modes.

Sampling from our posterior allows us to test the linearity of the parameter-to-QoI mapping, and this approximation is seen to be accurate with a moderately strong prior. However, even with the relatively modest problem sizes considered, testing the validity of our local Gaussian approximation of the posterior probability density would require sophisticated sampling methods which are beyond the scope of our study. It is worth noting, though, that one such method, stochastic Newton MCMC

The sensitivity of QoIs to small-scale variability is significant because not all glaciologically motivated QoIs are expected to have such sensitivities. For instance, the QoI considered by

A key difference between our approach and the control method inversions typically undertaken is the Euclidean inner product that appears in the misfit component of the cost function as opposed to an area integral of velocity misfit. As discussed in Sect.

Our study does not consider “joint” inversions, i.e., inversions with two or more parameter fields. With such inversions, complications can arise when both parameters affect the same observable, potentially leading to equifinality and/or ill-posedness. An example of such a pair is

Model uncertainty is not accounted for in our characterization of parametric uncertainty. In the expression for the posterior probability density (Eq.

A number of Hessian-based uncertainty quantification studies use the Gauss–Newton approximation to the Hessian (see Sect.

Our study does not consider time-dependent inversions, i.e., control methods wherein the cost function is time-dependent. While the majority of cost function inversions are time-independent, there are a growing number of studies carried out with time-dependent inversions

The

CK and JT developed the

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors acknowledge NERC standard grants NE/M003590/1 and NE/T001607/1 (QUoRUM).

This research has been supported by the Natural Environment Research Council (grant nos. NE/M003590/1 and NE/T001607/1).

This paper was edited by Alexander Robel and reviewed by two anonymous referees.