The friction coefficient and the base topography of a stationary and a dynamic ice sheet are perturbed in two models for the ice: the full Stokes equations and the shallow shelf approximation. The sensitivity to the perturbations of the velocity and the height at the surface is quantified by solving the adjoint equations of the stress and the height equations providing weights for the perturbed data. The adjoint equations are solved numerically and the sensitivity is computed in several examples in two dimensions. A transfer matrix couples the perturbations at the base with the perturbations at the top. Comparisons are made with analytical solutions to simplified problems. The sensitivity to perturbations depends on their wavelengths and the distance to the grounding line. A perturbation in the topography has a direct effect at the ice surface above it, while a change in the friction coefficient is less visible there.

The output of isothermal simulations of large ice sheets depends on the ice model, the topography, and the parametrization of the conditions at the base of the ice. The models are systems of partial differential equations (PDEs) for the velocity, pressure, and height of the ice. The boundary conditions of the PDEs are given by the topography and the friction model with its parameters. Of particular interest in the simulation of ice is the horizontal velocity and the height at the ice surface. In the inverse problem, the parameters at the base are inferred from data at the surface by solving adjoint equations and minimizing the difference between given data and simulated results. In this paper, we estimate the sensitivity of the surface observations to changes in the basal conditions by solving the adjoint equations to the full Stokes (FS) equations and the shallow shelf (or shelfy stream) approximation (SSA) (see

We are interested in the effect of perturbations of the topography and the slipperiness at the ice base on the velocity of the ice at the surface and its height. By solving the adjoint equations, we quantify the sensitivity to perturbations close to the grounding line and of different wavelengths. The sensitivity at the upper surface to perturbations in the basal topography and friction is different, and the separation of the two contributions appears to be difficult. The transfer functions between the perturbations at the base and the surface observations are more or less well behaved. A related problem is to infer the basal geometry and friction coefficients from observational data by inversion using the adjoint solution.

Most methods for inversion of ice surface data to compute parameters in the models at the ice base rely on a solution of the adjoint stress equation with a given fixed geometry of the ice as in

The conditions between the ice and the bedrock vary in time, and sometimes the friction parameter varies
several orders of magnitude in a decade (see, e.g.,

The forward advection equation for the height and the stress equations for the velocity for FS are here solved numerically in two dimensions (2-D) with
Elmer/Ice

There is a transfer matrix between the perturbations in the parameters at the base and the observations at the surface. Analytical expressions for time-dependent transfer functions for FS and SSA are derived in

The structure of the paper is as follows. The ice equations and the corresponding adjoint equations for FS and SSA are presented in Sect.

Vectors and matrices are written in bold as

The equations of two ice models and their adjoint equations are stated in this section.
The FS equations are considered to be an accurate model of ice sheets, and the SSA equations are an approximation of the FS equations suitable, e.g., for fast flowing ice on the ground and ice floating on water (see

The FS equations are a system of PDEs for the velocity of the ice

The domain of the ice is

The definitions of the strain rate

Let

We observe a quantity

The adjoint FS equations form a system of PDEs for the adjoint height

The perturbation of the observation in Eq. (

Only perturbations in

In the shallow shelf approximation of the FS equations, the velocity is constant in the vertical direction and the pressure is given by the cryostatic approximation (

It is sufficient to solve for the horizontal velocity

Let

The ice dynamics system is

The structure of the SSA system Eq. (

The adjoint SSA equations are derived in

The friction coefficients on the base and the lateral sides are perturbed by

In the 2-D model,

The adjoint variables

Perturbations

In order to simplify the notation, only a 2-D steady-state problem for the SSA model is considered here, but the analysis is applicable to 3-D steady-state problems as well as time-dependent problems with the FS or SSA models.

The time-independent perturbation of

The relation is discretized by observing

In the same manner, there are matrices

Consider the case when

The transfer functions in

The sensitivity problem and the inverse problem are related. Assume that there are

The relation between the transfer matrix and the inversion problem is illustrated by Eq. (

In the numerical experiments we use a 2-D constant downward-sloping bed with an ice profile from the MISMIP benchmark project in

The initial ice geometry with height

The initial configuration of the ice is a steady-state solution achieved by the FS model using Elmer/Ice (

The physical parameters of the ice.

Without losing the generality in the friction law and to investigate the relation between the basal velocity and the stress, the friction law exponent in the adjoint problem is assumed to be

A vertically extruded mesh is constructed for the given geometry with mesh size

The forward and adjoint FS problems are solved using the finite element code Elmer/Ice (

The time-stepping scheme for the forward and adjoint transient problems is the implicit Euler method with a constant time step

Both transient and steady-state simulations are run with pointwise observations of the horizontal velocity

The multiplier

Comparison of the weights

The adjoint solutions

The amplitude of the perturbation at the surface depends on the wavelength

The response at

We perform a pair of experiments to compare the results from perturbing the forward equation and the prediction by the adjoint solutions.
A relative

The changes on the horizontal velocity

The same MISMIP benchmark experiment as in Sect.

The numerical solution of the forward SSA equations Eq. (

Comparison of the steady-state numerical solutions of the SSA velocity

The weight functions

Comparison of the weights

The weight functions

Comparison of the weights

A close-up view of the steady-state weights in the lower panels of Fig.

The same perturbation on

The corresponding comparisons for the steady-state problem are made in Figs.

The rapid change in

The perturbations

All the predicted solutions from the adjoint SSA are in good agreement with the forward perturbation.

The inverse problem of the steady state for the friction coefficient may not be well-posed since the weights are all positive from

The singular values of the sensitivity matrices

The singular values of the transfer matrices

Good approximations of the sensitivity matrices

The changes in the horizontal velocity

The changes in the horizontal velocity

The changes in the horizontal velocity

The changes in the horizontal velocity

The solution of the adjoint equations is simplified in the comparison in Fig.

The changes in the horizontal velocity

The singular values of the transfer matrices corresponding to the two simplifications are displayed in Fig.

The singular values of the transfer matrices of SSA with simplifications from

A few issues are discussed here related to the control method for estimating the parameter sensitivity.

We solve the FS adjoint problem only one step backward in time to verify the numerical method due to limitations of the current framework of Elmer/Ice.
It is possible but more complicated and expensive to solve the adjoint problem numerically for a large number of time steps

The equations for the adjoints of FS and SSA in Eqs. (

The solutions of the horizontal velocity

There are many discussions regarding the choice of friction laws (see, e.g.,

The transfer relation

The transfer relation is computed by solving the forward problem once and then the adjoint problem for each one of the

The perturbations

Both the height and the stress equations and their adjoints are solved to find the weight functions here.
The inverse problem at steady state to infer

The sensitivity to perturbations

A weight is local if its extension in space is close to the observation point.
The weights on

The perturbations in

The transfer matrices from

Detailed derivations of the formulas are found in

The analytical steady-state solution to the forward Eq. (

The analytical steady-state solutions of the SSA adjoint Eq. (

If

The weights for

The SVD factorizes a matrix

The FS equations are solved using Elmer/Ice version 8.4 (rev. f6bfdc9) with the scripts at:

GC contributed most of the computations, and GC and PL contributed equally to the theory and the writing of the paper.

The authors declare that they have no conflict of interest.

Thomas Zwinger helped with the adjoint FS solver in Elmer/Ice. Comments by Lina von Sydow helped us improve a draft of the paper, and Murtazo Nazarov explained the algorithm for adaptive mesh refinement.

This research has been supported by a Svenska Forskningsrådet Formas grant (no. 2017-00665) to Nina Kirchner and the Swedish strategic research program eSSENCE.

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