Predictions of future mass loss from ice sheets are afflicted with uncertainty, caused, among others, by insufficient understanding of spatiotemporally variable processes at the inaccessible base of ice sheets for which few direct observations exist and of which basal friction is a prime example. Here, we present a general numerical framework for studying the relationship between bed and surface properties of ice sheets and glaciers. Specifically, we use an inverse modeling approach and the associated time-dependent adjoint equations, derived in the framework of a full Stokes model and a shallow-shelf/shelfy-stream approximation model, respectively, to determine the sensitivity of grounded ice sheet surface velocities and elevation to time-dependent perturbations in basal friction and basal topography. Analytical and numerical examples are presented showing the importance of including the time-dependent kinematic free surface equation for the elevation and its adjoint, in particular for observations of the elevation. A closed form of the analytical solutions to the adjoint equations is given for a two-dimensional vertical ice in steady state under the shallow-shelf approximation. There is a delay in time between a seasonal perturbation at the ice base and the observation of the change in elevation. A perturbation at the base in the topography has a direct effect in space at the surface above the perturbation, and a perturbation in the friction is propagated directly to the surface in time.

Over the last decades, ice sheets and glaciers have experienced mass loss due to global warming, both in the polar regions and also outside of Greenland and Antarctica

In computational models of ice dynamics, the description of sliding processes, including their parametrization, plays a central role and can be treated in two fundamentally different ways, viz. using a so-called forward approach on the one hand or an inverse approach on the other hand.
In a forward approach, an equation referred to as a sliding law is derived from a conceptual friction model and provides a boundary condition to the equations describing the dynamics of ice flow (in glaciology often referred to as the full Stokes (FS) model), which, once solved, render, e.g., ice velocities part of the solution. Studies of frictional models and resulting sliding laws for glacier and ice sheet flow emerged in the 1950s – see, e.g.,

Because few or no observational data are available to constrain the parameters in such sliding laws

Adopting an inverse approach, the strategy is to minimize an objective function describing the deviation of observed target quantities (such as the ice velocity) from their counterparts as predicted following a forward approach when a selected parameter in the forward model (such as the friction parameter in the sliding law) is varied.
The gradient of the objective function is computed by solving the so-called adjoint equations to the forward equations, where the latter often are slightly simplified, such as by assuming a constant ice thickness or a constant viscosity

Here, we present an analysis of the sensitivity of the velocity field and the elevation of the surface of a dynamic, grounded ice sheet (modeled by both FS and SSA, briefly described in Sect.

The sensitivity of the surface velocity and elevation to perturbations in the friction and topography is quantified in extensive numerical computations in a companion paper

In this section, the equations emerging from adopting a forward approach to describing ice dynamics are presented, together with relevant boundary conditions, for the FS (

The flow of large bodies of ice is described with the help of the conservation laws of mass, momentum, and energy

The upper surface of the ice mass and also the ice–ocean interface constitute a moving boundary and satisfy an advection equation describing the evolution of its elevation and location (in response to mass gain, mass loss, or/and mass advection). For ice masses resting on bedrock or sediments, sliding needs to be parameterized at the interface. The interface between floating ice shelves and sea water in the ice shelf cavities is usually regarded as frictionless.

We adopt standard notation and denote vectors in bold italics and matrices in three-dimensional space in bold, and we denote derivatives with respect to the spatial coordinates and time by subscripts

A schematic view of an ice sheet in the

The boundary

The vertical lateral boundary (in the

Before the forward FS equations for the evolution of the ice surface

Turning to the ice base, the basal stress on

With these prerequisites at hand, the forward FS equations and the advection equation for the ice sheet's elevation and velocity for incompressible ice flow are

The solution at the grounding line satisfies a nonlinear complementarity problem.
Let

The boundary conditions for the velocity on

The three-dimensional FS problem (

The simplifications associated with adopting the SSA imply that the viscosity (see

Above,

A flotation criterion determines the position of the grounding line (see, e.g.,

For ice sheets that develop an ice shelf, the latter is assumed to be at hydrostatic equilibrium.
In such a case, a calving front boundary condition

In this section, the SSA equations are presented for the case of an idealized, two-dimensional vertical sheet in the

We now discuss the steady-state solutions to the system (

Figure

The model parameters.

The larger the friction coefficient

Finally, it is noted that an alternative solution to (

The analytical solutions

In this section, the adjoint equations are discussed, as emerging in a FS framework (Sect.

On the ice surface

Examples of

The same forward and adjoint equations are solved both for the inverse problem and the sensitivity problem but with different forcing function

In this section, we introduce the adjoint equations and the perturbation of the Lagrangian function.
The detailed derivations of (

The definition of the Lagrangian

The adjoint viscosity and adjoint stress are

The perturbation of the Lagrangian function with respect to a perturbation

Let us now investigate the effect of time-dependent perturbations in the friction parameter on modeled ice velocities and ice surface elevation.
Suppose that the velocity component

The procedure to determine the sensitivity is as follows.
First, the forward equation (

We start by investigating the response of ice velocities to perturbations in friction at the base: when the slip coefficient at the ice base is changed by

Further, to investigate the response of the ice surface elevation,

In applied scenarios, friction at the base of an ice sheet is expected to exhibit seasonal variations.
These can be expressed by

Assume further that

Particularly in an inverse problem where the phase shift between

In another example, suppose that there is an interval with a step change of

To illustrate the phase delay in an oscillatory perturbation, a two-dimensional numerical example is shown in Fig.

Figure

The weight

The weight in (

Observations at

From a theoretical point of view, it is interesting to note that there is a relation between the sensitivity problem where the effect of perturbed parameters in the forward model is estimated and the inverse problem used to infer “unobservable” parameters such as basal friction from observable data, e.g., ice velocity at the ice sheet surface.
The same adjoint equation (

Let

It is shown in Appendix

In the inverse problem in

We have investigated the relation between the sensitivity problem and the inverse problem. By solving

A further theoretical consideration shows that the solution

The adjoint steady-state equation in a two-dimensional vertical ice in (

The analytical solution

With a small

Starting from (

The adjoint SSA equations read

From (

If the friction coefficient

Suppose that only

In a broader context, it is worth emphasizing that the adjoint equation derived in

The adjoint SSA equations in two vertical dimensions are derived from (

The steady-state solutions to the system (

In this section, the analytical solution to the adjoint equation (

For observations of

The relation in (

The perturbations

Finally, let us comment on other approaches to investigate the sensitivity of surface data to changes in

The analytical adjoint solutions

The analytical solutions of

The analytical solutions of

The weights

All perturbations in

The analytical solution of the weights

The analytical solution of weights

The following conclusions can be drawn from (

The closer perturbations in basal friction are located to the grounding line, the larger perturbations of velocity will be observed at the surface.
This is because the weight in front of

Variations in the observed velocity

When the variation in ice thickness is small compared to the overall ice thickness,

For an unperturbed basal topography, two different perturbations of the friction coefficient will result in the
same perturbation of the velocity. In other words, the perturbation

A rapidly varying friction coefficient at the base of the ice sheet will be difficult to identify by observing the velocity at the ice surface. In contrast, a smoothly varying friction coefficient at the base will be easily observable at the ice sheet surface. This is seen as follows:
Perturb

A perturbation in the topography with long wavelength is easier to detect at the surface than a perturbation with short wavelength.
If

In the case when

The second derivative term

The perturbation in

As in (

The contribution from the integrals in (

Finally, the time-dependent adjoint equation (

Figure

The adjoint weights for the observations at

The temporal variations of the adjoint weights at

The adjoint weights at

However, when

A reference adjoint solution at

Suppose that the temporal perturbation is oscillatory with frequency

The adjoint equations are derived in the FS and the SSA frameworks including time and the surface elevation equation.
Time-dependent perturbations

The perturbations in the observations are determined numerically in

In Sect.

For steady-state problems, and in an FS setting where

In this setting, a non-local effect of a perturbation in

In the inverse problems based on time-dependent simulations of FS and SSA, it is necessary to include the adjoint elevation equation.
If the perturbations in the basal conditions are time dependent and

Perturbations in the friction coefficient at the base observed in the surface velocity determined by SSA are damped inversely proportional to the wave number and the frequency of the perturbations in (

The adjoint viscosity

The adjoint friction in SSA in

The Lagrangian for the SSA equations is with the adjoint variables

Using the weak solution of (

Collecting all the terms in (

The FS Lagrangian is as in (

Define

The first variation of

Assume that

Consider a target functional

In a two-dimensional vertical ice with

If

Let

The forward and adjoint SSA equations in (

Multiply the first equation in (

The solutions

If

By Appendix

Combine (

The weight for

Use (

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

GC and PL designed the study. GC did the numerical computations. GC, PL, and NK discussed and wrote the manuscript.

The authors declare that they have no conflict of interest.

We are grateful to Lina von Sydow for reading a draft of the paper and helping us improve the presentation with her comments. We would also like to thank the editor Olivier Gagliardini and the two anonymous reviewers.

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 Olivier Gagliardini and reviewed by two anonymous referees.