A key challenge in modelling coupled ice-flow–subglacial hydrology is initializing the state and parameters of the system. We address this problem by presenting a workflow for initializing these values at the start of a summer melt season. The workflow depends on running a subglacial hydrology model for the winter season, when the system is not forced by meltwater inputs, and ice velocities can be assumed constant. Key parameters of the winter run of the subglacial hydrology model are determined from an initial inversion for basal drag using a linear sliding law. The state of the subglacial hydrology model at the end of winter is incorporated into an inversion of basal drag using a non-linear sliding law which is a function of water pressure. We demonstrate this procedure in the Russell Glacier area and compare the output of the linear sliding law with two non-linear sliding laws. Additionally, we compare the modelled winter hydrological state to radar observations and find that it is in line with summer rather than winter observations.

Subglacial hydrology is an important control on ice velocities at the margin
of the Greenland Ice Sheet. Observed seasonal acceleration of ice flow

Recent subglacial hydrology models have progressed to simultaneously
incorporating both distributed and efficient systems, explicitly treating the
interaction between the two

Initializing model parameters and state is necessary for applying a linked hydrology–ice dynamics model to the Greenland Ice Sheet. In contrast to the availability of measurements at the surface of ice sheets, however, the conditions at the ice–bed interface are poorly constrained. Some key challenges for modelling are the form of the sliding law which relates water pressures to basal drag, the values of the parameters in that relationship, and the values of water pressures at the ice–bed interface.

Inverse methods are an approach which can be used to constrain unknown
variables or parameters in an ice-sheet model. Inversions optimize the value
of an unknown to minimize the discrepancy between model output and observed
data. Since basal drag and the parameters of the sliding law are some of the
least constrained inputs to ice-sheet models, a common application of
inversions in glaciology is to determine the field of basal drag which best
reproduces observed surface velocities. A variety of inversion methodologies
have been applied in glaciology. These include iterative methods

In this study we develop an ice-sheet model and inversion code, which we apply
to the Russell Glacier region of Western Greenland in order to invert for
basal drag at the end of winter. The ice-sheet model uses the hybrid
formulation of

The ice-sheet model implemented is based on the hybrid formulation described
in

Following

Basal drag is defined by the sliding law. Three different sliding laws are implemented in the ice-sheet
model:

Following

The linear sliding law (Eq.

It is useful to represent the sliding laws in a common form:

The boundary conditions at the terminating margin of the ice sheet are

Two further boundary conditions are used in the ice-sheet model: a no-penetration condition at the margin of nunataks and a Dirichlet boundary condition at the lateral margins of the ice-sheet domain which are not the termination edge.

The equation for viscosity is

Viscosity is defined implicitly by Eq. (

The hybrid formulation of the conservation of momentum equations depend on
depth-integrated viscosity:

This integral, and others, is numerically integrated using the composite Simpson law.

Following

This integral can be used to define expressions for surface velocity in terms of basal velocity
and basal velocity in terms of depth-averaged velocity

Additionally, defining

As in

Equation (

Equation (

The Picard iteration linearizes Eq.

Evolution of surface geometry is not included in the ice-sheet model. This is appropriate since the ice-sheet model is applied on annual timescales, over which significant changes in ice-sheet geometry are not expected.

The ice-sheet model was tested against the ISMIP-HOM benchmark experiments A
and C

This section describes the details of an inversion code developed in
conjunction with the ice-sheet model. The methodology is based on

The cost function returns a scalar which measures the fit of the model to the observations. The cost function is defined as

The cost function defined above has two terms:

The control parameter refers to the variable which the inversion process
optimizes in order to best match model prediction and observations. Since our
aim is to determine the basal drag, the control parameter is a parameter in
the basal sliding law. For the linear sliding law,

The inversion process aims to determine the field of

The methodology to obtain the gradient

Consider the following model:

Further, define a function J:

The aim is to determine the gradient of the cost function J with respect to
the initial input

There are several observations about the TLM. First, the TLM determines the
perturbation of

The concept behind the adjoint model is that rather than determining how
changes in the input

Key observations about the adjoint model are as follows. (1) In contrast to the TLM,
which acts upon a perturbation, the adjoint model acts upon the sensitivity
of the cost function. (2) A single run of the adjoint model is sufficient to
determine the gradient

The adjoint model is generated based on AD (

Multiple methodologies exist for AD tools to generate the derivative code.
Previous applications of AD software to generate the adjoint in glaciology

Here, we apply the open source AD tool ADiGator

Pseudocode of the main ice-sheet model routine is shown in Algorithm
A1, and the corresponding code to calculate the adjoint is
shown in Algorithm A2 (see Appendix). Two new functions, S1
and S2, appear in the adjoint code. These encapsulate segments of code from
the forward model and can be processed by ADiGator. The function S2 contains
code which spans over two Picard iterations. The adjoint does not contain a
for loop corresponding to iterating through the Picard iterations in reverse
(cf.

The adjoint code explicitly calculates several Jacobian matrices (lines 15 to
23 in Algorithm A2). ADiGator is applied to the corresponding
functions to generate the Jacobian matrices, except the solution to the
system of linear equations, which requires special treatment. A counterpart
to the linear solve which returns the corresponding derivate is manually
programmed following the procedure detailed in the appendix of

Landsat 8 satellite image, band 2, showing the Russell Glacier area. Black box outlines the study area. Inset shows the location in reference to Greenland.

This implementation of the adjoint is equivalent to previously published
adjoint implementations

The gradient from the adjoint model is used to solve the optimization problem
which minimizes the cost function. The inversion code relies on minFunc

Performance of the inversion code was verified using a series of identical
twin tests

This subglacial hydrology model used is described in detail in

Both distributed and channelized flow are represented in the subglacial
hydrology model. Distributed flow is described by an average thickness and
flux over a representative area. As in

The Russell Glacier area is a land-terminating sector of the Greenland Ice
Sheet (Fig.

An outline of the study area is shown in (Fig.

The ice-sheet model–inversion code is applied to determine the basal
boundary condition at the end of the 2008–2009 winter season in the Russell
Glacier study site. The end of the winter season is assumed to be day 120 of
the year (30 April). Although the exact day is somewhat arbitrary, this day
was selected as it is shortly before surface runoff begins in the study area
and shortly before GPS records in the study site show enhanced motion

Applying the ice-sheet model–inversion code to the Russell Glacier area
requires a number of datasets. Mean winter surface velocities for 2008/2009
(Fig.

An important assumption made is that the mean winter velocities are
representative of both the beginning and end of winter. This assumption is
justified by observing published GPS records in southwest Greenland

Inversions are initialized using a basal drag set to the local driving stress
smoothed by a 3

The results of inversions depends on the relative values of the scaling
factors

Parameters for the ice-sheet model–inversion code are listed in
Table

The parameters for the subglacial hydrology are the result of an extensive
parameter search using a coupled ice-flow–subglacial hydrology model in

Constants used in the ice-sheet–inversion model applied to the Russell Glacier area.

Log–log plot for L-curve analysis of inversions of the Russell
Glacier area employing

Map of the log of the absolute difference between the observed
and modelled surface velocities for inversions using

A workflow is developed for incorporating modelled effective pressure into
inversions using non-linear sliding laws. This workflow is motivated by the
idea that both the subglacial hydrological system and ice flow are in
quasi-steady state during the winter. This allows us to invert for background
values of the constants in the sliding laws. The initial step is to invert
using a linear sliding law for the basal drag coefficient. Basal velocities
are calculated from modelled depth-integrated velocities
(Eq.

The subglacial hydrology model is then run for the winter season with the
basal drag and basal velocities from the linear inversion. The model is run
at 500

Finally, the non-linear inversions are run using the modelled water pressure
from the subglacial hydrology model winter run. Two sets of inversions are
conducted, one for the Budd sliding law and one for the Schoof sliding law.
The first set of inversions seeks to determine the distribution of

Inversion results for the three sliding laws. Subplots

Six inversions using the linear sliding law are run
(Fig.

A map of the difference between observed and modelled velocities shows the
highest difference occurs along the ice margin and in the vicinity of the
nunatak (Fig.

Basal melt during the winter is shown in Fig.

The subglacial hydrology model winter run evolves rapidly at the beginning of the run. However, by day 50 of the model run the rate of change is significantly reduced, and by day 240 of the run the model is in an approximate steady state.

The distribution of sheet thickness at the end of winter mirrors basal
topography, with the sheet thickest in topographic lows
(Fig.

An L-curve analysis (Fig.

Modelled basal melt rate using basal velocities from linear inversion.

Modelled state of the subglacial hydrology system
at the end of the winter.

Figure

Inversions of the Russell Glacier area are run with a constant creep
parameter

Basal velocities determined from the optimal inversion using a linear sliding law are input into the subglacial hydrology model. The distribution of basal velocities is used to calculate both the basal melt rate and the cavity space in the continuum sheet flow. Due to the selection of a creep parameter such that the sliding ratio is relatively high, it is likely that basal velocities are overestimated. This would result in an overestimate of water generated at the ice–bed interface and an overestimate of the capacity of cavity space. Application of a higher order ice-sheet model would be advantageous in these regards.

The pattern of basal drag inverted using the three different sliding laws
show limited differences. This is due to the fact that basal shear traction
must satisfy the global stress balance

Interpretation of radar lines in the Russell Glacier area suggests
significant winter storage of water along topographic highs, while
significant water flow through topographic lows occurs during the summer melt
seasons

The initialization procedure introduced is not capable of producing the
inferred year-on-year differences in the subglacial hydrological system at
the end of winter. Currently, the subglacial hydrology reaches an approximate
steady state by day 240 and is not particularly sensitive to the
initialization of the distributed sheet thickness. A full steady state takes
approximately 2 years

Other procedures for determining the background parameters of sliding laws
can likely be devised. Currently the procedure only uses mean winter
velocities. Using mean annual velocities may improve estimates of the sliding
law parameters by incorporating information from the melt season. A
subglacial hydrological model could be run for an entire year and basal
parameters determined from an annual average water pressure. A key difficulty
is running the hydrological model during the summer, as the development of
the system is known to depend on feedbacks with velocity

A new ice-sheet model and adjoint code are presented. The ice-sheet model is coupled to a recent subglacial hydrology model

All datasets used are publicly
available. BedMachine data are available from

We would like to thank Mathieu Morlighem and Ian Joughin for the BedMachine2 and MEaSUREs datasets, and Ian Hewitt for generously sharing the subglacial hydrology code. Conrad P. Koziol would like to acknowledge Robert Arthern for guidance on writing an ice-sheet model and inversion code, as well as Brent Minchew, Poul Christoffersen, and Daniel Goldberg for thoughtful discussions. Additionally, we are grateful to the scientific editor and two anonymous reviewers for their thorough scrutiny of the manuscript. C.P. Koziol was funded through St. John's College, Cambridge, and (in part) by NERC standard grant NE/M003590/1. Edited by: G. Hilmar Gudmundsson Reviewed by: two anonymous referees