We present a sensitivity analysis and discuss the probabilistic forecast capabilities of the novel sea ice model neXtSIM used in hindcast mode. The study pertains to the response of the model to the uncertainty on winds using probabilistic forecasts of ice trajectories. neXtSIM is a continuous Lagrangian numerical model that uses an elasto-brittle rheology to simulate the ice response to external forces. The sensitivity analysis is based on a Monte Carlo sampling of 12 members. The response of the model to the uncertainties is evaluated in terms of simulated ice drift distances from their initial positions, and from the mean position of the ensemble, over the mid-term forecast horizon of 10 days. The simulated ice drift is decomposed into advective and diffusive parts that are characterised separately both spatially and temporally and compared to what is obtained with a free-drift model, that is, when the ice rheology does not play any role in the modelled physics of the ice. The seasonal variability of the model sensitivity is presented and shows the role of the ice compactness and rheology in the ice drift response at both local and regional scales in the Arctic. Indeed, the ice drift simulated by neXtSIM in summer is close to the one obtained with the free-drift model, while the more compact and solid ice pack shows a significantly different mechanical and drift behaviour in winter. For the winter period analysed in this study, we also show that, in contrast to the free-drift model, neXtSIM reproduces the sea ice Lagrangian diffusion regimes as found from observed trajectories. The forecast capability of neXtSIM is also evaluated using a large set of real buoy's trajectories and compared to the capability of the free-drift model. We found that neXtSIM performs significantly better in simulating sea ice drift, both in terms of forecast error and as a tool to assist search and rescue operations, although the sources of uncertainties assumed for the present experiment are not sufficient for complete coverage of the observed IABP positions.

Large changes in the Arctic sea ice have been observed in recent decades in
terms of the ice thickness, extent and drift

Current short-term (i.e. within 10 days) sea ice forecasting systems
integrate either a stand-alone sea ice model (RIOPS,

Probabilistic forecasts, widely used in weather forecasting

This study concerns the probabilistic forecast capability of the sea ice
model neXtSIM

Our main research tool and object of study is the sea ice model
neXtSIM. The model neXtSIM is based on a Lagrangian
numerical scheme and on a continuous approach using a newly developed
elasto-brittle (EB) ice rheology. This mechanical framework is inspired by the
scaling properties of sea ice dynamics revealed by multi-scale statistical
analyses of observed sea ice drift and deformation
(

We concentrate here on the impact of the error from the wind field alone. The
reasons are twofold. First, the wind is the most influential external force
affecting sea ice motion. About 70 % of the variance of the sea ice motion
in the central Arctic can be explained by the geostrophic winds

Previous sensitivity analyses of the neXtSIM model have been
performed with respect to initial conditions and to some key sea ice
mechanical parameters (see Sect. 4 in

Systematic errors in the mean sea ice drift are evaluated by averaging
modelled and observed drift from the OSI-SAF dataset

Systematic errors in the neXtSIM ice velocities compared to
observations from the OSI-SAF dataset. Simulated and observed ice drift is
averaged over the period from 1 January 2008 to 30 April 2008. The cells with
less than 28 observations over the winter are masked. The colour scale
represents the velocity in km day

This paper is organised as follows: Sect.

In this section, we provide a general description of neXtSIM.
Deliberately, we choose to not go through all model equations here but
rather list those that are needed to get an overall understanding of how the
model works and that are relevant for the present study. For a more detailed
description of the model see

neXtSIM is a continuous dynamic–thermodynamic sea ice model. It uses
a pure Lagrangian advection scheme, meaning that the nodes of the model mesh
are moving at each time step according to the simulated ice motion. The model
mesh is therefore changing over time, is not spatially homogeneous and
can locally become highly distorted, that is, when and where the ice motion
field is showing strong spatial gradients. In this case, a local and
conservative re-meshing procedure is applied in order to keep the numerical
integrity of the model and the spatial resolution of the grid approximatively
constant during the simulation. The equations are discretised on a triangular
mesh and solved using the classical finite element method, with scalar and
tensorial variables defined at the centre of the mesh elements and vectors
defined at the vertices. The model uses a mechanical framework that has
been developed recently (

The evolution
equations for

The evolution equation for the damage is written as

The air and oceanic drags, respectively

Parameters used in the model with their values for the simulations performed for this study.

In this study, we perform a sensitivity analysis using a statistical approach
based on Monte Carlo sampling of the model inputs. We focus on the response
of the model to the uncertainties in the wind velocity field. In particular,
we are looking at the response of sea ice drift to wind perturbations
representing these uncertainties. Our methodology is based on simulating
Lagrangian trajectories of virtual buoys using an ensemble run of the
neXtSIM model forced by slightly different (i.e. perturbed) wind
forcing (see Sect.

The velocity of a given virtual buoy is calculated online, at each time
step, as a linear interpolation of the velocities simulated at the nodes of
the mesh element containing that buoy (see Lagrangian approach in
Sect.

For each ensemble member (trajectory), we define the following Euclidean distances

Furthermore, we define a two-dimensional time-dependent orthonormal basis,
centred on

From a

With the individual

It is finally worth observing that the two quantities,

Our domain of study is the region covering the Arctic Ocean. While the coasts
are considered as closed boundaries, open boundaries are set at the Fram and
Bering straits (see Fig.

Maps showing the Arctic domain considered for this study. The red lines are open boundaries, while the black coastlines are closed boundaries. The starting points of the virtual trajectories simulated with the neXtSIM and FD models are represented by the blue crosses.

The wind forcing is taken from the Arctic System Reanalysis (ASR)

Although the ensemble average of the perturbed

Figure

The ocean forcing comes from the TOPAZ4 reanalysis

Spatial distribution of the number of occurrences of free-drift events between 1 January and 30 April 2008. The temporal sampling frequency used is 1 day. These are the instances used for the optimisation of the air drag coefficient.

Scatter plots for the two components,

Our analysis is based on two periods of the year 2008: from
1 January to 10 May and from 1 July to 20 September, representative of the
winter and summer conditions, respectively. We have intentionally studied them separately
because winter and summer are characterised by significantly different sea
ice mechanical regimes and therefore drift responses. During the winter, the
whole Arctic basin is covered by ice and its concentration is close or equal
to 100 %; that is, the internal stresses in the ice, and the corresponding

We ran a total of 13 simulations in the winter and 8 in the summer during
successive, non-overlapping 10-day periods. Limiting the length of the
simulations to 10 days ensures that the sea ice state (thickness and
concentration) remains as realistic as possible in the free-drift simulation,
in which there are no physical limits to the amount of ridging and opening.
The starting positions are separated by 100 km and cover the domain
displayed in Fig.

As already stated, we compared neXtSIM with the so-called
FD model, so that all
simulations that follow have been carried out for the two models.
neXtSIM (Eq. (

In this section, the notations

Figures

Mean over the winter period of

Winter average of wind speed and ice thickness. Both maps are from the neXtSIM simulations, but a similar thickness field and the exact same wind speed field are obtained for the FD simulations.

Figures

From Figs.

For neXtSIM, the smallest values for the mean of

For FD, the mean values of

Mean over the summer period of

Summer average of wind speed and ice thickness. Both maps are from the neXtSIM simulations, but a similar thickness field and the exact same wind speed field are obtained for the FD simulations.

In both winter and summer, the time-averaged response of

As expected, for neXtSIM we observe an increase of

The absolute values of

Figure

Probability density function (PDF) of

Figure

In Fig.

Evolution of the spatial mean of

Means over the winter (top panels) and summer (bottom panels) of

Spatial domain average of the variance of the distances

Another important characterisation of the ensemble spread evolution can be
set by looking at the variance of the distance

We evaluate here how well the neXtSIM and FD models are able to
forecast real trajectories in hindcast mode. As a benchmark, we compare the
ensemble runs from both models to 604 (in winter) and 344 (in summer)
observed trajectories from the IABP dataset. The simulated trajectories of
both neXtSIM and FD are initiated on the same initial positions and
at the same time as the IABP buoys (displayed in
Fig.

Maps showing the positions (blue crosses; 603 during winter and 344 during summer) of the IABP buoy trajectory dataset used in this study as starting point of the ensemble trajectory simulations performed with the neXtSIM and FD models. The grey area marks the presence of the sea ice during at least 10 consecutive days (the length of the simulations) during the winter and summer periods.

As a metric for the models skill intercomparison, we use the
linear forecast error vector

We complete this evaluation comparing results from both models with those
from a single deterministic forecast in order to verify the advantage of
probabilistic forecasts. In this case, we run neXtSIM with
parameters found in

Mean of the absolute forecast error

Figure

The centre panel in Fig.

Finally, the right panel in Fig.

Overall, we conclude that the performances are significantly better for neXtSIM in winter but similar in summer, and this would likely remain so even after optimal tuning of the turning angle.

Comparing to a single deterministic neXtSIM forecast, we note that
the forecast error is close to the average of the probabilistic run but
larger in summer, reaching 34 km at 10 days. The main difference with the
probabilistic run is the poorer along-drift component

In

The probability of finding a drifting object inside the search area is referred to as the probability of containment (POC) and computed by counting the objects falling within the search area divided by the total number of objects. The POC may be interpreted as the ratio of the size of the search area to the square forecast error. Thus, a small forecast error compared to the search area leads to a strong POC; conversely, a small search area (ensemble spread) compared to the forecast error leads to a poor POC.

In order to evaluate the probabilistic forecast capabilities of neXtSIM and
the more classical FD model, the context of a search and rescue operation
is adopted. We assume that an IABP buoy has been lost for 10 days: its
initial position,

Due to a lack of related literature for search and rescue in sea ice, we consider the
values of open-ocean search areas and POC found in

Figure

On Fig.

Illustration of the forecast error and the anisotropic search area.
Blue dots represents the position of one member, while the barycentre of the
ensemble (its mean) is

The small values of the spread correspond to shorter forecast lead times (see
Fig.

It should be no surprise that the two models underestimate the errors since this is a common behaviour of probabilistic forecast systems, but the differences of the shape of the spread–error relationships indicate that the two models underestimate the errors for different reasons: the neXtSIM ensemble is lacking spread during the initial times of the forecast but the asymptotic convergence of the spread to the errors tends to be the cause of the constitution of the initial ensemble.

If one had considered the more linear spread–error relationship in the FD model alone, it would have been tempting to increase the variance of wind perturbation errors until a perfect match of the spread to the errors was obtained, but this would have over-tuned the variance of the wind and masked that the FD model suffers from unresolved physics.

Time evolution of the averaged ellipse areas for neXtSIM (solid lines) and FD (dashed lines) in winter (blue) and in summer (red).

Spread–error relationship for 12 h averages. The curves are based on the spatial mean of the forecast error shown as a function of the spread.

Whether a prediction model is too optimistic or too pessimistic may be equally problematic in view of search and rescue operations. In practice, the resources available for search and rescue operations are limited and only a given area can be covered, although the shape of the area (centre and eccentricity in the case of an ellipse) may not influence the cost significantly. Thus, rather than looking at the size of the search area as estimated from the ensemble model prediction, the search-and-rescue operation can be posed as follows: for a given area to be searched, which model forecast gives the ellipse that is most likely to contain the object?

Time evolution of POC according to the search area for
neXtSIM (solid lines) and FD (dashed lines) in winter

The ensemble forecast provides the expected position,

For each time

For both periods and both models, all curves exhibit a sigmoid shape with an
inflexion point, the position of which depends on the time horizon (higher POC and
larger search areas for longer drift duration). For a 7-day drift in winter
and a POC equal to 0.5, the area is around 300 km

Time evolution of the POC difference between neXtSIM and FD
for a search area equal to 50 km

How do the different models perform for different forecast time (i.e. drift
duration)? To answer this question, we study the time evolution of the
difference between the neXtSIM and FD POCs: when this difference is
positive (negative), neXtSIM (FD) is outperforming FD (neXtSIM).
The POC for both models is evaluated for a fixed search area – a vertical
section across the selectivity curves – equal to 50 km

The negative values for lead time shorter than 1 day in winter are again likely caused by the initialisation of the neXtSIM ensemble and another reason to constrain its initial anisotropy to observations.

The ensemble model sensitivity experiment carried out with neXtSIM
and with an FD model reveals the prominent role of the rheology, which marks
the key difference between the two models. On average over the whole Arctic
neXtSIM is less sensitive to the wind perturbations than the FD,
although large seasonal and regional differences are observed. This is
exemplified by the imprint of the ice thickness field in the ensemble spread
from neXtSIM and the much smaller sensitivity of neXtSIM in
winter than summer, in contrast to the FD model (Figs.

The two models have been tuned on different observations of ice, seen as in
free drift by each model, so that the different performances originate by the
differences in the resolved model physics at their best performance. The
diffusion regimes of neXtSIM and FD are very different in winter
(Fig.

Due to the dispersive properties of the sea ice, the shape of the ensemble of
simulated buoys positions is generally anisotropic. Such anisotropy is a
signature of the underlying mechanism that drives the dispersion of the
members, which is the shear deformation of the ice cover along active
faults/fractures in the ice. This mechanism is missing in the absence of
rheology (like in the FD model) and represents a clear strength in principle
for the EB rheology in neXtSIM, although with the
present ensemble initialisation it did not prove to be a practical
advantage. Other rheological models, such as the elasto-viscous plastic
model, also present some degree of anisotropy

The performance of the two models differs significantly when forecasting the
trajectories of IABP buoys. The ensemble mean position errors are larger in
the summer (5 km after 1 day and 12.5 km after 3 days of drift for
neXtSIM, about 16 % below the FD results) and consistent with the
values reported by

The model sensitivity to wind perturbations has been evaluated, yielding (for
10 days of drift) a spread from 5 to 10 km, for winter and summer, respectively,
but this is smaller than the corresponding errors (15 km from the barycentre
to the observations in Fig.

To further improve the spread–error relationship, alternative sources of errors should be considered such as model initial conditions and forcings (ice thickness, concentrations, damage, ocean currents). Since the errors are increasing faster in the first days of the simulations, the more likely source of local and short-term errors lies in the position and orientation of the sea ice fracture network, which is left unconstrained in any of the experiments presented here.

Although we would expect an increase of the ensemble spread if the ice thickness, concentrations and ocean currents had been taken into account in the ensemble initialisation, we do not believe it would lead to a much larger spread, especially in the winter. We suggest instead that, in the perspective of efficient sea ice forecasting, major efforts should be directed toward assimilating the observed fractures (as of satellite images). The assimilation of fracture (as objects rather than quantitative observations) represents a priori a challenging avenue in terms of data assimilation, which traditionally deals with quantitative scalar or vector observations; however, we envision that the damage variable in neXtSIM, showing localised features, can be constrained quantitatively to deformation rates as derived from observed high-resolution ice motions and serve as “object assimilation”.

In spite of the biases, the selectivity curves indicate that a probabilistic
forecast using neXtSIM is largely more skilful than the traditional
free-drift model, and it has the larger potential for practical use in search
and rescue operations on sea ice. Since the Arctic is not easily accessible,
forecast horizons of 5 to 10 days are probably the most relevant for
logistical reasons. On those timescales, the differences of POC shown in
Fig.

The physical consistency of the ensemble sensitivities is a necessary
condition to the success of ensemble-based data assimilation methods

Besides the potential use of observations of fractures, as mentioned above,
which is indeed another unique advantage of models such as neXtSIM,
ice drift data are also crucial. Observations of ice drift are still seldom
used for data assimilation, and when it is the case, the success is limited
by the lack of sensitivity of the sea ice model

The atmospheric reanalysis data used in this article are
available at

The sensitivity analysis was implemented and performed by MR. The results were analysed by MR, PR, AC and LB. The manuscript was written by MR and PR, then reviewed and improved with inputs from all authors.

The authors declare that they have no conflict of interest.

Matthias Rabatel, Pierre Rampal, Alberto Carrassi and Laurent Bertino have been funded by the Office of Naval Research project DASIM (award N00014-16-1-2328). Pierre Rampal, Alberto Carrassi and Laurent Bertino also acknowledge funding by the project REDDA of the Norwegian Research Council. Christopher K. R. T. Jones was supported by the Office of Naval Research (USA) under grants N00014-15-1-2112 and N00014-16-1-2325. The authors are grateful to Jean-Francois Lemieux, Helge Goessling and one anonymous reviewer for their insightful comments that helped improve the manuscript.Edited by: David M. Holland Reviewed by: Helge Goessling, Jean-Francois Lemieux, and one anonymous referee