We propose a method to reduce the error generated when computing sea ice deformation fields from synthetic aperture radar (SAR)-derived sea ice motion. The method is based on two steps. The first step consists of using a triangulation of the positions taken from the sea ice trajectories to define a mesh on which a first estimate of sea ice deformation is computed. The second step consists of applying a specific smoother to the deformation field to reduce the artificial noise that arises along discontinuities in the sea ice motion field. This method is here applied to RADARSAT Geophysical Processor System (RGPS) sea ice trajectories having a temporal and spatial resolution of about 3 days and 10 km, respectively. From the comparison between unfiltered and filtered fields, we estimate that the artificial noise causes an overestimation of about 60 % of opening and closing. The artificial noise also has a strong impact on the statistical distribution of the deformation and on the scaling exponents estimated with multifractal analysis. We also show that a similar noise is present in the deformation fields provided in the widely used four-point deformation RGPS data set. These findings may have serious implications for previous studies as the constant overestimation of the opening and closing could lead to a large overestimation of freezing in leads, salt rejection and sea ice ridging.

Sea ice motion can be retrieved from satellite synthetic aperture radar (SAR) images using
cross-correlation techniques and feature tracking algorithms

As deformation determines sea ice opening (i.e., positive divergence)
and closing (i.e., negative divergence), it may be used to estimate
important global quantities, such as the ice production in leads, with
some assumptions on sea ice growth and redistribution

In addition to essential information about sea ice opening and
closing, the analysis of sea ice motion and deformation also gives
a particular insight into the underlying physics controlling the sea ice
dynamics and provides precious information with which to validate sea
ice models.

The estimation of these global quantities (e.g., total
opening/closing) and statistical properties (e.g., spatial scaling
exponents) may be impacted by errors in sea ice deformation
data. Uncertainty on deformation is usually seen as a consequence of
motion tracking errors that depend on the algorithm and parameters
used.

However, two other sources of error can be identified. Both are linked
to the definition of the boundary of the cell (usually quadrangle)
over which deformation is computed.

This paper proposes a method to avoid unrealistic values and to
significantly reduce the noise obtained when computing sea ice
deformation from SAR-derived motion and presents an example of its
application to sea ice trajectories coming from the RGPS data set. The
complete method is described in Sect.

The method we developed is based on two steps. The first step consists of defining a mesh by doing a triangulation of a set of tracked points. For each individual triangular cell, the deformation is calculated using the motion of its three nodes estimated from the tracking procedure. The second step consists of applying a specific smoother to the obtained deformation fields to reduce the artificial noise.

Example of the divergence

In order to present the method, we first define a simple setup on
a square domain having a normalized area equal to 1. In this domain,
tracked points are distributed uniformly with a mean distance

In the first test case, a single crack is defined (black line in
Fig.

The first step of the method is to perform a Delaunay triangulation of
these points to generate a mesh on which deformation is computed. The
spatial derivatives of the displacement are obtained by calculating
the following contour integrals as in

This artificial noise generates an overestimation of the total opening
(and closing). In our test cases, the opening (and closing) error is
defined as the absolute difference between the
total opening (and closing) computed from the geometry of the problem and
the total opening (and closing) computed from the integration of the positive (and negative)
divergence given by the method. Repeating the slip line experiment 100 times, with

When repeating the same test case with quadrangles instead of triangles,
we found a rms error of about 18

In order to remove the artificial noise in the deformation fields, one
could apply an isotropic smoother over all the cells of the mesh.
We here denote

We propose a better method based on the fact that the deformation is
by nature constant along a linear kinematic feature. Averaging motion
derivatives along these features could then filter out the noise
without spoiling the information on the real deformation.
Contrary to the isotropic smoother presented here above, the mean value of the shear
obtained with the second method in the single-crack case
does not vary as a function of

Root mean square closing/opening error normalized by

To detect the cells that are involved in the mapping of each linear kinematic
feature, we define a threshold for total deformation
(

The proposed method relies on two parameters: the deformation
threshold that determines which cells are selected and parameter

The two other curves in Fig.

To get one crack opening while the other is closing,

Example of the unfiltered (left panel) and filtered (right panel)
divergence obtained for the double-crack test case at a normalized
resolution of 0.1. The domain is divided into three blocks. Points below the
principal crack are fixed. Points above the principal crack experience the
same displacement

This mixing of intersecting cracks explains why the normalized error
(triangle and square symbols in Fig.

The RGPS Lagrangian displacement product provides trajectories of sea
ice “points” initially located on a 10

Unfiltered divergence rate computed from the RGPS sea ice trajectory
data set and corresponding to the pair of images taken at

The RGPS Lagrangian deformation product provides the deformation of
each cell (which is quadrangle) of the original grid. The deformation
of a cell is updated each time the position of all its nodes are
updated. This method has a serious problem because cells may become
so distorted that spatial derivatives are ill-defined. As the RGPS deformation
data set does not
provide for each cell the position of its node, it is not possible to filter the data to avoid this
problem. This problem is specific to the RGPS deformation product
and would not appear if each pair of images was treated separately
with its own grid as in the GlobICE Image Pair product
(

To tackle these problems, we reprocessed the RGPS Lagrangian
displacement product to build a new deformation data set called the
RGPS Image Pair Product. We first identify the tracked points
corresponding to each pair of images (i.e., the set of points whose
position has been updated at the exact same date and with the same
time interval). We generate a Delaunay triangulation of these
points. Then we compute the deformation over what we consider as being
well-shaped cells, i.e., only for triangles having an area between 5
and 400

Using triangles instead of quadrangles roughly doubles the number of deformation
estimates and increases the resolution of the deformation product up
to 7

Unfiltered total deformation rate for the same example as in
Fig.

To apply the smoother, we first need to detect the cells that are
suspected to map the location of linear kinematic features.

Decreasing the threshold increases the number of selected cells and finally leads to excessive smoothing. Increasing the threshold reduces the number of selected cells and finally splits linear features into disconnected pieces for which the smoother is not efficient anymore. Indeed if a kernel only contains one cell, the smoother does not modify the value of the deformation over that cell.

To quantify the effect of this threshold on the quality of the
selection, we define an index based on the size of the smoothing
kernels

We explored the sensitivity of this quality index to the threshold
value for the entire winter season 2006–2007 and with the parameter

Filtered divergence rate after the application of the smoother to
the selected cells (see Fig.

Figure

In this section, we compare the original RGPS deformation data to the unfiltered and filtered versions of our RGPS Image Pair data set. A consistency check based on spatial scaling analysis is proposed and the differences between the three data sets in terms of spatial scaling and total opening/closing are discussed.

To compare the original RGPS deformation data with the unfiltered and
filtered deformation data produced by our method, we generate
composite pictures of the deformation rates for specific periods. The
periods have to be long enough to ensure a good spatial coverage but
not too long so as to avoid mixing incoherent information. For this study,
we select the data for which the time of the first and second images –
noted

Selected cells may cover the same area but correspond to different
dates and time intervals. This redundancy may impact statistical
distribution and scaling analysis, so we apply a second selection
step. We first define a regular grid at a resolution of
20

Composite picture of the divergence rate given by the RGPS deformation data set for the period 2–10 February 2007. RGPS cells are here represented by squares as their actual shape is not known.

Figure

To evaluate our method in a more quantitative way, we propose a metric
based on a spatial scaling analysis. Scaling analysis is a powerful
tool to characterize sea ice dynamical behavior and has been
successfully used in previous studies to reveal the power-law scaling
of sea ice deformations

Composite picture of the unfiltered divergence rate computed after the first step of our method for the period 2–10 February 2007.

Composite picture of the filtered divergence rate for the period
2–10 February 2007 obtained with a threshold parameter equal to 0.02 per day
and with the parameter

Scaling analysis: absolute divergence rate as a function of the
spatial scale, from the unfiltered (left panel) and filtered (right panel)
composite deformation field for the period 2–10 February 2007 (each color
corresponds to a different box size used for the coarse graining procedure).
The mean values

The spurious noise in the deformation fields corresponds to high
values of deformation and is potentially present for any active linear
kinematic features. This noise may then impact the distributions of
shear and absolute divergence and modify their mean
(first-order moment) but even more their standard deviation
(second-order moment) and skewness
(third-order moment). Moreover this noise is the
highest at the resolution of the data but rapidly decreases for larger spatial scales.
We thus expect that the presence of noise in the deformation fields
will have a strong impact on the result of the scaling analysis, especially
for the smallest scales and the highest-order moments of the distribution.
Indeed, we assume that the power-law model for the spatial scaling of
sea ice deformations has no physical reason to not hold over several
orders of magnitude. This assumption is based on

To perform the scaling analysis of sea ice deformation, we implemented
a coarse graining method similar to the one proposed by

Multifractal analysis: moments of the absolute divergence rates

Structure function

The filtered shear and absolute divergence
closely follow the power-law model for the spatial scaling as
their first-order moments are well aligned with the power-law fit
for the spatial scales ranging from 7 to 200

By definition, if a scaling holds for a given range of scales, it should be respected
for any pair of scales within this range.
To evaluate the deviation from the power-law scaling, we compute the power-law
exponents for each pair of successive spatial scales (i.e., from 7 to
14

To check that the reference values for the model parameters are well chosen,
we look at the deviation from the power law. This deviation is evaluated by the min–max error.
For each moment order, the min–max error is defined as the difference between
the minimum and maximum exponents obtained with any pairs of spatial scales
within the defined range. It other words, it is the length of the bar drawn in Fig.

Comparing the original RGPS deformation to the unfiltered deformation
allows us to evaluate the impact of using a triangulation to define
well-shaped triangular cells. As in

The simple fact of redefining a new mesh from the actual position of the
RGPS nodes allows us to avoid badly shaped cells and then to significantly
reduce the number and magnitude of extreme values.
For the same period, the composite picture obtained from the
unfiltered version of our RGPS Image Pair data set has maximum opening,
closing and shear rates equal to 0.63,

Cumulative probability functions – in other words, the probabilities
of exceedance – for the RGPS, unfiltered and filtered composite divergence
fields shown in Figs.

Comparing the filtered and unfiltered deformation allows us to analyze
the impact of the artificial noise. From the scaling analysis for the
total deformation, shear and absolute divergence, we found that the
scaling exponents estimated from the unfiltered fields are
systematically larger in magnitude by about 100

The impact of the artificial noise is also seen on the
structure function

Differences are also seen in the cumulative distribution of the
closing and opening rates (see Fig.

Finally, we compare the three data sets by computing the total area
that has been opened and closed. For the original RGPS deformation
data, 40 000

A method is proposed to derive accurate sea ice deformation fields from SAR-derived motion products. The first step of the method consists of a triangulation of the tracked points to generate a mesh of triangular cells on which a first estimate of deformation is computed. The second step consists of applying a smoother to the deformation fields. The method relies on two parameters: a deformation threshold and the size of the smoothing kernel.

By applying the method to idealized test cases, we show that using
triangles instead of quadrangles induces an increase of only about 10

The proposed method is used to produce a new deformation data set called RGPS Image Pair Product. Compared to the RGPS deformation data set, the RGPS Image Pair data set does not exhibit unrealistic large values caused by badly shaped cells. Moreover, our method drastically reduces the artificial noise arising along dynamic discontinuities.

By comparing the unfiltered and filtered deformation fields for winter
2006–2007, we estimate that this artificial noise may cause an
overestimation of the opening and closing of about 60

The findings of the present study indicate that errors in sea ice deformation fields retrieved from SAR-derived motion may have been strongly underestimated, leading to potential significant biases on the estimates of sea ice production, salt rejection and sea ice ridging that one may find in the literature.

The method proposed here is applicable to other sea ice drift
data sets, as provided, for example, by the GlobICE project. The method can
handle Lagrangian trajectories or displacement between pairs of
images. The same method could be applied to buoy trajectories when
their spatial resolution is high enough, as with nested arrays of
buoys

The method proposed here could be modified to better manage
intersecting cracks and to adapt its parameters depending on the local
fields. However, substantial improvements may also come by combining,
within tracking algorithms, the detection of dynamic discontinuities
and the computation of sea ice deformation as proposed by

S. Bouillon is supported by the Research Council of Norway through the post-doc project SIMech, Sea Ice Mechanics: from satellites to numerical models (no. 231179/F20, 2014–2016). P. Rampal is supported by the Kara Sea project sponsored by Total E&P. Many thanks to Tim Williams and Gunnar Spreen for interesting discussions. Edited by: D. Feltham