The objective of this note is to provide the background and basic tools to estimate the statistical error of deformation parameters that are calculated from displacement fields retrieved from synthetic aperture radar (SAR) imagery or from location changes of position sensors in an array. We focus here specifically on sea ice drift and deformation. In the most general case, the uncertainties of divergence/convergence, shear, vorticity, and total deformation are dependent on errors in coordinate measurements, the size of the area and the time interval over which these parameters are determined, as well as the velocity gradients within the boundary of the area. If displacements are calculated from sequences of SAR images, a tracking error also has to be considered. Timing errors in position readings are usually very small and can be neglected. We give examples for magnitudes of position and timing errors typical for buoys and SAR sensors, in the latter case supplemented by magnitudes of the tracking error, and apply the derived equations on geometric shapes frequently used for deriving deformation from SAR images and buoy arrays. Our case studies show that the size of the area and the time interval for calculating deformation parameters have to be chosen within certain limits to make sure that the uncertainties are smaller than the magnitude of deformation parameters.

Sea ice drifts under the influence of wind and ocean currents. Spatial gradients in the sea ice motion lead to distortion of the sea ice cover, termed deformation. The retrieval of sea ice drift vectors and deformation parameters from pairs or sequences of satellite synthetic aperture radar (SAR) images has gained increased attention during recent years because of the growing availability of suitable data (e.g., Stern and Moritz, 2002; Karvonen, 2012; Berg and Eriksson, 2014; Komarov and Barber, 2014; Lehtiranta, 2015; Muckenhuber et al., 2016; Demchev et al., 2017; Korosov and Rampal, 2017). Sea ice kinematics is also studied based on data from arrays of buoys or GPS receivers (e.g., Lindsay, 2002; Hutchings et al., 2008; Hutchings et al., 2012; Itkin et al., 2017), which in addition can serve as reference in comparisons to motion vectors obtained from SAR images. The knowledge of spatially detailed motion and deformation fields is potentially useful in ice navigation to locate divergent or compressive ice areas, as complementary information for operational sea ice mapping, for validation of models for forecasting of ice conditions, and for assimilation into ice models (Karvonen, 2012). Such practical applications require that the errors of the retrieved drift and deformation parameters are known. For buoys, errors in drift measurements depend on the accuracy of position and time readings. The accuracy of deformation parameters is not only affected by errors in drift magnitude and direction but also by the size and shape of buoy arrays (e.g., Hutchings et al., 2012; Griebel and Dierking, 2018). Drift vectors derived from pairs of satellite images are the result of correlation techniques or object detections, while deformation parameters are calculated from spatial arrangements of adjacent drift vectors surrounding the area of interest, in a manner that is independent of the coordinate system. This means that drift and deformation errors do not only depend on the geolocation accuracy and spatial resolution of satellite images but also on the reliability and robustness of the drift retrieval algorithm. In this technical note we focus on the estimation of statistical errors for ice velocity and deformation. The issue of error estimation was repeatedly addressed in the past, scattered in a number of publications and restricted to single aspects related to the respective analysis (e.g., Lindsay and Stern, 2003; Hollands and Dierking, 2011; Bouillon and Rampal, 2015; Hollands et al., 2015; Linow et al., 2015; Griebel and Dierking, 2018), and is also addressed in a more recent analysis by Bouchat and Tremblay (2020). Our motivation is to provide the mathematical background, together with examples of applications and discussions of validity, in a broader context. We emphasize that here we deal with statistical errors but not with boundary definition errors as described, e.g., in Lindsay and Stern (2003), Bouillon and Rampal (2015), and Griebel and Dierking (2018). Although this note is specifically focused on retrievals of parameters characterizing sea ice kinematics, the mathematical framework is also applicable to movement and deformation of ice shelves and glaciers, or for model simulations of sea ice, glacier, and ice sheet dynamics.

In Sect. 2 we summarize the basics and provide equations for calculating errors of drift and deformation parameters: divergence, vorticity, shear, and total deformation. The equations are used in Sect. 3 to quantify the influence of different parameters such as geolocation and tracking errors, or shape and size of buoy arrays and grid cells. Conclusions are presented in Sect. 4.

In this section, we provide a short description of the estimation of errors and the computation of strain rates and then derive the statistical errors for drift velocity, polygon areas, divergence, shear, vorticity, and total deformation. The statistical errors quantify uncertainties that are introduced by random fluctuations in the measurements. If the random fluctuations are small, data are measured with a high degree of precision but not necessarily with high accuracy. The latter requires that the measured value is close to the true value, whereas precision refers to the reproducibility of a measurement (Bevington and Robinson, 2003, chap. 1).

The formula for error propagation is based on the splitting method, i.e., the
decomposition of a measured variable

Deformation parameters are calculated from different combinations of the
components of the velocity gradient tensor (

The deformation of a region

The velocity vectors may be obtained from an array of buoys, where the buoys' positions are regarded as the vertices of a polygon. The displacement of a buoy is usually calculated from the distance between distinct positions, and the velocity is determined as the displacement divided by the time period between position fixes. When using pairs of satellite images, sea ice deformation is obtained from the displacements of recognizable structures or patterns in these images. These are referred to as ice structures from here on. In the reference image, a grid can be constructed by connecting the center positions of adjacent ice structures by lines. If movements of single ice structures differ between acquisitions of image 1 and image 2, the shapes and sizes of grid cells have changed in the second image. It is the presence of velocity gradients due to locally varying physical forces that causes the deformation. In practice the movement of sea ice is obtained using different methods (e.g., Holt et al., 1992; Stern and Moritz, 2002; Karvonen, 2012; Muckenhuber et al., 2016; Korosov and Rampal, 2017), which determine the spatial distribution and density of the displacement vectors. The vectors can be regularly spaced on the crossing points of horizontal and vertical grid lines as a result of pattern matching algorithms in an Eulerian approach, or they can be irregularly distributed, which is typical for the Lagrangian approach applied in feature or buoy tracking (see Fig. 1).

Eulerian grids

The errors discussed in the following subsections can be traced back to
errors in the position of reference points (i.e., vertices of a grid, or
buoys). Lindsay and Stern (2003) denote this error type as geolocation
error. On a horizontal plane two coordinates (e.g.,

In a SAR image, the geolocation (position) error is caused by the
inaccuracies of the parameters describing the satellite orbit as a function
of space and time. In general, the error caused by these inaccuracies is
uniform across the image with only small local variations. Hence the
assumption of independent geolocation errors is not valid if distances
between moving objects are small. Holt et al. (1992) give a correlation
length of 10 km for the uncertainty of the geolocation error,

The deformation is calculated from components of the velocity gradients
according to Eq. (5). Hence, we have to consider the uncertainty in the
measurements of velocity components

If, on the other hand, both components of the vector

The uncertainty of an area measurement is needed for application of Eq. (5)
and equations presented in the following sections. The starting point for
calculating the variance of error for the measurement of an area is the surveyor's area formula valid for a polygon with an outline consisting of

Application of Eq. (12) to different geometrical figures: rectangle, equal-sided right triangle, rhombus, regular hexagon, triangle, and quadrangle.

For an assessment on how the polygon shape affects the magnitude of
uncertainty we require that the enclosed area remains constant. The areas of
a square with side length

The question arises of how large the smallest detectable area change is in a
SAR image. To address this question, we assume a square grid cell with its
vertices on the positions of adjacent displacement vectors and its sides
parallel to the

We consider a grid with displacement or drift velocity vectors on the
vertices. For calculating the deformation parameters, we need the velocity
gradients

For an array of buoys, we have to consider errors of the area, the buoy
velocity components

For the uncertainty in

Equations (20) and (21) together with Eqs. (15) and (16) provided above indicate that statistical uncertainties are not only influenced by geolocation and tracking errors but also depend on the shape and size of grid cells and buoy arrays. In the following discussion we consider magnitudes of geolocation and tracking errors reported in the literature and selected squares and triangles as examples for grid cells in SAR images (Lindsay, 2002; Bouillon and Rampal, 2015) and for splitting large buoy arrays into smaller units (Hutchings et al., 2012; Itkin et al., 2017). The effect of combining several cells is investigated. Finally, we focus on the range of validity of the equations derived in Sect. 2 and alternative methods of analysis.

The statistical uncertainties have to be related to the typical magnitudes
of the deformation parameters. According to Leppäranta (2011, p. 70) the
total deformation of drifting ice typically varies between around 90 % d

In general, the uncertainty of the deformation parameters depends on the
ratio

The accuracy of time readings for the acquisitions of satellite images is on
the order of subseconds. The product of sea ice drift velocity and
uncertainty of time reading appears on the right-hand side of Eq. (6):

Another issue that has to be considered is the time synchronization between individual buoys in an array. Differences of a few seconds may be possible in practice. In the following discussion we assume that position data of all buoys are exactly synchronized but also discuss an example for which this was not the case in Sect. 3.5.

Here we first focus on the retrieval of deformation parameters calculated
from square grid cells in SAR images or from square-shaped buoy arrays. For
SAR images, we consider the case in which geolocation errors may have slight
variations, hence

Uncertainty of divergence and vorticity for a square in a
spatially varying velocity field with gradients

When ice drift is retrieved from images of modern SAR systems, the
contribution of those terms that depend on

According to Sect. 3.1, a value of

Magnitudes of terms 1 to 4 in Eqs. (23) and (24).

Hollands and Dierking (2011) found tracking errors between 0.8 and 1.6 pixels (their Tables 3 and 4, standard deviations), which corresponded to 20–40 m for IM (pixel size 25 m) and 120–240 m for WSM (pixel size 150 m).
With

Lindsay and Stern (2003) calculated deformation parameters for the RGPS
initial velocity grid (

At first sight, larger time intervals and grid cells seem to be advantageous to keep the uncertainties of deformation parameters at a low level. However, larger time intervals may cause problems in the retrieval of the ice drift field, since ice structures, which serve as reference for the retrieval, may change or even vanish with time. Larger grid cells may smooth out local variations of deformation.

If the first and second term in Eqs. (22) and (23) can be neglected, i.e.,
when magnitudes of deformation parameters are low (which is most likely for
measurements over larger spatial scales and for weak deformation events), we
can determine the minimum grid cell size that is required to keep the
uncertainties of divergence and vorticity below a given threshold. If we
assume an uncertainty threshold of 1 % d

Also triangles are used for calculations of deformation parameters in SAR
images (e.g., Bouillon and Rampal, 2015; Griebel and Dierking, 2018), and they
form the smallest units of buoy arrays (e.g., Hutchings et al., 2011;
Hutchings et al., 2012). Using the same approach as for the square above, we
obtain the following for a triangle with its base

Uncertainty of divergence for a triangle in a spatially
varying velocity field with gradients

The uncertainty of divergence for the equal-sided triangle (

For buoys, the tracking error is zero. Itkin et al. (2017) quoted 25 m as
geolocation accuracy for stationary buoys but used 50 m to account for
effects of buoy drift. One of us (Hutchings) analyzed the position errors of
GPS receivers in the Fairbanks (Alaska) region. The errors were normally
distributed for position data collected at the same location for several
days. The relative position error between pairs of GPS receivers, which has
to be used for deformation calculations, was 2 m over distances of 1–10 km.
Reported time intervals between acquisitions of buoy positions range from 10 s to 3 h (Hutchings, 2012; Itkin et al., 2017) with uncertainties
in time less than milliseconds (see above). Hutchings at al. (2012),
however, mention also a time error of 30 s, which was due to the
acquisition times of the buoys not being exactly time coincident. In such an exceptional case, the second term on the right-hand side of Eq. (8) may have
to be considered. If the ice drifts in the

In this section we ask how large the area of a triangle-shaped buoy array has to be chosen to keep the uncertainty for deformation below a given
threshold. We assume that the temporal sampling error can be neglected. The
time interval

The first two terms of Eqs. (25) and (26) require the knowledge of the sea
ice velocity field and its gradients. We will here focus on cases for which
these terms can be neglected. This requires that

Using only the third term

Since the area and shape of the triangle change under the action of
continuous stress, the uncertainty does not simply decrease by a factor of

The combination of grid cells or several buoys is one possibility to lower
the uncertainty of the area

Because buoy arrays rarely reveal simple shapes such as squares or right triangles, the uncertainties in area have to be calculated numerically using Eqs. (11) or (12). Hutchings et al. (2012), e.g., used 22 buoys, which they split into arrays of approximately equilateral triangles but also into arrays of 6, 9, and 12 buoys. Here we discuss combinations of squares and triangles.

First we investigate the effect of splitting a square or a right triangle
into smaller units. We start with a square window covering

Derivation of equation 30 in the

For a right triangle, we have only two contributions from the corners
instead of four as for the square (Fig. 5). In the

In SAR applications, the question is whether it is preferable to use, e.g.,
the smallest possible (“elementary”) square cell (determined by the
resolution of the ice drift field) with four drift vectors at the edges or
to combine adjacent cells. Formally, the uncertainty in area for the
elementary cell is

For buoy arrays it may be of advantage to use a larger number of buoys along
the outline of a polygon. Here we study the example of an isosceles triangle
with two sides of equal length (Fig. 6), which, e.g., comes closest to the
array/subarrays used by Hutchings et al. (2012). The term

Application of Eq. (12) on a triangle with two equal
sides for

If the shape of an array with many buoys approximately approaches the shape
of a circle with radius

It has to be kept in mind that the fundamental Eqs. (1), (2), (4), and (5) that we used for estimating the statistical uncertainties in the retrieval of deformation parameters are based on simplifying assumptions. Hence it is necessary to consider their range of validity when applying them.

The right-hand side of Eq. (5) for estimating

Higher-order estimates for

Equation (5) provides an area-averaged estimate of

What temporal sampling is necessary to resolve changes in the sea ice
velocity field? The velocity may be decomposed into a mean field and a
fluctuating part (Thorndike, 1986). Rampal et al. (2009) showed that the
variance of the fluctuating part has two regimes separated by a timescale
of

We have assumed that different error sources are uncorrelated, and hence we
have ignored the second term on the right-hand side of Eq. (2). While it is
often true that spatial errors are uncorrelated with temporal errors, it may
not always be the case that spatial errors are uncorrelated with each other.
For example, for the distance

When calculating uncertainties of deformation parameters, it is implicitly assumed that the sea ice velocity does not have discontinuities within the polygon in which the deformation is being estimated. This is because we use Eq. (5), which is based on Green's theorem. Numerous observations of the sea ice velocity field show narrow shear zones or “linear kinematic features” (e.g., Kwok, 2003; Marsan et al., 2004; Kwok, 2006) across which the velocity jumps abruptly, as a result of stresses in the ice that create leads and ridges. Some researchers, e.g., Griebel and Dierking (2017) have proposed methods to detect and isolate these discontinuities in the velocity field to avoid smoothing effects when averaging adjacent velocity vectors (e.g., for replacing outliers).

When applying Eq. (5) over an area with a discontinuity in the velocity
field, a step-like function occurring between two positions

In Sect. 2, the area-averaged velocity derivatives in a region are obtained by estimating contour integrals of the velocity around the boundary of the region. Two alternatives to this boundary integral (BI) method are briefly discussed here: the least squares (LS) method and the finite difference (FD) method.

In the LS method, the velocity components

The FD method provides an estimate of

In summary, the BI method provides area-averaged estimates of

For a rectangular region with velocities given only at the four corners, it
turns out that all three methods give the same estimates of

The LS method can be used as a diagnostic tool to determine whether the spatial resolution of the velocity data adequately captures the variability of the velocity field. Analysis of the spatial pattern of the LS residuals (errors) by standard methods (autocorrelation) reveals whether the linear velocity model is in fact a good fit to the velocity data or not. If it is a good fit, then the spatial resolution is adequate and the truncation error in the BI method is small. If it is not a good fit and sufficient data are available, the region should be divided into smaller pieces and the calculation repeated for each piece. The BI method should be used to calculate the actual (area-averaged) velocity derivatives, since it does not depend on a model that needs to be checked for goodness of fit.

In this study we derived equations for calculating the uncertainty of different deformation parameters for a given area, using displacement vectors retrieved from SAR images or buoy arrays. In the most general case, presented in Sect. 2.5, errors in measurements of position (geolocation error), velocity (determined from displacement), and area size have to be considered. Uncertainties in velocity and area size can be related to uncertainties in position measurements and (for velocity) time readings (Sect. 2.2 and 2.3). When retrieving displacements from pairs of SAR images, a tracking error has to be considered additionally.

In Sect. 3, uncertainties of divergence and vorticity are derived based on
the general equations introduced in Sect. 2, assuming squares and triangles
as outlines for the area over which deformation is calculated. We chose
these geometric shapes since they have been frequently used in past and
recent studies of deformation in sea ice. The major findings are as follows.

The equations reveal that the uncertainties in divergence and vorticity
increase with the magnitudes of the velocity gradients and with the
geolocation and tracking errors. They decrease with increasing size of the
area and the time interval

Since geolocation errors in SAR images are usually correlated over scales of

Geolocation errors in imaging modes of modern SAR systems are smaller than their spatial resolution (see Sect. 3.4.1). Errors in time readings of buoy positions and SAR image acquisitions are negligible in most cases. For buoy arrays, the magnitude of the position error may not be negligible. Here, the reader is advised to check the manual for the position sensor and pay attention to whether the error is given as standard deviation or in another format.

The tracking error that needs to be considered for displacement fields
retrieved from SAR images is on the order of the length of 1 pixel, as
several studies have shown. If the geolocation error can be neglected relative
to the tracking error, a good approximation for the uncertainty of
divergence and vorticity valid for a square with side

For a given threshold of acceptable uncertainty we estimated the necessary size of rectangular grid cells in SAR images and triangular buoy arrays, focusing on divergence and vorticity as examples (Sect. 3.4.2 and 3.5.2.). At larger temporal sampling rates, the areas can be made smaller.

The area uncertainty of the smallest possible (elementary) cell,
determined by the position of three or four adjacent displacement vectors at
the edges of a triangle or square, is smaller than for a group of adjacent
elementary cells with more displacement vectors on the perimeter around the
group (Sect. 3.6). If, on the other hand, for an area of fixed size a
variable number

In Sect. 3.7 and 3.8 we provided thoughts concerning the validity of the derived equations, which assume that the velocity field inside elementary cells is continuous and can be approximated by a two-dimensional linear function. By including second-order terms or carrying out least-square fits over subregions of the velocity field, the validity of linearity can be judged. In the former case the second-order terms need to remain below a certain threshold; in the latter, the correlation coefficient should be large. Discontinuities in the velocity field should be detected before deformation is calculated to allow their impact to be assessed and to consider appropriate strategies to alleviate their impact.

No data sets were used in this article.

WD and HLS derived equations and discussed the validity of the approach; WD and JKH collected and evaluated typical ranges of measurement parameters; all authors developed the concept of the study and worked on the text.

The authors declare that they have no conflict of interest.

We thank one anonymous reviewer and Amélie Bouchat for comments that led to the improvement of the manuscript.

The work of Wolfgang Dierking was partly funded by CIRFA (Center for Integrated Remote Sensing and Forecasting for Arctic Operations; RCN research grant no. 237906). Harry L. Stern was funded by NASA grant NNX17AD27G and Jennifer K. Hutchings by National Science Foundation grant NSF 1722729. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Michel Tsamados and reviewed by Amelie Bouchat and one anonymous referee.