SAR satellites are valuable for observing the Arctic sea ice because they offer high resolution, day-and-night coverage, and penetrate clouds. Hence, their images are a major source for ice charting. The main disadvantage of SAR images, however, is the ambiguity of the acquired backscatters signal, requiring experienced analysts to identify different ice types and open water , often at resolutions much coarser than the image resolution. Automated ice classification of SAR images has been an active field of research for the past two decades . Modern machine-learning based classifications aim to distinguish several ice classes based on ice age, thickness, and degree of deformation, as well as distinguishing various lead and open water types. Many different methods have been proposed e.g., but to date no comprehensive comparison and evaluation has been presented and no standard method has been established. Since our task at hand is simpler and limited to distinguish only the two classes of ice and open water, we decide to primarily use the backscatter signal to separate them. Depending on the incidence angle (the angle between the radar beam and the surface normal), open-water can appear brighter or darker than ice in SAR images . In our study period and region, lead fractions are consistently well below 5 % in multiple satellite products . Therefore, the SAR images in our analysis are mostly covered by ice, with only limited open water in leads. Thus, the mode of backscatter in a subdomain of the image (where variability in the incidence angle is negligible, e.g. in a 50×50 km² ROI) can be regarded as the typical backscatter of an ice surface for the given incidence angle at that position. Similarly, the standard deviation of the backscatter in this subdomain represents the spread in ice backscatter values. When the backscatter deviates from the average ice value of the subdomain by multiple orders of magnitude, it is unlikely that it originates from an ice surface and we classify the pixel as a lead, i.e. ocean. For this classification, the standard deviation and modal backscatter of the area surrounding each pixel are required, which we obtain by computing both in a window moving over the entire image. With the moving window, we also account for the varying backscatter signal of different ice types found in the image. In the study period, open water is only formed by deformation, as significant melting does not occur due to the cold temperatures. Therefore, we dynamically lower the thresholds in regions of divergence and shear (shear must be high to have an appreciable effect) motion. Finally, we manually tune the thresholds for each image individually to optimizing the open water classification visually optimized parameters are provided in the associated code repository.

In the following, we describe in detail the individual processing steps, starting with the original SAR image and ending with the ice-ocean classification (generating the so-called ice-ocean mask), as illustrated in Fig. . First, we use a bilateral filter to remove the speckle noise in the backscatter, similar to , and corrected for the varying incidence angles over the image (Fig. b). Next, we compute the mode I^(x,y) and standard deviation σI(x,y) of the backscatter signal over a moving window of size (50×50 km² or 1000×1000 pixels, respectively), which we henceforth refer to as running mode or running standard deviation. To reduce computation costs, we do not shift the moving window pixel by pixel to compute a true running mode, but shift the window by 500 pixel each step to compute the mode and standard deviations at anchor points resulting in half overlapping 1000×1000 pixel boxes. To obtain local backscatter mode and standard deviation at all pixels of the SAR image, we bilinearly interpolate these values from the anchor points (Fig. c, d). The region size was set to be large enough to average out the effect of lead systems on the standard deviation. Also, the divergence ϵ˙I=∂u∂x+∂v∂y and shear rates ϵ˙II=∂u∂x-∂v∂y2+∂u∂y+∂v∂x2 are computed from the drift product and interpolated bilinearly to each SAR pixel (Fig. e, f).

Also, the divergence ϵ˙I and shear rates ϵ˙II of the drift product are interpolated bilinearly to each SAR pixel (Fig. e, f). From the interpolated deformation rates we compute a scaling factor s(x,y) to reduce the threshold as follows, 1sI(x,y)=max⁡0.5,min⁡(ϵ˙I(x,y)-0.5d-1)/0.45d-1,1,2sII(x,y)=max⁡0.5,min⁡(ϵ˙II(x,y)-1.0d-1)/0.5d-1,1,3s(x,y)=max⁡0.5,min⁡(sI(x,y)×sII(x,y),1), which are the independent scaling factors for divergence and shear and their product is a joint scaling factor. Note that the scaling coefficient is capped by 0.5 and 1 for divergence and 0.75 and 1 for shear, such that the threshold is at most halved by the dynamic thresholding and does not automatically yield to an open-water classification of deforming pixels. The spatial variation of the resulting scaling factor is shown in Fig. g. In the next step, we compute the local backscatter anomaly, which describes the difference of the backscatter pixel value to the local backscatter mode in relation to the scaled local standard deviation, 4Ia(x,y)=I(x,y)-I^(x,y)s(x,y)σI(x,y). We identify pixels as open water if the backscatter anomaly is below a lower threshold Ia,min⁡ or exceeds an upper threshold Ia,max⁡. These thresholds were manually adjusted for each image to optimize the ice-ocean mask with the exact values documented in . As part of this manual adjustment also the deformation rate thresholds and scaling magnitudes in Eqs. ()–() were tuned, where the same thresholds were used for all images. In a last step, we apply a linear filter to the generated ice-ocean mask to damp noise at the pixel level , assuming that open water should be mainly found along leads in long and narrow strips rather than isolated openings in the ice cover. We apply a rotating linear kernel of length 7 at each pixel and determine the orientation of the lead as the kernel orientation with the lowest standard deviation. Along this orientation the ice ocean mask is averaged with the same linear kernel. We only classify pixels as open water if the filtered value is higher than 0.75 (Fig. i). The filter maintains open water points connected in lines and reduces the noise at the grid-scale speckle. The performance of this ice-ocean classification and potential misclassification of bright pressure ridges or frost flowers will be assessed and discussed in Sects.  and .

To date, satellite or airborne imagery in the visible spectrum are the major source of floe size and shape data e.g.. Therefore, these studies were limited by natural illumination and mostly represent broken-up summer sea ice conditions with lower ice concentrations . In these conditions, bright ice and dark ocean can easily be classified and subsequently different methods are used to delineate individual floes in the ice cover. While floes were first identified manually , recent methods are almost fully automated . presented a semi-automated approach to identify floes in SAR images. Despite the availability of SAR images during the Polar night, their algorithm has been applied to images of summer sea ice only. In consolidated ice conditions, as found in wintertime pack ice, floes can be in partial or complete contact with other floes and therefore are not necessarily surrounded by open water, which challenges the segmentation step of these algorithms, leaving us with no floe data from visible or SAR imagery for winter time. and have used satellite altimetry to measure the length of ice segments, so-called floe chords, along the satellite track to study winter time floes. Floe chords are one proxy for floe sizes, in addition to floe area, perimeter, or mean caliper radius . The main advantage of using floe chords is that they can be measured directly on the ice-ocean mask without identifying floes directly in an image. The FCD and FSD are related by statistics if the floe geometry is known, which we discuss in the following section. To cope with the consolidated ice conditions in our SAR images, we will use a floe chord proxy. We present a new approach of measuring the FCD and perform a statistical conversion to compute the FSD that is described in detail in the following. The development of this statistical conversion includes an in-depth study of floe geometries.

The heart of our analysis is bringing together the drift derived deformation of ROIs and the observed changes in the FSD. This procedure is illustrated in Fig. . We define a 50×50 km² (1000 × 1000 pixels) large area of ice, our ROI, that we follow over two consecutive days (Fig. a). We advect the outline of the ROI in the first satellite image with the drift information to obtain the outline of the ROI in the second image (Fig. b). Using a line integral approximation along the boundaries of the ROI we compute the shear and divergence rates from the displacements of the boundary, which describes the deformation of the ROI . In parallel, we classify ice and open water in both images from the backscatter and deformation information (Fig. a, b) and measure the FCD of the ROI for both days (Fig. e). We compute the FSD for both days from the FCD with the inverse method presented above. Finally, we compute the change in FSD over both days and relate it to the deformation (Fig. f).

In our example highlighted in Fig. , we observe major fracturing of the ice cover between the two days with the opening of four lead systems (Fig. c, d). These openings are well captured by our open water classification, where the additional information from the backscatter signal allows us to clearly define the outline of the leads (Fig. b). As expected over the fracture event, we find that the number of long floe chords is reduced and shorter floe chords is increased. The transition between regimes is located around a floe chord of ∼ 20 km. Converting this trend into floe size, we find that during the fracture event the probabilities of floes larger than ∼ 30 km is reduced while smaller floes become more prevalent. We note here that this increase in number density gets more pronounced the smaller the floe size. In the following we will apply now the presented approach to entire data set to explore an general link between the deformation caused by fracture and the floe size distribution.

The aim of this section is to test the robustness of our presented approach by evaluating the outcomes against independent observations where possible and perform physical consistency checks. As the uncertainties of the drift and deformation rates have been assessed in , we focus here on comparing the FCDs retrieved from SAR to independent estimates from ICESat-2 data and on testing how the opening recorded in the change in open-water mask relates to the divergence data derived from the drift product.

For the comparison of FCDs, we collect all available ICESat-2 data overlapping with the SAR images and use the SAR-derived drift information to correct for displacements caused by the different acquisitions times of Sentinel-1 and ICESat-2. Note that due to the 1 d temporal resolution of the drift data, this correction assumes homogeneous drift over 24 h and especially for strong drift regimes does not guarantee a reliable co-location at the pixel level. Therefore, we refrain from a pixel-based comparison of the SAR-derived open water mask with lead detections in ICESat-2 and focus on comparing the FCD for entire ROIs instead. For each ROI, where ICESat-2 data is available, we detect leads following a methodology akin to that of . Floe chords are measured as the distance between leads, and we compute the FCD for the ROI. Next, we measure floe chords in the ROI based on the SAR ice-ocean mask using the method outlined above. Rarely, do we have more than one overpass from ICESat-2 over an ROI, while for SAR images we sample floe chords along 30 000 lines. To compensate for the effect of differing sample sizes on the chord distribution, we randomly subsample the floe chords measured in the SAR images to equal the number of samples from ICESat-2 for the ROI. To get an estimation of the uncertainty originating from the limited sample size, we repeat this random subsampling of SAR chords 50 times. Figure a shows that the distributions from the two satellites agree well within the error margins of one standard deviation, with the strongest differences for floes larger than 20 km. The under representation of large floes in ICESat-2 might stem from the far smaller sample size of ICESat-2 chords. Both distributions show a power-law behavior as previously reported with well aligned power-law exponents. Their strong agreement provides us with confidence that the presented workflow is well suited to measure floe chords from SAR images for wintertime sea ice.

In the second part of our method evaluation, we aim to quantify how well the change in ice concentration, which we determine from the ice-ocean masks, fits the divergence rate of the ROI. As we assume thermodynamic ice growth is negligible over 1 d, divergence should increase the open water and reduce ice concentration in the ice-ocean mask. In Fig. b both quantities are compared in a scatter plot. While we observe a low precision and wide spread between the variables, they are clearly anti correlated as assumed and show a reasonable linear relationship close to the one-to-one line. We attribute the wide spread mostly to the uncertainty in the SAR-derived open water mask and to a lesser extent to the data quality (and lower resolution) of the drift product. Part of this spread could also originate from more complex ROI deformations including rafting, compensating concurrent small-scale ridging, and fracturing or shear motion. This comparison shows that we can link these two SAR derived data sources of deformation and ice-ocean masks, especially as we will mainly analyze strongly divergent drift regimes ϵ˙I>0.01 d⁻¹.

All ROIs in our analysis are located in consolidated pack ice with a mean ice concentration of 98 %. A summary of ice properties for all ROIs is provided in Fig. . We find that the ice concentration derived from our ice classification of SAR images agrees with AMSR2 ice concentration data ftp://ftp.awi.de/sea_ice/product/amsr2, last access: 9 October 2026 well within the uncertainty range of 5 % for this dataset and both SAR derived and AMSR2 concentration have the same mean. We determine the ice freeboard of the ROIs from co-located ICESat-2 measurements. The median freeboard is 36 cm while the mean freeboard is 41 cm. The differences between median and mean freeboard in the ROIs indicates a heavy tailed freeboard distribution typical of pack ice shaped by deformation. In general, the ice conditions derived from our SAR-based ice classification and deformation computations are representative and agree well with other measurements for the region of the MOSAiC drift . For both ice concentration and freeboard we observe a slight reduction between the consecutive SAR images matching the mean divergent ice motion over all ROIs.

We find on average a divergent ice regime (Fig. c) as the mean divergence is positive, while the majority of ROIs are undergoing convergent ice motion (negative median divergence). This highlights the presence of strong divergence events related to fracturing and lead opening accompanied by periods and areas of weaker convergent ice drift that closes leads. The observed distribution of shear rates (Fig. d) shows a typical heavy tailed distribution.

The FCDs and FSDs averaged over all ROIs can be approximated by power-law distributions (Fig. e, f). For small chords and sizes (< 250 m) the distributions show effects of the discrete character of measured chords originating from the pixel resolution of the SAR images. For chord lengths approaching the size of ROI, deviations from the power-law distribution are caused by the finite size effect. Using simple linear fits to the double logarithmic data, we estimate exponents around 0.7 for the FCD and 1.5 for the FSD. We acknowledge that we do not properly test if the underlying distribution is described by a power-law – a discussion that is beyond the point of this side analysis – but presenting the exponents allows us to put our findings in context of other studies. Our power-law exponent of 1.5 for the FSD is at the lower end of the spectra but well within the distribution of power-law exponents found in literature . A seasonal dependence of the FSD slope is well documented, with steeper exponents in fragmented summer conditions and shallower exponents in consolidated winter pack ice . Our shallow exponent is therefore consistent with the winter conditions analyzed here, in contrast to the summer conditions studied in most previous studies.

The main focus of our study is to analyze the change in the FCD and FSD in relation to fracture by deformation. In Fig.  these changes are shown as differences in the number density of floe chords and sizes for all ROIs. In the following, we will also use the term chord or size redistribution referring to the change in the number density between consecutive days. We find a wide spread in the observed redistribution functions, which we attribute to the chaotic behavior of ice fracture. The exact number and positions of failure lines within a deforming ROI strongly depends on the presence of defects and existing failure lines in the ice cover, which is described by the distribution of floe sizes and their specific geometry and arrangement. On top of this chaotic spread driven by local ice conditions, general patterns emerge when grouping by the mode of deformation. When averaging over all strong divergent (ϵ˙I>0.01 d⁻¹) ROIs in our analysis, we measure a reduction in large floe chords (c≳20 km) and an increase in medium and small floe chords (c≲20 km). Converted into floe sizes, this equates to a decrease in floes of sizes larger than ∼ 33 km and an increase in floes smaller than ∼ 33 km, below we refer to the zero of the redistribution function as the unchanged floe parameter. Notably, the increase in small floes follows a power-law distribution, highlighting the importance of ice fracture in creating the power-law distribution of floe-sizes. These are the first direct observations showing power-law characteristics of fracture related redistribution of floe sizes. With this we provide observational grounding to the previous hypothesis that fracture processes are a driving factor in the power-law shape of the FSD . For convergent ice motion, we find significantly lower changes in the number density of floe chords. The observed redistribution becomes clearer when converted to floe sizes. The amount of large floes (d≳10 km) increases alongside a reduction of medium size floes (300 m ≲ d ≲ 10 km) following the general intuition that convergent ice motion facilitates coagulation of smaller floes to larger ones. This process also takes place at lower convergence rates, where in addition to medium size floes also small floes (d≲300 m) contribute to coagulation. The strong convergent states, which are averaged in Fig. , however, show an additional increase in small floes (d≲300 m), a clear sign of ice fracturing in compressive ice states accompanied by strong shear motion taking place simultaneously with a closing of the ice cover described above.

In the following section we will study the average floe size redistribution as a function of the deformation rate with the aim to derive a functional form that can be used as a parameterization in numerical models for the evolution of floe sizes. As discussed in the previous section, different initial FSD and ice conditions drive a chaotic non-linear behavior resulting in a large spread of the observed floe size redistributions. Given that a very large data sample would be required to describe these dependencies directly, we consider the floe size redistribution as a boundary value problem dependent only on the deformation of the ROI but independent of the initial ice conditions. Hence, we average redistribtions of multiple ROIs binned by similar divergence rates. Given the floe definition of the numerical model and the fact that we are only able to observe floes surrounded by open water in the satellite data, we limit this analysis to ROIs with divergent ice motion.

The divergence bins are defined by sorting the ROIs by divergence rate and eliminating ROIs with divergence rates ϵ˙I<0.005 d⁻¹. Next, we create a bin centered at every second divergence value in our data ϵ˙I,ci and randomly draw up to 100 ROIs with similar divergence rates (|log⁡10(ϵ˙I,ci)-log⁡10(ϵ˙I,j)|≤0.2). The floe size redistributions of all ROIs within each bin are averaged and shown as a function of divergence rate in Fig. a. If there are less than 100 ROIs within a certain divergence bin, all ROIs within the divergence bin get averaged. In the next step, we look for a functional form that describes the variation of these averaged redistributions as a function of divergence rate.

Based on the general form of redistribution for divergent ice motion discussed in the previous Sect. , we design a function that consists of the product of two major parts: (1) a power-law function describing the observed scaling for newly fractured floes and (2) a hyperbolic tangent that shifts the power-law from increasing the amount of small floes to decreasing the amount of large floes. The change in the floe size number density ΔPd,ϵ˙I is therefore given by, 10ΔPd,ϵ˙I=a(ϵ˙I)de(ϵ˙I)tanh⁡b(ϵ˙I)d0(ϵ˙I)-d, where a(ϵ˙I) and e(ϵ˙I) are the amplitude and exponent of the power-law function respectively, b(ϵ˙I) is the transition width of the tangens hyperbolicus, and d0(ϵ˙I) is the intercept with the x-axis, which describes the unchanged floe size that is neither formed nor removed by the fracture event. All these parameters are assumed to be a function of the divergence rate.

We fit Eq. () to the average redistribution of each bin to determine its parameters. The functional form Eq. () has four parameters a(ϵ˙I), e(ϵ˙I), b(ϵ˙I), and d0(ϵ˙I), which need to determined by the fit. However, there is one additional constraint originating from the conservation of ice area, which reduces the amount of fitting parameters to three. As we only consider divergent ROIs, the ice cover only fractures and spreads out, while the overall area of ice within the ROI remains constant during the fracture event. Therefore the area gained by an increased amount of small floes need to be balanced by the ice area removed by decreasing the amount of large floes. This ice area conservation can be written as the following integral of the product of the observed FSD change with the floe area equating to zero, 11∫0∞ΔPd,ϵ˙I|a,e,b,d0πd24dd=!0. While we fit a, e, and b with a least squares fit, we use Eq. () to compute the unchanged floe size d0 with the fitted parameters. Given the non-linearity of the integral, we cannot reformulate Eq. () into an analytical solution for d0, but rather minimize the integral numerically within the fitting procedure. The fitted redistributions are given in Fig. b, showing good agreement with the observed redistributions. The fitted parameters as a function of divergence rate are provided in Fig. c, d, e, and f. To assess the uncertainty in the fitted parameters we repeat the random bin creation and fitting procedure 10 times. The resulting distribution of fitting parameters for each divergence rate bin is shown in Fig. c, d, e, and f. We find a large spread in the fitting parameter for small divergence rates ϵ˙I<0.01 d⁻¹ hindering the derivation of a general form in this range. Multiple reasons could be responsible for this finding: (1) at these low divergence rate the initial conditions (FSD, concentration) are more important than the strength of the fracturing itself, and (2) the motion associated with low divergence rates could be realized with relative motion between existing floes without major fracturing occurring. For higher divergence rates ϵ˙I>0.01 d⁻¹ we find a consistent evolution of all fitting parameters with increasing divergence rate indicating that fracturing of the ice cover is required to accommodate the ice motion in these strong divergence events, which leaves a distinct imprint on the observed redistributions. The dependence of the fitting parameters on the divergence rate can be approximated by linear functions, 12a(ϵ˙I)=a1ϵ˙I+a0,13e(ϵ˙I)=e1ϵ˙I+e0,14b(ϵ˙I)=b1ϵ˙I+b0. The amplitude a of the observed redistributions increases clearly with the divergence rate, while the power-law exponent e decreases with increasing divergence rate leading to a steepening of the power-law. The transition width b remains roughly constant with a slight decrease with increasing divergence rate. The unchanged floe size d0 determined to fulfill Eq. () also decreases linearly with increasing divergence rate mainly driven by the decrease of the power-law exponent e. The slope and intercept of these linear relationships (Eq. –) along with their uncertainty are summarized in Table .

In the process of implementing the parameterization given by Eq. () in a numerical model, a number of considerations are required to properly account for differences of spatial and temporal scales between the presented analysis and the model configuration used. First, the model could use a different choice of bins to discretize the floe size distribution. To account for this difference, the floe size redistributions (Fig. a and Eq. ) are normalized by bin width. The integral of the floe size redistribution ΔP over a specific floe size range d∈[di,di+1] represents the change in the number density for that specific floe size bin. This change then is representative of the typical ROI size (50 km × 50 km = 2500 km²) and a time step of 1 d. For different horizontal grid cell areas and time steps in the model, the computed redistribution of the number FSD needs to be scaled accordingly. Besides the number FSD also the divergence rate is scale dependent, which also needs to be considered if grid cell area and time step do not align with the ROI values. Known power-law scaling relationships both in the temporal and spatial domain can be used to adapt the divergence rate input in Eqs. ()–() accordingly. The floe size redistribution function formulated for a model with a spatial resolution of Δxm and a time step of Δtm for a floe size bin [dm,i,dm,i+1] is given by, 15ΔPmdm,i,dm,i+1,ϵ˙I,m=Δxm50km2Δtm1d∫dm,idm,i+1ΔPd,50kmΔxm-β1dΔtm-αϵ˙I,mdd, where α and β are the temporal and spatial scaling exponent respectively e.g. α=0.55 and β=0.49;, and the subscript m marks model quantities. The scaled divergence rate within the integral should be used to compute the amplitude, exponent, and transition width. The unchanged floe size needs to be determined following the conservation of ice area Eq. (). In this computation, it makes sense to account for floe size bins, where the computed redistribution exceeds the number of floes present in the bin.