In sea-ice-covered areas, the sea ice floe size distribution (FSD) plays an important role in many processes affecting the coupled sea–ice–ocean–atmosphere system. Observations of the FSD are sparse – traditionally taken via a painstaking analysis of ice surface photography – and the seasonal and inter-annual evolution of floe size regionally and globally is largely unknown. Frequently, measured FSDs are assessed using a single number, the scaling exponent of the closest power-law fit to the observed floe size data, although in the absence of adequate datasets there have been limited tests of this “power-law hypothesis”. Here we derive and explain a mathematical technique for deriving statistics of the sea ice FSD from polar-orbiting altimeters, satellites with sub-daily return times to polar regions with high along-track resolutions. Applied to the CryoSat-2 radar altimetric record, covering the period from 2010 to 2018, and incorporating 11 million individual floe samples, we produce the first pan-Arctic climatology and seasonal cycle of sea ice floe size statistics. We then perform the first pan-Arctic test of the power-law hypothesis, finding limited support in the range of floe sizes typically analyzed in photographic observational studies. We compare the seasonal variability in observed floe size to fully coupled climate model simulations including a prognostic floe size and thickness distribution and coupled wave model, finding good agreement in regions where modeled ocean surface waves cause sea ice fracture.

Earth's polar oceans are covered with sea ice: a thin, heterogeneous interface that plays an important role in the coupling between ocean and atmosphere. Sea ice is a collection of many individual pieces, called floes, which may be characterized in terms of a horizontal length scale, their “size”. On the large scales relevant to global climate modeling, the statistical variability of floe size is described using the floe size distribution

The FSD is an important property of the sea ice cover that influences the multiscale temporal and geographic variability of sea ice, akin to the grain size in sedimentology or particle size distribution in atmospheric chemistry. The scale of individual floes plays a role in many sea-ice-related processes: sea ice melt rate

Despite the potential relevance of sea ice floe size to polar climate evolution, there remain no climate-scale assessments of average floe size or the FSD. The observational record of floe statistics derives from visual imagery localized in space and time

Here we outline a method that exploits satellite radar altimetry to construct the FSD and its moments across polar regions with sub-kilometer spatial resolution, sub-daily temporal resolution, and spanning multiple orders of magnitude in size. Altimeters, like the ones carried on the Envisat, ICESat, CryoSat-2, and ICESat-2 satellites, make repeated, frequent passes over polar oceans, and substantial efforts have been made to process the satellite returns to discriminate between open water, floes, and leads. The altimetric returns have found many uses, including reconstructing the sea ice thickness field

One-dimensional measurements of sea ice properties, like along-track altimetric measurements of ice open water, have long been sought to describe the two-dimensional ice surface.

We outline the mathematical theory that allows for comparison of altimetric datasets and the FSD in Sect.

For an individual pass over sea ice by a polar-orbiting satellite altimeter, return waveforms along the satellite orbit track are assigned a surface type depending on the waveform shape and coincident sea ice concentration

For a domain of horizontal area

Bayes' theorem relates

This second probability distribution

To proceed and arrive at a concrete (although not general) realization of these functions, we will assume all floes are perfect circles. In assessments of the relationship between major and minor axes of individual floes, the “roundness” parameter for a floe is typically within 15 % of 1

Relating a floe chord to floe size for a circular floe. A satellite track (dashed black line) passes over a floe of radius

Consider the special case that all floes are perfect circles, illustrated in Fig.

The

These derived quantities are useful because they require no further information about the sea ice (such as its concentration) to compare against modeled FSDs. However, both

Suppose the FSD

Moments of a power-law tail can be evaluated explicitly (for

Constructing a FCD from altimetry.

We apply the analytic technique described in Sect.

Figure

The full CryoSat-2 dataset examined here spans the time period from October 2010 to November 2018, and floe chords measured using the above technique are binned into the CICE sea ice model's two-dimensional sea ice grid for each month and year to facilitate comparison with model products. This implies that we invoke the principles of isotropy, homogeneity, and stationarity of the FCD, required to produce such a distribution, on the length scale of the CICE model grid (O(25 km)) and timescale of a month. For every grid cell

Figure

The geographic variability of representative radius over the “early winter” (October–December) and “late winter” (February–April) periods is shown in Fig.

Given a collection of chord lengths, we would like to examine whether it is distributed according to a power law. Under the assumptions of Sect.

We note that a power law describes the scaling of a distribution's tail. Previous observational studies have discussed “double power laws”

Examining the power-law hypothesis.

The MLE method is a rigorous test of the power-law hypothesis that eliminates potential human bias when interpreting observational data. To illustrate why this is important, we first consider the entire set of 11 million chord lengths recorded in the Arctic in all months (October–April), spanning a length range from 300 m to 100 km. The histogram of these floe chords is the black line in Fig.

Examining the tail of the distribution in Fig.

Finding no statistical basis for a power-law fit to the tail in Fig.

Sea ice parameterizations that assume a power-law distribution may significantly bias sea ice statistics. The imposition of any fixed distributional shape, when FSD dynamics are scale-variant, leads to implicit nonlocal redistribution of sea ice between floe size categories

Segmenting the chord length data into individual months in the Arctic, there are none where

Top row: Temporal variability of power-law fits to Arctic FCDs.

While most of the Arctic has at least 1000 total measurements across all years, FCD tails (

Scaling coefficients can provide useful information about the distributional shape. In Fig.

A principal aim of this work is to allow model–data comparisons and facilitate testing rapidly developing FSD–FSTD models. Here we demonstrate how such a comparison can be made and provide useful information to modelers, even in the presence of the high uncertainties in this nascent FSD reconstruction technique. With the gridded data provided above, we may now directly compare development-stage sea ice models that incorporate FSD effects to observations. To do so, we use the

Geographic and climatological comparison of modeled and observed representative radii.

This FSTD model simulation is coupled to a slab ocean model and the WAVEWATCH III ocean surface wave model

Figure

Figure

Figure

Here we developed and demonstrated a method for deriving the statistics of the sea ice FSD from satellite radar altimeter measurements of chord length. This method provides the first pan-Arctic accounting of climate-relevant quantities derived from the FSD, permits testing of existing scaling laws previously used to characterize distributions of floe size, and allows for gridded comparisons between FSD models and observations. Using this new technique we produced climatological, annual-average, and geographic mean moments of the Arctic FSD across a range of resolved length scales from 300 m to 100 km.

With the combination of satellite altimetry and mathematical theory, we were able to rigorously examine the power-law hypothesis related to the FSD under simple assumptions about the underlying floe chord data and the fidelity of CryoSat-2 satellite retrievals. Segmenting measurements by geographic location, by month, and by year, we find limited statistical basis for a power-law scaling beginning below about 6.5 km. In a limited number of geographic locations, we find the observational data cannot rule out power-law scaling, except for typical sizes above about 6.5 km. Assuming a power-law floe size distribution can bias sea ice model output and conceptual understanding, the geographic variability and lack of consistent multi-scale behavior reinforces the need for sea ice models to account for floe-scale processes rather than diagnose a distributional shape.

Observations that span the polar regions and different years and seasons are valuable for future refinement of process-based models of the FSD. In Sect.

We emphasize strongly that refinement may be necessary to apply this method for operational purposes, trend analysis, and further model validation. This paper has focused on the framework for making altimetric measurements of the FSD and comparison to model output, but the obtained chord lengths and distributions have not been carefully validated against other observational methods, and this will be necessary before further application of this method. Before doing so, we have tried to outline the most significant uncertainties in the method. The typical assumptions of homogeneity, isotropy, and stationarity are invoked here at the length scale of the CICE model grid (O(25) km on each side) and timescale of 1 month. These statistical assumptions may not be satisfied if, for example, the number of measurements in a given region in 1 month is insufficient to sample the known anisotropy of the sea ice floe field, and additional passes change the mean chord length significantly (see Sect. S2 and Fig. S1). While we found little evidence for power-law scaling throughout most areas of the Arctic, this may be sensitive to the geographic (here the CICE model grid of approximately 25 km

While processed CryoSat-2 data have been validated against both visual imagery and ground-based observations, they were not designed with this application in mind – additional quality control may be necessary for climate studies of changing floe properties. The positive comparison between model and observation in Sect.

Even accounting for important caveats that arise from making satellite measurements, remotely sensing the sea ice FSD from altimeters at sub-daily resolutions can provide a significant increase in data for comparison and analysis of new sea ice models that parameterize the FSD. Previously the difficulty of making measurements of the FSD at relevant spatial and temporal scales has inhibited the widespread adoption of such floe-sensitive sea ice models. Understanding sea ice variability at the floe scale is also an important aspect of sea ice forecasting, and the ability to remotely assess the sea ice FSD at near-real time will allow for further improvement of operational forecasting networks.

CPOM sea ice data, including raw floe length data, are available through the CPOM data portal at

For generic probability distributions

Supposing

The real altimetric data product has a finite sampling resolution

We can compute the error function for any delta function distribution as

We note that increasing resolution of floe chords will result in tighter bounds on this error. When

We can explicitly solve Eq. (

Given a set of floe chords

The above analysis concerns the most likely

Due to limitations in the number of floe chords recorded at any particular location over time, we do not include all geographic locations when computing hemispheric means. Averaging is performed by including only geographic regions where there are at least 25 recorded floe chords. The area being averaged over is thus not fixed in time. For seasonal cycle plots, we only include months which have enough measurements for all fully sampled CryoSat-2 years (2011–2018). For annual averages, we include only those years where all CryoSat-2 months (excluding June–September) have enough measurements.

When masking additional regions to perform the model–observation comparisons in Fig.

The supplement related to this article is available online at:

CH derived the mathematical theory and wrote the paper. LR built and performed the climate model simulation. RT, AR, and AS provided and interpreted the CryoSat-2 data. KH, CG, CB, and BK contributed to the study design. All authors have participated in paper preparation.

The authors declare no competing interests.

CH was supported by the NOAA Climate and Global Change Postdoctoral Fellowship Program, administered by UCAR's Cooperative Programs for the Advancement of Earth System Science (CPAESS), sponsored in part through cooperative agreement number NA16NWS4620043, years 2017–2021, with the National Oceanic and Atmospheric Administration (NOAA) and the U.S. Department of Commerce (DOC). CH, CG, and KH thank the American Mathematical Society for their support through the Mathematics Research Community “Differential Equations, Probability, and Sea Ice”, funded by NSF grants 1321794 and 1641020. LR was funded via Marsden contract VUW‐1408 and the New Zealand Deep South National Science Challenge, MBIE contract number C01X1445. CMB was supported by the National Science Foundation grant PLR-1643431. BFK was supported by ONR grant N00014-17-1-2963 and NSF grant 1350795. RT, AR, and AS were supported by the UK NERC Centre for Polar Observation and Modelling and the European Space Agency.

This research has been supported by the National Oceanic and Atmospheric Administration, Climate Program Office (grant no. NA16NWS4620043), the National Science Foundation, Division of Mathematical Sciences (grant nos. 1321794, 1641020, 1350795, and 1643431), the Office of Naval Research (grant no. N00014-17-1-2963), the Marsden Fund (grant no. VUW‐1408), and the New Zealand Deep South National Challenge (grant no. C01X1445).

This paper was edited by Jennifer Hutchings and reviewed by Thomas Armitage and one anonymous referee.