The total Antarctic sea ice extent (SIE) experiences a distinct annual cycle, peaking in September and reaching its minimum in February. In this paper we propose a mathematical and statistical decomposition of this temporal variation in SIE. Each component is interpretable and, when combined, gives a complete picture of the variation in the sea ice. We consider timescales varying from the instantaneous and not previously defined to the multi-decadal curvilinear trend, the longest. Because our representation is daily, these timescales of variability give precise information about the timing and rates of advance and retreat of the ice and may be used to diagnose physical contributors to variability in the sea ice. We define a number of annual cycles each capturing different components of variation, especially the yearly amplitude and phase that are major contributors to SIE variation. Using daily sea ice concentration data, we show that our proposed invariant annual cycle explains 29 % more of the variation in daily SIE than the traditional method. The proposed annual cycle that incorporates amplitude and phase variation explains 77 % more variation than the traditional method. The variation in phase explains more of the variability in SIE than the amplitude. Using our methodology, we show that the anomalous decay of sea ice in 2016 was associated largely with a change of phase rather than amplitude. We show that the long term trend in Antarctic sea ice extent is strongly curvilinear and the reported positive linear trend is small and dependent strongly on a positive trend that began around 2011 and continued until 2016.

Much of the research on Antarctic sea ice variability focuses on the monthly, seasonal and interannual timescales

The dominant or primary characteristic of Antarctic sea ice variability is its annual cycle. Satellite-observed total Antarctic sea ice extent (SIE)
experiences a distinct annual cycle, peaking in September (19 million

The daily, annual cycle of SIE is traditionally calculated by simply taking the average (or the median value) for each day of the year. However,
satellite-observed SIE can vary widely from day to day. Some of this variation is due to the ice growth, melting, and divergence of the ice at the ice
edge and land spillover (coastal effect of mixed land/water grid cells), while some is due, for example, to transient effects of cloud and melt on
the ice surface

Our overarching aim in this research is not only to redefine the annual cycle but also to make a meaningful decomposition of the variation in the annual cycle of Antarctic SIE. We do so on the time dimension in such a way that each component can be interpreted individually, and when taken together all of the components give a complete picture of the variation in the sea ice. We consider the variation from the shortest timescale (instantaneous variation) and increasing the timescale sequentially we move through the day-to-day variation, the year-to-year (interannual) variation, and finally the longest timescale, the curvilinear trends of the multi-decadal variation. In the process, we make a number of technical contributions, most importantly to define complementary types of annual cycles that are meaningful in terms of this decomposition and also to the representation of volatility. We have deliberately chosen (time) dimensions based on their interpretability rather than solely statistical efficiency concerns. For example, the amplitude and phase components of the decomposition are much more interpretable than simple spectral components.

We begin by presenting a stochastic model for the sea ice extent that allows the annual cycle to be defined in flexible ways. This model can represent the real variability in SIE and reduces the contribution from the ephemeral effects described above. The model can account for the fact that the ice maximum is not achieved on the same day of the ice cycle each year. It also recognizes that the length of the ice cycle will vary and that the timing of advance and retreat of the ice varies from year to year. This means that the annual cycle is not constrained to a fixed cyclical pattern, rather it is a pattern that allows both temporal dilation and contraction as well as amplitude modulation.

To show the utility of the model, we develop several different annual cycles, including one that is invariant, one that is adjusted for phase only and
one that is adjusted for amplitude only. From the modeled annual cycles we define and extract the variability at the timescales mentioned. We conclude
with a decomposition of the variability of SIE during 2016, the year of anomalous decay of SIE. The data are described in Sect.

We used the Bootstrap Version 3 concentration fields

Figure

Recorded sea ice extent (SIE) (grey) for each year compared to a smooth annual cycle (red) over a 365 d period. The horizontal axis is the day of the cycle and the vertical axis is sea ice extent in millions of square kilometers.

In this section we give five ways to define an annual cycle in the sea ice extent. We start with the traditional definition of the annual cycle and progressively define annual cycles that are more sophisticated and that can represent more of the variation in the SIE over time. The second is an invariant annual cycle that retains the 365 d period of the traditional but incorporates the smooth functional form we might expect. The third adds amplitude variation to the invariant annual cycle so that the cycle itself varies from year to year with the amplitude of the year. The fourth adds phase variation to the invariant annual cycle, allowing it to capture the timing of the ice advance and retreat over each year. Finally, the fifth adds both amplitude and phase variation to the invariant annual cycle, allowing it to represent variation over time in both the amplitude and phase of the SIE.

Our decomposition of the sea ice extent starts with the traditional representation based on the annual cycle is as follows:

Within this representation, the annual cycle is traditionally estimated by

This traditional estimate (

It is possible that smoothing the data could be a solution to the statistical issues that arise from the way in which the traditional annual cycle is
calculated. To address this we define an invariant annual cycle,

The invariant annual cycle has the same motivation as the traditional annual cycle while being a clear statistical and conceptual improvement over the
traditional cycle. However, we argue that since it is also fixed by day of year, it may be too restrictive since it, like the traditional cycle, disguises the
contributions of both amplitude and phase to the annual cycle. To address this we define a complementary annual cycle that is deformed each year in
two ways. The first is amplitude in the sense that the yearly maximum and minimum extents may vary, but the shape of the daily extent
may be invariant. We enable the annual cycle to vary from year-to-year as a parameterized function of the annual cycle shape function. Specifically, we
define the amplitude-adjusted annual cycle,

Comparison of annual cycle estimates:

This annual cycle gives a different decomposition of the extent to the invariant annual cycle as it captures variation due to amplitude
variation. Specifically, adjusting for amplitude results in a 55.2 % improvement in the MSE compared to the traditional cycle (see
Table

Another component of the annual cycle that is important is the phase. This is the timing of the maximum and minimum extents. It is important because
it determines the length of the annual cycle and influences its shape. We enable the annual cycle to vary from year-to-year as a parameterized function
of the phase of the annual cycle shape function, defining the phase-adjusted annual cycle,

Here

The phase-adjusted annual cycle gives a different decomposition of the extent to the invariant annual cycle as it captures variation due to phase
variation. It allocates the component of the variation in extent due to phase variation to the annual cycle rather than the residual
term,

Surprisingly, the adjustment for phase shows even more improvement (63.9 %) in the MSE than that for the amplitude-adjusted annual cycle, indicating that the phase contributes more to the variability of the annual cycle of SIE than the amplitude. Most studies of Antarctic sea ice variability focus on the amplitude at maximum and minimum extents, but this analysis indicates that the phase (the timing of these extrema) is at least as important a contributor to the variability.

Finally, we can combine the amplitude and phase adjustment ideas to define an annual cycle that jointly adjusts for both. We define the
amplitude- and phase-adjusted annual cycle (APAC),

Figure

Comparison of the various proposed annual cycles in terms of how well they explain the variation in daily SIE. Values are given as percentages of mean-squared error and the root-mean-squared error (RMSE).

The discussion above describes several different ways of defining the annual cycle of SIE. While an annual cycle adjusted for phase or amplitude only would not be the best estimate for the data, differences between them and the optimal estimated annual cycle (i.e., APAC) could reveal sources of variability in the daily SIE.

Estimating the annual cycle using our model allows us to calculate statistics that reveal the underlying variability in the daily SIE. Below we decompose the sea ice variation on the time dimension, moving up the temporal scale from the very short term (the instantaneous variation) to the day-to-day variation, followed by the interannual variation and finally the multi-decadal variation, i.e., the trend.

The recorded sea ice extent will deviate from the true sea ice extent. This may be due to some combination of weather, artifacts of the satellite
algorithm used for retrieval, and the electromagnetic spectrum across which the device or satellite is measuring, among other things. To represent this,
we write the recorded SIE,

Here we introduce the term volatility to describe the instantaneous variation (or precision) in the recorded SIE as an approximation for the extent. Such variation may be due to ephemeral effects like those mentioned above.

Volatility of the recorded SIE for the NIMBUS-7 era (26 October 1978 to 20 August 1987) and the DMSP era (21 August 1987 to 2018). It is averaged over each day of the cycle in these eras. The units are given in millions of square kilometers. The purple curve is the day-to-day change in SIE from Fig.

Normally the standard deviation of the residual,

To model the volatility, we specify a generalized autoregressive conditional heteroskedasticity (GARCH) model

Figure

It is useful to know the daily rate of change of SIE because it gives insight into the daily timing of growth (advance) and melt (retreat) of the sea
ice. It is also an expression of the phase of the annual cycle. Contemporary trends in Antarctic sea ice are shown to be linked to the changes in the
timing (phase) of advance and retreat

The SIE minimum (day 0, Julian day 46) is coincident with the minimum growth rate. The ice advances, reaching the maximum growth rate by day 81 and
maintaining this maximum growth rate for approximately 40 d before slowing to a minimum growth rate by day 225 (late September) of the cycle. Sea
ice retreat begins at approximately day 225 and occurs quite rapidly compared to the advance, reaching a maximum rate at day 308 (late December) before
slowing to a stop at day 365 (Julian day 46 or mid-February). The rates of advance and retreat of the ice are not constant over the annual cycle. The
maximum rate of retreat of the ice is more than twice the maximum rate of advance. Figure

Day-to-day change in the annual cycle of sea ice extent for the traditional (orange) and invariant (black) annual cycles. The horizontal axis is the day of the cycle, and the vertical axis is change in sea ice extent in millions of square kilometers.

Taken together, the daily rate of change and the volatility (Figs.

The detection and analysis of anomalies (deviations from the annual cycle) is essential to the understanding of contributors to variability. Here we
discuss three different but related types of anomalies. First there is the true anomaly, represented by

Comparison of anomalies from three annual cycle estimates for 2014–2018: the raw anomaly from the traditional annual cycle (black), the estimated anomaly from the invariant annual cycle (blue), and the estimated anomalies from the amplitude- and phase-adjusted annual cycle (red). The vertical axis is the anomaly in millions of square kilometers.

We estimate the true anomaly by using Eq. (

Figure

The trends in SIE for both the Arctic and Antarctic have been the subject of much study. Most studies assume a linear trend and employ a linear model
of the monthly data to estimate those trends

Three estimates of the trend in the recorded SIE represented in terms of the amount of SIE associated with the change. The blue line is the linear trend estimated for data from 1 January 1979 to 31 December 2017. The red line is the linear trend estimated for data from 1 January 1979 to 31 December 2018. The black line is the curvilinear trend estimated for data from 26 October 1978 to 31 December 2019. The dashed lines are the 95 % pointwise confidence bands for the smooth curvilinear trend.

The curvilinear trend in SIE for 1979–2015 and 1979–2018 derived using this method is illustrated in Fig.

Even within the context of nonlinearity, the anomalously low SIE represents a dramatic negative adjustment to Antarctic SIE

The decomposition shows that the curvilinear trend (green) for 2016 is small and positive early in the cycle and becomes strongly negative later in the
year, making a large contribution to the negative anomaly during this time of rapid change, as identified in Fig.

Variability in the annual cycle of Antarctic sea ice extent is dominated by the amplitude and phase of the cycle. In this study, we examined the variability in the annual cycle of total Antarctic sea ice extent (SIE) in detail at timescales ranging from instantaneous to day-to-day, interannual, and multi-decadal trends, thus offering a complete picture of the temporal variation in the sea ice. To facilitate this analysis, we developed first a statistical and mathematical model of the annual cycle in which the amplitude and phase, the two major contributors to its variability, are allowed to vary. This is contrary to traditional methods that restrict the variation in amplitude and phase, thus limiting their contribution to the variability. We define a number of complementary annual cycles – the invariant, which is an optimally smoothed annual cycle with no adjustments for phase or amplitude, an annual cycle that adjusted for phase only, another adjusted for amplitude only, and one that is adjusted for phase and amplitude (APAC). Each of these annual cycles represent clear conceptual and statistical improvements over the traditional method of calculating the annual cycle, with the APAC showing the most improvement. We propose the APAC as a substitute for the traditional method. However, the differences between the other annual cycles and the APAC reveal sources of variability in the daily SIE. For example, comparing the annual cycles adjusted for phase only and amplitude only revealed that the phase contributes more to the variability in the annual cycle than the amplitude.

The timescales into which the variability of SIE was decomposed allow useful interpretations of the factors that give rise to the variability. Using the volatility defined and described here for the first time, we show how much of the total SIE is due to transient effects (such as satellite effects and algorithmic artifacts). We also show how those effects vary over the annual cycle, and in the process we note that there are differences in the volatility (and hence uncertainty) that arise because of sensor type. The daily rate of change in SIE allows a precise definition of the timing and rate of advance and retreat of the sea ice, a quality that is very important given that much of the contemporary variability in Antarctic sea ice occurs at sub-monthly scales. Combination of the information given by the volatility and daily rate of change suggests that the volatility is lowest when the sea ice is at its minimum and highest during the time of the maximum rate of retreat. Given that the rapid rate of retreat of the ice has been associated with dynamic processes, this suggests that the peak in volatility at the end of the cycle is due to ephemeral effects associated with dynamic forcing.

To look at the interannual timescale, we defined and estimated several different but related anomalies, i.e., measures of deviation from the annual cycle, that may be used to evaluate the contributions to Antarctic sea ice variability from sources (local, oceanic and atmospheric) other than the large-scale sources that control cyclical, amplitude and phase changes. These show that our proposed annual cycle, the APAC, is a better fit to the recorded SIE.

We established that the trend in daily total Antarctic SIE over time is strongly nonlinear and that the linear estimates are weak and dependent on a positive trend that began in 2011 and ended in 2016. Interestingly, our decomposition of the annual cycle of 2016 into the components of variation defined in this paper shows that the main contributor to the anomalous SIE was the phase. That is, the anomalously low SIE was due mainly to the fact that the retreat began earlier than normal and was faster than normal. The amplitude made a much smaller negative contribution that did not vary much over the year.

We used the daily total Antarctic SIE in this analysis. However, sea ice variability around Antarctica is strongly regional, and the annual cycles of these regions are markedly different from each other and are also changing. The model-estimated annual cycles and the timescale decomposition presented here will facilitate examination of the regional variability of Antarctic sea ice. Finally, although our method was developed on Antarctic SIE, this decomposition methodology is applicable to a wide range of climatic variables (e.g., temperature, Arctic sea ice extent) that experience an annual cycle.

The data used to generate the sea ice extent are freely available from the National Snow and Ice Data Center (NSIDC)

MNR conceived the idea for this study. MNR and MSH developed the statistical methodology and analyzed the data equally. MSH wrote the software and process of the data. Both authors assisted in writing and editing the manuscript.

The authors declare that they have no conflict of interest.

The authors wish to thank Will Hobbs and Laura Landrum for their insight.

This research has been supported by the National Science Foundation, Office of Polar Programs (grant no. NSF-OPP-1745089).

This paper was edited by Ted Maksym and reviewed by Walter Meier and one anonymous referee.