Fully understanding how glaciers respond to environmental change will require new methods to help us identify the onset of ice acceleration events and observe how dynamic signals propagate within glaciers. In particular, observations of ice dynamics on seasonal timescales may offer insights into how a glacier interacts with various forcing mechanisms throughout the year. The task of generating continuous ice velocity time series that resolve seasonal variability is made difficult by a spotty satellite record that contains no optical observations during dark, polar winters. Furthermore, velocities obtained by feature tracking are marked by high noise when image pairs are separated by short time intervals and contain no direct insights into variability that occurs between images separated by long time intervals. In this paper, we describe a method of analyzing optical- or radar-derived feature-tracked velocities to characterize the magnitude and timing of seasonal ice dynamic variability. Our method is agnostic to data gaps and is able to recover decadal average winter velocities regardless of the availability of direct observations during winter. Using characteristic image acquisition times and error distributions from Antarctic image pairs in the ITS_LIVE dataset, we generate synthetic ice velocity time series, then apply our method to recover imposed magnitudes of seasonal variability within

Earth-observing satellites have been in orbit for over half a century, but it was only in 2011 that a sufficient quantity of data had been collected to complete the first pan-Antarctic map of ice velocity

One of the most direct and insightful ways to understand how ice moves, what controls its flow, and how it responds to changes in its environment is to observe dynamic variability under a wide range of periodic forcings. Long-term trends and interannual variability

It is certainly understood that many mountain glaciers speed up and slow down throughout the year

In this paper, we describe a precise method of measuring the average seasonal cycle of ice flow dynamics, with the ultimate goal that our method may be used to map the typical magnitude and timing of the seasonal glacier dynamics worldwide. Our study is primarily focused on Antarctica, where seasonal variability is poorly understood and where data limitations currently present the greatest challenges to making such measurements. We test the sensitivity of our method on several thousand synthetic ice velocity time series, then validate it by applying the method to satellite data covering Russell Glacier in Greenland, where we compare our results to GPS observations that show persistent seasonality.

Example scenario:

The method we present applies to ice velocity datasets such as GoLIVE

This paper focuses primarily on the Landsat image pairs that populate the ITS_LIVE dataset, in part because their record extends back as far as 1985 in some locations. The long Landsat record may help ascertain a climatological seasonal cycle of ice dynamics at any given location; however, we face the limitation that optical satellites like Landsat do not collect data throughout the dark winters in polar regions, where land ice is most prevalent (see Fig.

Antarctic image pair acquisition times. Over 1.8 million Landsat image pairs provide Antarctic ice velocities in the ITS_LIVE dataset.

ITS_LIVE velocity data statistics. Maps of

For any given

The first step toward quantifying seasonal variability for any pixel is to remove any interannual variability from the time series. Interannual variability can be determined by smoothing the velocity data using any of several common methods. A polynomial fit is robust and computationally efficient but requires choosing a polynomial order, which is subjective and can lead to overfitting or underfitting the data. Alternatively, a moving average makes no prior assumption about the shape of the velocity curve and can adapt to any arbitrary interannual variability. After exploring several approaches, we find the best results by first detrending the time series with a low-order polynomial, then using a hybrid of a moving average and a spline fit, which we describe below.

When assessing interannual variability, we temporarily ignore the duration over which each image pair measured ice displacement and simply assign the average velocity to the center date

Velocity time series for an ITS_LIVE pixel. The clustering of these 14 208 measurements taken near the grounding line of Byrd Glacier typifies ITS_LIVE image pair data, with short-

After detrending the time series with a weighted polynomial fit, we characterize any residual interannual variability with a spline fit to the mean velocities of each year. We take the weighted mean velocity of all measurements whose

In the method described thus far, most summer image pairs whose

We characterize the magnitude and timing of seasonal variability as a simple sinusoid that can be applied in the

After characterizing interannual variability, we subtract it from the velocity time series at the center date of each image pair. The residuals

We seek to define seasonal velocity variability

For a more robust least-squares solution, we employ a trigonometric identity to rewrite Eq. (

Again, we cannot solve Eq. (

By employing Eq. (

If our first estimates of

Figure

To assess uncertainty in our method, we generate synthetic random continuous velocity time series, then artificially sample them using random subsets of image acquisition times from Landsat image pairs that cover Antarctica. We then apply the methods described in Sect.

Any synthetic velocity time series used for testing should resemble the true nature of ice dynamics in its variability on all timescales. Accordingly, we create realistic interannual variability by matching the distribution of the standard deviations of velocities in each grid cell in the ITS_LIVE annual velocity mosaics from 2013 to 2018. Figure

To create each synthetic time series of interannual variability, we generate uniformly distributed random values centered about 0 at daily temporal resolution. We apply a first-order low-pass Butterworth filter with a cutoff period of 548 d to each random time series to ensure that no annual cycles are present, and then we discard the first and last 548 values of each time series to eliminate edge effects of the filter. We then scale the remaining time series such that its standard deviation matches a prescribed level of interannual variability.

Our handling of interannual variability is an attempt to mimic the observable ways in which Antarctic glaciers speed up and slow down from year to year. Ideally, we would also carry forth in a similar manner for seasonal variability, imposing cyclic behavior that matches the true character and distribution of the types of seasonal variability that exist in nature. However, as the intent of this paper is to develop the methods that will be necessary to understand where and how ice velocities vary on seasonal timescales, we cannot at present create synthetic seasonal-variability distributions to match what truly exists in nature. Instead, to each synthetic time series we add a sinusoid with a period of 365.25 d, a random phase, and a random amplitude between 0 and 100 m yr

Each synthetic time series is artificially sampled using characteristic acquisition times and error distributions from Antarctic Landsat-derived ITS_LIVE image pairs. In Fig.

Sensitivity test parameters. To understand how various factors influence measurement sensitivity, we isolate and vary a number of key characteristics of the synthetic time series and the data analysis method. In the first test, we vary only the number of iterations described in Sect.

RFD indicates

We conducted several tests to determine the accuracy with which we can recover the amplitudes and phases of seasonal cycles in synthetic velocity time series. The parameters of each test are detailed in Table

In the first test, we sought to understand how many iterations are necessary to achieve a stable solution. In this test, we generated 10 000 synthetic velocity time series, each having interannual variability with a standard deviation of 4.2 m yr

Sensitivity test results. The tests outlined in Table

We conducted a second test to determine how many image pairs are necessary to detect seasonal cycles in a velocity time series. We found that with just 32 image pairs, we were able to recover phase with sufficient accuracy to correctly identify the season of maximum velocity (Fig.

In a third test, we found that one factor more than any other threatens the accuracy of seasonal-variability detection. Given a velocity time series sampled by 1153 image pairs, seasonal-amplitude error increases approximately linearly with the level of background interannual variability (Fig.

The final four panels (g–j) of Fig.

Thus far we have determined that 10 iterations are more than sufficient for the model of seasonal variability to converge. We have also found that our measurements can be considered agnostic to the timing of ice velocity variability and to the timing of satellite image acquisitions when applied to at least 500 to 1000 image pairs. With this understanding, we now apply our method to 100 000 synthetic time series that typify the interannual ice velocity variability and satellite image acquisitions that have been measured across the Antarctic continent as a whole. Interannual variability in the synthetic time series was randomly sampled from the distribution shown in Fig.

Pan-Antarctic seasonal signal recovery. Following the parameters of the final test in Table

To confirm that we can extract seasonal ice velocities from feature-tracking data, we compare an in situ GPS position time series against results obtained by applying our method to ITS_LIVE velocities that were generated from 15 m resolution optical Landsat 7 and 8 imagery. We focus on Russell Glacier in Greenland, where GPS data provide more than a decade of observations and where seasonal velocity variability has previously been reported

GPS validation at Russell Glacier, Greenland. Hourly time series of

Figure

We compare the GPS velocity time series to velocities obtained from 5189 ITS_LIVE image pairs in the pixel closest to the fixed median location of the GPS receiver throughout the course of its data collection. Key findings are listed in Table

ITS_LIVE analysis compared to GPS. The secular mean velocities and

The largest disagreement in Table

The method we present in this paper requires a multiyear record with at least several hundred image pairs to confidently identify the amplitude and phase of seasonal variability. Some key regions of interest, such as Totten Glacier in Antarctica, do not yet offer enough cloud-free images to meet this threshold, so a few more years of data collection may be required before our methods can be successfully implemented there. In addition, it may be difficult or impossible by our methods to detect seasonality in places of interest such as Pine Island and Thwaites glaciers, which are currently undergoing dramatic interannual change

Nonetheless, Fig.

The most significant limitation of the method we present may lie in our approximation of seasonal variability as a sinusoid. Where seasonal dynamic variability has previously been documented, it has been found that in some cases a sawtooth function or other higher-order fits might match seasonal variability more closely than a sinusoid

Despite the tendency of a sinusoid to oversimplify complex time series, we contend that no other description of seasonal variability is as elegant or robust over decadal timescales and that understanding seasonal ice dynamics must begin with a zeroth-order description of the amplitude and phase of ice velocity variability. While it is true that a glacier can accelerate in response to a transient event and return to an equilibrium velocity within just a few days

The approach we present captures the fundamental mode of subannual variability in ice velocity and conserves all ice displacement that occurs throughout the year. The simple two-term explanation of amplitude and phase provides a robust description of seasonality that is less prone to error than higher-order fits such as three-term sawtooth functions. By providing a simple measure of amplitude and phase, we offer a straightforward method to compare how neighboring glaciers respond to a common seasonal forcing or investigate how dynamic signals propagate upstream or downstream in a given glacier.

In Sect.

The methods presented in this paper have focused primarily on optical satellite data because no other type of sensor provides such a long record of ice velocity. As more radar data become available, particularly since the launch of Sentinel 1a/b, the problem of missing winter data will be eliminated, but the methods presented in this paper will still hold. When an abundance of feature-tracked velocities from radar become available, Eq. (

Given a relatively continuous time series of at least 500 to 1000 image pairs, our method can extract the amplitude of seasonal variability with a precision on the order of about 1 m yr

With the method we describe, we may begin to map seasonal ice dynamic variability on a global scale in a consistent and meaningful manner, and by providing a method that can be employed independently in the two dimensions of Cartesian coordinates, we hope to gain a more complete understanding of how dynamic signals propagate through the world's ice.

Phase and amplitude uncertainty in Sect.

In addition to the tests described in Sect.

Letting phase uncertainty of 45 d be the threshold indicating whether the season of maximum ice velocity can be accurately determined, in Fig.

Interannual variability and seasonal signal detection. The dark-green region of this plot indicates scenarios in which phase can be accurately detected with 1153 image pairs. When seasonal amplitudes are above the noise floor of about 1 m yr

The values of seasonal amplitude and phase uncertainty listed in Table

We use hourly position data from the PROMICE KAN_L GPS station on Russell Glacier

Data analysis in this paper relied upon Antarctic Mapping Tools for MATLAB

The supplement related to this article is available online at:

CAG conceived of and carried out the study with guidance and technical assistance from ASG. LCA postprocessed and interpreted the GPS data. CAG wrote the manuscript with input from ASG and LCA.

The authors declare that they have no conflict of interest.

Thanks to Brent Minchew and the two anonymous reviewers for providing feedback on this paper. Data from the Programme for Monitoring of the Greenland Ice Sheet (PROMICE) and the Greenland Analogue Project (GAP) were provided by the Geological Survey of Denmark and Greenland (GEUS) at

The authors were supported by the NASA Postdoctoral Program, the NASA Cryospheric Sciences program, and the NASA MEaSUREs program.

This paper was edited by Etienne Berthier and reviewed by two anonymous referees.