Climate change has reduced global ice mass over the last 2 decades as enhanced warming has accelerated surface melt and runoff rates. Glaciers have undergone dynamic processes in response to a warming climate that impacts the surface geometry and mass distribution of glacial ice. Until recently no single technique could consistently measure the evolution of surface flow for an entire glaciated region in three dimensions with high temporal and spatial resolution. We have improved upon earlier methods by developing a technique for mapping, in unprecedented detail, the temporal evolution of glaciers. Our software computes north, east, and vertical flow velocity and/or displacement time series from the synthetic aperture radar (SAR) ascending and descending range and azimuth speckle offsets. The software can handle large volumes of satellite data and is designed to work on high-performance computers (HPCs) as well as workstations by utilizing multiple parallelization methods. We then compute flow velocity–displacement time series for glaciers in southeastern Alaska during 2016–2021 and observe seasonal and interannual variations in flow velocities at Seward and Malaspina glaciers as well as culminating phases of surging at Klutlan, Walsh, and Kluane glaciers. On a broader scale, this technique can be used for reconstructing the response of worldwide glaciers to the warming climate using archived SAR data and for near-real-time monitoring of these glaciers using rapid revisit SAR data from satellites, such as Sentinel-1 (6 or 12

The magnitude and direction of glacier flow adjust in response to the warming climate, leading to changes in seasonal flooding and droughts, landscapes and habitats, and ultimately sea level variations. Surface flow is a key variable for determining glacier mass balance

Modern techniques and platforms used for monitoring glacier flow include synthetic aperture radar (SAR)

The SAR-derived displacements for a single epoch can be transformed into 3D (north, east, vertical) displacements by either combining multiple datasets or assuming various model constraints

Historically, three components of mean glacier velocity were computed from DInSAR and/or range offsets by introducing a surface-parallel flow (SPF) constraint. This approach was used for 3D mapping of Greenlandic

In contrast to MSBAS-based techniques,

Here we focus on dynamic changes along six land-terminating glaciers in southeastern Alaska during 20 October 2016–21 January 2021: Agassiz, Seward, Malaspina, Klutlan, Kluane, and Walsh glaciers. This technique can be used to analyze 3D flow velocities of glacier surfaces over large regional scales using nearly 3 decades of archived SAR data and for near-real-time monitoring of these glaciers using rapid revisit SAR data.

Schematics of the simplified case described by Eq. (3). Ascending and descending SAR acquisitions at time

Results of numerical simulations demonstrating the ability of this technique to reconstruct input signal in one of components. Equations of input signals are shown in corresponding subfigure legends;

The inversion technique described below utilizes ascending and descending range and azimuth speckle offset products computed from SAR data using a speckle offset tracking algorithm implemented in GAMMA software

The 3D displacement time series are computed by inverting a set of linear equations, first solving for the north, east, and vertical flow velocities

Equation (

The need for regularization arises because SAR images from different tracks are acquired at different times, which results in more unknowns than equations, producing a rank-deficient, underdetermined problem. When solving a set of linear equations in general there can be three possible scenarios: the number of equations can be less than, equal to, or greater than the number of unknowns. In the equal case, the matrix is square and no regularization is required (but can still be applied). In the greater case, the least square solution is found using singular value decomposition (SVD); this scenario is common in 1D MSBAS, wherein usually there are more interferograms than single-look complex (SLC) images. In the lesser case, as always in 2D and 3D MSBAS, the solution is found using either the truncated SVD or the zeroth-order Tikhonov regularization. The higher-order regularizations must be applied if the objective is to fill the temporal gaps due to missing data, which results in smoothing and the interpolation of missing values in the temporal domain. We observe that the first- and second-order regularizations work equally well in this case, probably because of slowly changing velocities.

In the

The regularization matrix

The structure of

Assuming that

Results of numerical simulations demonstrating the ability of this technique to reconstruct complex uncorrelated and correlated input signals in all three components. Equations of input signals are shown in corresponding subfigure legends;

Sentinel-1 SAR data used in this study;

Outlines of four areas of interest (AOIs) in southeastern Alaska are shown in red. AOI1 covers Agassiz (AG), Malaspina (MG), and Seward (SG) glaciers. AOI2 covers Klutlan Glacier (KG). AOI3 covers Walsh Glacier (WG). AOI4 covers Kluane Glacier. Flow lines in black and red were computed using Open Global Glacier Model (OGGM) software

Correlation (first matrix in cell) and covariance (second matrix in cell) matrices of north, east, and vertical components of velocity for six synthetic tests shown in Fig.

The 3D flow displacement time series are then computed as in Eq. (

Tikhonov regularizations of various orders can be applied during the inversion, resulting in temporal smoothing. The zeroth-order regularization is effectively the constant displacement constraint. The first-order regularization is effectively the constant velocity constraint, and the second-order regularization is effectively the constant acceleration constraint. The first- and second-order regularizations both produce good, virtually indistinguishable results. The example above in Eq. (3) uses first-order regularization. Zeroth- and second-order regularizations are explicitly shown in

MSBAS methodology has been developed for computing multidimensional time series by combining multiple DInSAR data acquired at different times and in various observational geometries. The 2D (east and vertical) method was described in

We used synthetic tests with the actual transform matrix

Without added noise the reconstructed output signal is practically identical to the input signal; with added noise, the reconstructed signal still resembles the input signal very well. For a quantitative assessment, we computed correlation and covariance matrices between three vectors comprising east, north, and vertical components of velocity at each observation epoch. Six correlation and covariance matrices are presented in Table

In the covariance matrices, diagonal elements are variances of north, east, and vertical components of the velocity. They reflect variability due to a true signal and noise. Potentially, an input model can be subtracted to compute variances due to noise; however, it is not a goal of this test. Instead, we are interested in covariance (i.e., non-diagonal) terms of the covariance matrix. They are expected to be small (comparable) in comparison to diagonal terms in the case of the uncorrelated (correlated) signal. In the correlation matrices, it is expected that non-diagonal terms should be small (close to one) in the case of the uncorrelated (correlated) signal. Indeed, this pattern is clearly observed in both cases of uncorrelated (Table

Overall these tests indicate that the ascending–descending geometry is sufficient for a full reconstruction of 3D motion. This can also be inferred theoretically by computing a rank of the transform matrix in the case of one ascending and one descending pair acquired at the same time, which would be equal to 3.

Spatial and temporal baselines of Sentinel-1 pairs used in this study. Mean
temporal resolution, i.e., mean temporal spacing between consecutive SAR
acquisitions regardless of orbit direction, computed as duration divided by
the number of SAR images (

Magnitude of mean 3D flow velocities plotted using logarithmic scale. For
regions P1–P4 at Malaspina and Seward glaciers, P5–P6 at Klutlan Glacier, P7–P8
at Walsh Glacier, and P9 at Kluane Glacier time series are provided in
Figs.

Southeastern Alaska has experienced significant ice mass loss and retreat over the last 50

We focus on studying the dynamic changes along six land-terminating glaciers in southeastern Alaska during 20 October 2016–21 January 2021: Agassiz, Seward, Malaspina, Klutlan, Kluane, and Walsh glaciers (Fig.

In this study, we used 218 ascending (track 123) and 232 descending (track 116) Sentinel-1 interferometric wide (IW) single-look complex (SLC) images with 2.3

3D flow displacement time series for regions P1–P9, the locations of which are shown in Figs.

3D flow velocity time series for regions P1–P9, the locations of which are shown in Figs.

The magnitude of the mean 3D linear flow velocities plotted for the entire region using a logarithmic scale is shown in Fig.

For each AOI, the SAR backscatter intensity images show the six glaciers in detail: Agassiz, Malaspina, and Seward (AG, MG, and SG; Fig.

The direction and magnitude of the mean linear flow velocities sampled along central flow lines from Malaspina and Seward, Klutlan, Walsh, and Kluane glaciers are shown in Figs.

Examples of 3D flow displacement and velocity time series for the

Regions P5 and P6 are located on Klutlan Glacier at elevations of about 1900
and 1500

The technique presented in this study is a viable solution for computing 3D
flow displacement time series from ascending and descending range and azimuth
SAR measurements. Synthetic tests (Figs.

The reported precision of the individual offset maps computed using the SPO technique is

One of the practical computational challenges of the SPO technique is the selection of pixels, the offsets of which are computed with high confidence. After multiple tests, we determined that the SNR function works very well for this purpose but only when the search window is large. However, such a large window applied to the medium-resolution SAR data limits the spatial resolution of the results. It is possible to use high-resolution SAR data and the

We compared the magnitude of mean linear horizontal flow velocities along the four profiles with the results presented in

The overall direction of vertical flow is down along almost the entire length
of the Seward and Malaspina glaciers (Fig.

Velocities along Klutlan Glacier vary in more complex ways, with multiple zones
of upward and downward flow observed (Fig.

The Walsh Glacier is another surge-type glacier with recent surge
activity. Using optical Landsat data,

A surge of the Kluane Glacier has previously been detected using RADARSAT-2
SAR measurements

These six in-depth-analyzed glaciers were selected from the regional results shown in Fig.

We presented a flow displacement technique to observe variations in glacier surface flow in 3D using ascending and descending SAR scenes. The 3D flow displacement (and/or velocity) time series computed allowed us to map in unprecedented detail the state and the temporal evolution of six glaciers in southeastern Alaska during 20 October 2016–21 January 2021. On a broader scale, this technique can be used for reconstructing the historic response of worldwide glaciers to the warming climate using over 30

The range and azimuth offsets computed from Sentinel-1 data as well as all derived products and processing software used in this study can be downloaded from Mendeley Data at

The animations of flow velocities for studied glaciers (files movie_malaspina.gif, movie_klutlan.gif, movie_walsh.gif, movie_kluane.gif) are provided. Comparisons between the magnitude of mean linear horizontal flow velocities along the four profiles with the results presented in

The supplement related to this article is available online at:

SeS was responsible for conceptualization, data curation, formal analysis, investigation, methodology, project administration, resources, software, visualization, and writing (original draft; review and editing). KT was responsible for investigation, formal analysis, methodology, and writing (review and editing). RC was responsible for investigation, formal analysis, methodology, and writing (review and editing).

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank the European Space Agency for acquiring and the National Aeronautics and Space Administration (NASA) and ASF for distributing Sentinel-1 SAR data. Figures were plotted with GMT and Gnuplot software. The work of Sergey Samsonov was supported by the Canadian Space Agency through the Data Utilization and Application Plan (DUAP) program. The work of Kristy Tiampo was supported by CIRES, University of Colorado Boulder.

The work of Ryan Cassotto was supported by NASA (grant no. 80NSSC17K0017).

This paper was edited by Joseph MacGregor and reviewed by Brent Minchew and two anonymous referees.