Ground-Based Radar Interferometry of Sea Ice

: In light of recent Arctic change, there is a need to better understand sea ice dynamic processes at the ﬂoe scale to evaluate sea ice stability, deformation, and fracturing. This work investigates the use of the Gamma portable radar interferometer (GPRI) to characterize sea ice displacement and surface topography. We ﬁnd that the GPRI is best suited to derive lateral surface deformation due to mm-scale horizontal accuracy. We model interferometric phase signatures from sea ice displacement and evaluate possible errors related to noise and antenna motion. We compare the analysis with observations acquired during a drifting ice camp in the Beaufort Sea. We used repeat-scan and stare-mode interferometry to identify two-dimensional shear and to track continuous uni-directional convergence. This paper demonstrates the capacity of the GPRI to derive surface strain on the order of 10 − 7 and identify different dynamic regions based on sub-mm changes in displacement. The GPRI is thus a promising tool for sea ice applications due to its high accuracy that can potentially resolve pre- and post-fracture deformation relevant to sea ice stability and modeling.


Introduction
The Arctic sea ice has undergone widespread decline in recent years [1] and a near seasonally ice-free Arctic Ocean is expected within three decades [2]. Sea ice retreat has implications for climate, marine life, and human activities [3] by increasing accessibility for ocean navigation [4][5][6]. Recent changes in sea ice have also led to more challenging or even hazardous conditions for over-ice travel [7][8][9] on an often thinner and less stable ice cover. Increasing our understanding of sea ice stability, fracturing, and failure processes is needed to mitigate risk [10], extend windows of operations, and improve pan-arctic and regional sea ice modeling and prediction.
Interferometry is a technique for measuring the difference in returned phase between two radar scenes commonly used for deriving surface topography or sub-pixel, wavelengthscale surface displacements. Interferograms can be generated from either synthetic aperture radar (SAR) or real aperture radar using a similar processing chain including co-registration, multi-looking, interferogram formation, and adaptive phase filtering [30,31]. The resulting interferogram consists of phase change values wrapped between −π and π. The phase change, ∆Φ, represents a combination of several factors: ∆Φ disp represents the phase change due to surface motion between temporally separated acquisitions. ∆Φ topo is the phase change due to topography if acquisitions are acquired from different antenna locations (dual receiving antennas permit this as standard practice with the GPRI). These terms can provide information about the motion or morphology of a surface. However, it can be challenging to accurately extract the magnitude of motion or topography in the case of significant phase contributions due to atmospheric Remote Sens. 2021, 13, 43 3 of 18 effects, ∆Φ atm , noise, ∆Φ noise , or positioning errors, ∆Φ error . ∆Φ atm originates from the different effective path lengths resulting from changes in atmospheric path delay between acquisitions and is known to impact InSAR and GNSS measurements [32]. The GPRI signals exclusively penetrate air close to the ground, which can hold more moisture than air aloft. Therefore, although the GPRI signal penetrates less atmosphere than spaceborne SAR, ∆Φ atm can be substantial [33]. With that said, the moisture content will strongly depend on temperature, hence for sea ice, ∆Φ atm is expected to be most pronounced during spring/summer. The geometry used for data acquisitions is illustrated in Figure 1. The interferometric phase change is given by the expression: Here, is the wavelength of the signal (1.7 cm) and ∆( + ) represents the change in signal path length from the transmitting antenna to the scatterer (i.e., phase center of the backscattered signal) ( ) and back to the receiving antenna ( ). The sensitivity to vertical and horizontal surface motion depends on the incidence angle of the system, . For applications over relatively level terrain like sea ice, incidence angles typically exceed 85° at ranges beyond approximately 100 m, meaning that sensitivity to horizontal motion is much greater than to vertical motion. The sensitivity can be calculated through and : representing horizontal and vertical motion resulting in line-of-sight (LOS) displacement of /2 (one full phase cycle). We can further derive the ambiguous velocity by dividing by the temporal baseline between acquisitions, , e.g., in the horizontal: The phase-derived velocity can be calculated, e.g.,: = ∆Φ 2 (5) Figure 1. Geometry illustration of the GPRI mounted on a ridge. The three antennas are each associated with a height above the level ice surface, h, a range to a specific target, r, and an incident angle, θ. The GPRI is free to tilt at an angle, α, relative to a level reference point due to ocean waves with amplitude, A, and wavelength, . Figure 1. Geometry illustration of the GPRI mounted on a ridge. The three antennas are each associated with a height above the level ice surface, h, a range to a specific target, r, and an incident angle, θ. The GPRI is free to tilt at an angle, α, relative to a level reference point due to ocean waves with amplitude, A, and wavelength, λ w .
Here, λ is the wavelength of the signal (1.7 cm) and ∆(r 1 + r 2 ) represents the change in signal path length from the transmitting antenna to the scatterer (i.e., phase center of the backscattered signal) (r 1 ) and back to the receiving antenna (r 2 ). The sensitivity to vertical and horizontal surface motion depends on the incidence angle of the system, θ. For applications over relatively level terrain like sea ice, incidence angles typically exceed 85 • at ranges beyond approximately 100 m, meaning that sensitivity to horizontal motion is much greater than to vertical motion. The sensitivity can be calculated through d a h and d a v : representing horizontal and vertical motion resulting in line-of-sight (LOS) displacement of λ/2 (one full phase cycle). We can further derive the ambiguous velocity by dividing by the temporal baseline between acquisitions, B t , e.g., in the horizontal: The phase-derived velocity can be calculated, e.g.,: The phase changes measured by the GPRI due to topography, ∆Φ topo , can be expressed by: where r 2 -r 3 is the difference in LOS distances from the phase center of the backscattered signal to the receiving antennas. We can estimate the height sensitivity by the ambiguity height. For a vertically mounted antenna (α = 0), h a can be expressed [24]: where the second and third terms are often negligible. Here B s is the spatial baseline between the GPRI's receiving antennas. Interferometric coherence, γ, is a useful parameter in addition to ∆Φ in interferometric processing. γ describes the correlation between the signals from the two acquisitions, which can be impacted by spatial (γ sp ), temporal (γ te ), thermal (γ th ), and process (γ pr ) decorrelation [34]: γ th is impacted by the signal-to-noise ratios of the two acquisitions (SNR 1 and SNR 2 ): γ pr is controlled by errors in the interferogram processing. γ sp is impacted by the spatial baseline between acquisitions, resulting in an altered scattering response from the surface due to a different incidence angle. γ te is affected by changes in the scattering medium between acquisitions and is generally expected to be minimal over short timescales. Reduced coherence introduces noise in the interferogram, ∆Φ γ , with standard deviation (Cramer-Rao bound): (10) here, N L is the independent number of looks and should exceed N L ≥ 4 [35]. The phase noise can in turn be used to describe the expected error of the derived displacement: and similar for topography, which describes the displacement or height value corresponding to a standard deviation of the phase noise.

Interferometric Processing with the Gamma Portable Radar Interferometer (GPRI)
The Gamma portable radar interferometer (GPRI) is a fully coherent radar operating at Ku-band (λ~1.7 cm) capable of interferometric measurements [36]. The system consists of three slotted wave guide antennas mounted on a tripod. The antenna configuration is interchangeable; in this case the transmitting antenna is situated 35 cm and 60 cm above the two receiving antennas (Figure 1). The transmit antenna generates a fan-beam spreading 35 degrees in the vertical (zenith) and 0.4 degrees in the horizontal (azimuth) [36] resulting in a variable azimuthal pixel ranging from 0.14 m in near range (20 m) to about 30 m in far range (5 km). The bandwidth of 200 MHz results in a consistent 0.75 m pixel spacing in the range direction.
The GPRI can operate in dynamic or stare modes. In the dynamic mode, the antennas rotate azimuthally, collecting data over a partial or near full 360-degree scan. A single scan can be used to map surface topography by using both receiving antennas from one acquisition. A single-scan interferogram has a zero temporal baseline, which eliminates ∆Φ disp (Equation (1)). Repeat scans can be collected with as short as minute-scale temporal baselines, enabling derivation of the dynamic change in between acquisitions. If the temporal baseline of repeat-scan interferometry is kept to minutes, ∆Φ atm (Equation (1)) and decorrelation contributing to γ sp and γ te (Equation (8)) will be low. When the GPRI is left in place between acquisitions, repeat images do not require coregistration reducing Remote Sens. 2021, 13, 43 5 of 18 the chance of significant ∆Φ error . However, ∆Φ error can still result from small-scale antenna movements during the time between acquisitions (e.g., antenna tilting). If the same receive antenna is used during data collection, ∆Φ topo is eliminated.
Stare-mode enables continuous data collection in one direction only. Each pulse transmitted at 500 Hz is treated as an individual acquisition enabling the formation of interferograms with temporal baselines as short as 2 milliseconds (with no averaging). These short temporal-baseline interferograms between pulses can be used to derive near instantaneous motion and thus ice acceleration (Equation (4)). Alternatively, as done in this work, the first pulse acquisition can be used as the master for all subsequent pulses, resulting in progressively larger temporal baselines for the derivation of accumulated displacement.

Deriving Phase Sensitivity to Motion and Topography of Sea Ice
We derived ambiguity displacements, and to investigate the level of sensitivity to horizontal and vertical ice movement (Equation (3)). These values depend on the incidence angle, which is determined by sensor elevation and range. Examples if stationed on level ice (0 m), a ridge (5 m), building (20 m), and higher (50 m) are illustrated in Figure  3a. Here, increases nearly linearly with range and decreases quickly with range down to /2 (dashed and solid lines respectively). The sensitivity to horizontal and vertical motion is inversely proportional to and . We can thus identify a rule of thumb, that the GPRI is more sensitive to horizontal motion past a range distance approximately equivalent to the elevation of the sensor (crossing between dashed and solid lines in Figure 3a).
We also investigate the phase response to potential mm-scale antenna shifts. A likely cause of such antenna motion is a tilt caused by buckling or ocean waves resulting in a Repeat-scans were collected of nearly the full 360-degree scene around the radar at temporal baselines ranging from minutes to hours to investigate the application of the GPRI for examining km-scale deformation processes at these timescales. Stare-mode acquisitions were collected along the red line in Figure 2b to investigate the possibility of resolving continuous ice deformation and short-lived transient sea ice processes. These stare-mode acquisitions enabled the formation of interferograms with temporal baselines ranging between 0.01 (0.05 after coherently averaging 5 acquisitions) and 30 s representing the accumulated phase change from the time of the first signal in a stack of 3000 independent acquisitions.
In the work presented here, we analyze one repeat-scan and one set of stare-mode interferograms. The GPRI system was stationary between and during acquisitions eliminating the need for coregistration. We used only the upper receiver antenna in order to achieve a zero spatial baseline and eliminate sensitivity to topography. No ground validation was collected for these acquisitions. However, interferometry has been validated over sea ice in the past and shown high accuracy in comparison with in-situ strain measurements [10].

Deriving Phase Sensitivity to Motion and Topography of Sea Ice
We derived ambiguity displacements, d a h and d a v to investigate the level of sensitivity to horizontal and vertical ice movement (Equation (3)). These values depend on the incidence angle, which is determined by sensor elevation and range. Examples if stationed on level ice (0 m), a ridge (5 m), building (20 m), and higher (50 m) are illustrated in Figure 3a. Here, d a v increases nearly linearly with range and d a h decreases quickly with range down to λ/2 (dashed and solid lines respectively). The sensitivity to horizontal and vertical motion is inversely proportional to d a h and d a v . We can thus identify a rule of thumb, that the GPRI is more sensitive to horizontal motion past a range distance approximately equivalent to the elevation of the sensor (crossing between dashed and solid lines in Figure 3a).  Table 1 for distinct range values. Here, noise and tilt errors are listed for lower coherence values ( = 0.7) more representative of level ice and subtle tilt values = 3.6 × 10 −3 degrees and = 0. This tilt corresponds to the tilt of the ice surface due to infragravity waves with A = 1 cm and = 1 km. This is also equivalent to the tilt if one leg of the antenna tripod settles with ~4 mm (assuming the tripod height is 1.5 m and the legs 22.5° from vertical). Such tilt will result in and equal 0.43 mm and 6.3 cm respectively at 1 km range (Table 1).   We also investigate the phase response to potential mm-scale antenna shifts. A likely cause of such antenna motion is a tilt caused by buckling or ocean waves resulting in a tilted Remote Sens. 2021, 13, 43 7 of 18 ice surface or tilting of the antenna or scaffolding used to mount the tripod (Figure 2a). A tilt with angle α will result in a phase error, σ ∆φ α : where h 1 and h 2 are the heights of the transmitting and receiving antennas above the tilting surface (e.g., if placed on a ridge, the height of antenna plus the sail height). β is the angle of tilt relative to the look direction. This approximation is accurate to~0.7 mm in the near range, but quickly increases in quality to <0.03 mm at 100 m range. A potential tilt error will lead to errors in respective displacement (e.g., in horizontal displacement, σ d hα ): The relationship between errors and range is illustrated in Figure 3b. It may be possible to distinguish sensor tilt from an actual deformation signal; if interpreted as lateral movement, the contribution will be largely independent of range. If interpreted as vertical motion, it may lead to implausible values in the far range due to the large d a v . Ambiguity values are listed in Table 1 for distinct range values. Here, noise and tilt errors are listed for lower coherence values (γ = 0.7) more representative of level ice and subtle tilt values α = 3.6 × 10 −3 degrees and β = 0. This tilt corresponds to the tilt of the ice surface due to infragravity waves with A = 1 cm and λ w = 1 km. This is also equivalent to the tilt if one leg of the antenna tripod settles with~4 mm (assuming the tripod height is 1.5 m and the legs 22.5 • from vertical). Such tilt will result in σ d h and σ d v equal 0.43 mm and 6.3 cm respectively at 1 km range (Table 1).

Horizontal
Vertical Elevation Parameters include θ = incidence angle, d a h and d av = horizontal and vertical ambiguity displacement, h a = ambiguity height, σ d hγ , σ dv γ , and σ hγ = phase noise errors, and σ d hα , σ dv α , and σ hα = tilt errors. We assume the lower antenna is situated 1.5 m above a 5 m ridge. The distance between the transmitting and receiving antenna for the derivation of motion is 60 cm. The distance between the two receiving antennas is also 60 cm for the derivation of topography.
The interferometric phase is less sensitive to typical sea ice topography than motion. h a increases nearly linearly and approaches 30 m at a range of 1 km (Figure 3c; Table 1; Equation (7)). A large h a can be advantageous for assessment of topographic reliefs of hundreds of meters. However, for sea ice, where elevations do not typically exceed 10 m [37], this can result in large (cm-to m-scale) relative errors. For single-scan interferometry, height estimates are susceptible to offsets in the sensor orientation. The error is illustrated in Figure 3d and can be described [24]: Phase noise is another source of error in the phase estimates of elevation and displacement. The error depends on the number of looks, SNR, and interferometric coherence (Equation (10)). Figure 3b,d exhibit expected errors based on N L = 4 independent looks and γ ≥ 0.9. Example values are also listed in Table 1 (Equation (10)). σ d hγ is limited to less Remote Sens. 2021, 13, 43 8 of 18 than 1 mm while σ d vγ linearly increase and exceed 14 cm at 1 km (γ = 0.7). For topography, σ h γ exceed 3 m by 1 km (Table 1).
Based on this sensitivity analysis, the GPRI shows promise for deriving horizontal deformation of sea ice based on a small d a h~8 .5 mm. The expected errors are also small (~1 mm) relative to typical strain rate values in the Arctic on the order of 10 −7 s −1 (1 mm s −1 at 10 km) [38]. This suggests that the GPRI may be able to resolve fracturing both on the cm-scale (e.g., thermal cracks) and mm-scale (e.g., microcracks in the structure of the ice). Furthermore, the GPRI is less sensitive in the vertical and implications of this in terms of deriving vertical displacement and topography are examined in the discussion. We therefore focus the remaining analysis on examining how we may recognize and characterize lateral (horizontal) deformation based on both modeled and observed interferograms.

Modeling Repeat-Scan Interferometric Response to Sea Ice Strain
The repeat-scan mode is suited to assess lateral strain development based on its high accuracy and the ability to capture multiple acquisitions within minutes. To investigate strain applications, we modeled strain fields and associated interferometric phase response to shear and convergence. In this model, the GPRI sensor is stationed on sea ice and the ice within the footprint of the GPRI undergoes uniform displacement (left column in Figure 4). The interferometric phase is only sensitive to the displacement relative to the GPRI. This has two consequences: (1) only the motion field in the reference frame of the GPRI is observed (center column in Figure 4) and (2) only radial changes in motion results in a phase change. The consequence is that there will be zero phase change along the green axis superimposed on Figure 4b,e. On the other hand, maximum phase change will occur along the blue axes.

Modeling Stare-Mode Interferometric Response to Sea Ice Strain
We further modeled the interferometric phase response for the same convergence field as in Figure 4 in stare-mode. Here, the GPRI is oriented in the direction of motion (dashed line in Figure 4d). The uniform convergence leads to consistent velocity over time that decreases towards the sensor (displayed in Figure 5a in the time-range geometry). The cumulative motion is displayed in the same coordinate system in Figure 5b indicating increasing strain with time. The resulting interferogram is displayed in Figure 5c. The The interferometric phase value represents 1-dimensional information as only motion towards or away from the antenna is measured. For InSAR, where the look angle is fixed, the interferometric fringes are ambiguous between shear and convergence. This can make it challenging to distinguish between these two deformation modes [39]. This is not the case for GPRI measurements as the variation in look angle during repeat-scan GPRI measurements results in different fringe patterns for shear and convergence. However, with ground-based radar, there are directional ambiguities both within shear and convergence that can make it challenging to determine the exact direction of deformation, which is not the case with airborne or spaceborne InSAR. For instance, in the case of dextral (right-lateral) shear (Figure 4a), the fringe pattern takes the form of a cross (Figure 4c), identical to a case of perpendicularly oriented sinistral (left-handed) shear. Similarly, the direction of convergence cannot be resolved based on Figure 4f alone as convergence in the opposite direction of Figure 4d will also lead to the same relative motion (Figure 4e) and interferogram.

Modeling Stare-Mode Interferometric Response to Sea Ice Strain
We further modeled the interferometric phase response for the same convergence field as in Figure 4 in stare-mode. Here, the GPRI is oriented in the direction of motion (dashed line in Figure 4d). The uniform convergence leads to consistent velocity over time that decreases towards the sensor (displayed in Figure 5a in the time-range geometry). The cumulative motion is displayed in the same coordinate system in Figure 5b indicating increasing strain with time. The resulting interferogram is displayed in Figure 5c. The modeled temporal baseline and deformation field is kept the same as in

Observing Shear Strain Using Repeat-Scan Interferometry
We evaluated repeat-scan GPRI imagery acquired during the U.S. Navy's ICEX 2020 operations on 10 March between 05:00 UTC and 19:30 UTC. The backscatter image indicates a complex ridge system to the west and smoother ice with scattered ridges to the east (Figure 6a) also visible in the optical imagery (Figure 2b). The areas of deformed ice retain generally good coherence in between acquisitions particularly in the near to midrange (Figure 6b). The interferometric phase can thus be reliably interpreted in these areas. The resulting interferogram (Figure 6c) has a temporal baseline of 14.5 h and a fringe pattern resembling the modeled shear in Figure 4c.
This interferogram could represent either of two orthogonal shear fields due to the directionally ambiguous nature of the phase values. It could either represent dextral (right-lateral) shear along an axis oriented 150 degrees from North, or sinistral (left-lateral) shear along an axis oriented 240 degrees from North. We therefore need to consider other dynamical factors in order to resolve this ambiguity. At the time of the GPRI observations, the ice camp was approximately 100 km north of the Alaska coastline, which is oriented In stare-mode, the look angle of the GPRI is constant, similar to InSAR, resulting in ambiguities between shear and convergence. The rate of absolute deformation is also not known as the stare-mode can only be used to directly derive the minimum displacement. In the case of the GPRI being oriented along the dashed line in Figure 4d, the actual deformation can be derived based on the phase values. However, if the GPRI was oriented along the dotted line in Figure 4d, the derived motion would only represent a radial component of the absolute motion.

Observing Shear Strain Using Repeat-Scan Interferometry
We evaluated repeat-scan GPRI imagery acquired during the U.S. Navy's ICEX 2020 operations on 10 March between 05:00 UTC and 19:30 UTC. The backscatter image indicates a complex ridge system to the west and smoother ice with scattered ridges to the east (Figure 6a) also visible in the optical imagery (Figure 2b). The areas of deformed ice retain generally good coherence in between acquisitions particularly in the near to mid-range (Figure 6b). The interferometric phase can thus be reliably interpreted in these areas. The resulting interferogram (Figure 6c) has a temporal baseline of 14.5 h and a fringe pattern resembling the modeled shear in Figure 4c.

Observing Compressional Strain Using Stare-Mode Interferometry
We evaluated a series of stare-mode acquisitions from 4 Mar at 23:57 UTC. Due to the predominately level ice and large incidence angle, the ridges are clearly identifiable (e.g., s5-s7 in Figure 6a) as bright bands in the backscatter imagery (Figure 7a). Due to the high SNR of these locations, they also feature high coherence (Equation (9); Figure 7b) giving us significant confidence in the interferometric phase (Figure 7c). We evaluated the interferometric phase-derived motion in the line-of-sight direction at seven distinct points: s1-s5 on multi-year ice in the near range, and s6-s7 on first-year in the far range (Figure 8a). The selection was based on γ > 0.98 and a large spatial spread from 72 m to 2.1 km in range. Through these points, we identified multiple displacement components of the stare-mode signal: First, we analyzed the trends in the data. All points exhibit a near constant component of motion towards the GPRI over the 30 s interval, resulting in a linear decrease in range over time (Figure 8a). The amount of displacement for all scatterers with coherence >0.98 (not just s1-s7) was approximately proportional to the distance from the GPRI (Figure 8b), indicating convergent strain in the look-direction. The linear fit in Figure 8b suggests a strain of ~4.6 10 (slope of the line), but is not perfect. Further examination of the strain observed at each scatterer indicates an asymptotic-like decrease in strain magnitude with distance ( Figure 8c). This can be explained by tilting of the ridge where the antenna is mounted resulting in apparent horizontal motion of the antenna not related to ice deformation. This shift is constant with range and thus dominating strain values in near range where relative displacement values are lower. We fit a hyperbola to the data y = −338 10 /x − 4.45 10 . This function indicates two things, that the ridge gradually tilted resulting in a horizontal antenna shift of ~0.3 mm and that the ice was compressing with a total strain of ~4.5 10 during the 30 s of acquisitions. The stare mode tracks continuous deformation. Hence we can also calculate the strain rate (change in strain per This interferogram could represent either of two orthogonal shear fields due to the directionally ambiguous nature of the phase values. It could either represent dextral (right-lateral) shear along an axis oriented 150 degrees from North, or sinistral (left-lateral) shear along an axis oriented 240 degrees from North. We therefore need to consider other dynamical factors in order to resolve this ambiguity. At the time of the GPRI observations, the ice camp was approximately 100 km north of the Alaska coastline, which is oriented approximately ENE/WSW. A microweather station at the camp reported an average windspeed of 3.8 m s −1 from a westerly direction, while the reanalysis ERA5 from the European Centre for Medium Range Weather Forecasts (ECMWF) indicate the regional winds over the southern Beaufort Sea were northwesterly. Position data from satellite tracked buoys at the camp show that the GPRI observations coincided with a period of near-zero ice drift. Between 05:00 and 07:00 on 10 March 2020, the ice camp drifted approximately 60 m southward (toward land) and then effectively came to rest for the next two days.
The near absence of motion indicates the ice was not in free drift and was therefore subject to internal stresses. Although the local stress field at the scale of the GPRI observations will have been affected by variations in ice thickness and strength, we expect that the NW regional winds would have imposed right-lateral stresses on the ice between the camp and Alaska coastline. We therefore consider the solution representing right-lateral shear along an axis approximately parallel to the wind (135 degrees from North) to be the most likely. There is no similar ambiguity in deformation rate. Hence, we are able to estimate the strain (i.e., magnitude of ice deformation relative to the distance from the sensor). We first match the observed interferogram (Figure 6c) with modeled shear interferograms. The strain is then calculated from the motion field used to produce the best match in terms of fringe spacing. We derived a strain of~10 −5 meaning approximately 1 cm of motion per km. Note that the interferogram does not exactly match a shear strain field. Other deformation modes may play a role and result in inaccuracies in this strain estimate.

Observing Compressional Strain Using Stare-Mode Interferometry
We evaluated a series of stare-mode acquisitions from 4 Mar at 23:57 UTC. Due to the predominately level ice and large incidence angle, the ridges are clearly identifiable (e.g., s5-s7 in Figure 6a) as bright bands in the backscatter imagery (Figure 7a). Due to the high SNR of these locations, they also feature high coherence (Equation (9); Figure 7b) giving us significant confidence in the interferometric phase (Figure 7c). We evaluated the interferometric phase-derived motion in the line-of-sight direction at seven distinct points: s1-s5 on multi-year ice in the near range, and s6-s7 on first-year in the far range ( Figure 8a). The selection was based on γ > 0.98 and a large spatial spread from 72 m to 2.1 km in range. Through these points, we identified multiple displacement components of the stare-mode signal: Second, we detrended the stare-mode displacement data to remove the uniform c pression and tilt allowing us to examine motion on timescales shorter than 30 s. The m prominent residual motion is synchronous, periodic oscillations with a frequency and plitude of approximately 4 Hz and 0.2 mm. We attribute this motion to wind-indu antenna vibrations and remove it using a 1-s running average. The detrended and filte data also indicate near synchronous motion of all scatters towards and away from antenna by around 0.2 mm during the 30 s stare interval (Figure 9a). Although the m nitude of this motion varies slightly between scatterers, this variation does not appea be related to range. Such motion could therefore be related to periodic widening and rowing of a dislocation fracture close to the antenna or periodic tilting of the antenn adjacent first-year ice floe that contained scatterers s6 and s7 (Figure 2b). Hence, correlation between residual phase variations of scatters may be a useful way of identifying mechanical discontinuities in an apparently continuous ice pack.  First, we analyzed the trends in the data. All points exhibit a near constant component of motion towards the GPRI over the 30 s interval, resulting in a linear decrease in range over time (Figure 8a). The amount of displacement for all scatterers with coherence >0.98 (not just s1-s7) was approximately proportional to the distance from the GPRI (Figure 8b), indicating convergent strain in the look-direction. The linear fit in Figure 8b suggests a strain of~4.6 ×10 −6 (slope of the line), but is not perfect. Further examination of the strain observed at each scatterer indicates an asymptotic-like decrease in strain magnitude with distance ( Figure 8c). This can be explained by tilting of the ridge where the antenna is mounted resulting in apparent horizontal motion of the antenna not related to ice deformation. This shift is constant with range and thus dominating strain values in near range where relative displacement values are lower. We fit a hyperbola to the data y = −338 ×10 −6 /x − 4.45 × 10 −6 . This function indicates two things, that the ridge gradually tilted resulting in a horizontal antenna shift of~0.3 mm and that the ice was compressing with a total strain of~4.5 ×10 −6 during the 30 s of acquisitions. The stare mode tracks continuous deformation. HHHence we can also calculate the strain rate (change in strain per second) to~1.4 × 10 −7 s −1 near a typical 1 × 10 −7 s −1 strain rate for Arctic sea ice [38].
Second, we detrended the stare-mode displacement data to remove the uniform compression and tilt allowing us to examine motion on timescales shorter than 30 s. The most prominent residual motion is synchronous, periodic oscillations with a frequency and amplitude of approximately 4 Hz and 0.2 mm. We attribute this motion to windinduced antenna vibrations and remove it using a 1-s running average. The detrended and filtered data also indicate near synchronous motion of all scatters towards and away from the antenna by around 0.2 mm during the 30 s stare interval (Figure 9a). Although the magnitude of this motion varies slightly between scatterers, this variation does not appear to be related to range. Such motion could therefore be related to periodic widening and narrowing of a dislocation fracture close to the antenna or periodic tilting of the antenna or the ridge upon which it was installed.

Geometrical Constraints of Sea Ice Strain Measurement
This work demonstrates the applicability of the GPRI instrument for the assessment of sea ice strain at scales from tens of meters to kilometers. As we have shown, the accu- Lastly, we evaluated small variation in apparent residual displacement between scatterers by subtracting the residual motion measured at scatterer s1 from the motion measured at other scatterers (Figure 9b). This calculation demonstrates that there may still be some remaining independent motion of each scatterer. Although it is near the measured noise level (Figure 9c), its spatial smoothness and consistency between s2 and s7 indicates that it contains useful information. In particular, we note that the residuals for ridges s2-s4 are highly correlated with each other over time and bear little similarity to the residual for ridges s5-s7. Regardless of the cause of this small residual motion, this result suggests that scatterers s2-s4 are mechanical coupled to each other, while scatters s5-s7 are able to move more independently. As it happens, scatters s2-s4 were all located on the same multiyear ice floe as the GPRI, while s5 corresponds to a small ridge at the transition to an adjacent first-year ice floe that contained scatterers s6 and s7 (Figure 2b). Hence, correlation between residual phase variations of scatters may be a useful way of identifying mechanical discontinuities in an apparently continuous ice pack.

Geometrical Constraints of Sea Ice Strain Measurement
This work demonstrates the applicability of the GPRI instrument for the assessment of sea ice strain at scales from tens of meters to kilometers. As we have shown, the accuracy of derived sea ice motion can be high, but depends on the acquisition geometry and the ice surface itself. The horizontal sensitivity is larger if the system is placed near sea level, largely eliminating sensitivity to vertical motion. However, a near 90-degree incidence angle can cause large specular reflection of the radar signal for level ice resulting in low SNR, coherence, and accuracy. Ridges can retain high coherence even in far range, but the ability to derive a continuous motion field over level ice can often be restricted to near-to mid-range and will depend on the presence of cm-scale ice roughness. Also, depending on the surrounding ice morphology, a near 90-degree incidence angle can result in significant areas subjected to radar shadowing behind large ridges. It can therefore be advantageous to place the GPRI system at higher elevations of meters to tens of meters. If placed on a ridge, this may expose the GPRI to increased unwanted motion from elevated tilt at higher elevation. Therefore, grounded ridges may represent an optimal ice surface to mount the GPRI if present.

Sensitivity to Vertical Ice Motion
GPRI is not only sensitive to horizontal motion, but also to vertical, especially at near range. Therefore, the GPRI can potentially have applications for deriving near range vertical motion. For a system stationed on a ridge, the accuracy in high-coherence areas is at the mm-scale out to roughly 100 m where the ambiguity is 12 cm in the vertical. We thus speculate that the system may be able to characterize cm-scale vertical ice motion as a result of ocean waves. The accuracies here are based on 4 independent looks. Vertical deformation or displacement can occur smoothly across scales of 10 1 to 10 3 m; hence accuracy could potentially be significantly improved through increased multi-looking and filtering. This could possibly enable the recognition of vertical motion in mid to far range in the absence of horizontal motion. It may also be possible to change the observation geometry to increase sensitivity to vertical motion including higher elevation of the GPRI system and larger spatial baseline between receiving antennas. The baseline is relevant if interpreting the signals from both antennas to attempt to separate LOS motion into vertical and horizontal components due to a slightly different θ. This would further be improved by using a synchronous second GPRI system placed at a different elevation. The strong dependence of d a v on range could also likely be used to determine the nature of the different motion signals in many cases. Vertical displacement as a result of underlying ocean waves is an example of a smooth signal that could potentially be identified in the near range part of the image scene.

Sensitivity to Antenna Movement
In addition to vertical ice motion, other factors may also impact a horizontal strain analysis including sensor tilting, changing sensor elevation, and changing atmospheric conditions. We modeled antenna positioning changes to investigate the resulting magnitude and phase patterns. The directions are relative, but we refer to north as towards the top of the page. First, we modeled the phase change from a 1 cm eastward sensor shift in between acquisitions for repeat scan (Figure 10a) and stare-mode interferometry with a west-facing antenna (Figure 10b). The shift leads to zero phase change in N-S direction and a gradual phase change towards the E-W that is independent of range in mid to far range. Second, we modeled the phase change in response to a 5-cm increase in sensor elevation. This leads to a more localized effect in near-range independent of look angle (Figure 10c,d). This signal is equivalent to a tidal signal if the sensor is stationed on shore and the temporal baseline is on the order of hours. Otherwise, these signals do not resemble our observed or modeled ice deformation signals as presented in Figures 4 and 5.

Sensitivity to Topography
The GPRI can also be used to derive topography using both receiver antennas. However, the application for deriving sea ice morphology is significantly hampered by a relatively small 60 cm of the standard GPRI configuration resulting in low sensitivity to topography (large ℎ ). This results in large relative contributions of noise, which cannot easily be filtered due to the often-large spatial variability of sea ice morphology in contrast to an often smoothly varying strain field. This can in turn lead to large dm-to m-scale expected errors. More work is therefore needed to explore whether the GPRI could be useful for assessing sea ice topography with appropriate accuracy [29]. Here, possibilities to improve accuracy include (1) extending the vertical baseline by utilizing a larger antenna tower, (2) utilizing multiple acquisitions in the absence of ice motion for improved SNR and accuracy, and (3) acquiring bistatic acquisitions using a configuration of two GPRI systems.

Conclusions
This work evaluates the Gamma portable radar interferometer (GPRI) as a tool to assess the dynamics and morphology of sea ice. We examined the sensitivity of the GPRI to horizontal and vertical motion as well as topography. The GPRI is highly sensitive to In this work, we attributed the horizontal sensor movement to tilting of the ridge where the GPRI is positioned. This apparent cyclic tilting back and forth could be wave induced in which the tilt corresponds roughly to waves with an amplitude A~1 cm and wavelength λ w~0 .5 km. In the case of wave motion, a large part of the signal will be due to antenna tilt resulting in synchronous peaks in the signal (similar to Figure 9a). We also expect part of the signal to be due to the vertical motion of the ice surface, which will result in an offset between the peaks. However, there is no such identifiable range-time delay in the signal. Here, more work is needed to explore how to effectively identify and track waves and other types of vertical motion, e.g., buckling using stare-mode interferometry.

Sensitivity to Topography
The GPRI can also be used to derive topography using both receiver antennas. However, the application for deriving sea ice morphology is significantly hampered by a relatively small B s ≤ 60 cm of the standard GPRI configuration resulting in low sensitivity to topography (large h a ). This results in large relative contributions of noise, which cannot easily be filtered due to the often-large spatial variability of sea ice morphology in contrast to an often smoothly varying strain field. This can in turn lead to large dm-to m-scale expected errors. More work is therefore needed to explore whether the GPRI could be useful for assessing sea ice topography with appropriate accuracy [29]. Here, possibilities to improve accuracy include (1) extending the vertical baseline by utilizing a larger antenna tower, (2) utilizing multiple acquisitions in the absence of ice motion for improved SNR and accuracy, and (3) acquiring bistatic acquisitions using a configuration of two GPRI systems.

Conclusions
This work evaluates the Gamma portable radar interferometer (GPRI) as a tool to assess the dynamics and morphology of sea ice. We examined the sensitivity of the GPRI to horizontal and vertical motion as well as topography. The GPRI is highly sensitive to lateral sea ice displacement with an 8.5 mm ambiguity displacement. Expected errors are also relatively low. Examples are 0.3 mm expected noise error for high-coherence areas (>0.95 e.g., rough ice and ridges) and 0.4 mm in the case of underlying ocean waves or antenna settling. The high sensitivity to lateral sea ice motion makes the GPRI a promising tool for examining sea ice dynamics at the km-scale. There is much unknown about floe-scale sea ice rheological behavior partly because of the challenge in obtaining observations with the required accuracy and spatial and temporal coverage. Here, we demonstrated the GPRI as a tool to augment existing methods with potential to improve our understanding of sea ice dynamics.
By modeling the interferometric phase, we investigated the ability to resolve 2dimensional strain fields in repeat-scan mode. We demonstrated this application by identifying~10 −5 shear strains in repeat-scan data out to a range of~4 km. We also operated the GPRI in stare-mode, which enabled tracking of near instantaneous sea ice strain rates of~10 −7 s −1 and identification of subtle dynamical boundaries through strain variability. Based on these preliminary results, we speculate that the GPRI may be suitable to identify fracture mechanisms by evaluating emerging spatial discontinuities in continuous repeat-scan acquisitions. Centimeter-scale fracturing mechanisms may also be tracked through stare-mode interferometry by analyzing stark temporal discontinuities in lateral strain. In the case of gradual development of multiple mm-scale cracks, the fracture mechanisms may also be evaluated by estimating strain and yield stress. We also speculate that the GPRI data may further help determine the nature (i.e., elastic, plastic, and viscous deformation) of pre-and post-fracturing mechanisms.
Further investigations into the use of GPRI to evaluate sea ice rheological behavior will depend on more acquisitions and could benefit from improved observations and analysis, including: