C-Band Polarimetric Coherences and Ratios for Discriminating Sea Ice Roughness

. The rapid decline of sea ice in the Arctic has resulted in a variable sea ice roughness that necessitates improved methods for efficient observation using high-resolution spaceborne radar. The utility of C-band polarimetric backscatter, coherences, and ratios as a discriminator of ice surface roughness is evaluated. An existing one-dimensional backscatter model has been modified to two-dimensions (2D) by considering deviation in the orientation (i.e., the slopes) in azimuth and range direction of surface roughness simultaneouslyasanimprovementinthemodel.Itisshowntheoreticallythatthecircularcoherence( 𝜌 RRLL )decreasesexponentially with increasing surface roughness. The crosspolarized coherence ( 𝜌 HHVH ) is found to be less sensitive to surface roughness, whereas the copolarized coherence ( 𝜌 VVHH ) decreases at far-range incidence angles for all ice types. A complete validation of the adapted 2D model using direct measurements of surface roughness is suggested as an avenue for further research.


Introduction
Arctic sea ice is going through a rapid decline [1,2].Thinner first-year ice (FYI) is replacing multiyear ice, leaving an ice cover, which is more sensitive to deformation and changes in atmospheric and ocean forcing.Increased open water and marginal ice zones (MIZs), due to the enhanced mobility of a relatively thinned pack ice, are further susceptible to increases in surface roughness and greater surface roughness variability [3].Greater surface roughness in the MIZ is of importance due to higher rates of heat flux [4] and momentum [5] exchanges occurring across the ocean-sea iceatmosphere interface, greater biological productivity [6], and potential limitations imposed on ship navigation.Although the literature contains information on how the MIZ responds to wind and wave forces, it is necessary to investigate the electromagnetic (EM) response of the MIZ to facilitate satellitebased observations.Satellite-based observation is necessary due to the scarcity of surface observations in a MIZ, as well as the difficulties in collecting physical measurements due to the instability and roughness of the ice floes.
The use of polarimetric synthetic aperture radar (pol-SAR) represents a promising approach for satellite-based monitoring of surface roughness and, concurrently, discriminating sea ice types within a MIZ.A pol-SAR records the amplitude and phase information of backscattered energy for four transmit-receive polarizations (HH, HV, VH, and VV), thereby facilitating the derivation of the full polarimetric response of the target.It is recognizable that the diversity in polarization achievable by pol-SARs or even by dualpolarization SAR systems provides more complete inference of target features (e.g., sea ice) than conventional, single channel SARs.Furthermore, recently launched pol-SARs are capable of higher spatial resolution (<10 m) imaging, leading to enhanced potential for monitoring complex ice environments.
Discrimination of ice types using SAR has been conventionally achieved by utilizing different combinations of linearly polarized backscattering coefficients [7][8][9].Multiyear ice, smooth FYI, rough FYI, and new ice/open water in the Beaufort and Chukchi Seas during March have been identified using a single-polarization SAR image intensitybased classification scheme [9] while others used singlepolarization SAR image texture analysis to discriminate new ice, FYI, and multiyear ice during the month of March in the Beaufort Sea and the Mould Bay, respectively [10,11].Dual International Journal of Oceanography copolarized backscattering coefficient differences in HH and VV have been used to discriminate FYI, multiyear ice, and lead areas in the Beaufort Sea during March [12].However, the complexities in polarimetric signatures associated with the dynamic mixture of surface roughness and ice type conditions in an MIZ during fall freeze-up remain to be examined.Such an examination requires utilizing polarimetric radar backscatter, so that the material (dielectric) and geometrical properties of the surface, which influence backscatter, may be individually assessed.
In this study, ship-based observations of co-(linear) and crosspolarized backscatter, circular polarimetric coherences ( VVHH ,  HHVH , and  RRLL , resp.), and copolarized and crosspolarized polarization ratios ( co and  cross , resp.), are used to evaluate their utility for ice surface discrimination capabilities using a polarimetric radar operating in C-band (5.5 GHz).Characteristics of these polarimetric parameters for a variety of ice types in an MIZ during fall freeze-up are assessed with the following objectives: (1) to investigate the performance of polarimetric  RRLL for sea ice surface roughness discrimination by adapting the one-dimensional backscatter model of [13] to two dimensions and introducing roughness as deviations in range and azimuth directions, (2) to evaluate the utility of C-band polarimetric backscatter, coherences, and polarization ratios as a discriminator of surface roughness or ice type in a MIZ during fall freeze-up.

Study Area.
The study area is located in the southeastern Beaufort Sea and Amundsen Gulf regions in the western Canadian Arctic (Figure 1).The seasonal Cape Bathurst Polynya forms in the region and hosts a number of flaw leads during the winter [14].During fall freeze-up, this area contains a variable mix of ice types under various stages of formation, for example, new ice, pancake ice, frost flowers, deformed ice, gray ice, and nilas (Figure 2).The photographs in Figures 2(a [15].Ancillary meteorological data were collected through a ship-based AXYS Technologies Inc. (Sydney, BC, Canada) Automatic Voluntary Observing Ships (AVOS) system.This system was mounted approximately 20 m above sea level on the wheelhouse to minimize the ship's influence and could measure air temperature and wind speed.

Theoretical Formulation.
Sea ice is a distributed radar target, and the conditions of stationarity and homogeneity seldom hold for dynamically changing ice in a MIZ.The radar backscattering is therefore analyzed using temporally and spatially varying stochastic processes.Backscatter from sea ice is incoherent and either partially or completely polarized, as described by the polarimetric covariance matrix.The electric field vector of an incident () and scattered () EM wave can be given by E  =   ĥ +   k, where  − / term accounts for wave propagation effects in amplitude and phase.If the orientation of a surface such as sea ice in azimuth direction is rotated by an angle, the corresponding new backscatter matrix can be constructed as provided by Lee et al. [16].
The coherency matrices can be derived as copolarized (3), crosspolarized (4) and circular (RRLL: right-right left-left rotation of the electric field vector about the line of sight) (5) coherences in magnitude form [13,16] as (for derivation of  RRLL , see Appendix A), where  is the complex scattering matrix; an asterisk ( * ) represents the complex conjugate.The brackets ⟨⋅⟩ represent ensemble averages of the observed data.There were approximately 34 pulses sent per incidence angle.An ensemble average was performed on those 34 pulses.Raw data were processed into range profiles and were averaged in the azimuth for each measured incidence angle.Polarimetric ratios  co and  cross are simply power ratios of backscattered energy.Polarimetric coherences and polarization ratios have demonstrated utility in reducing the ambiguities caused by the nonlinearity between system response and target properties.Regarding Arctic sea ice, some literature is available on the use of  VVHH ,  RRLL , and  co at different EM frequencies.C-band backscatter coefficients (HH, HV, and VV) and  VVHH have been used to characterize various FYI types (compressed, rubble and ridge, and smooth) and multiyear ice [17].Thin sea ice has been effectively discriminated from FYI using C-band  co ratio [18]. VVHH and  co have been used to discriminate Arctic leads using L-band radar signatures [19].In a similar study, Wakabayashi et al. [20] described polarimetric characteristics of different FYI types (thin ice, smooth, and rough) using L-band  RRLL and  co and showed the utility of coherences and ratios in discriminating ice types.Nakamura et al. [21] discriminated ice surface using  co ratio in an observational study of lake ice using airborne L-and X-band SAR.These studies lack a holistic overview of the utility of different polarimetric coherences and ratios to discriminate thin FYI types in a MIZ.

Active Microwave Backscattering
Data.C-band polarimetric backscattering data were collected using a completely stationary ship-mounted scatterometer system developed by ProSensing Inc., (Amherst, MA, USA) and mounted 7.56 m above the mean sea level on the port side of the Amundsen (Table 1).The system acquires backscatter and phase data in terms of the combinations of linear transmit-receive polarization combinations, HH, HV, VH, and VV at incidence angles 20-60 ∘ (5 ∘ increments) over a 60 ∘ azimuth range.The calibration of the instrument was performed through the methods given elsewhere [22,23].Polarimetric backscattering data were collected from homogeneous samples of snowcovered (dry and fresh) first-year ice (SCFYI), deformed FYI (DFYI), consolidated pancake ice (PI), snow-covered frost flowers (SCFF), and dense frost flowers (DFF) on different dates during November 2007.Data from each ice type sample comprised three to four contiguous scatterometer scans, which took up to 35 minutes to complete.The scatterometer had a footprint of 1.1 m 2 in the range direction at a 45 ∘ incidence angle [23] with the footprint increasing in size with incidence angle [22].Towards objective 2, scan data for each ice type were grouped by incidence angle representing near (20-25 ∘ ), mid (35-40 ∘ ), and far (55-60 ∘ ) range groupings.These groupings best represent the diversity of scattering mechanisms available across the acquired incidence angle range.In the near range, surface scattering is expected to dominate the measured C-band backscatter, while surface-volume scattering is increasingly expected to influence C-band backscatter beyond approximately 30 ∘ , that is, mid to far ranges [24].Furthermore, combining data from adjacent incidence angles doubled the number of samples from 8 to 18 depending on ice type, although at the expense of range resolution.Scatterometer data had unequal number of data points in each range group, which does not fulfil parametric ANOVA requirements for statistical significance testing.Polarimetric coherences and ratios of ice types were tested for independence from each other for each incidence angle grouping.Testing was done using the nonparametric Kruskal-Wallis statistic, with  = 0.01 significance level (one tailed) used as the threshold for statistical independence.

Surface Roughness and Circular Coherence.
In pursuit of objective 1, a polarimetric backscattering model is used which is mainly a Bragg backscattering (coherent scattering) model modified for surface roughness considering the surface slope by slightly changing the tilt of the surface from the horizontal.Microwave measurements of surface roughness using co-or crosspolarization backscattered power are most successful in flat areas.In sea ice microwave remote sensing, the dielectric constant and topography (slope in range and azimuth) are important.According to (22) in the one-dimensional scattering model of [13], the circular coherence is only sensitive to surface roughness.Surface roughness has been considered as a change in the slope of ice in azimuth and ground range directions [13,16,25].This is implemented mathematically in the Bragg backscattering model by considering roughness as a depolarizer which conforms to reflection symmetry; that is, the backscattering properties are identical on either side of the plane of incidence and HV = VH [25,26].The distribution of azimuth slope angles  1 is considered as one-dimensional Gaussian distributed [13].
The rotation matrix [16] and the coherency matrix [13] are calculated after introducing the rotation in azimuth anticlockwise about range direction.In this case,  RRLL is derived as [13] where   1 is the standard deviation of the orientation angle distribution in azimuth direction and  1 is slope angle in azimuth direction.From (8), the  RRLL is only dependent on the orientation of ice surface in the range direction, or the standard deviation of the orientation angle distribution (i.e., surface roughness).Here, the surface roughness is introduced through rotation by angle  2 in the range direction anticlockwise about azimuth direction (Figure 3).Angle  2 is not shown in Figure 3 due to complexity of the geometry.In this case also, the corresponding distribution of shift in orientation angle is Gaussian distributed.The new rotation matrix  2 is given by The new averaged coherency matrix over the Gaussian distribution ( 2 ) can be calculated as where ( 2 ) = ∫cos 2 2 ( 2 ) 2 and ( 2 ) = ∫cos 2 2 2 ( 2 ) 2 ., a part of an element of coherency matrix, is defined according to scattering matrix,  [13]. * is the conjugate of .Both  * and  are not used in the computation of  RRLL .
Given the above, the  RRLL is dependent on the standard deviation of the orientation angle distribution in range and the dielectric constant of the surface.Thus, it is shown that the new  RRLL is exponentially changing with the change in orientation angle in the azimuth direction, but it behaves in a way given by ( 12) and is dependent on both surface roughness (standard deviation) and the dielectric constant (scattering matrix) of the surface when roughness in two directions is considered.In our model, when two-dimensional roughness is considered, circular coherence is observed to be sensitive to both surface roughness and dielectric constant, thus, making it difficult to differentiate roughness.2D model being more realistic requires further considerations of separating dielectrics from roughness.Now, the slope-induced roughness is examined in the range direction only.Lee et al. [16] gave a relationship between slope in azimuth, slope in ground range, radar look angle (), and rotation in azimuth.Schuler et al. [13] expressed this relationship in terms of root mean square (rms) surface height () and correlation length (), assuming that the range slope and orientation in azimuth are small perturbations around their means, Figure 4 shows the incidence angle dependence of  RRLL by varying the  2 / 2 ratio.As the roughness increases,  RRLL decreases.For  ≫ , that is, the surface is very smooth, the maximum value of  RRLL approaches unity. RRLL decreases exponentially from unity to a fixed value of  2 / 2 ratio at a particular incidence angle.A rough surface yields a smaller  RRLL , which increases with increasing radar look angle.The range of  2 / 2 for the presented ice classes is expected to lie between 0.001 and 0.1 [27].
The relationship between slopes in azimuth and range direction is further demonstrated.Corresponding shifts and radar incidence angle are given by (see Appendix B), where tan  is azimuth slope, tan  is range slope,  1 and  2 are the perturbations in orientation in azimuth and range directions, respectively, and  is radar look angle.Figure 4 represents the case when orientation shift in the range direction is observed.In a sea ice remote sensing context, both surface roughness and the dielectric constant of ice affect  RRLL when slope is changed in azimuth direction, whereas only surface roughness affects  RRLL when slope is changed in range direction.

Sea Ice Type Discrimination (Coherences and Ratios).
The date and hour of scatterometer data acquisitions corresponding to each sea ice type, as well as coincident meteorological parameters, namely, wind speed, air temperature, and relative humidity, are provided in Table 2.The photographs of the selected ice samples are shown in Figure 2.With the exception of wind speed, there is negligible variation in meteorological conditions between ice type scans.As such, it is expected that between-scan, temperature-induced effects on the dielectric properties, and backscattering intensities from the different ice types are negligible.
Figure 5 shows backscattering coefficients for co-(HH and VV) and crosspolarization (HV) configurations of each ice type.The two frost flower cases (DFF and SCFF) are plotted separately to exemplify differences in backscattering behavior on the basis of their different frost flower concentrations.The DFF and SCFF have a visually measured concentration of approximately >95% and 20%, respectively.While SCFYI is visually separable using HH, HV, and VV polarizations at all incidence angles (low backscatter), PI and DFYI signatures overlap and are difficult to separate from each other.This may be indicative of PI geometry within the scatterometer footprint, as PI comprises a series of upturned edges and flat areas of ice (see Figure 2).The curvature of upturned PI edges causes a backscatter response similar to that caused by the deformations (upturned ice) in the DFYI.DFF and SCFF are differentiable at HV and VV polarization at mid to far incidence angles.Mean coherences and polarization ratios for each ice type as a function of incidence angle grouping are documented in Table 3.All sea ice types show high  VVHH , indicating low depolarisation and primarily single (surface) backscattering.The  HHVH for DFF is notably higher than that from the other ice types, which points to strong depolarisation caused by the frost flower structures.As shown in the previous section, a low value of  RRLL indicates a rougher surface.At mid to far ranges in Table 3, the  RRLL for DFYI is the lowest while for SCFYI it is the highest, which is consistent with the roughest and smoothest ice types, respectively.Furthermore, for frost flower-covered surfaces, that is, SCFF and DFF, the lower magnitude of  RRLL is consistent with the higher concentration of frost flowers.At near-incidence angle range, the SCFYI shows higher roughness (i.e., lower  RRLL = 0.47, Table 3) compared to that of PI (0.61).This may be due to the fact that the snow is dry and has low salinity, which allows EM waves to penetrate through the snow.This is likely to provide roughness of snow-ice interface rather than air-snow interface.At mid incidence angle range, as expected, SCFYI shows lower roughness (i.e., higher  RRLL = 0.65, Table 3) compared to that of PI (0.61).Mid incidence angles are well suited for differentiating ice roughness/types using  RRLL .
Looking at polarization ratios in Table 3, the  co increases rapidly with incidence angle and is the highest at the far range for SCFYI.The  co ratio is also high for DFF, but it remains fairly constant across all incidence angles.High  co is also representative of saline ice surface (FYI in this case) or surface scattering.The presence of dry snow (∼1-2 cm) allows the EM waves to penetrate through snow, which causes reflection from the ice-snow interface.The  co behavior of SCFYI is consistent with that of a surface, which is very smooth (i.e., a Bragg surface), where the ratio between backscattered H and V is only dependent on incidence angle and dielectric constant [28].On the other hand, the  co behavior for DFF is consistent with that of a rough surface exhibiting backscatter from features with preferential vertical orientation [22].Including the  cross ratio in this comparison further supports the distinction in backscattering mechanisms.The near range  cross ratio is much smaller for SCFYI than DFF, indicating it to be much smoother.The DFYI and DFF show the highest overall  cross , due to multiple scattering within deformities for DFYI and depolarisation caused by frost flowers for DFF. Figure 6 shows box plots of coherences and polarization ratios of each ice type.Table 4 provides the significance values resulting from statistical tests for independence between each ice type based on a given coherence or ratio.All data in Figure 6 and Table 4 are based on the aforementioned incidence angle groupings from near to far range and, together, facilitate a conceptual approach to assessing the utility of each parameter for distinguishing ice types within an MIZ.Summarizing Figure 6 and Table 4, the near range  HHVH and Significance values are provided in Table 4.
cross provide the greatest separation between classes, while the far range  VVHH and  co provide the greatest separation.By combining  co (far) with either of  HHVH or  cross (near), all ice types are independent of each other.From Figure 4 it is known that a lower  RRLL is associated with a rougher ice surface.
It is demonstrated using theory that lower values of  RRLL indicate a rougher ice surface.Referring to Figures 4 and 6, International Journal of Oceanography  RRLL is high for increasing incidence angles and for low surface roughness.This is only true for SCFYI and SCFF.In the presence of dry and fresh snow the volume contribution from FYI can be ignored, in which case  RRLL dictates surface roughness of the snow-ice interface rather than air-snow interface.The coherence estimates are negligibly affected by the signal-to-noise ratio (typically >10 dB) during the processing of scatterometer data.These coherences can also be computed using polarimetric observations from spacebased platforms.

Summary and Conclusions
The one-dimensional backscatter model of Schuler et al. [13] was modified to two dimensions of surface roughness by considering deviation in the orientation angles (i.e., the slopes) in azimuth and range direction simultaneously as an improvement in the model.Parameters derived from the fully polarimetric C-band microwave backscatter response from sea ice targets were demonstrated to have utility for small-scale (cm level) sea ice roughness identification.Circular coherence has been investigated for its usefulness in discriminating surface roughness among other polarimetric parameters.Circular coherence is theoretically shown to detect measurement sensitivity to surface roughness.
The conclusions with reference to objective 1 are as follows.It was shown theoretically that the  RRLL decreases exponentially with increasing surface roughness.However,  RRLL responds to both roughness (standard deviation) and dielectric constant (scattering matrix) of the surface in the case when the orientations of the ice target in azimuth direction are changed.It remains challenging to separate roughness effects from the dielectric effects using C-band backscatter measurements. RRLL independently does not provide a robust sea ice roughness discrimination scheme.However,  RRLL provides an improved insight of sea ice surface roughness combined with other polarimetric coherences and channel ratios in the chosen samples.The experimental data also show that rougher ice surface exhibits lower mean value of  RRLL (Table 3, Figure 4), though a complete validation of the effect of changing orientations of ice floe on  RRLL is required.This would require polarimetric backscattering data and surface roughness information to be acquired at different lines of sight (i.e., orientation of ice floes).Unfortunately, difficulties associated with extreme weather conditions and limitations to navigation in the Arctic restrict such detailed data acquisition; however, a tank experiment could be a useful alternative.
The utility of C-band polarimetric coherences and ratios is addressed in the light of objective 2 as follows: for coherences,  VVHH is smaller at far range incidence angles for all ice types. HHVH is less sensitive to roughness and is not a good discriminator of roughness.Regarding channel ratios, based on Kruskal-Wallis test,  co is more sensitive to increasing surface roughness compared to  cross and demonstrates utility for separating ice types compared to the other observed parameters.
The knowledge obtained through surface-based polarimetric coherences and ratios can readily be extended to discriminate sea ice roughness on small scales using Cband microwave satellites (currently in orbit RADARSAT-2, RISAT-1).Future work will be to develop an algorithm combining all polarimetric coherences and ratios to discriminate individual ice type in a MIZ.These observations may become particularly useful for satellite measurements once planned SAR constellations (Sentinel series) systems are available, as currently planned by National Aeronautics and Space Administration and European Space Agency.In (B.6), if the perturbation in orientation in range direction is zero, that is,  2 = 0, it reduces to equation given by Lee et al. [16].

Figure 1 :
Figure 1: Geographic map of study area showing sampling locations.

Figure 3 :
Figure 3: Illustration of scattering plane geometry with slight deviations in the orientation angles in azimuth ( 1 ) and range directions ( 2 : not shown), respectively, as means of two-dimensional surface roughness.

Figure 4 :
Figure 4:  RRLL varying with squared ratio of rms surface height and surface correlation length;  RRLL decays exponentially; however, it decays faster at steep incidence angles.

Figure 6 :
Figure 6: Box plots of coherences and polarization ratios of ice types based on near, middle, and far range incidence angle groupings.Significance values are provided in Table4.

Table 1 :
Technical properties and specifications of C-band scatterometer.

Table 2 :
Meteorological parameters associated with each ice type on different dates.
# Bold numbers indicate important significant values.