The physical model for sea ice includes both internal structure (such as salinity, brine and bubbles, including their size, distribution, and concentration) and external characteristics (such as surface roughness, thickness, and related characteristics) that influence the microwave scattering characteristics of sea ice as illustrated in Fig. 1. The incident angle and transmitted angle of the electromagnetic wave are θ and θ_t, respectively. The transmitted wave is scattered by the scatterers embedded in the sea ice at the air–ice interface, and the corresponding incident and scattering angle are represented as θ₀ and θ_s, respectively. The thickness of the sea ice is d.

Fig. 1.

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	Fig. 1. Physical and microwave scattering model of sea ice, showing (i) top surface scattering, (ii) volume scattering, (iii) bottom surface scattering, and (iv) scattering due to surface–volume interactions.

The sea ice electromagnetic scattering includes several components: (i) top surface scattering from the air–ice interface, (ii) volume scattering from the scatterers in the sea ice, (iii) bottom surface scattering from the ice–water interface, and (iv) scattering from the interaction of the surfaces and the scatterers as shown in Fig. 1 clearly. Therefore, according to these four components, the total bistatic scattering coefficients of sea ice can be described as follows:^[29] 1 where σ⁰ represents the total bistatic scattering coefficient, σ_ts the top surface scattering, σ_vol the volume scattering, and σ_in the interaction term.

Here, σ_ts is derived from IEM and has the following form:^[27] 2 where k is the wave number, k_x = ksinθ cosϕ, k_y = ksinθ sinϕ, k_z = k cosθ, k_sx, k_sy and k_sz have similar definitions by replacing θ and ϕ with θ_s and ϕ_s, pp represents VV or HH polarization, S(θ, θ_s) is the shadowing function defined by Sancer^[31] (Appendix A), σ is the RMS height of the surface, Wⁿ is the Fourier transform of the n-th power of the surface correlation coefficient (Appendix B), and the expression for is given in Appendix C.

Top surface scattering can be expressed as 3 and bottom surface scattering has the following form:^[27] 4

The volume scattering term is derived from the RT model:^[27] 5 where μ = cosθ, μ_s = cosθ_s, μ₀ = cosθ₀, μ_t = cos θ_t, κ_s and κ_a are volume scattering and absorption coefficients, respectively. In terms of the above parameters, the extinction coefficient, κ_e, the albedo, a, and the optical depth, τ are found as follows: κ_e = κ_a + κ_s, a = κ_s/κ_e, and τ = κ_ed, respectively. Detailed calculations of the Rayleigh parameters are given in Appendix D. Here, T is the refraction coefficient and P_pp represents the Rayleigh phase term,^[27] 6 7

The interaction term is also derived from the RT model,^[27] 8 For the backscattering case, θ = θ_s, ϕ = 0, and ϕ_s = π. Thus, the backscattering coefficients of sea ice can be calculated from Eq. (1) to Eq. (8).

The polarization ratio of sea ice is defined as follows: 9 The normalized form of the PR_i is 10 From Eq. (1) to Eq. (10), the backscattering coefficients and corresponding polarization ratio of sea ice can be calculated. Next, numerical experiments are carried out to study the polarization ratio characteristics of seawater, bare and snow-covered sea ice, for both L-band backscatter and C-band backscatter.

The ability of microwave radar to penetrate sea water is weak, and the penetration depths at C-band and L-band are about 0.4 cm and 3 cm, respectively.^[29] Therefore, microwave electromagnetic scattering mainly acts on the sea surface, generating surface scattering, which can be calculated by the integral equation method (IEM). The electromagnetic scattering of seawater is mainly related to incident angle, electromagnetic wave frequency and wind speed. The wind speed affects the sea surface roughness (characterized by the RMS height and the correlation length), which in turn affects the radar backscattering cross-section.

For L-band and C-band radars, the variations of the polarization ratio of seawater with the RMS height of the sea surface are shown in Figs. 2(a) and 2(b), respectively. The C-band and L-band polarization ratio of seawater increase with incident angle and decrease with RMS height of the sea surface. By comparison, the polarization ratio is more sensitive to RMS height for C-band than that for L-band. At small-to-medium incident angles, the L-band seawater polarization ratio hardly varies with the RMS height. As shown in Fig. 3, the seawater polarization ratio remains almost constant as the correlation length of the sea surface increases, at both C and L band. In addition, we also find that the L-band seawater polarization ratio is greater than the C-band sea ice polarization ratio.

Fig. 2.

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	Fig. 2. Variations of polarization ratio of seawater with RMS height of sea surface for (a) L-band and (b) C-band. The correlation length is 5 cm, the dielectric constant of seawater for L-band and C-band are 82.70+j23.88 and 64.21+j37.52, respectively.

Fig. 3.

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	Fig. 3. Variations of polarization ratio of seawater with correlation length of sea surface for: (a) L-band and (b) C-band. RMS height is 0.6 cm, dielectric constant of seawater for L-band and C-band are 82.70+j 23.88 and 64.21+j 37.52, respectively.

In this subsection, the polarization ratios of bare sea ice under different physical conditions are simulated. Simulation parameters used in the model are given in Table 1.

Table 1.

Table 1.

Table 1. Simulation parameters for bare sea ice. . Variable Parameter Simulation value f frequency 1.25 GHz for L-band, 5.3 GHz for C-band σ1 RMS height of air–ice interface 0.4 cm σ2 RMS height of ice–water interface 0.2 cm L1 correlation length of air–ice interface 5 cm L2 correlation length of ice–water interface 3 cm εr dielectric constant of scatterers in the sea ice 82.70+j23.88 for L-band, 64.21+j37.52 for C-band εri dielectric constant of sea ice 3.4+j0.22 εrw dielectric constant of seawater 82.70+j23.88 for L-band, 64.21+j37.52 for C-band d sea ice thickness 20 cm fv volume fraction of scatterers 10% r scatterer radius 0.25 mm

Table 1. Simulation parameters for bare sea ice. .

As shown in Fig. 4(a), the L-band bare sea ice polarization ratio increases with incident angle increasing and decreases with RMS height of the ice surface increasing. When incident angles are smaller than about 45°, the C-band bare sea ice polarization ratio hardly changes with RMS height increasing; when incident angles are greater than 45°, the polarization ratio decreases with the RMS height increasing (Fig. 4(b)). For both C and L-bands, the bare sea ice polarization ratio hardly changes with correlation length (Fig. 5).

Fig. 4.

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	Fig. 4. Variations in bare sea ice polarization ratio with RMS height of ice surface for: (a) L-band and (b) C-band.

Fig. 5.

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	Fig. 5. Variations in bare sea ice polarization ratio with correlation length of ice surface for: (a) L-band and (b) C-band.

The variations in bare sea ice polarization ratio with ice thickness at three typical incident angles are shown in Figs. 6(a) and 6(b). The former decreases as ice thickness increases and remains constant after the ice thickness has reached a critical value, which is mainly related to the penetration depth of electromagnetic wave. Because the penetration depth of L-band for sea ice is greater than that of C-band, the L-band sea ice polarization ratio is more sensitive to change in sea ice thickness than the C-band sea ice polarization ratio. At small incident angles, electromagnetic scattering is dominated by surface scattering, and the polarization ratio hardly changes with ice thickness. As the incident angle increases, the sea ice polarization ratio also increases; that is, the difference between VV polarization and HH polarization increases, which is mainly caused by bottom surface scattering and volume scattering. The corresponding variations in backscattering coefficient with ice thickness are shown in Figs. 6(c) and 6(d), which have similar trends to the variation of the polarization ratio with ice thickness. The backscattering coefficients of bare ice decrease with the incident angle for both L- and C-bands, due to the dominant specular scattering mechanism, over the large angular domain.

Fig. 6.

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	Fig. 6. Variations in bare sea ice polarization ratio with ice thickness at three typical incident angles for (a) L-band and (b) C-band, and corresponding variations of backscattering coefficients for (c) L-band and (d) C-band.

The scatterers in the sea ice (mainly brines and bubbles) are the source of the volume scattering and bottom-volume scattering interaction. Since the interaction is weak, only ‘single-bounce’ interaction is considered. In the experiment, we only consider brines. Figures 7 and 8 show the polarization ratios as a function of the volume fraction and radius of scatterer, for different incident angles and radar bands. In both bands, the polarization ratio decreases slightly with volume fraction and radius of the scatterer. The influences of these two parameters on the bare sea ice polarization ratio are negligible. This indicates that the volume scattering has little effect on bare sea ice polarization ratio for both L-band and C-band.

Fig. 7.

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	Fig. 7. Variations of the bare sea ice polarization ratio with brine volume fraction at three typical incident angles for: (a) L-band and (b) C-band.

Fig. 8.

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	Fig. 8. Variations of the bare sea ice polarization ratio with scatterer radius at three typical incident angles for (a) L-band and (b) C-band.

Snow may have a significant influence on the polarization ratio characteristics of sea ice. Similarly, the total backscattering now consists of the top surface scattering from the air-snow interface, volume scattering in the snow, bottom surface scattering from the snow-ice interface and bottom-volume scattering interaction. To study the effect of snow on the sea ice polarization ratio, numerical experiments were carried out based on the above scattering model. The parameters set in the model are shown in Table 2.

Table 2.

Table 2.

Table 2. Simulation parameters for snow-covered sea ice. . Variable Parameter Simulation value f frequency 1.25 GHz for L-band, 5.3 GHz for C-band σ1 RMS height of air-snow interface 0.4 cm σ2 RMS height of snow-ice interface 0.2 cm L1 correlation length of air-snow interface 5 cm L2 correlation length of snow-ice interface 3 cm εr dielectric constant of scatterers in the snow 4.13+j0.38 εri dielectric constant of sea ice 4.13+j0.38 εrs dielectric constant of snow 1.6+j0.55 d snow thickness 20 cm fv volume fraction of scatterers 20% r scatterer radius 0.5 mm

Table 2. Simulation parameters for snow-covered sea ice. .

As shown in Fig. 9(a), when there is snow cover on the sea ice, the polarization ratio of the L-band radar for sea ice increases with the incident angle increasing, and decreases with the RMS height of the snow surface increasing. When the incident angle is less than 60°, the decrease can be neglected (less than 1 dB); that is, the L-band sea ice polarization ratio does not vary with the RMS height of snow surface. In Fig. 9(b), when the incident angle is smaller than 50°, the polarization ratio of the C-band radar scattering on the sea ice increases as the incident angle increases; and when the incident angle is larger than 50°, it decreases as the incident angle increases. When the incident angle is less than 60°, the sea ice polarization ratio increases with the RMS height of snow surface increasing. When the incident angle is greater than 60°, the sea ice polarization ratio decreases with the RMS heightening. Comparing with C-band radar, the effect of the snow surface RMS height on the L-band sea ice polarization ratio is negligible. It can be seen from Fig. 10 that the influence of the correlation length variation on the polarization ratio of the sea ice for C-band and L-band are negligible.

Fig. 9.

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	Fig. 9. Variations of snow-covered sea ice polarization ratio with RMS height of snow surface for (a) L-band and (b) C-band.

Fig. 10.

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	Fig. 10. Variations of snow-covered sea ice polarization ratio with correlation length of snow surface for (a) L-band and (b) C-band.

The variations of sea ice polarization ratio with snow thickness at three typical incident angles are shown in Fig. 11. The L-band sea ice polarization ratio does not vary with snow thickness while the C-band sea ice polarization ratio decreases with snow thickness increasing. At small-to-medium incident angles, the influence of snow thickness on C-band sea ice polarization ratio is negligible due to the relevant mechanism for the dominant surface scattering in this angular region. For L-band, the backscattering coefficient of snow-covered sea ice shown in Fig. 11(c) does not change with snow thickness either. For C-band, in Fig. 11(d), with the increasing of snow thickness, the backscattering coefficient of sea ice remains constant at small-to-medium incident angle and increases at large incident angle. The backscattering coefficient of snow-covered sea ice decreases with incident angle increasing. The reasons for these results have been described earlier.

Fig. 11.

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	Fig. 11. Variations of snow-covered sea ice polarization ratio with snow thickness at three typical incident angles for (a) L-band and (b) C-band, and corresponding variations of backscattering coefficients for (c) L-band and (d) C-band.

As shown in Fig. 12(a), at three typical incident angles, the L-band sea ice polarization ratio does not change with scatterer volume fraction. At a small incident angle (20°) and medium incident angle (40°), the C-band sea ice polarization ratio hardly changes with the scatterer volume fraction; at a large incident angle (60°), the C-band sea ice polarization ratio decreases with scatterer volume fraction. The variation of the sea ice polarization ratio with scatterer radius is shown in Fig. 13. The L-band sea ice polarization ratios do not change with the scatterer radius in three cases, while the C-band sea ice polarization ratios do not change with the radius of scatterer at a small incident angle (20°) and decreases with the scatterer radius at a medium incident angle (40°) and large incident angle (60°).

Fig. 12.

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	Fig. 12. Variations of snow-covered sea ice polarization ratio with scatterer volume fraction at three typical incident angles for: (a) L-band and (b) C-band.

Fig. 13.

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	Fig. 13. Variations of snow-covered sea ice polarization ratio with scatterer radius at three typical incident angles for (a) L-band and (b) C-band.

For the L-Band, according to this analysis, we find that the seawater polarization ratio only depends on incident angle, while the sea ice polarization ratio is mainly affected by incident angle and ice thickness. The influence of other factors on it is negligible. Figure 14(a) shows the variations of L-band polarization ratio with incident angle for sea water, bare sea ice with different thickness and snow-covered sea ice. It can be seen that there is obvious difference between the sea water polarization ratio and the sea ice polarization ratio in Fig. 14(b). The larger the incident angle and the ice thickness, the greater the difference between them will be. When the incident angle exceeds 40°, the polarization ratio can be used to distinguish between sea water and sea ice. When snow covers the sea ice, the difference between the polarization ratios of sea ice and sea water increases, making it easier to distinguish between sea water and sea ice.

Fig. 14.

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	Fig. 14. For L-band, (a) variations of the polarization ratio with incident angle for seawater, bare and snow-covered sea ice and (b) their difference varying with incident angle.

For the C-band, the polarization ratio for radar scattering on bare sea ice is mainly related to the incident angle, RMS height and ice thickness, and decreases with the ice thickness. When the ice thickness reaches 10 cm, the sea ice polarization ratio no longer changes because the ice thickness is generally more than 10 cm in polar areas, and the effect of ice thickness is negligible. Therefore, it is only necessary to compare the sea ice polarization ratios under different incident angles and RMS heights. As shown in Fig. 15, the solid line indicates the sea ice polarization ratio, and the dashed line refers to the seawater polarization ratio. Figure 15(b) shows the mean PR difference between sea water and sea ice for C-band. When the incident angle is less than 40°, the mean PR difference is quite small; when the incident angle exceeds 40°, the mean PR difference for C-band increases with the incident angle. Therefore, for C-band, sea ice and sea water can be discriminated by the polarization ratio at medium-to-large incident angles.

Fig. 15.

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	Fig. 15. For C-band: (a) Variations of polarization ratio for seawater, and sea ice with incident angle and (b) their difference varying with incident angle.