Backscatter model for the unusual radar properties of the Greenland Ice Sheet

. A number of planetary objects exhibit large radar reflectivity and polarization ratios, and more recently, a similar behavior has been observed over a vast portion of the Earth's surface: the percolation facies of the Greenland Ice Sheet. Surface-based ranging radar data and snow stratigraphy studies demonstrated that the radar properties of that portion of Greenland are caused by enhanced scattering from massive, large, solid-ice bodies buried in the top few meters of the dry, cold, clean snowy surface of the ice sheet and created by seasonal melting and refreezing events. Here, we model the icy inclusions as randomly oriented, discrete, noninteracting, dielectric cylinders embedded in a transparent snow medium. An exact analytical solution is used to compute the scattered field from the cylinders. Using this model, we correctly predict the polarimetric radar observations gathered by an airborne imaging system at three wavelengths (5.6, 24, and 68 cm), between 19 ø and 65 ø incidence angle. The diameter and number density of the cylinders that are inferred from the radar data using the model are consistent with in situ observations of the icy inclusions. The large radar reflectivity and polarization ratios are interpreted as arising from internal reflections of the radar signals in the icy inclusions that first-order external reflection models fail to predict. The results compare favorably with predictions from the coherent backscatter or weak localization theory and may provide a complementary framework for interpreting exotic radar echoes from other planetary objects.

. It had been known for several years that the percolation facies exhibited an unusual strong type of radar backscattering [Swift et al., 1985], but it was not until 1991 that calibrated radar data could be gathered in that region, at three different wavelengths, multiple incidence angles, and most important, with the full polarimetry, using the NASA/Jet Propulsion Laboratory AIRSAR airborne synthetic-aperture radar imaging system [•an Zyl et al., 1992]. The AIRSAR results showed that the circular and linear polarization ratios of the Greenland percolation facies are extremely large and comparable in magnitude to the largest ratios recorded for EGC. These icy formations are well known to glaciologists [Benson, 1962;Pfeifer et al., 1991;Jezek and Gogineni, 1992;Echelmeyer et al., 1992]. They form in the top few meters of the snowy surface of the ice sheet as a result of seasonal melting and refreezing events. They differ from the glacial ice, 50 to 100 m underneath the surface, that results from diagenetic processes transforming snow into solid ice. The physical processes leading to the formation of the icy inclusions have been studied in great detail [Benson, 1962;Pfeifer et al., 1990]. At these high elevations (>_2000 m above mean sea level) and high northern latitudes (_>63øN) snow remains at temperatures <OøC throughout the summer, except at point locations where meltwater can percolate downwards, along active channels, through much of the previous winter's accumulated snow. Meltwater refreezes at depth (_<1 m) when it encounters a discontinuity in hydraulic conductivity associated with a fineto-coarse grain size transition [Pfeifer and Humphrey, 1992]. When .... • activ. e, the percolati0n channels appear slushy. When refr•'•/'zing, they form a network of ice pipes, lenses and layers that distribute laterally, sometimes over great distances. Ice lenses are lens-shaped layers which pinch out laterally, parallel to the firn (old snow) strata; while ice pipes are pipelike, vertically ex-tending masses reminiscent of the percolation channels which feed ice lenses and layers. Ice layers are typically several millimeters to a few centimeters thick and extend over several tens of meters. Ice pipes (Figure 1) and ice lenses are 2-20 cm wide and 10-100 cm long [Jezek et al., 1994].
Ice layers also form at lower elevations, in the socalled soaked•snow facies [Benson, 1962], but the snow there reaches 0øC in the summer and is therefore moist and not transparent to the radar signals. Hence, the radar signals cannot interact with the buried ice bodies. In winter, the melted snow refreezes to form a superimposed ice zone which acts as a continuous, thick, impermeable horizon of low radar reflectivity and polarization ratios. Conversely, summer melting rarely occurs at higher elevations, in the dry-snow facies [Benson, 1962]. No icy formations are found in the top meters of the snowy surface and the radar reflectivity and polarization ratios are as low as in the soaked-snow facies.
Using the icy Galilean satellites as an analogy, the coherent backscatter effect was suggested as a possible l•igure 1. Photograph of an ice pipe found at 1.80 m depth in the firn at Crawford Poin• on June 11, 1991. The ice pipe is 70 cm long, with a diameter varying between 3 and 10 cm (court. esy of K. Jezek, OSU).  -•  24-cm   ,,I,,,,,,,,,I,,,,,,,,,I,,,,,,,,,I,•, [Ostro et al., 1992]; and /•oe for EGC at 13 cm only [Ostro et al., 1980]. The data points for EGC, arbitrarily placed at 11 ø incidence (3.5 cm) and 69 ø incidence (13 cm), are disk-integrated values. explanation for the radar properties of the Greenland percolation facies [Rignot et al., 1993]. The upper few meters of the ice sheet are sufficiently transparent to allow long photon path length and higher order scattering, and contain an abundance of solid-ice scatterers comparable in size to the radar wavelength. These properties suggest that coherent backscatter could dominate the radar echoes. Here, we present a revision of our earlier hypothesis that the Greenland radar echoes are explained by the coherent backscatter effect. Our approach is based on the modeling of the subsurface configuration responsible for the Greenland radar echoes and on an examination of the radar properties of the icy scatterers. The results are used to address the following three issues: Is there a backscatter model that can quantitatively mimic the radar properties of Greenland at several radar frequencies, all polarizations, and various incidence angles? What kind of geophysical information can we extract from the radar data using this model? What can we learn from this study in terms of interpreting unusual radar echoes from other planetary surfaces?

Radar Observations
The NASA/Jet Propulsion Laboratory airborne SAR (AIRSAR) operates a synthetic-aperture imaging radar simultaneously at three wavelengths (A = 5.6 cm (C band); 24 cm (L band); and 68 cm (e band)), recording the complete scattering matrix at each wavelength by alternatively transmitting and receiving vertical and horizontal-polarized radar signals. With an operating altitude of about 9000 rn above ground and a 10-km swath, the radar system collects images where the incidence angle of the illumination varies typically between 19 ø in near range (closest point to the radar) to 65 ø in far range (furthest point from the radar). The data are The polarization ratios are defined such that for pure reflection off a perfectly smooth dielectric surface •uc -0 and •UL --0 because a pure reflection reverses the handedness of the hellcity of the incident circular polarization (hence RR or LL -0), but preserves the orientation of the incident linear polarization (hence HV or VH = 0). In the case of volume scattering from randomly distributed dipoles, we have •uc -1 and •uoe -1/3 [Long, 1965]. For pure double reflections off a perfectly smooth dielectric dihedral whose lower face is horizontal, pc -c• (because RL -0) and poe -0 (because -0). Figure 2 are the disk-integrated measures of the radar reflectivity and polarization ratios of EGC at 3.5-and 13-cm from Ostro c! al. [1980,1992].  Table 1. Both situations are expected to yield large radar reflectivity and polarization ratios via multiple scattering interactions. In broadleaf-upland tropical rain forest in Belize (-17.58øN, 89.0øW) [Freeman e! al., 1992], tr•) c is lower than that recorded for Greenland at the same incidence (Figure 2), and Pc m 1 and PL m 1/3 at 24 and 68 cm. These values of pc and poe are consistent with scattering dominance by the volume of tree branches and foliage of the forest canopy which act as randomly distributed thin cylinders or dipoles. At 5.6 cm, •uc < 1 and •uoe < 1/3 because the branches are no longer thin compared with the observing wavelength. There are, however, numerous cases of forested areas where •uc > 1 at the longer wavelengths. For instance, in palm-tree communities of the Manu National Park tropical rain forest, in Peru (-11.98øN, 70.8øW) AIRSAR measured Pc > 1.5 and •uoe < 0.1 at 68 cm (Table 1), with er•c much lower than that for Greenland. Similarly, in the flooded floodplain forests of the Bonanza Creek Experimental Forest (64.75øN,-148øW), near Fairbanks, Alaska, •uc > 1 at 24-and 68-cm, •9c is large, and •UL < 1/3. We interpret this behavior as being due to double-bounce reflections of the radar signals from the tree trunks to the wet ground back to the radar direction. Doublebounce scattering increases with increasing tree height [  (hence is largest for tall forests), increasing wetness of the ground layers and/or of the tree trunks (hence largest for flooded forests), and increasing radar penetration (hence largest at the longer wavelengths and/or for sparse forests). Double-bounce reflections yield Pc >> i and poe = 0 unless the tree trunks are slanted or damaged [ . These examples illustrate that situations where Pc > i are not uncommon in forested areas and are explainable in terms of trunk-ground scattering interactions. The observed radar reflectivity remains, however, much lower than that recorded in Greenland. The CM number refers to the nomenclature used to archive the AIRSAR images, A (cm) is the observing wavelength, and 0 is the incidence angle. The box of image pixels used to extract the average radar characteristics in each AIRSAR image is indicated in the first colum in (row:column) format. For SP flow, a•c is omitted (NA, not applicable) because it could not be calibrated with confidence, but the polarization ratios are correct. The 5.6-cm SP flow data were corrected for an erroneous antenna pattern correction introduced during the original processing of the SAR data.

OF GREENLAND ICE SHEET 9393
Enhanced radar backscatter and strong depolarizalion of the radar signals may also occur on surfaces that are very rough at the scale of the radar wavelength, for instance through multiple reflections of the radar signals on the large facets of blocky structure of the surface. Several authors [Fahnenstock et al., 1993;Jezek et al., 1994] argued that the unusual radar properties of the Greenland percolation facies are caused by surface scattering from the rough ice layers. To determine whether this is a valid explanation, we examined the radar response of several types of very rough surfaces and experimented with theoretical backscatter models.
Lava flows are good examples of rough terrestrial terrain. Table 1 (Table 1). These large polarization ratios cannot be caused by the coherent backscatter effect because the refractive index of rock in air is too large to yield coherent backscattering [Peters, 1992;Mishchenko, 1992bMishchenko, , 1992c. A more likely explanation is that scattering is dominated by multiple double-bounce reflections on the dihedrals formed by the large facets of the blocky structure of the surface. As It• > 1/3 at 5.6 and 24 cm, the dihedrals must be systematically randomly oriented, otherwise Itoe -0 (see (5)). The lower values of Itc and Itoe at 68 cm are consistent with block sizes of less than 1 m. Similarly, the HH/VV ratios are close to i (Table 1), as predicted from double-bounce scattering from randomly oriented dihedrals (see (7)).  Figure 3, illustrate the incompatibility of the model predictions with the AIRSAR observations. The contrast between 5.6-and 24-cm echoes is overpredicted and the modeled radar .reflectivity at small incidence is several decibels below that recorded for Greenland.
Hence, the IEM model predictions, together with numerous radar observations of rough terrestrial surfaces (Table 1), strongly suggest that surface scattering from the ice layers cannot explain the radar characteristics of Greenland.

A Backscatter
Model for the Percolation Facies One common deficiency of many backscatter models is that they are only approximations to the exact solution of the scattered field from the scattering objects.
Higher-order modes of interactions of the radar signals with the objects are simply ignored. Although this simplification is justified for most natural targets, it is not the case for Greenland, where higher-order internal reflection terms are predominant. oriented in the horizontal plane, we have a o -Because the ice pipes and lenses are typically separated by much more than one wavelength, the scattered field from these objects is uncorrelated. As a result, total radar backscatter from a distribution of discrete, dielectric cylinders is computed as the incoherent sum of the scattered field from the noninteracting, discrete cylinders.
The absorption properties of dry, cold snow are assumed to be negligible. Snow is nearly transparent to radar signals at those wavelengths. Snow, however, steepens the incidence angle of the radar illumination through refraction of the radar signals at the Mr-snow interface, reduces the dielectric constant of water-ice in dry air (½ = 3.2) to a lower value corresponding to water-ice in dry snow (½ = 1.78 for a snow density of

kg/m a [Tiuri e! al., 1984]), and reduces the effective
wavelength of the radar signals by v/7, here 25%.

The exact scattering matrix for a dielectric cylinder of infinite length is given by Bobten and Huffman, [1983].
The analytical solution for a finite cylinder is computed by scaling the solution for the infinite cylinder by a shape factor, f, kh .

f---Slnc(k h cos O) (2)
where k is the wavenumber in the propagation medium, h is the cylinder length, 0 is the incidence angle, and sine(x) =sin(x)/x. If the cylinder length varies randomly by a quantity +½n, the average solution for the scattered field intensity is obtained by averaging f2 between h -½n and h + ½n. When ½n >> ,X, the effective value of f2 is 1 < f2 >= 2•r 2 cos 2 0' which is independent of both h and cu. This result means that when the cylinder length fluctuates by --, ,• or more, the mean cylinder length has no influence on the radar properties of the cylinder. Given the typical sizes of ice pipes and lenses, this condition is easily satisfied at 5.6 and 24 cm. We assume it also applies at 68 cm. Hence, the shape factor f2 is only a function of the incidence angle.
We now examine how to compute the scattered field from randomly distributed discrete, dielectric cylinders, given the solution for one cylinder.
where a and b are complex numbers whose magnitude and phases are functions of the cylinder dielec-tric constant and diameter [Bohren and HuJ•man, 1983].  (a-b)/(a + b)12. Hence, vertical cylinders do not generate any cross-polarized intensity unless they are systematically randomly oriented in the vertical plane. We assume that the randomness in orientation of the ice pipes reflects spatial irregularities in shape and orientation of the ice pipes along their longest dimension at a scale comparable to or larger than the wavelength (Figure 1). Figures 5a-5c show the radar reflectivity and polarization ratios of various sized cylinders as a function of the size parameter (k . r), where k is the wavenumber and r is the cylinder radius. Narrow peaks in and /•L observed for particular values of (k . r) coincide with a 4-180 ø phase difference between HH-and VV-polarization. These peaks are caused by internal reflections of the radar signals in the cylinders, which include the glory ray effect [Bohren and Huffman, 1983] as well as other types of internal reflections included in the exact analytical solution to the scattered field. Internal reflections are the only type of returns that would cause a large phase difference between H-polarized and V-polarized radar signals. As the refractive index of the cylinder increases, the magnitude of these internal reflections is expected to decrease, as shown in Figures 5d-5f for water-ice cylinders in vacuum (refractive index ~1.8). Hence, the refractive index of the cylinders needs to be small enough (typically <1.6) to yield strong internal reflections and /•c > 1, and the cylinders need to be randomly oriented to yield large values of/•.

Using this backscatter model, we predict large polarization ratios for the icy inclusions. To illustrate this result,
Model predictions for an ensemble of horizontal and vertical cylinders are shown in Figure 6 along with the AIRSAR data. Several model parameters were tuned to best fit the model predictions with the AIRSAR data.
• The pa-These parameters are rh, r•, Nh, N• and a o. rameter tuning was done as follows. Test values for rv were selected among those producing a peak in polarization ratios (Figure 5) because the model showed that the value of r• controls the polarization ratios of the ensemble of horizontal and vertical cylinders at high incidence. Since /•c and/• are low at low incidence, the value of ra is less critical. We chose ra = r• = r to simplify the procedure. The selected average cylinder radius was then used to generate a normal distribution of cylinder radius, with a relatively small standard deviation (see caption to Figure 6). The best results were usually obtained with r ~ ,X/2, which is not surprising, since it corresponds to wavelength-sized objects.
Given r, the number densities were selected to best fit rr•c because the model showed that rr•) c is proportional to Na at small incidence and to N• at high incidence. Note here that with vertical or horizontal cylinders only (N• = 0 or Na = 0), the model could not predict the correct trends in /•c and /• versus the incidence angle. Both horizontal and vertical cylinders are needed v to match the AIRSAR observations. The angle a o was finally selected to properly balance/•L and •c, as the model showed that these ratios vary in opposite directions when a• changes. Because of the large number of constraints provided by the multiparameter radar data, • values was found to only one set of r, N•, Na and a o yield a good agreement with the AIRSAR data, at each radar wavelength. The HH/VV ratio of the ensemble of cylinders is predicted to be 1 (see (7)), in agreement with the 5.6-and 24-cm data (Table 1), and is therefore not discussed. Finally, the HH-VV phase difference (fifth independent parameter of the covariance matrix) is predicted to be close to zero by the model, but with a large standard deviation caused by internal reflections (Figure 5), which is consistent with the AIRSAR observations of large standard deviations of the HH-VV phase difference (Table 1) At 24 cm, we found r-8.9 cm, Nv -i cyl/m 2, ct o -50 ø, and Nn -3 cyl/m 2. Field observations suggest this value may be at the limit of being too large to represent an ice pipe. We could not match the AIRSAR data at both 5.6 and 24 cm using a single normal distribution of the cylinder radius with a large standard deviation. The model requires a bimodal distribution of r to yield a good agreement with both wavelengths. We do not know whether a bimodal distribution of the cylinder radius is realistic. The 24-cm radar signals probably interact with more than one layer of ice bodies, yielding more complex interactions than those accounted for in our model. Unfortunately, no surface-based radar ranging data were gathered at that wavelength to determine whether the dominant scatterers are still localized in the first annual layer of ice bodies.
At 68 cm, the polarization ratios and radar reflectivity are always low for r between 3.1 and 8.9 cm and would only be large for r ~ 15-20 cm. Icy inclusions are never this large. Both the model and the 68-cm radar observations are therefore consistent with the typical size of the inclusions. An example of model prediction at 68 cm is shown in Figure 6 using the model parameters optimized at 24 cm. The agreement with the AIR-SAR data is reasonable, except for the radar reflectivity.
One possibility is that (3) does not apply at 68 cm (~68cm fluctuations in length of the ice pipes is excessive).
The polarization ratios are unaffected, but the radar reflectivity could be overestimated. Another possibility is that scattering is of a different nature at 68 cm. This result is suggested by HH/VV >> 1 and the low value of the standard deviation of the HH-VV phase difference at 68 cm (Table 1). As ice becomes nearly transparent at that wavelength, radar signals probably interact with much deeper layers of solid ice.

Discussion
The modeling results demonstrate that internal reflection of radar signals in horizontal and vertical, discrete, solid-ice inclusions buried in the snowy, radartransparent, surface of the ice sheet can explain the extraordinary radar properties of the Greenland percolation facies. The agreement with the AIRSAR data is excellent at 5.6 cm. At the longer wavelengths, the source of scattering is less certain in the absence of surfacebased radar ranging measurements. Nevertheless, the There is no requirement on the geometrical shape of the scatterers. The present model also requires scatterers of low refractive index embedded in a weakly absorbing medium, but not necessarily closely spaced. In situ observations of the subsurface configuration of the ice sheet show that icy inclusions are often sepa-rated by many wavelengths, in which case the coherent backscatter effect should not take place. In addition, model predictions from the coherent backscatter theory using spherical scatterers suggest that /•c and /•L should decrease with an increasing incidence angle (Figures 6,9,15,18