Coherent backscatter enhancement in bistatic Ku-/X-band radar observations of dry snow

The coherent backscatter opposition effect (CBOE) enhances the backscatter intensity of electromagnetic waves by up to a factor of two in a very narrow cone around the direct return direction when multiple scattering occurs in a weakly absorbing, disordered medium. So far, this effect has not been investigated in terrestrial snow in the microwave spectrum. It has also received little attention in scattering models. We present the first characterization of the CBOE in dry snow using ground-based and space-borne bistatic radar systems. For a seasonal snow pack in Ku-band (17.2 GHz), we found backscatter 5 enhancement of 50–60% (+1.8–2.0 dB) at zero bistatic angle and a peak half-width-at-half-maximum (HWHM) of 0.25◦. In X-band (9.65 GHz), we found backscatter enhancement of at least 35% (+1.3 dB) and an estimated HWHM of 0.12◦ in the accumulation areas of glaciers in the Jungfrau-Aletsch region, Switzerland. Sampling of the peak shape at different bistatic angles allows estimating the scattering and absorption mean free paths, ΛT and ΛA. In the VV polarization, we obtained ΛT = 0.4±0.1m and ΛA = 19±12m at Ku-band, and ΛT = 2.1±0.4m,ΛA = 21.8±2.7m at X-band. The HH polarization 10 yielded similar results. The observed backscatter enhancement is thus significant enough to require consideration in backscatter models describing monostatic and bistatic radar experiments. Enhanced backscattering beyond the Earth, on the surface of solar system bodies, has been interpreted as being caused by the presence of water ice. In agreement with this interpretation, our results confirm the presence of the CBOE at Xand Ku-band frequencies in terrestrial snow.


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The scattering of electromagnetic waves in any type of medium can be used to characterize some of its structural properties. In radar remote sensing, the scattering characteristics of snow have been intensely studied to derive properties of the snowpack.
However, an important effect, the coherent backscatter opposition effect (CBOE), can enhance the radar backscatter return by up to a factor of two. This effect has rarely been considered in descriptions of the backscatter return from snow in monostatic radar experiments (where the transmitter and the receiver are co-located), because even though the CBOE is present, its mag-20 nitude cannot be quantified without a bistatic reference measurement (where the transmitter and the receiver are at separate locations). To fully characterize the CBOE, bistatic radar experiments need to be performed.
In the radio-frequency spectrum, the CBOE is mostly discussed in connection with snow and ice deposits where microwave absorption is relatively weak (Warren and Brandt, 2008;Mätzler and Wegmüller, 1987a, b). In planetary science, it has been proposed as an explanation for the unusually high radar cross-sections of surfaces of various solar system bodies (Muhleman et al., 1991;Black et al., 2001;Hapke et al., 1993). It has also been discussed as a potential cause of the unusual radar echoes from the Greenland ice sheet (Rignot et al., 1993), although internal reflections were proposed as an alternative explanation (Rignot, 1995). In both of these contexts, additional measurements at small but nonzero bistatic angles were desired (but not 60 feasible), as they would have provided a way to more easily and robustly characterize the effect (Hapke, 1990;Rignot, 1995).
Bistatic radar measurements of surfaces of solar system bodies are possible by using an orbiting spacecraft in combination with the deep space network receivers on Earth (Simpson, 1993;Palmer et al., 2017). However, such experiments require a very specific geometric alignment of the spacecraft's orbit with respect to the Earth and are thus not common. Nevertheless, several experiments were carried out with the Moon as the target (Yushkova et al., 2018): the Clementine bistatic radar experiment 65 observed an opposition peak in certain areas of the lunar surface. This peak has been suggested to be attributable to the CBOE which implied the existence of ice deposits on the surface (Nozette et al., 1996), though other work has called the interpretation of the Clementine data into question (Simpson and Tyler, 1999). More recently, the Mini-RF instrument of the Lunar Reconnaissance Orbiter, in concert with the Arecibo Observatory's radio telescope acting as transmitter, detected the opposition surge in certain areas of the lunar surface, again attributed to the presence of near-surface deposits of water ice 70 (Patterson et al., 2017).
In many of these experiments, observation of a backscatter enhancement peak at radio-frequencies has been interpreted as the CBOE and has been used to infer the possible existence of water ice on the surface of the corresponding solar system bodies. Other works considered the CBOE in microwave scattering models of terrestrial snow (Tan et al., 2015) but could not analyze the peak shape of the CBOE. In this work we demonstrate that the existence of snow on the Earth can indeed cause 75 a CBOE. We present a sampling of the peak shape at Ku-and X-band radio wavelengths with ground-based and space-borne imaging radars.

Methods
To characterize the angular peak of backscatter enhancement effects in the radio-frequency spectrum, we used two bistatic radar systems, the ground-based system KAPRI and the space-borne satellite formation TanDEM-X. For both systems, the 80 transmitter and receiver are placed on independent platforms and thus the bistatic angle can be varied. The bistatic angle β is defined as the angle between the transmitter, the observed location, and the bistatic receiver. In the exact direct return direction, the bistatic angle is zero and the scattering alignment is called the monostatic configuration.

Ground-based observations -KAPRI
KAPRI (Baffelli et al., 2017) is a ground-based Ku-band FMCW real-aperture radar system developed by Gamma Remote The antennas are oriented towards the snow covered north-west face of the mountain Rinerhorn, Switzerland. The region of interest (ROI) for the winter and summer seasons is shown in blue and red respectively. Their overlap is shown in purple. C is a reference point for the orientation of the bistatic receiver (see Fig. 4). The hollow line slicing through the ROI masks out metallic beams from a ski-lift on the slope.
Satellite imagery data: Sentinel-2 on 20 February 2021. Modified Copernicus Sentinel data 2021/Sentinel Hub. devices with different antenna configurations (Stefko et al., 2021b), it offers coverage of areas hundreds of meters wide at a range of several kilometers. It operates at central frequency 17.2 GHz (λ = 1.74 cm), with 200 MHz bandwidth. The system and the processing pipeline for bistatic KAPRI datasets are described in detail in (Stefko et al., 2021b). Details of the antenna configuration while using a cable synchronization setup can be found in (Stefko et al., 2021a, Fig. 2 For the ground-based experiment (map shown in Fig. 1), the observed region of interest (ROI) was located on the northwestern face of Rinerhorn peak near Davos, Switzerland. Both devices were located on the valley side opposite the peak, at 46.763 N, 9.788 E (Fig. 1). The radar location features a straight and relatively flat segment of road approximately 200 m long with unobstructed view of Rinerhorn. The devices were placed at approximately 1620 m altitude, while the ROI altitude 95 spans from 2050 m to 2270 m. With this upward-looking observation geometry the vast majority of the ROI area is observed under a shallow local incidence angle larger than 70°. Problems with multipath interference arising from the upward-looking observation geometry while employing a fan-beam radar system (Lucas et al., 2017) are avoided by placing the instruments on the opposing side of the valley.
We performed two experiments: in summer (05 August 2020), the ROI was covered by low grass. In winter (18 February 100 2021), the area was completely covered by approximately 1.5 m of seasonal snow. Each measurement began at approximately 08:00 local time, and the total duration of the observations was 3.5 hours in summer, and 5.5 hours in winter. In winter, a snow pit revealed snow temperatures of −10°C at the snow surface and −0.2°C at the bottom of the snowpack (Fig. 2, left). The classical snow grain size, measured as the mean maximum extent of snow crystals (Fierz et al., 2009), was D max = 0.3 mm at the surface and 1.5 mm at the base (Fig. 2, right).

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To select the ROI, a mask fulfilling the following three conditions was applied for each season: 1) Include only terrain higher than the tree line at 2050 m altitude. 2) Exclude areas containing man-made structures (metallic support beams, metal ropes, buildings, corner reflectors). 3) Exclude areas affected by radar shadow and exclude areas outside of the main beam of the secondary antennas -these areas were detected by applying a threshold on the magnitude of the single-pass interferometric coherence γ of the secondary receiver in the VV channel. In every acquisition in the summer dataset, pixels with γ < 0.85 110 were masked out, in the winter dataset pixels with γ < 0.80 were masked out. A sliding window of 5 × 3 (range × azimuth) pixels was used for coherence estimation. The winter threshold is lower due to lower overall coherence as opposed to summer.
The masks defining the ROI for each season are shown in Fig. 1. They cover practically the same region of the hillside. The acquired calibrated SLC datasets were spatially multi-looked using a 5×3 window to obtain the intensity images, and analyzed in the radar polar geometry (range×azimuth angle). The intensity valueÎ(β) (the hat symbolˆindicates it's a measured quantity) was computed for every acquisition by averaging the measured intensities of all pixels within the ROI. We analyzed only the VV and HH polarization as the cross-polarized signal was too close to the noise floor to provide reliable data.

Device configuration and measurement procedure
The primary (monostatic) transmitter-receiver remained stationary during the experiment (Fig. 3, top) and performed azimuthal sweep acquisitions of the observed area at a range of approximately 2.5 km. The secondary device (bistatic receiver) was moved 120 step-wise to sample bistatic angles between 0.04 • and 1.92 • . In winter, the secondary device was mounted on a large sledge (Hornschlitten, Fig. 3, middle). In summer, a wheeled cart was used as a movable radar platform (Fig. 3, bottom). The trajectory of the secondary device is visualized in Fig. 4.
The bistatic angle β was calculated for each position of the secondary receiver S as (1)

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The length of the bistatic baseline b is given by the length of the vector between the primary and secondary radar's positions, after projecting it into the plane orthogonal to the line of sight. The line of sight is the vector between the primary radar P and the reference point C in the ROI (see Fig. 4). Its length is d PC = 2500 m.
To ensure optimal overlap of the antenna patterns in the ROI, the secondary device was leveled and oriented manually in each position. The pitch and roll angles of the mobile platform with respect to the true vertical direction were measured with a 130 digital spirit level at each measurement point, and did not exceed 2°in either direction. In azimuth, the device was oriented with a compass and optical viewfinder using a reference point in the center of the ROI (Point C in Fig. 1). The estimated precision is 1°.
The transmit antennas on the primary KAPRI device have a physical horizontal length of 2 m (Fig. 3, top), and thus the bistatic angle at 2.5 km range differs by ∼ 0.05°between the two edges of the transmit antenna. This imposes a practical limit 135 on the resolution of the sampling of the intensity curve: any variation of intensity within bistatic angles of less than 0.05°will be smeared out by the non-zero size of the transmit antennas.

KAPRI: radiometric precision
Three main factors were identified which can affect the radiometric precision of the measurements: temporal drift of the scattering properties in the ROI, the radiometric stability of the bistatic KAPRI system, and the pointing precision of the 140 secondary receiver's antennas.
The trajectory of the bistatic receiver ( Fig. 4) was designed to repeatedly increase and decrease the absolute value of the bistatic baseline. This allows detection of any temporal drift of the scattering intensity over the course of the measurement (i.e. on the order of minutes to hours). Drifts would be detected by the different shape of the left and right wing of the intensity curveÎ(β).  The radiometric stability of KAPRI can be assessed by investigating the monostatic scattering intensity observed by the monostatic device from a reference target (a corner reflector). The maximal detected variation was observed in the HH channel in the winter season, with standard deviation of 16 % relative to the mean value.
For each individual measurement the beam pointing direction of the secondary receiver differed by less than 1 • in azimuth from the ideal central pointing direction towards point C. Due to the antenna pattern of the secondary receiver (Stefko et al.,150 2021b, Fig. 7), an azimuthal misalignment of 1 • can reduce the signal intensity by not more than ∼ 1 dB (25%) at the edge of the "ideal" antenna pattern footprint covering the ROI. However, when considering the total received backscatter from the ROI, this reduction is partially compensated, since the observed backscatter intensity from the other edge of the ROI would necessarily increase.
Due to the limited radiometric stability and the beam pointing uncertainty the observed backscatter intensity can thus be ex-155 pected to vary stochastically with estimated standard deviation of approximately 20 %, affecting each individual measurement by a significant amount. These effects are difficult to compensate for, since there were no reference targets in the scene with a sufficiently high and stable bistatic radar cross-section. For this reason, no a-posteriori radiometric calibration was applied to the data. However, the two effects are stochastic in nature, and uncorrelated between individual receiver positions, and thus with a sufficiently high number of acquisitions, the enhancement peak should still be detectable, albeit with lower radiometric 160 precision.

Satellite observations -TanDEM-X
The TanDEM-X satellite formation is the first space-borne bistatic radar system with an adjustable bistatic baseline. The formation consists of two free flying synthetic aperture radar (SAR) satellites, TerraSAR-X and TanDEM-X, orbiting the Earth in about 514 km height in a helix-like formation (Krieger et al., 2007). The two radar instruments operate at X-band at a 165 central frequency of 9.65 GHz (λ = 3.11 cm). Depending on the acquisition mode, both satellites can act as either transmitter or receiver or both. In the bistatic mode, the transmit-receive satellite operates in a monostatic observation geometry, and the receive-only satellite operates in a bistatic observation geometry.
Since the launch of TanDEM-X in June 2010, the distance between the two satellites has been varied by several kilometers.
The largest (and smallest) distances were obtained during the TanDEM-X science phase between Oct 2014 and February 2016 170 (Hajnsek et al., 2014). To find an area best suited for observation of the CBOE in X-band, we searched the entire TanDEM-X archive for areas that are covered by deep snow and where long acquisition time series with large bistatic angles are available.
Unfortunately, near the poles, bistatic angles are relatively small, making a sufficient sampling of the CBOE peak difficult. At the equator, the largest bistatic angles of up to 0.35 • are available but snow is naturally rare. As a best compromise, we selected the Jungfrau-Aletsch region in Switzerland and the Teram-Shehr/Rimo glacier in the Karakorum.

Primary observation site: Jungfrau-Aletsch region
The Jungfrau-Aletsch region has been selected as a TanDEM-X super test site with the aim to acquire as many acquisitions as possible, and to explore the scientific value of the bistatic radar mission. 118 bistatic acquisitions in the two polarizations (VV, HH) were acquired between 2011 and 2019, most of them during winter (Fig. 5). For 104 acquisitions, TerraSAR-X acted as transmitter, for 14 TanDEM-X. Bistatic baselines between 65 and 2100 m are available, corresponding to β = 0.005 − 0.21°.

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The incidence angle at the scene center is θ = 32°(orbit 154, descending). Time-averaged backscatter images of the study area are provided in the supplementary figures S2 and S3. Interferometric and polarimetric properties of the dataset were analyzed by Leinss and Bernhard (2021).
The Jungfrau-Aletsch region is highly glaciated with multiple peaks reaching above 4000 m. Cold firn, several tens of meters deep, is likely present throughout the year: depending on exposition, the transition to temperate firn is at 3400-4000 m while 185 the upper 15 m of firn experience seasonal temperature cycles and can freeze in winter (Suter et al., 2001;Suter and Hoelzle, 2002;Jun et al., 2002). In March 2021, firn temperatures of −11 ± 3 • C in the upper two meters, and −4 ± 2 • C at -8 m, were measured by Bannwart (2021) at two sites, one on 3380 m (46.5525 • N / 8.0286 • E) and on 3350 m (46.5483 • N / 8.0323 • E).
The region contains Great Aletsch Glacier (46.50 • N, 8.03 • E), the largest glacier in the European Alps. Its equilibrium line altitude, above which accumulation dominates, is at˜3000m (Zemp et al., 2007). In the ablation area below, seasonal snow is 1. The accumulation area of glaciers with altitudes above 3500 m. These areas are at or above the temperate-to-cold firn transition and we assume that firn conditions did not change too drastically from winter to winter. To ensure refreezing 195 of firn after summer, and to avoid snow melt in spring, we restricted the model parameter estimation on data acquired between 01 December and 31 May (gray shading in Fig. 5). The dry, deep firn acts as a thick medium with multiple scattering in the volume but low absorbing.
2. The ablation area of Great Aletsch Glacier with altitudes below 2700 m. Field measurements indicate a seasonal snow cover of 0-3 m on the glacier tongue during winter (Leinss and Bernhard, 2021). The seasonal snow acts as a thin 200 medium of volume scatterers with low absorption.
3. Forested areas with at least 7 m height, mainly conifer forest located in the Rhone valley and the Grindelwald region.
The Forest acts as a medium where volume scattering can occur but absorption is strong.

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The equilibrium line altitude of Teram-Shehr glacier, an eastern tributary of Siachen glacier in the Karakorum, is at approximately 5250 m (Agarwal et al., 2017). Compared to the Jungfrau-Aletsch dataset, considerably fewer acquisitions are available here and bistatic angles are not significantly larger: for orbit 98 (β = 0.04-0.23 • , θ = 39 • , asc.) nine acquisitions showed dry firn and 17 acquisitions wet or partially wet firn. For orbit 75 (β = 0.04-0.19 • , θ = 43 • , desc.) we found 18 acquisitions with dry firn and 25 with wet firn. Both datasets contain the HH polarization only. To ensure refreezing of firn after summer, we 210 restricted the analysis to data between 01 December and 30 June.

TanDEM-X: bistatic angle
For TanDEM-X, the bistatic angle β = b/R was determined from the average slant-range distance R to the scene center and from the bistatic baseline b, derived from the orbit coordinates. To compute b, the distance between the two satellites was decomposed into the along-track baseline B AT , the across-track baseline B XT and the parallel, or line-of-sight baseline baseline 215 B LOS . The bistatic baseline b, perpendicular to the line-of-sight direction, is given by (2) Figure 5 shows time series of b, B AT , and B XT . Because of the bistatic acquisition geometry, where the phase center of the bistatic receiver is located in the midpoint between the transmitter and the receiver (Duque et al., 2012), the across and alongtrack baselines used in Eq.
(2) are a factor of two larger than the effective interferometric across-and along-track baselines 220 given in the acquisition's meta-information (Leinss and Bernhard, 2021, cf. Fig. 2).
Even though we refer in the following to the monostatic acquisition, we note, that the orbital velocity of v = 7.6 km s −1 results in a small, velocity induced, bistatic angle of β v = 0.003 • for the monostatic receiver, because the satellite moves 30 m between transmission and reception of a radar pulse. The high orbital velocity also decreases (increases) the along-track baseline B AT by 30 m when the bistatic receiver follows (is ahead of) the transmitter. We considered this in the analysis but 225 found the effect negligible.

TanDEM-X: radiometric calibration and computation of backscatter ratios
Resolving the peak shape of the CBOE with a maximum expected peak height of 3 dB requires a precise radiometric calibration of the bistatic dataset. To avoid any terrain or incidence angle dependent calibration, we analyzed the ratio between the backscatter intensityÎ bist observed by the bistatic receiver and the intensityÎ mono observed by the monostatic transmitter-230 receiver: The index r,0 indicates taking the ratio relative to I(β = 0). Averaging ratios likeÎ r,0 would result in a biased estimate. To estimate unbiased spatially or temporally averaged ratios we first applied the averages onÎ bist andÎ mono and then computed the ratioÎ r,0 . We useÎ to refer to the radar brightness commonly denoted by β 0 (Raney et al., 1994) to avoid confusion with the 235 bistatic angle β. The hat symbolˆindicates measured quantities.
For every orbit and each polarization channel, we coregistered time series of the interferometric TanDEM-X CoSSC (Coregistered Single look Slant range Complex) acquisition pairs Duque et al., 2012). To obtain the intensities Î mono andÎ uncal. bist , we detected the temporally coregistered CoSSCs, applied 10 × 10 pixels multilooking, and downsampled the data by a factor of 10.
bist /Î mono showed, therefore, differences of 10-30% between the bistatic and the monostatic receiver. While in the VV polarizationÎ uncal.
r,0,VV showed spatially relatively constant values at small bistatic angles,Î uncal. r,0,HH showed terrain-independent trends of a few percent, presumably due to different antenna patterns (supplementary Figs. S4 and S5). To compensate for these patterns, we calibrated for each polarization the intensityÎ uncal. bist with the ratio of the pixel-wise temporal 245 mean · temp. of 17 scenes with β < 0.033 • . This threshold for β was chosen small enough to avoid any significant bias from backscatter enhancement. The bistatic intensity after antenna calibration iŝ To obtain the calibrated intensityÎ bist , we compensated in each acquisition pair for the remaining spatially constant offset between the monostatic and bistatic data. For this we multipliedÎ ant.cal.
bist with the ratio of the monostatic and bistatic radar 250 brightness, spatially averaged, as indicated by · spat. cal.area , over a pre-defined calibration area: For calibration of the Jungfrau-Aletsch dataset we used areas that showed a temporally stable and baseline-independent backscatter ratioÎ r,0 . These areas were defined using two iterations. In a first iteration, we masked out very dark areas, possibly affected by noise, such as shadow (Î mono < −14 dB) and also very bright areas such as layover and strong local scatterers 255 (Î mono > +1 dB) through thresholding the temporal mean of the backscatter intensity. We also masked out the ROIs later analyzed, by masking elevations above 3000 m where multi-year firn occurs, regions covered by forest, as well as the ablation area of Great Aletsch Glacier. After using the remaining pixels for calibration, in the second iteration we masked out additionally areas whereÎ r,0 , computed pixel-wise using the temporal means ofÎ mono andÎ bist from 43 acquisitions with bistatic angles smaller than 0.04 • (cf. Eq. (4)), differed more than 5% from unity. Such deviations appeared in areas of low radar backscatter 260 and areas not directly affected by layover but next to layover in the far-range direction. The deviations might originate from bright azimuth or range-ambiguities or areas affected by sidelobes of layover (also visible in supplementary figures S4 and S5).
We also removed areas where the pre-calibrated backscatter ratioÎ r,0 showed a temporal standard deviation larger than 0.08 (supplementary figures S6 and S7). Finally, to avoid that the CBOE or possibly the SHOE affect the calibration, we masked out areas that showed more than 5% enhanced scattering in the direct return direction in the large-baseline acquisitions B ⊥ > 2 km.

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In total, we masked out approximately 50% of pixels from the scene (supplementary Fig. S8) and used the remaining pixels, mainly grassland, rock and the ablation areas of glaciers for calibration in Eq. (5). In this data-driven calibration we assume that the regions selected for calibration show an equal backscatter intensity for the monostatic and bistatic receiver.
The Teram-Shehr dataset did not provide enough suitable acquisitions for antenna calibration according to Eq. (4). For calibration with Eq. (5), we used the glacier tongue (ablation area, below 5000 m).
To determine the backscatter ratio for the ROIs (see Sect. 2.2.1 and Sect. 2.2.2), we used Eq.
(3) withÎ bist andÎ mono averaged over the ROI. In addition, the mean backscatter intensityÎ mono was used as a proxy to differentiate between dry and wet snow for snow covered areas.
To display imagery of I r,0 with sufficient radiometric resolution, we applied additional 4 × 4 px multilooking to the downsampled backscatter imagery, corresponding to an effective multilooking operation of 41 × 41 pixels. This value was chosen 275 to keep the standard deviation σ = I/ √ N of the multilooked intensity I sufficiently low (Oliver and Quegan, 2004). N is the number of looks. Given that adjacent pixels are statistically not completely independent (the SLC data is oversampled by a factor of 1.3 in slant range and and 2.9 in azimuth, resulting in 3.73 pixels per look) we obtain a value of N = 41 2 /3.73 = 450 looks which corresponds to a radiometric accuracy (standard deviation) of 0.2 dB (5%) at an intensity of I = −5 dB.

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For interpretation and modeling of our results we considered the book from Hapke (2012). In Chapter 9, Eqs. 9.40 and 9.44, as well in (Akkermans et al., 1986, the peak shape of the coherent backscatter enhancement is described for non-absorbing or absorbing media with the equation: where B C (β) is the magnitude of the coherent backscatter intensity enhancement at bistatic angle β ≈ sin β relative to the 285 incoherent background I 0 . Depending on absorption, B C (0) has a height between 0 and 1 and B C (∞) = 0 (Fig. 6). The background intensity I 0 is determined by the incoherent scattering properties of the medium so that describes the total backscatter intensity I(β) in the proximity of several degrees from the direct backscatter direction. K is the medium porosity coefficient (see below). For notational simplicity we defined In this equation λ is the wavelength in the medium, Λ T is the transport mean free path, and Λ A is the absorption mean free path. The length of the transport mean free path Λ T is a measure of the medium's scattering properties, and corresponds to the scattering mean free path Λ S for particles that scatter EM radiation isotropically (Hapke, 2012, Eq. 7.24b). Correspondingly, Λ A is a measure of the medium's absorbing properties. For negligible scattering, Λ A describes the absorption length where 295 the incident radiation is reduced to 1/e; a non-absorbing medium would have Λ A = ∞. Similar, for negligible absorption, Λ T describes the one-way penetration depth where the incident radiation has been reduced to 1/e by side-way scattering.
The porosity coefficient is K = 1 for weakly scattering media in which Λ T is much longer than the size of scatterers. This follows from Hapke (2012, Eq. 7.44) where the product of the extinction coefficient E = Λ −1 S + Λ −1 A and the distance between scattering particles L must be EL 1. For dry snow at X-and Ku-band wavelengths where the one-way penetration depth Figure 6. Modeled peak shape of the CBOE for different scattering mean free paths ΛT given in multiples of the wavelength λ and for different absorption mean free paths ΛA (in multiples of ΛT ) in a medium with small scattering particles (porosity coefficient K = 1). For a non-absorbing medium (ΛA = ∞) a very sharp peak can be observed. Already with a weak absorption ΛA = 30ΛT (blue) the peak height is reduced to 50% and the peak becomes much rounder. For comparable scattering and absorption lengths the peak is not noticeable (red).
(≈ Λ S ) is on the order of meters, where absorption is negligible (Λ A Λ S ), and where the particle (or lattice) distance L can be approximated by the snow grain size r g (millimeters), we obtain EL ≈ Λ −1 S r g ≈ 10 −3 1 which justifies K = 1. The peak shape, as drawn in Figure 6, is determined by the ratio of scattering mean free path Λ T to the wavelength λ and by the probability distribution of scattering path lengths in the medium. A (monostatic) scattering path begins at the first scattering event in the medium, travels along multiple scatter events with mean distance Λ T , and ends when the radiation is 305 scattered back out of the medium in the direct return direction (Hapke, 2012, Chapter 9.3). In the monostatic configuration, radiation traveling along such a path interferes constructively with radiation propagating along the time-reversed counterpart, thus causing the backscatter intensity enhancement. Long scattering paths, consisting of multiple scattering events, have a longer distance between the path's start and end point and cause a narrow peak, while short scattering paths cause a broad peak. The final peak shape is determined by the sum of all occurring peak shapes of different widths, weighted according 310 to their occurrence probabilities. The more absorption occurs, the shorter are the scattering paths that can contribute to the coherent peak, and the lower is the probability for the occurrence of higher order scattering, hence the peak becomes rounder  , Fig. 7). Long scattering paths can also be suppressed by a finite sample thickness, which also causes a rounding of the peak (Van Der Mark et al., 1988, Fig. 20). Figure 6 shows the shape of the CBOE peak for a range of values of Λ T , Λ A given in multiples of λ. Longer scattering lengths Λ T cause a narrower peak, while the peak height and the curvature 315 near β = 0 is determined by Λ A . Hence, when characterizing the full peak shape, Λ T and Λ A can be determined.

Application to KAPRI data
With the ground-based KAPRI instrument, the benefit of the flexible configuration allows us to sample the intensity peak up to relatively high value of bistatic angle β, and thus the flat region of the intensity curve (I(β → ∞) → I 0 ) should be observable.
However, the very top of the peak is difficult to sample due to the non-negligible size of the primary device's antennas, as well 320 as the possibility of the devices obstructing each other's view when placed very close together. Because of this, for analysis of KAPRI data, we use the intensity ratio I r,∞ (β) which is normalized to the incoherent background intensity I(∞), and can be expressed with aid of Eq. (7) as: To calculate the intensity ratio of Eq. (9) from the actual observed mean ROI intensityÎ(β), we approximate I(∞) as the

Application to TanDEM-X data
With TanDEM-X we measured the intensity ratioÎ r,0 (β) between the bistatic receiverÎ bist =Î(β > 0) and the monostatic 335 receiverÎ mono (β v = 0.003 • ) ≈ I(0) (Sect. 2.2.3). This approximation is well justified considering that the expected width of the peak is at least one order of magnitude larger than the small bistatic angle of the monostatic receiver (cf. Fig. 6) and that rounding of the peak tip due to weak absorption can be expected. The TanDEM-X measurement can, therefore, be described by Eq. (7) as:

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The intensity ratio I r,0 (β) is 1.0 for β = 0 and reaches its minimum 0.5 at β → ∞ when absorption is negligible. With increasing absorption the contrast lowers and I r,0 (β → ∞) increases from 0.5 to eventually 1.0.
A lower limit for the enhancement B C (β = 0) can be quickly estimated: because B C (β max ) > B C (∞) = 0 it follows from Eq. (11) that i.e. the enhancement B C is at least as large as the relative difference between the monostatic and the bistatic backscatter,Î mono andÎ bist , at the largest available bistatic angle β max .
To fit the model, we used winter data from the accumulation area (Sect. 2.2.1). From these, we removed 14 acquisitions for which TanDEM-X instead of TerraSAR-X acted as transmitter, resulting in slightly different antenna pattern that could not be compensated through the calibration, especially in the HH polarization. To determine the optimal value of the parameter pair 350 (Λ T , Λ A ), and the 95% confidence intervals, we used the TRF method (Sect. 2.3.1), and set the starting parameter of (Λ T , Λ A ) to (2 m, 20 m). However, a sampling of the RMSE[Î r,0 (β) − I r,0 (β)] in the parameter space of Λ T , Λ A around the optimal value revealed that the global minimum is weakly constrained, and solutions across a large span of values of Λ A provide an acceptably low RMSE value. Thus, to explore multiple parameter pair values, we sampled a range of values of Λ A , and used a downhill simplex method implemented in the amoeba IDL function (Nelder and Mead, 1965) to determine the corresponding 355 Λ T .

Jungfrau-Aletsch region
The large number and coverage of the TanDEM-X scenes allows an analysis of the dependency ofÎ r,0 on β for different land cover types. In Fig. 9, where the color of data points refers to I mono (Fig. 10)  approximately 20% compared toÎ mono . For wet snow no reduction ofÎ r,0 is observed at β max . In contrast, neither the ablation area of Great Aletsch Glacier, Fig. 9(c), nor areas covered by conifer forest, Fig. 9(d), show any significant dependence ofÎ r,0 on β.
To investigate the spatial distribution of areas that show enhanced backscattering, Fig. 11 shows imagery of the monostatic-380 to-bistatic backscatter ratioÎ −1 r,0 together with the radar brightness I mono for a series of three acquisitions with β = β max at the onset of snow melt in April/May 2015: On 24 April (Fig. 11a, d), backscatter enhancement is visible for a considerable amount of the area, corresponding to glaciers at high altitude (> 3000 m). On 05 May (Fig. 11b, e), the backscatter enhancement is limited to high altitudes, because snow melt is occurring up to an altitude of˜3300 m. On 07 June (Fig. 11c, f), snow melt reaches the peaks of the highest mountains (4274 m) and no enhanced backscattering is detectable at any place.

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To estimate the scattering and absorption parameters Λ T and Λ A , as well as the peak width and the backscatter enhancement, we fitted the model, Eq. (11), to the dry firn data of the accumulation area, constrained to winter acquisitions (the selection is   lists for each solution the modeled peak characteristics and the RMSE with respect to the measured data. Figure 12(b) illustrates how the different model solutions asymptotically reach the incoherent background I r,0 (β → ∞) at large bistatic angles. The sampling of larger bistatic angles would reveal whether significantly lower values of I r,0 (β) than observed exist, and would, therefore, allow for a better constraint of the model parameters.
A contour map of the RMSE between the measured and the modeled values (VV) is shown in Figure 13. While the shallow 400 global minimum (RMSE = 0.0106) is located at the optimal solution Λ T = 2.12 m, Λ A = 21.8 m, multiple other solutions exist, that show only slightly higher RMSE values between 0.011 and 0.015 (see also Table 1). This set of possible parameter Figure 11. (a-c): monostatic-to-bistatic backscatter ratioÎ −1 r,0 , observed by TanDEM-X at the largest available bistatic angles βmax = 0.2 • before and during snow melt. (d-f): radar brightness for the same dates. Areas covered by wet snow appear dark. Great Aletsch Glacier is flowing clockwise from top to bottom. In (a, b), high altitude areas (above 3000 m and above 3400 m) show backscatter enhancement. In (c) wet snow is present in the entire scene and absorption prevents the CBOE. In (f) an increase of the backscatter intensity becomes visible on the tongue of Great Aletsch Glacier and on nearby vegetation covered slopes, indicating that in these areas all snow has melted. Images are shown in slant range/azimuth coordinates.
pairs (Λ A , Λ T ) forms a non-linear curve (1-dimensional manifold) in the 2-dimensional parameter space (red "+"-symbols in Figure 13).  Table 1. Scattering length ΛT , peak height BC (0), and peak width (HWHM) for a set of chosen absorption lengths ΛA determined from the TanDEM-X dataset (VV) of dry firn in the high altitude accumulation area. The bold lines indicate the optimal parameter pair for the VV and HH polarization. RMSE is the root mean square error between the measured and the modeled value of Ir,0.

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Due to the limited amount of acquisitions of the Teram-Shehr and Rimo glacier we only briefly summarize here some observations and refer to the supplementary figures S9-S11 for details. In the accumulation area > 6100 m, we observed backscatter ratiosÎ r,0 (β) between 0.72 and 0.77 in five large-baseline acquisitions with β = 0.193 • from orbit 75 between 11 May and Figure 13. Contour map of root mean square error (RMSE) between measured and modeled data for TanDEM-X (Fig. 12) for different pairs of ΛA, ΛT . The plot shows a weak global minimum because acquisitions at sufficiently large β, that could better constrain ΛA, were not currently available. The model parameters and RMSE values for the red "+" symbols are given in Table 1. For these points, the parameter value of ΛT was estimated by nonlinear least squares minimization for different choices of ΛA.
24 June 2015. For orbit 98 with β = 0.231 • , we observed ratiosÎ r,0 (β) between 0.74 and 0.81 between 01 May and 03 June 2015. With Eq. (12) these values allow estimating an lower limit for the backscatter enhancement B C and correspond to an 410 enhancement of at least 23-39% (+0.9 to +1.4 dB). In comparison, for the Jungfrau-Aletsch region, we obtain a lower limit of 19-27% (+0.8 to +1.0 dB). Due to weak model constraints in the Teram-Shehr dataset, we do not provide parameters estimated by model fits. In both Ku-band and X-band observations of terrestrial snow, we observed narrow intensity peaks with an angular width of a fraction of 1°. These peaks are clearly attributable to the CBOE as opposed to the SHOE, which follows from a comparison of the effects' properties as described by Hapke (2012, Chapter 9): Firstly, the SHOE requires that the scatterers are much larger than the wavelength of the incident radiation so that they can cast sharp shadows. This requirement can hardly be fulfilled at radar wavelengths of several centimeters, since ice particles 420 in snow have average diameters on the order of millimeters (Kuga et al., 1991). A grain size of 0.3 − 1.5 mm was observed in the seasonal snowpack studied by the KAPRI experiment (Fig. 2). The snowpack did not show any cm-sized ice structures that could have been caused by strong melt-events. Furthermore, the narrow peak width is in agreement with characteristics of the CBOE, while a SHOE peak usually has a width of several degrees or tens of degrees, depending on particle size distribution. Finally, SHOE is only present in media where single scattering is dominant. Multiple scattering processes decrease 425 the amplitude of the SHOE and increase amplitude of the CBOE. Since dry snow is a weakly absorbing medium for microwaves where multiple (i.e. volume) scattering is considerable (Kuga et al., 1991), the CBOE is expected to be the dominant effect. Figure 7 shows a statistically significant enhancement peak for the winter acquisition, and a lack of such a peak for the 430 summer dataset, which was acquired using an identical target region of interest, identical acquisition procedure (except platform substitution to allow movement on snow/road), and identical processing pipeline. The summer dataset thus serves as a useful control which ensures that the detected enhancement peak is not an erroneous artifact of the bistatic data processing pipeline, and it also indicates that the enhancement peak is indeed caused by the snow layer present on the hillside. The absence of an enhancement peak in summer leads (in accordance with the red line in Fig. 6) to estimates of comparable values Λ T ≈ Λ A . In 435 the context of the model, an absorption length as short as the scattering length suppresses higher-order scattering paths, and thus is equivalent to a dominant single-scattering process.

Ground-based observations -KAPRI
The best-fit values of model parameters in Fig. 7 indicate a scattering mean free path value Λ T between 30 and 50 cm, and an absorption length Λ A between 6 and 24 m for both polarizations. No statistically significant difference of the parameter estimates is observed between the HH and VV polarizations. This is well-aligned with the theoretical model of Mishchenko 440 (1992), in which there is only a very small difference between the co-polarized backscatter enhancement factors for these two polarizations. The HWHM of the angular peak of ≈ 0.25°is sufficiently wide so that KAPRI's transmit antennas' non-zero size, which limits the angular resolution to 0.05 • (Sect. 2.1.2), has only a very limited effect on the precision with which the width and height of the peak can be determined.
The thickness of the snow layer during the winter acquisitions was measured on-site as approximately 1.5 m (Fig. 2), and 445 thus the estimate of Λ A is several times higher than the snow layer thickness. The extremely shallow local incidence angle (above 70°for the vast majority of the ROI area) and short transport mean free path Λ T would likely lead to longer trajectories of the radiation through the snow medium before reaching the ground layer. Nevertheless, the snowpack's thickness of only 3-4 scattering mean free paths could result in higher order scattering being limited by the snow depth. Missing higher order scattering, in turn, is an explanation for a rounding of the peak shape (Akkermans et al., 1988, Fig. 7). During the field 450 experiment, a corner reflector at the bottom of a 1.55 m deep snow pit was still visible, indicating that at least a fraction of microwaves reached the ground, thereby limiting higher order scattering.

Satellite observations -TanDEM-X
A significant dependence of the backscatter intensity on the bistatic angle,Î r,0 (β), is only visible in the accumulation zones of the Jungfrau-Aletsch region with altitude H > 3500 m (Fig. 9a, b). Above this altitude, a firn layer with below freezing snow 455 temperatures is present to a depth of several tens of meters (Haeberli and Alean, 1985;Suter et al., 2001). This thick and cold firn layer represents a disordered medium where multiple scattering is possible and at the same time microwave absorption is weak, because liquid water is absent. The existence of the CBOE in dry firn is further supported by the spatial and temporal distribution of an enhanced brightness ratioÎ r,0 . Spatially, the enhancement matches to the accumulation area of high altitude glaciers -in the Jungfrau-Aletsch region (Fig. 11) as well as for Teram-Shehr/Rimo glacier the Karakorum (Figs. S10 and S11).

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Temporally, the backscatter enhancement vanishes in these areas when snow melt sets in and thus the scattering predominantly takes place at the snow surface.
The rounding of the peak shape in Fig. 12(a) indicates that either weak absorption or a limited thickness of firn is present in the accumulation area. Field measurements indicate cold firn in at least the upper 8 m (Bannwart, 2021) and literature data indicate that temperate firn might be present at around 15 m below the surface (Suter et al., 2001). The global minimum at 465 Λ A = 21.8 m in Fig. 13 might therefore provide a realistic estimate for Λ A as larger absorption lengths are difficult to conform to field measurements.
On the tongue of Great Aletsch Glacier, where a seasonal snowpack is present during winter, no backscatter enhancement was observed in X-band (Fig. 9c). As seasonal snow is younger than multi-year firn, smaller snow grain sizes are expected, resulting in longer scattering lengths compared to the accumulation area where we observed Λ T = 2.1 m. Considering the 470 thickness of the seasonal snowpack of 0-3 m, these longer scattering lengths would correspond to on average less than 2-3 scattering events for the two-way travel through the snowpack. In consequence, the single scattering at the snow-ice interface at the bottom of the snowpack can remain the dominant scattering process. The low average number of scattering events in the seasonal snow volume is, therefore, not sufficient for the CBOE to occur on the ablation area of Great Aletsch Glacier.
In forest covered areas, no significant dependency ofÎ r,0 on β is visible. We think the reason is that, compared to dry snow, 475 multiple scattering at X-band is reduced in forest due to absorption of microwaves, hence the CBOE is prevented.

Comparison of bistatic measurement geometries
The KAPRI experiment sampled a larger range of bistatic angles (up to 1.92 • ) so that the flat incoherent intensity background I 0 could be sampled. Therefore, both the width and the height of the enhancement peak in winter can be constrained much better than with the TanDEM-X observations where β max ≈ 0.2 • . This, in turn, translates to better constrained estimates of parameters 480 Λ T and Λ A as illustrated by the clearly visible global minimum in the plot of the RMSE value in Fig. 8 as compared to Fig. 13.
Compared to the KAPRI experiment, the bistatic angles sampled by TanDEM-X are relatively small, making it possible that not the entire peak of the CBOE has been sampled. In consequence, the bistatic data at β max = 0.2 • might still be affected by the CBOE. Missing measurements at larger bistatic angles result in a weak constraint of the parameter pair (Λ T , Λ A ), permitting a range of value pairs that each can fit the data (Table 1 and Figs. 12 and 13). To better constrain the observed values of Λ T , Λ A , bistatic angles of at least β = 0.5 • would need to be sampled by TanDEM-X. However, such larger angles are currently not available.

Impact of the CBOE on backscatter observations
Generally, the existence of a narrow backscatter enhancement peak around the monostatic direction needs to be kept in mind by all researchers performing backscatter measurements of snow, regardless of whether the used sensors are considered monostatic 490 or bistatic. On one hand, for truly monostatic sensors the CBOE is strongest. On the other hand, some radar sensors are considered as monostatic even though they have a small but non-zero spatial separation between the transmitting and receiving antennas. Due to this bistatic baseline, the detected backscatter intensity value could be significantly reduced compared to the value that would have been detected by a truly monostatic sensor. When prior estimates of Λ T and Λ A over a particular medium are available, Eq. (6) could be used to estimate the width and height of the peak, which can subsequently be used to estimate 495 bistatic angle values where the CBOE affects the measurements.
As an example of the necessity to precisely align the measurement geometry to the expected width of the peak, Tan et al. (2015) compared modeled results to active and passive microwave measurements at X-to Ka-band performed in Finland, Sodankylae as part of the NoSREx field experiment (Lemmetyinen et al., 2016). The active measurements were performed with the SnowScat instrument . The model, which includes backscatter enhancement into the DMRT 500 theory, might provide a significant step forward for modeling of the radar backscatter signal from snow. However, if the peak width of the CBOE in the NoSREx experiment is comparable to the narrow observed peak widths in our study, the bistatic angles of the SnowScat measurements would actually be one order of magnitude too large to observe the CBOE. This follows from the instrument height of 9.6 m, the incidence angle range of 30-60° (Lemmetyinen et al., 2016), and the antenna separation of 72 cm , resulting in bistatic angle values β = 1.8 − 3.7 • .

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Except for possible extreme cases of a medium causing an extremely narrow enhancement peak (with width on the order of thousandths of a degree), the velocity-induced bistatic angle β v of moving radar platforms is negligible in the context of the CBOE. For the side-looking geometry, the bistatic angle β v caused by platform motion with velocity v can be calculated as β v = 2v/c, where c is the speed of light. Thus, for all conventional sensors -even for satellite platforms in low Earth orbit moving at speeds of 6-8 km s −1 -the resulting value of β v is on the order of thousandths of a degree or less.

Applications based on the CBOE
In TanDEM-X data we have observed a backscatter enhancement of at least 1.3 dB for firn covered areas of the European Alps and in the Karakorum, while for firn free areas no backscatter enhancement could be observed. This suggests that detection of deep firn with X-band is possible when large enough bistatic angles β > 0.2 • are available.
For seasonal snow, we observed a clear CBOE peak (∼ 1.8 dB of backscatter enhancement) at Ku-band using KAPRI 515 whereas in X-band we were not able to observe an enhancement. The higher sensitivity of high-frequency systems (Ku-or possibly Ka-band) to detect the CBOE in seasonal snow results from the shorter scattering length, since sufficiently highorder scattering events can occur within the snow layer of limited thickness. Furthermore, the difference between KAPRI observations in summer (no CBOE from vegetation) and winter (snow-induced CBOE) demonstrates how Ku-band bistatic observations could potentially provide a mean to discriminate dry snow from vegetation and therefore provide a mean to map 520 snow cover extent. Beyond that, the frequency dependency of the scattering lengths makes a characterization of the intensity of the CBOE at multiple frequencies possible which, in turn, could allow for a quantitative characterization of the height or water equivalent of seasonal snow. Area covered by snow, snow depth, and snow-water equivalent are considered key data products for the snow Essential Climate Variable (Belward, 2016). Bistatic missions characterizing the CBOE occurring in snow can thus be an asset in mapping these data products.

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In terms of polarimetric measurements, the results of this study, as well as theoretical models (Mishchenko, 1992), indicate that the effect is present predominantly in co-polarized channels, and the effect is equally strong in both horizontal and vertical polarizations. Nevertheless, in further studies the use of full-polarimetric radar systems can still be advantageous, e.g. to decisively differentiate the CBOE and the SHOE based on their different impact on linear and circular polarization ratios (Rignot, 1995;Hapke, 2012, Sect. 9.4).

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Existing bistatic ground-based SAR sensors (Pieraccini and Miccinesi, 2017;Wang et al., 2019), and to a certain extent also airborne bistatic SAR sensors (Dubois-Fernandez et al., 2006;Meta et al., 2018) could be employed to study the effect locally with a high temporal resolution and to observe temporal variations of the effect. Space-borne platforms, while limited by orbital mechanics and repeat-intervals, can provide a means to sample and to characterize the CBOE on the global scale.
Finally, our characterization of the CBOE at X-and Ku-band in terrestrial snow could inspire future inter-planetary missions, 535 aiming to search for water ice and possibly other types of snow, to employ bistatic radar measurements.

Conclusions
In this paper we presented the first observations of the coherent backscatter opposition effect (CBOE) and the sampling of its angular peak shape at radio wavelengths within the Earth's cryosphere. The existence of the peak was confirmed in seasonal snow cover at Ku-band wavelengths by the ground-based bistatic radar system KAPRI. With the bistatic satellite formation 540 TanDEM-X, the effect was also confirmed at X-band within the accumulation zone of high glaciers in the European Alps and the Karakorum.
The observability of the CBOE in bistatic radar measurements of snow presents an opportunity for future satellite missions aiming to derive snow properties from synthetic aperture radar data on the global scale. The radiometric precision requirement for such a spaceborne radar system is demanding, since the theoretical maximal amplitude of the effect is 3 dB -in this study, 545 we were able to characterize the peak using TanDEM-X and data-driven radiometric calibration. Deployment of such bistatic systems -at bistatic angles up to one or two degrees, and covering the entire CBOE peak including the incoherent background -would open up a new pathway to characterize snow through microwave scattering.
The Ku-band observations presented in this paper suggest that the CBOE can be used as an indicator for presence of seasonal snow cover. At X-band, the CBOE could be applied to detect dry snow thicker than several meters, e.g. multi-year firn in accumulation areas of glaciers. Furthermore, through analysis of the angular width and height of the enhancement peak,