The sensitivity of satellite microwave observations to liquid water in the Antarctic snowpack

. Surface melting on the Antarctic Ice Sheet has been monitored by satellite microwave radiometery for over 40 years. Despite this long perspective, our understanding of the microwave emission from wet snow is still limited, preventing the full exploitation of these observations to study supraglacial hydrology. Using the Snow Microwave Radiative Transfer (SMRT) model, this study investigates the sensitivity of microwave brightness temperature to snow liquid water content at frequencies from 1.4 to 37 GHz . We ﬁrst determine the snowpack properties for 8 selected coastal sites by retrieving proﬁles 5 of density, grain size and ice layers from microwave observations when the snowpack is dry during winter time. Second, a series of brightness temperature simulations is run with added water. The results show that: i) a small quantity of liquid water ( ≈ 0.5 kg m − 2 ) can be detected, but the actual quantity cannot be retrieved out of the full range of possible water quantities, ii) the detection of a buried wet layer is possible up to a maximum depth of 1 to 6 m depending on the frequency (6–37 GHz ) and on the snow properties (grain size, density) at each site, iii) surface ponds and water-saturated areas may prevent melt 10 detection, but the current coverage of these water bodies in the large satellite ﬁeld of view is presently too small in Antarctica to have noticeable effects, iv) at 1.4 GHz , while the simulations are less reliable, we found a weaker sensitivity to liquid water and the maximal depth of detection is relatively shallow (<10 m ) compared to the typical radiation penetration depth in dry ﬁrn ( ≈ 1000 m ) at this low frequency. These numerical results pave the way for the development of improved multi-frequency algorithms to detect melt intensity and the depth of liquid water below the surface in the Antarctic snowpack.

PAPER under pre-evaluation for The Cryosphere presence of underlying water, because of the microwave penetration. For this reason, it is inappropriate to interpret the liquid water detected by microwave observations as "surface melting" strictly (Torinesi et al., 2003a). Nevertheless, it is not precisely known up to what depth water can be detected. This depth is likely to change during the melt season due to snow metamorphism (i.e., grain coarsening, densification, formation of ice layers). A better estimation of this depth is required, particularly in the context of multi-frequency sensor missions that could provide more advanced information on the fate of meltwater, percolation, refreezing and runoff in and on the snowpack (Leduc-Leballeur et al., 2020;Mousavi et al., 2021). 3) Most algorithms produce a "surface melting" binary indicator, but there is also a great interest in the quantification of meltwater volume and of the melt rate (Trusel et al., 2013) as predicted by surface energy budget and climate models (Kuipers Munneke et al., 2012;Fettweis et al., 2011). However, the possibility of retrieving such advanced information from microwave observations is debated, and 65 limitations and possible accuracy need to be assessed. 4) When the surface becomes extremely wet (i.e. a saturated water layer, running water, surface ponding), the microwave brightness temperature decreases -because open water surfaces have a low emissivity (Comiso et al., 2003) -up to the point that melt detection may become impossible despite the surface being obviously wet. To our knowledge, existing products do not take this limitation into account. More specifically, the areal fraction of a pixel covered by supraglacial lakes, above which melt detection becomes impossible, remains to be quantified. 70 This study works to fill the four knowledge gaps outlined above by presenting a series of sensitivity analyses using microwave radiative transfer modeling. The focus is on the multi-frequency sensors (or combination of sensors) found in operational and in-preparation radiometry missions such as SMOS (Kerr et al., 2010), SMAP (Entekhabi et al., 2010), AMSR-E, AMSR2, AMSR3 (Kasahara et al., 2020) and CIMR (Kern et al., 2020). The overarching goal is to refine the interpretation of the datasets produced with the existing algorithms, and to pave the way to new, more advanced, melt detection algorithms. This 75 study is particularly aimed at users who need a better and more quantitative understanding of the various microwave melt products for detailed investigations and thorough firn model evaluations.
To conduct realistic simulations with wet snow, a pre-requisite is an accurate representation of the snowpack at the onset of the melt season. Given the lack of adequate in situ snow measurements in the Antarctic coastal marginal areas, our approach is to first retrieve the snowpack properties from the microwave observations. This is only possible when the snowpack is dry, 80 before or after the melt season. Our method builds on ideas from Mote and Anderson (1995) and Brucker et al. (2010), but with a more advanced setup. It provides a simplified but realistic dry snowpack that can be then modified with added water in different proportions and depths to investigate the microwave sensitivity to liquid water.
The paper is structured as follows: Section 2 describes the microwave observations and the test sites. Section 3 outlines the radiative transfer model, the approach to retrieve the snowpack properties and the conducted simulations with added water 85 in varying amount and depth. Section 4 presents the simulation results. Section 5 addresses the implication of the sensitivity analysis for the users of melt products and for algorithm developers.
PAPER under pre-evaluation for The Cryosphere Discussion PAPER under pre-evaluation for The Cryosphere 2 Study sites and observations Eight Antarctic coastal sites have been selected to investigate the sensitivity of microwave data to meltwater ( Figure 1 and Table   1). Five of them (Maudheimvida, Halvfarryggen, Larsen C, Larsen B and Roi Baudouin) are chosen due to the availability of 90 detailed meteorological observations and melt estimates (Jakobs et al., 2020). Three other sites are located on major ice shelves and have a wide variety of conditions: Wilkins (high accumulation, presence of aquifer, Montgomery et al. (2020)), Amery (low accumulation, presence of supraglacial lakes, Spergel et al. (2021)) and Shackleton (high accumulation, Saunderson et al. (2022)). All the sites are on ice shelves except Maudheimvida and Halvfarryggen. With firn and ice deeper than 100 m, the influence of the underlying substrate is only significant at L band in dry conditions and is treated appropriately in our modeling 95 setup.
Passive microwave observations acquired by the Advanced Microwave Scanning Radiometer 2 (AMSR2) on-board Japan's Global Change Observation Mission 1st -Water "SHIZUKU" (GCOM-W1) satellite are used for the retrieval, general statistics and melt detection. Data at 6, 10, 19, and 37 GHz at vertical and horizontal polarizations are extracted from the National Snow and Ice center (NSIDC) AMSR-E/AMSR2 Unified Level 3 daily product, version 2. The product has a resolution of 25 km 100 at 6 and 10 GHz and 12.5 km at 19 and 37 GHz. The typical brightness temperature accuracy is ±1 K. Observations from the Soil Moisture Ocean Salinity (SMOS) from the European Space Agency (ESA), the Centre National d'Études Spatiales (CNES) and the Centro para el Desarrollo Tecnológico Industrial (CDTI) are also used to provide L-band data at 1.4 GHz (Kerr et al., 2001). We use the Level 3 product (Bitar et al., 2017) downloaded from the Centre Aval de Traitement des Données SMOS (https://www.catds.fr/, last access: 14 March 2022) and selected the nearest pixel of each site from the EASE-Grid 2 105 in equatorial projection, which tends to be distorted in the polar regions (around 100 km in the meridian direction and 6 km in the zonal direction). The daily brightness temperature at 50-55°viewing angle is extracted at both vertical and horizontal polarizations. The typical brightness temperature accuracy is ±2 K In this paper, the melt is detected both in the simulations and in the observations when the brightness temperature exceeds a threshold value calculated as the June to September mean brightness temperature (referred to as "dry brightness temperature" 110 hereafter) + 20 K. More elaborated algorithms exist (e.g. Mote and Anderson, 1995;Torinesi et al., 2003a;Johnson et al., 2020) but we choose the same fixed offset at all frequencies for sake of simplicity and reproducibility of the results. The value of 20 K is lower than that used by Zwally and Fiegles (1994) but corresponds to a typical value of the adaptive algorithm by Torinesi et al. (2003a). The simulation results and analysis code will be made publicly available for further experiments with other algorithms (see Data Availability Section).  (Picard et al., 2018).
This model is one-dimensional, and it represents the snowpack with horizontal layers. In each layer, the snow microstructure PAPER under pre-evaluation for The Cryosphere Discussion PAPER under pre-evaluation for The Cryosphere   (Torinesi et al., 2003a), ice thickness is from BEDMAP2 (Fretwell et al., 2013)  Kim (2021) for general random porous media and applied to snow in Picard et al. (2022b). This theory has the advantage of treating the ice and air components of the snow microstructure in a symmetrical way, so that the scattering function is continuous over the whole range of ice fractions (0-1). Before the use of this method, these functions were discontinuous when the ice components become preponderant with respect to air in the firn (i.e. around a density of 917/2 = 457 kg m −3 ).
In a last step, SMRT solves the multi-layer radiative transfer equation using the discrete ordinate method (Picard et al., 2018).

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Atmospheric absorption and emission are neglected here due to the low frequencies and the relatively cold and dry Antarctic atmosphere. The output is the brightness temperature of the snowpack at two polarizations (vertical and horizontal) and at 55°i ncidence angle for both AMSR2 and SMOS, close to the Brewster angle. Most simulations are run for a single point, but some are run at multiple points to account for the heterogeneity within large pixels. The resulting brightness temperature over such a pixel is the average of the results at all points.

Wet snow permittivity
The main effect of the liquid water in snow with respect to microwaves is to increase both the real and imaginary parts of the complex effective permittivity. Hence, estimating the effective permittivity of the water, ice and air mixture is an important step to investigate the sensitivity of microwaves to the water content, and especially to determine the minimal detectable amount of water. However, no perfect permittivity formulation exists despite tremendous efforts in the 1970's and 1980's (Tiuri and 140 Schultz, 1980;Sihvola et al., 1985;Hallikainen et al., 1986;Mätzler, 1987, and references therein).
In a nutshell, two main regimes of wetness must be distinguished (Colbeck, 1980) depending on the low or high volumetric water content. In the pendular regime, the volumetric water content is low (<3-7% of the total snow volume) and the water appears as isolated inclusions in the pore space (i.e. between the ice grains), usually at the joints and in the necks, where the surface tension energy is minimal. A possible representation in this regime is to consider isolated water inclusions mixed in a 145 dry snow background. In this case, Maxwell-Garnet (MG) mixing formula applies (Sihvola, 1999). However, a delicate choice remains for the shape of these inclusions that controls the depolarization factor in this formula (Colbeck, 1980;Mätzler, 1987).
In the funicular (> 3-7%) regime, water forms a continuous shell around the ice and the air pores become isolated (a non continuous phase). By assuming spherical ice grains coated by a thin water shell, the MG mixing formula applies if the water is the background and ice is the inclusions (Chopra and Reddy, 1986). This result is counterintuitive since the water is usually 150 in minority, but it can be understood by the fact that the electromagnetic waves mainly interact with the outer part of the grains, that is with the thin water shell. This model coincidentally has the advantage to well describe saturated snow, as water actually occupies the background, ice grains are isolated and air is virtually absent in this case. For testing, we also consider the MG mixture of water droplets in an ice background (called here water pocket model). In both cases, to account for the air, the Polder and van Santen formula (Polder and van Santen, 1946) can be used to mix the air and one of these "wet ice" mixtures.  Countless other mixing strategies and empirical fits are possible, resulting in significantly different permittivity formulations.
A number of them have been added in SMRT for this study. For a few of these formulations, Fig. 2 shows the real and imaginary parts of the permittivity as a function of the liquid water content ranging from zero (dry snow) up to the saturation point (no air) for the various sensitivity tests. All formulations agree on the fact that the permittivity (for both real and imaginary parts) increases with the water content due to the significantly higher water permittivity with respect to that of ice and air. However, 160 differences between the formulations are large, for both the real and imaginary parts.
The formulations shown with circles in Fig. 2 are based on dielectric measurements of wet snow while the formulations shown with solid lines are solely based on theoretically mixing ice, air and water with spherical inclusions. The formulation for the pendular regime by Colbeck (1980) is designed for small amounts. Likewise, the formulations by Wiesmann and Mätzler (1999) and by Hallikainen et al. (1986) (used here with the version updated by Ulaby and Long (2015)) respond quasi-linearly 165 to the water content, because they were fitted with experiments including small water contents only. Their validity for high water contents is unknown. MEMLS v3 (Mätzler and Wiesmann, 2007) extends the formulation by Wiesmann and Mätzler (1999) for high contents. A first group of formulations features similar behavior for small water contents, it comprises Colbeck (1980) pendular, Wiesmann and Mätzler (1999), MEMLS v3 and the coated spheres. At the other end of the range, near the saturation point, the coated spheres, the two-step mixing (ice+water)+air, and both Colbeck (1980) models converge towards 170 similar values (≈ 14 + j16). The water pocket model is an outlier, which is expected given that melting snow is unlikely to be made of water inclusions in the ice crystals.
The fundamental reason for these large differences and lack of consensus is the extreme sensitivity of the effective permittivity to the depolarization ratio, which itself depends on the shape of the inclusions. Water inclusions in the pendular regime PAPER under pre-evaluation for The Cryosphere Discussion PAPER under pre-evaluation for The Cryosphere can be very elongated depending on the ice matrix. Until new permittivity and microstructure measurements are made, new developments are unlikely. A pragmatic choice between these formulations remains the only option.
In this study, we selected the MEMLS v3 formulation for the reference simulations because it is based on actual measurements, and has an intermediate behavior. We also considered the formulation by Hallikainen et al. (1986) and show that it can not reproduce melting snow observations.

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The effective snowpack properties are retrieved at all test sites independently, before and just after the melt season. We call these two periods 'winter' (June-September) and 'autumn' (April-May) for convenience. The retrieval is done by searching the optimal profiles of snow properties which lead to the best agreement between the microwave observations and SMRT output. We do not consider the temporal variations during the periods, only the averaged observations are used as input and the output is a single "average" snowpack for each period. This retrieval approach builds upon previous work for the dry snowpack 185 on the Antarctic plateau (Brucker et al., 2010), and is also somewhat comparable to Mote and Anderson (1995) where the scattering of the snowpack is first optimized in winter to then better detect summer melt. Our approach differs mainly due to our consideration of a more complex snowpack. Specifically, here, the unknown snow properties include the correlation length, snow density and ice layer number density (given in number per meter). This triplet is able to drive the most dominant pattern of variations in brightness temperature. Density mainly controls the absorption and scattering, grain size controls the scattering, 190 and the ice layer number controls the difference between the horizontal and vertical polarizations (H-pol and V-pol hereinafter).
The terrain is assumed flat and the surface roughness is neglected after preliminary tests performed with SMRT and the IEM rough surface model (Brogioni et al., 2010) to determine the sensitivity of the brightness temperature in wet conditions to this snowpack characteristic. The profiles of the properties are assumed to be piece-wise linear functions with four tie points at depths of 0, 3, 8 and 20 m, chosen to cover the range of e-folding depths at 6 GHz and higher frequencies. These tie points are 195 the main unknowns of the optimization problem. At depths >30 m, the density is set constant to a typical value of 912 kg m −3 for deep firn (Burr et al., 2019) down to the base of the ice shelf (ice thicknesses are given in Table 1), where a saline water interface is added.
The temperature profile is prescribed for each season and site. It is approximated by superposing the seasonal temperature cycle penetrating down to d = 2 m depth (Picard et al., 2009) and the overall steady gradient within the ice shelf resulting To compute the optimal values and uncertainties at the tie-points, we use Bayesian inference (Martin, 2022). The reason of this choice is first because we have only 8 observations (6, 10, 19 and 37 GHz at two polarizations, as L-band data is 205 unused at this stage) for 12 unknowns (3 properties at 4 tie-points). The minimization problem is under-determined, and the set of optimal properties is not unique. The second motivation is to account for the uncertainties in the model results with respect to the observations. We proceed as follows: each unknown is given a prior distribution and the goal is to compute PAPER under pre-evaluation for The Cryosphere Discussion PAPER under pre-evaluation for The Cryosphere the joint posterior distribution of the unknown parameters of the model given the microwave observations. For the prior, we consider that little is known about the snowpack in these regions, and choose uniform prior distributions with very wide ranges 210 (also called uninformative priors) to avoid constraining the results with incorrect assumptions. The prior tie-point densities are sampled in the ranges 200-700, 300-800, 400-910, 400-910 kg m −3 respectively at the four depths; the grain sizes in the ranges 0-1.5, 0-2.5, 0-2.5, 0-2.5 mm respectively at the four depths; and the ice layer number densities in the range 0-7 m −1 at all depths. For the densities, we also add a constraint, the profiles with decreasing densities with depth are attributed a lower probability than those with increasing densities. The likelihood is set by assuming that the observations are normally distributed iterations and the first 50% samples are discarded ("burn-in"). The output is an ensemble of 16000 tie-point sets following the posterior probability. However, the MCMC methods tend to produce auto-correlated samples and in our case, the effective sample size as defined in Martin (2022) is estimated ≈100. To assess the MCMC convergence, we calculated the potential scale reduction factor (R) (Gelman and Rubin, 1992) as provided by the ArviZ software (Kumar et al., 2019). Overall we 225 obtainR values lower than 1.2 for all parameters and all sites (in the burned-in ensemble) which is acceptable. For each site, the most probable posterior set (Maximum A Posteriori, MAP) is selected in the ensemble and used in the following sensitivity analyses. We also conduct some simulations with a large number of samples to investigate the impact of retrieval uncertainty on the sensitivity analysis.
In summary, after the optimization phase, we have, for each site and each season, many parametrized snowpacks to simulate 230 the microwave observations and one of them is the optimal. While it is tempting to analyze the retrieved properties of those snowpacks, it is important to recall that the problem is under-determined and the snowpack representation simplified. Many equifinal sets of parameters give similar brightness temperatures, despite they may depict quite different snowpacks from one another, and from the real snowpack as well (Beven and Binley, 1992).

Wet snow simulations 235
To investigate the sensitivity of satellite microwave observations to liquid water, we ran simulations with the optimal dry snowpacks (for the winter season, unless otherwise stated) to which water was added in various quantities and at various depths. Water was always added by filling the air pores, which means that the ice mass (i.e. the dry snow density) is kept constant. In addition to generate a temperature profile representative of the summer season, we apply the same method as in winter except that T s is set to 273 K.  The general trend is decreasing brightness temperatures with increasing numbers of melt days, which is particularly clear at the highest frequencies (19 and 37 GHz). Amery, despite a moderate number of melt days, stands out with the lowest brightness temperature among all sites. It may be related to its distinctively low SMB (Table 1). The H-pol follows in general the V-pol For instance, at H-pol, the brightness temperature is low and close to that at 6 GHz on Amery and Larsen C, whereas it is much higher and close to that of 37 GHz in the other sites. The reason for this is not clear.
PAPER under pre-evaluation for The Cryosphere In general, V-pol brightness temperature at Brewster angle is mainly driven by volume scattering and snow temperature.
The observed decreasing trend with the number of melting days (e.g. a 40 K decrease at 37 GHz) suggests that scattering is increasing with the number of melting days, probably because scattering is driven by the snow grain size, and grain coarsening in the presence of water is much faster than in dry conditions (i.e. wet snow metamorphism (Colbeck, 1982)). It must also be taken into account that fresh snowfalls counterbalance this coarsening by renewing snow at the surface with small grains. This 265 may explain why high-accumulation areas (Shackleton and Wilkins) feature slightly higher brightness temperatures despite a large number of melting days, and conversely why Amery has low brightness temperatures.
In general, the H-pol brightness temperature is more complex because it is in part controlled by snow scattering and snow temperature (exactly as V-pol) and in addition, it is sensitive to the surface density and the vertical density fluctuations in the snowpack (layering). The ice layers decrease the brightness temperature at H-pol due to the reflections on the high dielectric 270 contrast between snow and ice in the upper part of the firn (Montpetit et al., 2013). The variations in V-pol and H-pol are correlated and of similar amplitude only if the ice layer effect is negligible. Here we find that at the highest frequency (37 GHz), the H-pol variations are close to that at V-pol. The reason is that the microwave e-folding depth is about one meter (e.g. 1.3 m for Halvfarryggen, and 0.75 m for Roi Baudouin) and only a limited number of layers are crossed by the upwelling radiation over such a small depth. In contrast, at the lowest frequencies, the e-folding depth is much larger (e.g. 11.5 m for Halvfarryggen, 275 and 5.2 m for Roi Baudouin at 6 GHz). The H-pol signal is lower than the V-pol signal due to the cumulative number of layers crossed by the upwelling radiation emitted in the snowpack at depth. The H-pol brightness temperatures and the H/V ratios are therefore generally low at low frequencies.
The simulations with the optimal parameters for each site produce a good match with the observations at AMSR2 frequencies (dots in Fig. 3). The Root Mean Square Error (RMSE) is low; 1.3, 1.3, 1.7, 2.0 K at 6, 10, 19 and 37 GHz respectively, 280 accounting for both polarizations. These errors are comparable to the standard deviation σ estimated by the algorithm (all-site posterior mean of 2.5 K).
In contrast, the results for L band (1.4 GHz) are very different. L-band simulations shown in Fig. 3 were obtained with a modified grain size profile because the simulations with the optimal parameters (not shown) present very large errors (33 K RMSE considering all sites and both polarizations). This may be partly explained by our choice to exclude the SMOS observa-285 tions from the optimization. Nevertheless, the problem is more profound. To reproduce the SMOS observations, it is necessary to significantly increase scattering at 1.4 GHz (to simulate the low observed brightness temperature). This could be done by increasing the grain size (we found that an overall factor of ≈ 2.8 is necessary) but the collateral impact is degraded brightness temperatures at the higher frequencies. The snow grains are indeed small with respect to the wavelengths (Rayleigh scatterers), implying that their scattering spectral response is strongly increasing with the frequency (≈ 4th power of the frequency). brightness temperatures observed at L band compared to those at higher frequencies are an indicator of the presence of large scatterers in the snowpack, i.e. probably ice nodules or pipes (Jezek et al., 2018). We chose not to add such scatterers in our snowpack representation because it would require an increased number of unknown parameters that would not be compensated by the additional observations provided by SMOS, making our estimation problem even more under-determined than it is. PAPER under pre-evaluation for The Cryosphere Instead, we devised a pragmatic approach. The grain size is multiplied by 2.8, for all the sites when only simulating the L-band brightness temperature (value obtained by successive tests), which leads to a reduced, but still relatively high, RMSE of 25 K.
These results are shown in Fig. 3. We hypothesize that this approach works for the purpose of this study based on the fact that absorption becomes the dominant process over scattering when water is added. However, in general, the L-band results must be taken with caution. For this reason, these results are addressed in a dedicated section (Section 4.6) after the AMSR2 frequency 300 results.
The mean retrieved parameters for all the sites are shown in Fig. 4. Some general observations can be made despite the risk of compensation between parameters (equifinality). It is also worth noting that the properties at 20 m are not always constrained by the observations, as for instance when the e-folding depth is only 5.2 m at Roi Baudouin at 6 GHz, the lowest frequency used in the optimization. In this case, the method returns a virtually random value for this depth. Nevertheless, we observed that 305 the grains are systematically smaller near the surface and generally increase in size with depth, which is a consequence of the observed brightness temperature dependence on the frequency as explained in Brucker et al. (2010). This vertical distribution is expected because fresh, small-grained, snow is present in winter, overlying the wet-metamorphosed, coarse-grained, snow from the previous summer at depth. The sites with high brightness temperatures (Halvfarryggen and Maudheimvida) have distinctly smaller grains at 3 m depth, which is consistent with the lack of melt at those locations. The density increases with depth. The terms of melt and brightness temperature. The explored range is 0-20 kg m −2 , which corresponds to 0-20% in volume in the wet layer, and about 0-50% in mass (the surface density of the best parameter set is 220 kg m −3 ). Fig. 5 shows the results.
The microwave signal is affected by the presence of liquid water in the snowpack according to two different microwave emission regimes. In the first regime, the sudden apparition of water at the surface of the ice crystal sharply increases the snow absorption because the imaginary part of the water permittivity is extremely high compared to that of ice (0.0017 at 325 19 GHz, Mätzler et al., 2006) (Fig. 2). As a result, the brightness temperature increases because the snowpack becomes a nearblack body. At V-pol, the brightness temperature reaches 271 K, which is very close to the perfect back body (273 K). This is a consequence of the observing configuration close to the Brewster angle, the vertically polarized emitted radiation is transmitted PAPER under pre-evaluation for The Cryosphere Discussion through the surface without any loss. In reality, such high brightness temperatures are rare, for example they can be found only 347 times in the full daily AMSR2 19 GHz records gridded at 12.5 km over 9 summers (2012)(2013)(2014)(2015)(2016)(2017)(2018)(2019)(2020)(2021). For comparison, melt is 330 detected 2.5 × 10 6 times on this same grid and for this same period. In contrast, the H-pol brightness temperature only reaches 240-250 K depending on the frequency, because the surface reflects part of the snowpack emission back down. Despite this lower maximum, the amplitude of the increase is larger at H-pol than at V-pol (≈60 K versus 55 K) which is also the case in the observations (Fig. 5c) and is the reason why most single-channel detection algorithms use the H-pol.
Another important information for the user of melt products is the water content threshold from which melt can be detected 335 by the microwave sensor. This information is crucial for the validation of firn models for instance. We find that very small amounts of water of 0.11, 0.07, 0.05 and 0.06 kg m −2 can be detected (assuming a +20 K H-pol increase) at 6, 10, 19, 37 GHz respectively (values for the Roi Baudouin site). The sensitivity is slightly higher at 19 and 37 GHz because of the short wavelengths. These values are similar to those reported in previous work  using the coated permittivity model (the strongest absorption) and using MEMLS (Figure 1 in Tedesco et al., 2007). When the amount of water further increases, the brightness temperature reaches a maximum and then slowly decreases ( Fig. 5b). In this second regime, the absorption is still very strong, but the effect of the high real part of the water permittivity becomes significant (Fig. 2). Note that the small slope changes (e.g. around 17.5 kg m −2 at 19 GHz) is a numerical artifact due to this increasing strong permittivity and how SMRT deals with the refraction in the DORT method (details in Picard et al., 2018). The wet surface becomes more and more reflective at H-pol (0.02 at 0% and 0.27 at 20% at 19 GHz) reducing the 345 brightness temperature (Naderpour et al., 2017;Leduc-Leballeur et al., 2020). Then, the Brewster angle increases (from 49°at 0% to 63°at 20%) so that the V-pol brightness temperature at 55°starts decreasing as well, though in a lesser proportion than at H-pol. The same behavior is observed on wet soils and is at the basis of soil moisture retrieval algorithms for SMOS (Kerr et al., 2001). In principle, snow wetness could be retrieved in this regime using H-pol brightness temperature or H/V ratio, as investigated by Naderpour and Schwank (2018).

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The transition between the first, "absorption" regime and the second, "reflective" regime takes place around 0.75 and 1.75 kg m −2 at 37 and 6 GHz respectively. Shi and Dozier (1995) reported a slightly higher value of 3 kg m −2 for radar at C band (5.6 GHz). Using a different formulation of the wet snow permittivity would change these values. For instance the coated sphere model (results not shown) has a higher imaginary part of the permittivity, which leads to even more rapid increases in the first regime, and a reduced decreasing rate in the second regime, because the absorption dominates even more 355 than the scattering and reflection mechanisms.

Intra-pixel variability
Because of the coarse resolution of the microwave radiometers (≈ 25 km), it is likely that the presence and quantity of liquid water in the snowpack is variable within a given pixel. Combined with the highly non-linear response shown in Fig. 5, this variability is able to affect the sensitivity analyzed in the previous section. To simulate such an heterogeneous pixel, we assume that the liquid water content follows a normal distribution, with negative values set to 0 kg m −2 (dry snow) and compute the 370 brightness temperature for 10000 liquid water values sampled from this distribution. The resulting brightness temperature over such an heterogeneous pixel is the average of all these brightness temperatures. at Roi Baudouin when the liquid water content is heterogeneous. σT W C is the standard deviation of the normal distribution used to generate local total liquid water content (negative values set to 0). Fig. 5. The water content threshold from which melt can be detected is about 0.9 kg m −2 for a 2 kg m −2 standard deviation, a ten-fold increase compared to 0.08 kg m −2 when the pixel is assumed homogeneous.
However, little is known about the distribution of surface melt rate and of the liquid water content over a 25-km pixel in reality. It is difficult to conclude whether σ T W C ≈ 2 kg m −2 is a realistic value or not. The sources of liquid water content 380 variability include the surface roughness, the large scale terrain topography, the wind and cloudiness, dust deposition, snow density, and snow grain size, etc. Future research should investigate the distribution of melt in more detail. Meanwhile, this result demonstrates that the uncertainty in the liquid water content heterogeneity is a major contributor to the uncertainty associated with relating brightness temperature values to liquid water content. Indeed, for a given observed brightness temperature value, many different mean liquid water contents are possible, reducing the possibility of retrieving accurate values of liquid 385 water content from microwave radiometry. is reduced below 20 K, in line with Tedesco (2009). This variability in the sensitivity motivated the development of a more advanced algorithm (Tedesco, 2009) where a radiative transfer model (MEMLS) estimates the brightness temperature threshold for a user-defined fixed water content depending on the snowpack characteristics.

Impact of the permittivity formulation
The results presented in the previous sections rely on the selected permittivity formulation (MEMLS v3). To illustrate the 405 importance of the formulation, we run simulations with the H86 formulation (Hallikainen et al., 1986;Ulaby and Long, 2015) which has a relatively extreme behavior compared to the selected formulation. The results in Fig. 9  formulation features a particularly low imaginary part while the real part is not different from that of the other formulations.

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When the snowpack is transitioning from dry to wet, the scattering is enhanced by the rapidly increasing real part (i.e. snow grains becomes more reflective) while the absorption remains weak due to the low imaginary part. The consequence is a decreasing brightness temperature with increasing water content until the absorption starts increasing significantly offsetting the scattering effect. The "scattering" regime dominates up to about 2.5 kg m −2 when the brightness temperature recovers to the winter value, and then further increases in the "absorption" regime as found with the MEMLS v3 permittivity formulation.

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To assess the relevance of H86 formulation, we explored a large number of timeseries of AMSR2 brightness temperatures around the Antarctic and never noticed such a decrease of brightness temperature caused by weak melt events. We therefore conclude that the imaginary part of H86 formulation is certainly inadequate to model wet snow.

Variations with the thickness of the wet snow layer -Experiment 2
The previous results were obtained with a fixed wet snow layer of 10 cm. Figure 10 reports a few results for different thicknesses 420 of wet snow h w (for Roi Baudouin and at 6 and 19 GHz only) considering two predictor variables (i.e. the variable on the xaxis): the total liquid water content T W C (in kgm −2 or the volumetric liquid water content θ i (a fraction, no unit, usually expressed in percents). These variables are related by: T W C = θ i ρ water h w where ρ water is the water density. We find that varying the thickness of the wet snow layer has little influence on the results if the total amount of liquid water is fixed (Fig.   10a), and in contrast, very large variations are observed if the volumetric liquid water content is fixed (10b). This result is 425 expected given that the brightness temperature is mainly driven by the absorption of the upwelling radiation through the wet PAPER under pre-evaluation for The Cryosphere Discussion snow layer, and that this absorption is quasi proportional to the total mass of water, at least for small amounts of water (i.e. the imaginary part of the effective permittivity is linear for small amounts, see Fig. 2). It means that the results in the previous section generalize well if the layer thickness differs from 10 cm as long as the total liquid water content is used in the xaxis. Despite this, many authors usually report results in terms of volumetric liquid water content (e.g. Tedesco and Kim,430 2006; Naderpour and Schwank, 2018) because it is a measurable quantity and it naturally appears in metamorphism models.
However, in this case, our results also show that it is crucial to clearly report the thickness of the wet layer.
The case of thin wet snow layers (i.e. 1 and 2 cm) is particular. Fig. 10a depicts a smaller increase in brightness temperature (especially at H-pol and 6 GHz) for the thinnest layers, compared to the thicker layers. We explain this specific finding by the high volumetric liquid water content in the thin surface layer, which leads to a high permittivity constant, a high reflectivity, 435 and in turn a low emissivity.
In conclusion, plotting brightness temperature variations as a function of the total liquid water content (kg m −2 ) allows for a better generalization of our results for other thicknesses than if the volumetric liquid water content was used. For this reason, we generally present our results in terms of total liquid water content.

Influence of the depth of the wet snow layer -Experiment 3 440
Because of refreezing at night or during/after snowfall, it is frequent that a wet layer is covered by dry snow. How the brightness temperature is affected and how deep wet snow can be detected at a given frequency is an important question when investigating refreezing (Leduc-Leballeur et al., 2020). Fig. 11 shows the brightness temperatures when a 10-cm wet snow layer (1.5 kg m −2 ) PAPER under pre-evaluation for The Cryosphere Discussion sinks into the dry snowpack at Roi Baudouin. If we exclude a local maximum near the surface at H-pol (due to the impedance matching reducing the reflectivity when dry snow covers the wet snow layer), the brightness temperature decreases as a function 445 of depth and reaches an asymptotic value corresponding to the dry snowpack. The rate of decrease depends on the frequency mainly. The 37 GHz signal reaches the asymptotic value above 2 m, and this depth is about 4, 8 and 10 m at 19, 10 and 6 GHz.
These depths are consistent with the typical e-folding depths estimated in the Antarctic dry snowpack (Surdyk, 2002;Picard et al., 2009) and the decreasing trend with frequency is explained by the increasing absorption and scattering coefficients with frequency. Fig. 12 illustrates this behavior with observations from the Wilkins site. This site was selected because refreezing (or 450 percolation) of meltwater is slow and regular, leading to a clearly-visible slow relaxation of the brightness temperatures at 10 and 6 GHz (decreasing in exponential shape). In contrast, 19 and 37 GHz brightness temperatures rapidly decrease as soon as the surface meltwater refreezes in March and then slowly increases from April, especially at 37 GHz, due to the accumulation of fresh snow with fine grains over the summer layer composed of metamorphosed snow, with coarse grains. This area is known for the presence of aquifers (Montgomery et al., 2020;van Wessem et al., 2020) 455 If we consider a detection threshold of 20 K above the dry mean H-pol brightness temperature, Fig. 11 indicates that wet snow can be detected up to 5.6, 2.8, 1.4, 0.6 m at Roi Baudouin for 6, 10, 19 and 37 GHz respectively. These values only partially follow the variations of e-folding depths with frequency, because the detection is also dependent on the brightness PAPER under pre-evaluation for The Cryosphere Discussion temperature amplitude between dry and wet snow. This amplitude is lower at lower frequencies, and the detection is less sensitive in general.

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Interestingly these detection depths are variable from site to site and depend on the season as depicted in Fig. 13. The site dependence seems to be influenced by the dry brightness temperature (Amery and Larsen C feature the lowest winter brightness temperatures at the two/three lowest frequencies and the deepest possible detection) rather than by the e-folding depth which is expected to be reduced when the brightness temperature is lower due to coarser snow grains. At 37 GHz the pattern is less marked, probably because of the reduced penetration depth, because the snow in the topmost meter is new in winter and thereby 465 not influenced by the previous year melt.
The maximum detection depth is generally reduced in the autumn snowpack (dark colored bars in Fig. 13) compared to that in winter (light shaded bars). Meltwater profoundly transforms the upper snow layers by wet metamorphism and formation of ice nodules. Larger grains and nodules lead to enhanced scattering (when the snowpack becomes dry) which reduces the e-folding depth. The lower frequencies are less affected because these processes are most active near the surface, and also 470 because of the lesser role of scattering. This result implies that the detection at 37 GHz may be more challenging, with the necessity to adapt the threshold during the course of the melt season.
No maximum detection depth can be estimated for Maudheimvida and Halvfarryggen (the two driest sites) because the brightness temperature variations with the depth of the wet snow layer are much weaker than the threshold of 20 K and weaker than at the other sites (Fig. 14 for Halvfarryggen), probably because the dry snowpack is already close to a black body (small 475 grains, high density, according to our retrieval). With a smaller threshold at these two sites, the melt could be detected in Hpol at low frequencies according to SMRT but not at high frequencies. This unexpected behavior is confirmed by the observed PAPER under pre-evaluation for The Cryosphere Discussion  In Experiment 1, we evaluated the influence of the water content for small to moderate fractions (up to 20 kg m −2 which is equivalent to 20% water volume) and showed a decrease of H-pol brightness temperature in the reflective regime. In the case of intense melt and multiple freeze-thaw cycles, an impermeable ice horizon may form at or below the surface. By preventing percolation, this horizon allows meltwater to accumulate, leading to high fractional volumes of water, which is often referred 485 to as 'slush' in the literature and is frequent in Antarctica . In the most extreme case, a supraglacial lake may form (no ice crystals are present at the surface). As soon as the thickness of such a saturated layer exceeds a quarter of the wavelength (Wiesmann et al., 1998), i.e. a few millimeters depending on the frequency, the brightness temperature strongly decreases due to the high reflectively of this layer. Here we investigate the risk of misclassifying such a pixel as not melting, despite the obvious presence of meltwater. Two cases are studied: a snow layer saturated with water ('slush') and supraglacial 490 lakes.
Saturated snow is modeled as a mixture of ice scatterers in a water background. Fig. 15 shows the variations of brightness temperatures when the proportion of water increases from pure ice to pure water. When this layer is exactly at the surface (solid PAPER under pre-evaluation for The Cryosphere Discussion lines) the brightness temperature first increases (due to the increasing absorption), and then decreases to very low values (due to the increasing reflectivity of the wet interface). Analysis of the AMSR2 dataset suggests that such values, which are much 495 lower than the winter average, have never been observed on glacial ice (it is however common on sea-ice), which questions how realistic this simulation is.
Another set of simulations with the saturated layer overlaid by a thin moderately-wet snow layer (5 cm thick and total water content of 5 kg m −2 ) was run (dashed lines in Fig. 15). The signal is completely different to the previous simulation because the wet snow layer is a strong absorber, so no radiation emerges from the saturated layer below and its emission is close to that 500 of a black body, so it has a high brightness temperature. These simulations highlight the critical role of the surface conditions.
If the snow is wet but not saturated in the upper snowpack (a thickness of a quarter wavelength is sufficient), this layer and everything below has no influence on the brightness temperature.
Supraglacial lakes correspond to a water fractional volume of 100% in Fig. 15, characterized by very low brightness temperatures. However, the coverage of supraglacial lakes is never complete at the scale of the microwave radiometer resolution 505 (≈12-50 km) in Antarctica; a pixel contains some fraction of ponded lake water and the remaining area is snow. To investigate this situation, which is frequent on the ice shelves, we simulate the pixel-averaged brightness temperature by linear combination of a simulation with a flat, pure-water surface and a simulation with a wet snowpack with 1.5 kg m −2 total water content as in Experiment 1. The results in Fig. 16 show a strong negative trend in brightness temperature as a function of supraglacial lake fraction at both polarizations. When this trend intersects the threshold of detection (shown as a dashed horizontal line for 510 each frequency and polarization, assuming the threshold at dry brightness temperature +20 K), the mixed pixel is detected as non-melting, which is a false negative. This occurs for lake fractions larger than 42-48% at 6 GHz and 57-60% at 19 GHz, PAPER under pre-evaluation for The Cryosphere Discussion the lower and higher bounds corresponding to the H-pol and V-pol respectively. Such high areal lake coverage fractions have rarely been observed in Antarctica (Arthur et al., 2022). For example, even before the near-complete collapse of the Larsen B Ice Shelf in 2002, which has been attributed to the hydrofracture drainage of about 3000 lakes (Banwell et al., 2013;Robel and 515 Banwell, 2019), the fractional lake coverage reached a maximum of only 10% (Banwell et al., 2014). Searching for a significant decrease in brightness temperature in the AMSR2 dataset have been unsuccessful even on shelves subject to ponding, as for instance on the north George VI Ice Shelf where a maximum of 15% lake coverage was observed , or on the eastern Roi Baudouin Ice Shelf .
We can conclude that false negative melt detection due to supraglacial lake is unlikely at present time in Antarctica. In 520 contrast, the Greenland Ice Sheet is subject to more intense surface melting in the ablation area (Bell et al., 2018). For example, a recent case of flooding on the ice sheet was reported in August 2021 (Box et al., 2022) after an atmospheric river, combined with rain, triggered a massive melt event. The decrease of brightness temperature at 19GHz is visible although the surface was probably wet and remained wet for many days after this event.
4.6 The sensitivity of the L band to liquid water 525 Using the grain size adjusted snowpack, we assess the sensitivity of the L band to liquid water, first in the top layer with a similar setup as Experiment 1, except that the wet layer is 30 cm thick to account for the much longer wavelength at L band.
The analysis is conducted at Roi Baudouin where the dry snowpack simulations are fairly good (Fig. 3). Fig. 17 highlights the overall lower sensitivity at L band compared to the higher frequencies; the increase in brightness temperature is lower (15-25 K) and the maximum at V-pol is reached for a larger water amount (23 kg m −2 ). The H-pol reaches its maximum around 530 6.5 kg m −2 which is ten times the amount observed for the higher frequencies. This weaker sensitivity explains why lower thresholds have been used for the detection of melt with SMOS, as well as why a slightly adapted algorithm has been used PAPER under pre-evaluation for The Cryosphere Discussion ( Leduc-Leballeur et al., 2020). This lower threshold does not imply larger detection errors because the brightness temperature during the dry periods is very stable throughout the year at L band (Macelloni et al., 2016;Leduc-Leballeur et al., 2020).
Hence even a change of ≈10 K is significantly larger than the noise level and the temperature-induced variations of brightness 535 temperature.
According to the very low values of ice absorption at L band (Mätzler, 1996;Macelloni et al., 2016Macelloni et al., , 2019, a wet snow layer could contribute to the radiation emanating from the snowpack up to depths of hundreds of meters. This is in apparent contradiction with the results in Fig. 18 showing the brightness temperature at L band with a 30 cm thick wet snow layer (total water content of 14 kg m −2 ) at increasing depths (Experiment 3). The brightness temperature decreases within a few meters 540 and reaches an asymptotic value. This value is relatively close to the dry brightness temperature (211 K vs 207 K), making the melt detection difficult unless a very low threshold is used. This rapidly vanishing wet signature is due to the fact that the firn at L band at depth is only weakly scattering, so that both the wet snow layer and the firn at depth behave as a black body at a similar temperature, i.e. they emit the similar radiation flux. In the case where the firn temperature is significantly lower than 273 K (cold site), the asymptotic value may be higher than the winter (cold) brightness temperature and the detection at greater 545 depths may be easier. Fig. 18 also shows a rapid increase of brightness temperature at H-pol between the top wet surface (213 K), as in Experiment 1, and just below the surface (235 K) (i.e. covered by a dry snow layer). The impedance matching is the cause (i.e. the reflectivity of the surface is lower when covered by dry light snow) and the consequence is that wet snow may be easier to detect when slightly buried than when right at the surface.

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Overall, the detection of liquid water at L band is more difficult than for higher frequencies, but possible at greater depths.
PAPER under pre-evaluation for The Cryosphere Discussion PAPER under pre-evaluation for The Cryosphere 5 Discussion and conclusions From the perspective of the algorithm developers and the users of melt products, three important knowledge gaps can be addressed with our results and analyses.
The minimum liquid water content required for microwave radiometry to detect the presence of meltwater is generally small 555 compared to the melt amount produced over a typical summer day in Antarctic coastal areas (Jakobs et al., 2020). Even a small melt event is likely to be detected, especially with AMSR-E or AMSR2 which have a near local noon overpass time (Picard and Fily, 2006). In contrast, the detection at L-band requires more water over a greater depth of snow. This result is confirmed in the observations (Leduc-Leballeur et al., 2020) but is subject to caution, because our simulations of the dry snowpack are not very accurate for L-band, due to a poor representation of large scatterers in the snowpack (probably ice pipes).

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For a fine analysis of the melt products, estimation of a precise minimum liquid water content for detection might be necessary. Unfortunately, this threshold depends on many variables: the algorithm parameters (e.g. the brightness temperature threshold), the wave frequency and polarization, the preexisting snowpack properties (density and grain size near the surface) and more importantly the spatial distribution of liquid water within the sensor field of view (melt heterogeneity). Some of these variables are known a priori (e.g. sensor configuration), and their effect can be simulated by our modeling framework, but some 565 others are subject to high uncertainties, resulting in uncertain minimum liquid water content values. To evaluate firn models against microwave-derived melt products, the usual current practice is to assume a fixed minimum water content (Fettweis et al., 2011;Kuipers Munneke et al., 2012). Instead, we suggest the exploration of a range of minimum water content between e.g. 0.2 and 1 kg m −2 to test the significance of the results. For instance, a model showing a systematic positive bias for a large range of minimum water contents is clearly positively-biased. On the other hand, a bias with changing sign when the minimum 570 water content is varied would indicate that the model is not biased and is as good (or uncertain) as the melt product. For the algorithm developers, we suggest to provide outputs with varying parameters, as for instance in Banwell et al. (2021), where the brightness temperature threshold used to detect melt was varied in a reasonable range (i.e. from 2.5 to 3.5 × the standard deviation of the winter brightness temperature) to give a rough uncertainty estimations for the annual number of melting days.
The maximum snow depth for liquid water detection is another important parameter to enable the accurate use of microwave 575 radiometer melt products. For instance to compute the "total" liquid water in a firn model, outputs must be evaluated from the surface to this depth. The main factor controlling this depth is the frequency, with variations from 0.1-0.2 m at 37 GHz to about 10 m at 1.4 GHz. Taking advantage of this large depth range to provide rich information on surface refreezing and percolation is an important research avenue (Leduc-Leballeur et al., 2020;Miller et al., 2020;Colliander et al., 2022). The preexisting snowpack properties (density, grain size and ice layers) are a secondary factor modulating this maximum depth, and since all 580 these properties evolve rapidly in wet conditions, the depth is subject to seasonal variations. We have shown that the retrieval of these three snow properties is feasible before and after the melt season, but the wet snowpack remains opaque during the melt season, forbidding the retrieval of any information under the wet layer. Additionally, as a general rule, the maximum depth of detection decreases slightly over the course of a melt season due to grain coarsening, densification, and ice layer formation that occur in wet conditions. PAPER under pre-evaluation for The Cryosphere The two regimes of variations in the brightness temperature results in non-monotonic variations as a function of liquid water content. This could be an issue for melt detection as very wet snow has a low brightness temperature (especially at H-pol), potentially below the melt detection threshold. Our results show that the required amount of water concentrated in the upper layer is large and is not typical of the Antarctic environment at the present time. While the H-pol is often favored over the V-pol for melt detection (Zwally and Fiegles, 1994;Torinesi et al., 2003b), future algorithms could assess 590 the benefit of using V-pol brightness temperature to mitigate this problem, versus the disadvantage of the lower V-pol contrast between dry and wet conditions. Finally, in the reflective regime, for liquid water contents higher than 2-5 kg m −2 , the H and V-pol signals can be used to retrieve liquid water content (Naderpour and Schwank, 2018;Houtz et al., 2021), but we have shown that appropriate prior knowledge of the surface density and absence of water at depth are essential.
Geographically, our study relies on specific sites in Antarctica's coastal areas, mainly ice shelves. However, most results 595 show relatively small inter-site variations, which leads us to conclude that the benefit of this study is more general and contributes to the knowledge of the Antarctic Ice Sheet at large. Applicability to the accumulation area of the Greenland Ice Sheet (where the glacial ice is far below the microwave penetration depth) and where the number of melting days is similar, is probably acceptable. In addition, despite using AMSR2 and SMOS observations, our results also apply to any radiometric sensor operating at frequencies of 1.4 -37 GHz and with incidence angles of 50-60°. This includes for instance SMMR, SMAP, 600 SSM/I, SSMIS and CIMR. This work has provided the basis for developing new advanced melt detection algorithms, a logical future avenue of research.
However, we stress that fundamental knowledge is still missing regarding the wet snow permittivity, the small scale liquid water variability, and the snowpack properties in the coastal Antarctic environment, particularly on the ice shelves. Future work should focus on filling these gaps given the growing importance of the ice-sheet hydrology as the climate warms.

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Code and data availability. The SMRT model code, including the new permittivity formulations, is available from https://github.com/ smrt-model/smrt [last access 8 Nov 2022] (Picard et al., 2022c). The code to produce the figures is available from https://github.com/ smrt-model/smrt_liquid_water_paper [last access 8 Nov 2022] (Picard, 2022). This repository also includes the retrieved snowpack properties and the observed and simulated brightness temperatures at the select study sites.
Author contributions. GP conducted the study, implemented the permittivity equations in SMRT and run the simulations. ML, GM, AFB, or opinions expressed herein, are those of the authors and do not necessarily reflect those of NOAA or the U.S. Department of Commerce.
Competing interests. LB is a member of the editorial board of The Cryosphere. Other authors declare no competing interests.