Microwave radiometers measure the naturally emitted thermal radiation, called the brightness temperature (TB), by the firn as observed in the microwave portion of the electromagnetic spectrum. It is related to the emissivity e and the effective physical temperature Tphy of snow/firn/ice media for a given frequency f, polarization p, and incidence angle θ. If firn were vertically homogeneous or isothermal, the TB could be found according to Rayleigh–Jeans approximation (Ulaby and Long, 2014): 1TB(f,p,θ)=eTphy. However, firn is not a vertically homogenous medium. Both the emissivity and temperature vary with depth. As a result, the TB is given by a depth-integrated product of physical temperature and emissivity, weighted by the emissive, absorptive, and scattering properties of the snow/firn/ice layers (Jay Zwally, 1977), which is strongly dependent on the frequency of observation.

To account for the depth dependencies of snow and ice properties, firn is regarded as a complex multilayer dense medium. For each layer, an effective physical temperature and permittivity are determined from firn absorptive and scattering properties. Then, the microwave emission and its propagation are typically modeled using an equation of radiative transfer (RT). Considering firn to be a stack of N plane-parallel layers consisting of isotropic and homogeneous material in each layer, the RT equation can be given as (Jin, 1994, 1997; Picard et al., 2013; Tsang et al., 2000) 2cos⁡θddzTBz,θ,∅=κaTphyzI-κeTBz,θ,∅+∫0π2∫02πsin⁡θ′dθ′d∅′P(θ,∅,θ′,∅′)⋅TBz,θ′,∅′. Here, TBz,θ,∅ denotes the vertically and horizontally polarized brightness temperatures at depth z propagating along a direction characterized by θ (zenith angle) and ϕ (azimuth angle). κe, κa, and κs are the extinction, absorption, and scattering coefficients, respectively, representing medium properties. For an isotropic medium, the extinction coefficient can be described as κe=κa+κs. θ′ and ϕ′ are slant angles, and P is the bistatic scattering phase function. Tphyz is the physical temperature of snow at depth z, and I is a unit vector. Thus, the first term on the right-hand side of Eq. (2) represents the microwave emission (TB) of snow/fir/ice from depth z, and the second term denotes the extinction (attenuation) of the emission due to absorption and scattering. The third term represents the sum of total scattered emission in the direction of the receiver (as specified by θ and ϕ). Equation (2) is solved analytically or numerically subject to boundary conditions at each layer interface and at the top and bottom of the medium.

The extinction coefficient, κe, is a function of the effective dielectric constant of the layer and frequency of the observation. Thus, the overall TB is given by the depth-integrated profiles of the effective physical temperature and dielectric constant of each layer. Therefore, penetration depth plays a key role in determining the variability in TB, especially in low-frequency bands. For a low-loss medium such as firn, the penetration depth can be approximated as (Elachi and Zyl, 2021) 3δ=cε′2πfε′′, where c is the speed of light and ε′ and ε′′ are the real and imaginary parts of the dielectric constant of the firn. As shown, δ is inversely proportional to both f and ε′′. The L-band signal thus penetrates a significantly thicker layer than the higher frequency, such as the Ka-band signal. Liquid water markedly increases ε′′ (compared to ε′ in proportion), decreasing the penetration depth for any frequency. For a typical snow density (measured for dry snow) in the percolation zone, it can be more than 4 m for an average LWC of less than 1 % with the Ulaby and Long (2014) model's wet snow dielectric constant, decreasing exponentially with the LWC. Thus, for an average LWC of 3 % and higher, it is around 1 m and less. The average LWC in the percolation zone is typically not higher than 4 %, except for extraordinary melt years (such as 2012, not included in the study), and typical infiltration of liquid water is also generally within the upper 4 m (Samimi et al., 2020, 2021).

There are two types of scattering processes in the snow/firn medium affecting the propagation: surface scattering and volume scattering. The relative size of the scatterers compared to the wavelength determines the degree and types of scattering. For high-frequency bands (> 10 GHz), the impact of volume scattering is critical because the fractional volume of scatterers (snow/firn) is significant. This is why the high-frequency signals interact more with fresh snow, grain size, and roughness at the surface. Low-frequency signals (<10 GHz) are relatively insensitive to volume scattering from snow grains because the size of the scatterers is much smaller than the wavelength. Surface scattering occurs due to surface irregularities at the interface between layers of different dielectric constants, affecting all the frequencies when present. Horizontal and vertical ice layers (strata) are formed at various depths in the firn primarily from the refreezing of seasonal snowmelt. Over time, older ice layers move downward due to the snow accumulation, while new ice layers are formed for subsequent melts at the top layers, creating a complex set of stratigraphy and significantly influencing the L-band signals from the deeper layers. Therefore, L-band TB is determined by the subsurface temperature, stratigraphy, and LWA.

L-band TB exhibits some distinct spatial features over the GrIS during a frozen season. Along a typical west–east transect, TB is the highest in the ablation zone, then it gradually decreases to its lowest value in the percolation zone, followed by a gradual increase towards a moderate value in the upper accumulation zone. A mirror image is seen on the eastern side of the ice sheet. The spatial features of H-pol TB are similar to V-pol TB, but they are more affected by subsurface layering. This is illustrated in Fig. 1 with V- and H-pol mean frozen-season TB values and their normalized polarization ratio (NPR; defined as NPR = (TBV - TBH) / (TBV + TBH)). The ablation zone is characterized by exposed glacial ice with a higher density and internal temperature than the ice sheet towards Greenland's interior. It is soaked and swept by a large amount of meltwater every year. During the frozen season, the L-band emission has a high effective emissivity, radiating the warmer physical temperature of the deeper layers. In the percolation zone, on the other hand, moderate but varying melt occurs almost each year or every few years and percolates down and refreezes at different depths, forming discrete ice layers and ice pipes and causing substantial scattering of mean TB (Jezek et al., 2018). High NPR values highlight the area with dense ice layers (strata). The upper accumulation zone experiences light or no melt but accumulates snow, resulting in less density variation compared to the percolation zone. For detecting melt and quantifying LWA, we used vertically polarized TB (V-pol TB) considering its lower sensitivity to subsurface stratigraphy.

During the summer season in presence of melt, the L-band TB generally decreases in the ablation and upper accumulation zones compared to the frozen season, while it increases significantly in the percolation zone. Figure 2 illustrates this for a sample summer day (31 July 2019) when melt was detected across the K transect (∼67° constant latitude; see the red line in Fig. 2a). The melt flags (square symbols over the dashed line) specify the TB samples for which melt was detected (Sect. 3.2). The presence of LWC in the snow and firn increases the absorption and emission in turn (Mote and Anderson, 1995). However, at the lower elevation around the ablation zone, the TB decreases from its very high level (∼260 K) as the LWC of the seasonal snow layer increases. This is because, when the LWC in the snow layer exceeds a threshold, snow becomes saturated and creates a reflective boundary at the ice and snow interface, suppressing the emission from the ice layer and resulting in overall lower TB. This is caused by intense melting common in the ablation zone (Fig. 2b). The percolation zone experiences moderate melt, making the snow and firn highly absorptive during melt season. As a result, the TB gradually increases from its winter references (Fig. 2b), making the L-band sensitive to the total amounts of melt. In the upper accumulation zone, melt seldom occurs. However when it does occur, it may percolate and refreeze quickly in the colder snow, creating ice layers that cause reflection and reducing L-band TB signals.

Figure 3 shows the L-band V-pol TB time series during March–October 2016 at the DY2 AWS, a location representative of the percolation zone. During the frozen season, the L-band TB is relatively lower and stable. During the melt season, it captures the diurnal signals during melting phases (melt generation). However, it diminishes as the melt percolates to deeper layers. From the onset through the end of the melt season, the density and grain size increase in the snow and firn layers due to melt (Vandecrux et al., 2022). Although the L-band TB is relatively insensitive to the grain growth, the post-melt TB level may still decrease because of increased reflection from newly formed ice layers. This effect is pervasive, especially across the accumulation zone, justifying a dynamic threshold in threshold-based melt detection algorithms.

A locally calibrated and validated EBM (Ebrahimi and Marshall, 2016; Samimi et al., 2020, 2021) was used as the primary reference for comparison. The EBM was initialized with ice core density profiles, stratigraphy, and subsurface temperature profiles (Vandecrux et al., 2023) and was forced with the hourly surface forcing from PROMICE and GC-Net AWSs. The model first calculates the net energy balance from the surface forcing by combining the energy fluxes towards the surface layer. Then, it runs a subsurface model to calculate heat conduction and melt rates in the upper 20 m of the snow/firn by resolving the profile into 43 vertical layers, with gradually decreasing thickness near the surface.

When the surface temperature reaches the melting point and the net energy is positive, melting occurs. Conversely, if net energy is negative and the surface layer is at the melting point, any existing liquid water will freeze, releasing latent heat and causing the surface layer to cool until all liquid water is refrozen, depending on the energy balance. When surface layer temperatures are below the melting point, and there is either an excess or deficit of energy leading to warming or cooling, the energy balance within a one-dimensional model of subsurface temperature evolution determines the subsurface temperature and density profiles. The model determines hydraulic conductivity and permeability after Meyer and Hewitt (2017), while thermal conductivity was modeled following Calonne et al. (2019). The profile then governs the availability of local water at any level for the next time stamp. The model relates to a basic approach to how meltwater flux percolates downward using Darcy's law. The local water balance is determined by mass conservation in each subsurface layer. Once a layer becomes temperate, it can retain liquid water within its pore space or allow it to percolate deeper (Coléou and Lesaffre, 1998). The subsurface model is coupled with a hydrology model that redistributes the meltwater; depending on the subsurface temperature profile, the meltwater may refreeze. Due to refreezing, density may increase, and ice layers may form that may reduce or completely block meltwater infiltration. The firn densification was modeled as in Vionnet et al. (2012). We henceforth refer to this model as the EBM for simplicity. To evaluate the LWA retrieval, we calculate the daily average LWA from the hourly EBM output.

We used output from GEMBv1.0 as a secondary source of comparison. It is a module in the Ice-sheet and Sea-level System Model (ISSM; https://issm.jpl.nasa.gov/, last access: 20 September 2024) that models the ice sheet surface energy and mass exchange and snow/firn state in a one-dimensional column over time (Gardner et al., 2023). It has more than 100 vertical layers with <5 cm thickness in the top layers and employs spatially variable grid size based on the ice sheet dynamics. GEMB formulates irreducible water content according to Colbeck (1973) and uses a bucket scheme (Steger et al., 2017) for liquid water infiltration. Parameterization of firn densification and thermal conductivity follows Herron and Langway (1980) and Sturm et al. (1997), respectively. Readers are referred to Gardner et al. (2023) and references therein for further details. The model was forced with the same hourly surface forcing from PROMICE and GC-Net AWS, but gap filled with ERA5 (Hersbach et al., 2020) atmosphere and radiation conditions, after the methods described by Paolo et al. (2023). The ERA5 surface temperature and downwelling longwave radiation forcing were spatially bias-corrected for each month, such that all values were adjusted by the difference between the RACMO2.3 (Noël et al., 2016) and the ERA5 1980–2015 monthly means. GEMB outputs include temperature, density, and LWC profiles.

To compare SMAP daily LWA time series with corresponding LWA from the EBM and GEMB models, we considered the standard evaluation metrics, including mean difference, standard deviation (SD), mean absolute difference (MAD), Pearson linear correlation coefficient (r), and root-mean-square error (RMSE), for summer seasons (1 June–31 October 2021–2023). We also compared the day of melt onset (the first day of summer melt) and melt freeze-up (the last day of summer melt), the summer melt duration (difference of melt onset and freeze-up), the maximum summer LWA, and the annual sum of daily LWA (LWA_YS). To determine the day of melt onset and freeze-up, we only considered melt events with LWA >2 mm, to avoid any spurious melts that may result from any instrumental noise or other sources. The LWA_YS is the sum of daily LWA over 1 year. It is a measure of the total seasonal LWA, but it does not represent the total surface melt generated over 1 year. This is because SMAP observes the instantaneous LWA, the net water balance, which is the cumulative sum of surface melt, refreezing, and runoff over SMAP footprint. When the net water balance remains positive overnight, it can be considered multiple times in the total integrated LWA as long as it persists.

Figure 5 shows a comparison of the SMAP-retrieved LWA with the LWA derived from the EBM at six different PROMICE and GC-Net AWS sites for the 2023 summer season (1 June–31 October). The melt season at the CP1 site (Fig. 5a) began in the fourth week of June, according to both SMAP and the EBM, and continued through the first week of September according to SMAP, while it extended through the end of September in the model estimate. Shortly after the complete refreezing of the first melt event in late June, SMAP resumed recording LWA in the first week of July. Both SMAP and the EBM closely agree in both phase and magnitude of LWA during first half of July. Afterwards, the EBM reports overall higher LWA for the rest of the season, and it seemed to retain liquid water for an elongated period when SMAP showed a fully refrozen firn. The overall agreement is given by the Pearson linear correlation coefficient (r) of 0.79 and root-mean-square difference (RMSD) of 19 mm. The onset of the melt event at the KAN_U site (Fig. 5b) is concurrent to CP1 in accordance with the EBM. However, SMAP did not record melting at this site until the first week of July. Unlike the CP1 site, SMAP reports persistent LWA through the first week of October, whereas EBM shows complete refreezing by the second week of September. Both SMAP and the EBM captured a smaller LWA at the KAN_U site compared to the CP1 site. This is somewhat counter-intuitive because the KAN_U site is located at a lower elevation than the CP1 site (see the elevation in Fig. 5d). In fact, KAN_U is characterized by having a lower accumulation and higher melt rate every year (MacFerrin et al., 2019; Machguth et al., 2016). However, excessive melt has also created thick ice slabs in this location (MacFerrin et al., 2019; Machguth et al., 2016). As a result, liquid water cannot percolate to the deeper layers and run off horizontally. The model excludes this liquid water in the form of “drainage”, and SMAP only sees the existing meltwater in its field of view. At the DY2 site (Fig. 5c), LWA estimated by SMAP and the EBM are more closely resembled both in phase and magnitude (except the difference in timing of complete freeze-up). This is reflected by a nearly perfect correlation (r=0.98) and a small overall RMSD (4 mm) as shown.

SMAP LWA also closely aligns with the EBM at the NSE site in magnitude and duration of liquid water presence (r=0.98 and RMSD = 3 mm), although SMAP seemed to miss the late August small melt event (Fig. 5e). The agreement, however, exhibits the greatest deficiencies at the SDL site for this melt season (Fig. 5f). Although the timing of the melt onset and late August secondary melt event matches precisely, the EBM reports an overall higher LWA and an extended summer melt duration at this location. This is manifested in the performance metrics shown by a relatively higher RMSD (24 mm) and comparatively lower correlation coefficient (0.77). The performance at the SDM site is generally good (r=0.92 and RMSD = 6 mm), except the EBM demonstrates a delayed refreezing compared to SMAP (Fig. 5g).

It is pertinent to highlight that, while in situ LWAs at all these AWSs were derived from the energy balance model forced by the pointwise measurements at the AWS locations, the SMAP retrievals estimated a spatially averaged LWA corresponding to the ∼30 km effective resolution of the enhanced-resolution TB. Approximately during the first half of the melt season, the LWA is primarily determined by meltwater generation in response to the net radiation flux at the surface, whereas, roughly during the second half, when the net radiation flux remains negative, refreezing becomes the dominant process. Hence, the model's representation of the surface melt infiltration, heat transfer, and other physical processes plays a significant role, posing additional uncertainties. The AWS measurements used to run the model also add some inherent uncertainties. Therefore, assessing relative accuracy is not straightforward. Nevertheless, the general agreements between the model- and SMAP-retrieved LWA in magnitude and phase at these locations suggest that the spatial heterogeneity of melt processes is not acute in these areas.

We performed a pairwise comparison among SMAP, EBM, and GEMB models (Fig. 6) for the 2021, 2022, and 2023 melt seasons (based on available meteorological data) at the six AWS locations. Performance metrics are documented in detail in Table 1 (mean difference, SD, mean absolute difference, Pearson linear correlation coefficient, and RMSD) and Table 2 (melt onset, freeze-up, duration of summer melt, maximum summer melt, and annual sum of daily LWA). Because of the SMAP outage for summer 2022, performance metrics in Table 1 only considered the operational part of SMAP. Table 2, however, excludes SMAP for the 2022 melt season except in the melt onset information, as the other metrics were impacted by the outage.

At the KAN_U site, the overall agreement between SMAP and the EBM was determined to be better (r>0.75) than the agreement between SMAP and the GEMB model (r<0.55) for the 2021 and 2023 melt seasons (Fig. 6a–c). All the AWS data required to run the EBM for the 2022 melt season were not available. GEMB used ERA5 data to gap-fill this period, and SMAP LWA closely aligns with GEMB estimates for the first part of the summer season until the outage. However, the GEMB model demonstrates earlier melt onset in the 2021 and 2023 melt seasons compared to both SMAP and the EBM. SMAP estimated a maximum summer melt of 56 mm at this site in the 2021 melt season, while both the EBM and GEMB models recorded a maximum summer melt of 30 and 45 mm, respectively, in the 2023 melt season. No pair shows consistent superiority at the CP1 site (Fig. 6d–f).

SMAP LWA generally aligns closer with the GEMB model LWA in the 2021 melt season and with the EBM LWA in the 2022 melt season, whereas, in the 2023 melt season, the EBM and GEMB model match closer to each other than to SMAP. DY2 lacks AWS forcing during the 2021 melt season. Therefore, EBM results are missing for this melt season. Between SMAP and the GEMB model, the latter estimates more LWA overall (LWA_YS 1196 vs. 2101 mm), but there is a reasonable alignment between the peaks of the two LWA time series (Fig. 6g). The overall RMSD was found to be 10.67 mm. For the other two melt seasons (Fig. 6h–i), SMAP and the EBM results show superior agreements (r>0.96 and RMSD ∼4 mm). The GEMB model reports slightly higher LWA in 2023, both in magnitude and duration (4214 mm LWA_YS compared to 2893 mm (SMAP) and 2608 mm (EBM)), resulting in a higher overall RMSD (>12 mm) with the other two.

As per maximum summer melt and LWA_YS, SDL, SDM, and NSE sites experienced the highest LWA in the 2023 melt season compared to the other two melt seasons under consideration (Fig. 6j–r). SMAP did not record any LWA in any of these sites during the 2022 melt season, when the EBM (except NSE where AWS data were not available) and GEMB models also reported the smallest LWA in three melt seasons (Fig. 6k, n, and q). In the 2021 melt season, SMAP estimated an overall lower LWA and shorter summer melt duration than that of the EBM and GEMB models in these sites, but the agreements between EBM and GEMB models are in the same orders (see Tables 1 and 2), with both exhibiting delayed refreezing consistently compared to SMAP.

Figure 7 shows the SMAP-retrieved LWA time series at the abovementioned six AWS locations on the southwestern and southeastern sides of the GrIS percolation zone. The time series does not include results during the 2019 and 2022 outages. As evidenced, the AWS sites in the southwestern sites (Fig. 7a–c) experienced more average LWA and longer summer melt duration than the AWS sites in the southeastern sites (Fig. 7d–f). SDL, SDM, and NSE witnessed an insignificant LWA (<10 mm) during the 2015–2020 melt seasons. However, it was found to be increasing in recent years (Fig. 7d–f). SMAP recorded the highest LWA in the 2023 melt season during 2015–2023 at all the AWS locations, except at KAN_U, where 2021 marked the highest melt season.

Figure 8 illustrates the annual sum of daily LWA (LWA_YS) for 2015–2023. Here, we masked the area where melt is detected by decreasing summer TB (compared to winter reference). As mentioned in Sect. 2.2.3, the current LWA quantification algorithm applies to increasing TB only. This excluded the melt flags in the ablation zone and upper accumulation zone, as indicated by gray shades in Fig. 8. There were also some occasions when summer TB decreased below the winter threshold in the percolation, too. Those anomalies were probably caused by short-lived melt events that refroze between SMAP passes and impacted TB. These anomalies are also masked and not included in the results. As depicted, SMAP captured the similar spatial trends of LWA distribution across the percolation zone of the GrIS as reported by previous studies (van den Broeke et al., 2016; Houtz et al., 2021). In the time frame under consideration, the 2023 melt season (Fig. 8i) had the highest LWA_YS (2634 mm on average for the percolation area), while 2017 (Fig. 8c) had the lowest value (757 mm on average for the percolation area). In 2015 (Fig. 8a), the northern ice sheet exhibited a relatively high LWA_YS; similar intensity and extent were also recorded for 2023 (Fig. 8i). Notably, the melt extended to upper elevations in the dry snow zone in 2021 and 2023. Unfortunately, SMAP outages in 2019 (Fig. 8e) and 2022 (Fig. 8h), led to incomplete coverage for those years and lower LWA_YS. It is worthwhile to reiterate that the integrated LWA is a measure of the total seasonal LWA in the specified area.