Radar at high frequency is a promising technique for fine-resolution snow water equivalent (SWE) mapping. In this paper, we extend the Bayesian-based Algorithm for SWE Estimation (BASE) from passive to active microwave (AM) application and test it using ground-based backscattering measurements at three frequencies (X and dual Ku bands; 10.2, 13.3, and 16.7 GHz), with VV polarization obtained at a 50° incidence angle from the Nordic Snow Radar Experiment (NoSREx) in Sodankylä, Finland. We assumed only an uninformative prior for snow microstructure, in contrast with an accurate prior required in previous studies. Starting from a biased monthly SWE prior from land surface model simulation, two-layer snow state variables and single-layer soil variables were iterated until their posterior distribution could stably reproduce the observed microwave signals. The observation model is the Microwave Emission Model of Layered Snowpacks 3 and Active (MEMLS3&a) based on the improved Born approximation. Results show that BASE-AM achieved an RMSE of

Every year, snow and ice cover about 50 % of the land surface in the Northern Hemisphere (Brown and Robinson, 2011), reflects back up to 80 % of the solar radiation, cools the earth's surface (Flanner et al., 2011), and provides water for about

Active microwave radar at the X and Ku bands shows great promise for high-resolution snow depth (SD) and SWE mapping (Rott et al., 2012; Tsang et al., 2022). This technique is based on detecting changes in volume scattering from the snow medium and thus builds on the heritage from passive microwave remote sensing (Tsang et al., 2022). Active microwave remote sensing can achieve far higher spatial resolution than passive microwave via synthetic aperture radar (SAR) processing. The existence of snow on the ground and its volume scattering generally increases the backscattered radar signal as compared with that of bare soil. Multiple satellite missions have proposed use of this technique, but so far, none have been selected for space-borne operations. The Snow and Cold Lands Processes (SCLP) mission proposed to NASA, the Cold Regions Hydrology High-Resolution Observatory (CoReH

Algorithm development for Ku- and X-band SAR retrievals is of vital importance. The radar backscatter from snow is sensitive to SWE but is complicated by confounding factors including snow microstructure, the backscatter from the substrate beneath snow, and forest cover (Tsang et al., 2022). Recent advances have begun to resolve the substrate issue, specifically by subtracting the contribution of the rough surface scattering at the snow–soil interface (e.g., Zhu et al., 2018) and using passive microwave measurements (Zhu et al., 2021). Forests pose an important limitation on the applicability of the technique, and recent studies have helped refine estimates of forest conditions under which SWE can be estimated (e.g., Macelloni et al., 2012; Lemmetyinen et al., 2022). In this paper, we focus on retrieval issues posed by snow microstructure.

The complexities of retrieving SWE from radar measurements derive from fundamentals of snow physics and electromagnetic physics. Radar backscatter is highly sensitive to snow microstructure, commonly characterized by the size of the individual snow crystals (e.g., Xu et al., 2010; King et al., 2018; Rutter et al., 2019). Because grain shape is highly irregular and exhibits significant spatiotemporal variability, and because grains are oftentimes well bonded within a snowpack, we often refer to “snow microstructure” or to the microstructure correlation length rather than grain size (Picard et al., 2022).

The snow correlation length can be considered as the length scale describing the auto-correlation function (ACF) of the ice–air medium, signifying the distance within which this medium can still be considered correlated (Mätzler, 1997). In this study, we specifically estimate the exponential correlation length of the snow microstructure. The distinction lies in how the correlation length is determined. While correlation length is fitted from the ACF near the origin, the exponential correlation length is fitted from a longer range of two-point distances in the medium (Mätzler, 2002). High values of correlation lengths generally correspond to high values of grain size. Pan et al. (2017) explored the relationship between grain size and correlation length for the Nordic Snow Radar Experiment (NoSREx) dataset. Radar backscatter is quite sensitive to snow microstructure and the dependence is highly non-linear. These complexities have led algorithm developers to introduce a priori information on grain size to help constrain the retrieval problem (Tsang et al., 2022), which in turn makes the retrieved SWE accuracy dependent on the unbiasedness of the prior grain size, at least to some extent. The CoReH

While past work focused on an error propagation approach to infer the required precision for an equivalent grain size, retrieval algorithms that explicitly statistically model each unknown term in the retrieval problem have not been explored in the literature. Here, we extended the Bayesian-based Algorithm for SWE Estimation (BASE) (Pan et al., 2017) to active microwave (AM) application. Unlike a simple steepest descent algorithm or Newton's method, a Markov Chain Monte Carlo (MCMC) method is used in BASE-AM, providing posterior distributions of several variables at the same time from observations and prior distributions, without any assumption of linear error propagation. The MCMC method looks for a global optimization instead of a local optimization. We tested BASE-AM SWE retrieval using ground-based radar measurements from the NoSREx (Lemmetyinen et al., 2016a). We used the Microwave Emission Model of Layered Snowpacks 3 and Active (MEMLS3&a) (Proksch et al., 2015) based on the improved Born approximation (IBA) as the observation model, and we consider a two-layer snow structure composed of a surface layer and a bottom layer. We iteratively updated the snow (snow layer thickness, exponential correlation length, density, and temperature), soil (soil temperature, roughness, and total water content), and model variables in MEMLS3&a to build MCMC chains. The model variable iterated is Q, a semi-empirical parameter to separate the total backscattering into co- and cross-polarization components. The exponential correlation length is the snow microstructure parameter specifically used in MEMLS3&a. We deliberately chose a biased monthly SWE prior from land surface model simulations compared with the in situ observations and thus implicitly tested whether radar data can overcome such biases. Acknowledging the challenge of obtaining appropriate snow microstructure priors, we chose a fixed and nearly uninformative prior for exponential correlation length, which followed a normal distribution as

If this SWE retrieval algorithm for radar using the MCMC approach successfully estimates SWE, then the two-layer approach with a biased prior on SWE and an uninformative prior for snow microstructure contains adequate information to estimate SWE in support of three-frequency radar observations, thus providing a new perspective on the need for a priori information as outlined by Rott et al. (2012) and Rutter et al. (2019). If the algorithm is unsuccessful, we will have found that a precise prior for snow microstructure is required, in agreement with previous literature. We hypothesize, based on previous results for passive microwave (Pan et al., 2017) at this site, that the radar retrieval algorithm will also successfully estimate SWE using the same generic prior.

From 2009 to 2013, continuous snow radar experiments were conducted at the Intensive Observation Area (IOA) (67.362° N, 26.633° E) located in the Finnish Meteorological Institute Arctic Research Centre (FMI-ARC) in Sodankylä, Finland, during the NoSREx campaign (Lemmetyinen et al., 2016a). The IOA is located in a clearing of a typical Scots pine (

The goal of NoSREx was to observe the backscattering coefficient (

Concurrently, snowpits were excavated near the SnowScat twice per week for the first season and weekly for the following seasons. Snow depth, snow stratigraphy, and geometric grain size of each snow layer were measured; snow temperature and snow density were measured by 10 and 5 cm steps, respectively. There were also several automatic sensors installed at IOA to provide additional information. Continuous SWE measurements were available from the Gamma Snow Instrument (GWI), although it tends to be contaminated by high-frequency noise at short timescales. Soil temperature and soil liquid water content were measured by the Delta-T Devices ML2x sensor at 2 cm in four IOPs, and by the Decagon 5TM sensors at 5, 10, 20, 40, and 80 cm in IOP3 and IOP4. Soil at the IOA has a texture of 70 % sand, 29 % silt, and 1 % clay, as well as a bulk density of 1300 kg m

Figure 1 shows the measured backscattering coefficients at VV polarization at a 50° incidence angle, with the measured SWE, geometric grain size (

Concerning other variables, Fig. 1 shows that the IOPs with a higher maximum SWE tend to have a higher average snow density. It also agrees with the snow compaction theory according to snow process models (Jordan, 1991). The underlying soil was frozen for most of the snow season; however, it was unfrozen and had a large soil water content in the early and late snow seasons. In almost all IOPs, we observed a decreasing trend in backscattering at all frequencies at the beginning of the snow season caused by soil freezing processes. Figure 2c and d show more details with respect to the measured backscattering coefficient at 10.2 GHz. In mid-snow season, IOPs with a lower liquid water content or a higher average snow density tend to have higher backscattering at 10.2 GHz. The significantly different

Measured backscattering coefficient at VV polarization (

Relationships of the measured backscatter signals with the snow and soil parameters: the measured backscattering ratio between 16.7 and 10.2 GHz, vs. snow depth (SD)

In this paper, we adapted the Bayesian-based Algorithm for SWE Estimation (BASE), used for inverting passive microwave radiometer measurements in Pan et al. (2017), to radar data. The Markov Chain Monte Carlo (MCMC) method (Gelman et al., 2003) is a numerical realization of the Bayes theorem. Starting from a prior distribution of predicted variables, MCMC randomly searches within the minimum to maximum range of each variable and picks estimates that are both close to the prior and to the observations through the radiative transfer model. The likelihood ratio (

At each iteration, if the MCMC algorithm tries to change the value of an estimated variable from

The MCMC applied here differs from that of Pan et al. (2017) in the following aspects:

The observations and observation model were changed from passive to active (radiometer brightness temperature to radar backscatter).

We used basically the same generic prior for SWE estimation in Pan et al. (2017). However, the prior distributions were changed from lognormal distributions to normal distributions, because a lognormal-distributed prior leads to a lognormal-distributed posterior distribution, which has a skewness that is challenging to interpret.

Snow layer thicknesses were independently estimated in Pan et al. (2017), whereas in this study, we estimate the bottom-layer thickness and the relative ratio of the surface layer thickness to the bottom-layer thickness. This is needed because we found that the radar at these frequencies has a small sensitivity to volume scattering from the surface snow layers of small grain sizes. The use of the layer thickness ratio predetermined the existence of the surface layer and it was assumed to follow

A flowchart of BASE-AM.

Table 1 provides a comprehensive list of variables estimated in the MCMC algorithm, along with their respective priors. The snow and soil priors are generally aligned with the generic prior used in Pan et al. (2017) at the same site. The main distinction lies in the standard deviations of priors, which were halved compared with those of Pan et al. (2017). This adjustment was made because the boundaries of normal distributions are less constrained than lognormal distributions. As in Pan et al. (2017), the SWE prior was a multiple-year average of monthly-mean SWEs from global 2° resolution VIC model simulations (Nijssen et al., 2001). The snow density and snow temperature priors were from the taiga snow type in Sturm's snow classes (Sturm et al., 1995). The prior for snow exponential correlation length was uninformative, following

Summary of priors and boundaries for each estimated variable.

The forward observation model to calculate the snow backscattering was the Microwave Emission Model of Layered Snowpacks 3 and Active (MEMLS3&a) (Proksch et al., 2015) based on the improved Born approximation. MEMLS3&a for active backscattering simulation is a semi-empirical model that converts passive microwave reflectivity of the snowpack to backscattering. It assumes that the distribution of the diffuse part of the bistatic scattering coefficient is Lambertian. In this model, first the snow reflectivity calculated by passive MEMLS (

After

MEMLS3&a requires the total and specular part of soil reflectivity, instead of soil backscattering. We utilized the QHN model with frequency-independent parameters (QHNfi) developed in Montpetit et al. (2015) to calculate the total soil surface reflectivity. The specular part (

The soil dielectric constants were calculated using a revised generalized refractive mixing dielectric model (GRMDM) (Mironov et al., 2004) adapted for frozen soil. The frozen soil is considered as a mixture of dry solids, bound water, transient water, and ice as in Mironov et al. (2017). The model utilized the same bound water content as in Mironov et al. (2004), whereas it fitted temperature-dependent bound water dielectric constants and other important variables using a frozen soil dataset measured at the Beijing Normal University (see Appendix A). The soil texture was set the same as the measurements from NoSREx (Lemmetyinen et al., 2016a).

Figures 4–6 use the snowpit measured on 13 March 2012 (hereafter referred to as “pit 49”) during IOP3 as an example to show how MCMC works. Figure 4 shows the simulated backscattering coefficient at each iteration, whereas Figs. 5 and 6 show the variations of all estimated variables at each iteration.

Figure 4 shows that the simulated backscattering coefficients at each iteration in the chain (after the burn-in period) are close to the observations, which is one of the key characteristics of MCMC-based retrieval. The mean bias is 0.23, 0.68, and

Figure 5 shows the MCMC chains (left) and the posterior distributions (right) of the snow variables. The posterior distributions of layer thickness (first row) have lower uncertainty than the priors and the mean is different from that of the prior. For the estimation of exponential correlation length (

Figure 6 shows the MCMC chains (left) and the posterior distributions (right) of the soil and model variables. The first and fourth rows show that the backscattering coefficient is not sensitive to soil temperature or to the model variable,

MCMC chain of simulated backscattering coefficients (lines) compared with the measured backscattering coefficients (diamonds) for pit 49:

MCMC chain of layered snow properties (first column) and their posterior distributions compared with the prior distributions (second column) for pit 49. Lyr1 is for the bottom layer and Lyr2 is for the surface layer.

MCMC chain of other soil and model variables (first column) and their posterior distributions compared with the prior distributions (second column) for pit 49.

Figure 7 shows the MCMC-estimated snow depth (SD) (Fig. 7a) and SWE (Fig. 7b), from the averages of MCMC chains after the burn-in period. The BASE-AM algorithm corrects the underestimation of VIC priors: the original biases for SWE are

BASE-AM estimated snow depth (SD) versus observed SD at Sodankylä

Summary of root-mean-squared error (RMSE) for SD

Figure 9 shows a comparison between the MCMC-estimated exponential correlation length (

Figure 9a shows that when

Time series

Figure 10 uses 2D distribution maps to show the relationship between layer thickness and exponential correlation length (

A 2D histogram of probability between layer thickness and exponential correlation length (

The backscattering coefficient at 10.2 GHz (

As for the other two soil variables, the results in Sect. 4 are based on the default BASE-AM algorithm configuration to estimate total soil water content and soil roughness simultaneously. To further explore the algorithm, we conducted an additional experiment estimating only the soil moisture using a fixed soil roughness of 1 mm. We found that when both of these soil parameters were estimated, the RMSE of the simulated backscattering coefficient in MCMC (0.43 dB) was closer to the observations (0.52 dB) . In addition, the accuracy of MCMC-estimated SD and SWE is slightly higher, with the RMSEs for all snowpits being 10.2 cm and 28.67 mm, respectively, compared with 10.71 cm and 30.14 mm, respectively. However, Fig. 11 shows that, when the soil roughness is fixed, the temporal variation of estimated soil liquid water content matches better with the sensor measurements, which means the soil liquid water content becomes retrievable. This suggests a possible strategy where the soil roughness is estimated early in the season, and the result would be used for the rest of the period if there is a desire to better estimate soil moisture dynamics from the radar data.

Comparison of MCMC-estimated soil liquid water content (

Figure 12 shows the MCMC-estimated SD and SWE when the snow is assumed to have a single layer. The same snow and soil priors were used. When the one-layer snow assumption is used, BASE-AM cannot fully correct the underestimation of the SWE prior. This occurred because the presence of a surface snow layer was overlooked. This layer generates very small volume scattering, rendering it nearly transparent to radar. Therefore, it is crucial to acknowledge the existence of such a surface layer; otherwise, the total SD will be underestimated. At the same time, because the equivalent one-layer

Figure 14 shows a comparison of the SD–

Therefore, as a short summary, we recommend considering two layers since radar tends to overlook the surface snow layer of small grain size. Additionally, based on the balance between the current numbers of radar observations and predicted variables, introducing more layers provides little or no improvements in SWE estimation, unless a reliable prior for snow stratigraphy detail becomes available.

BASE-AM estimated snow depth (SD)

BASE-AM estimated

Comparison between the SD and profile-average

Summary of SD, SWE, and profile-average exponential correlation length (

In this paper, we developed a Bayesian-based Algorithm for SWE Estimation for Active Microwave (BASE-AM) to retrieve the snow and soil parameters from a site in Sodankylä, Finland, based on X- and dual Ku-band VV polarization backscattering coefficients using biased SWE and uninformative snow microstructure priors. Results show that by predetermining the snowpack to have two layers, SD can be retrieved with an RMSE of 10.2 cm in 0–1 m range, and SWE can be retrieved with an RMSE of 28.7 mm in 0–200 mm range. The radar backscattering observations can correct the bias of SD prior from

By iteratively updating several snow and soil variables in the MCMC chain and comparing the prior and posterior distributions, we showed that the key variables required to be estimated in BASE-AM are layer thickness, layer microstructure parameter, and soil liquid water content. The backscattering coefficients in snow and soil are not sensitive to temperature. Backscattering intensity at low frequency is sensitive to snow density, but density cannot be easily retrieved. The polarization splitting parameter (

Overall, our results indicate that active remote sensing observations coupled with generic priors and a two-layer retrieval scheme can support estimation of SWE. Moreover, it is essential to note that the setting of the prior is not a fixed component of the MCMC algorithm. For application in other regions, additional research may be necessary to reset the prior for

The soil dielectric constant model utilized in this paper was developed from the Mironov et al. (2004) model and revised according to the soil dielectric constant measurement experiment conducted by Jinmei Pan using the Agilent vector network analyzer at the Beijing Normal University in China. An introduction of the experiment can be found in Wu et al. (2022). Here, a brief introduction of this model is provided (with more details to be published at a later date). The measurements utilized to develop this model cover a wider soil texture than that in Wu et al. (2002), from loamy sand (77.27 % sand, 16.02 % silt, and 6.71 % clay) to silty clay loam (17.49 % sand, 49.68 % silt, and 32.82 % clay). The gravimetric soil water content of all samples varies from 3 % to 60 % and the soil temperature varies from

Sources or equations of

Simulated and observed real part of soil dielectric constants (

The real and imaginary part of the squared root of soil dielectric constants (

The key to the soil dielectric constant model is to model

The dielectric constants of water components can be modeled by Debye's equation as

For each component, it is required to determine

Figure A1 shows an example of the model utilized to predict the measured soil permittivity (real part of dielectric constants) at 100 MHz by the Decagon 5 TM sensor in Sodankylä. The simulation utilized the measured liquid water content and measured soil texture. The total soil water content comes from the measured liquid water content before freezing. To calculate soil permittivity, we used a fully independent model as compared with the Decagon 5 TM sensor. Figure A1a shows that our model is able to predict the change in soil permittivity with changes in soil temperature. Figure A1b shows that the simulated soil permittivity is highly consistent with the observations at 10–80 cm depth below the soil surface. The mean bias is 0.0838, with an RMSE of 0.0569. In addition, the model overestimates the measured permittivity for the single top layer at 5 cm, which is influenced by air above the soil and also organic matter. Including the 5 cm layer, the mean bias is 0.129 with an RMSE of 0.198. It indicates that the soil dielectric constant model described here is suitable to be used as part of the forward model described in this paper.

Please refer to data availability of Lemmetyinen et al. (2016a) to acquire the NoSREx datasets. The source code of this work has been uploaded and is now available on Github (

JL provided the NoSREx datasets, and preliminary analysis of these datasets was conducted by JP, MD, and JL prior to retrieval. MD and DL provided the MCMC algorithm ideas and tools, which were later implemented and revised by JP to conduct the SWE retrieval experiments. All co-authors participated in the analysis of the MCMC results and collaborated on writing and revising the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors thank the NoSREx team for their hard work and dedicated effort in completing this snow experiment and providing the great dataset to support our study. We also thank Kimmo Rautiainen for providing the soil frost depth measurement for us to do more analysis, and we thank Simon Yueh and Richard Kelly for providing valuable comments when we were studying the first MCMC outputs. We dedicate this study to the memory of Joshua King, who passed away 21 February 2023. Josh's pioneering measurements and keen insight into the estimation of snow water equivalent from radar observations were an inspiration for us and for the entire community, and he will be deeply missed.

This research has been supported by the National Natural Science Foundation of China (grant no. 42090014), the National Key Research and Development Program of China (grant no. 2021YFB3900104), NASA (grant no. 80NSSC17K0200), the European Space Agency (ESA Contract no. 22671/09/NL/JA), and the Research Council of Finland (grant no. 325397). It also received financial support from the China Scholarship Council and the OSU Presidential Fellowship.

This paper was edited by Homa Kheyrollah Pour and reviewed by Jiyue Zhu and two anonymous referees.