Using Robust Regression to Retrieve Soil Moisture from CyGNSS Data

Abstract

Accurate global soil moisture (SM) data are crucial for modeling land surface hydrological cycles and monitoring climate change. Spaceborne global navigation satellite system reflectometry (GNSS-R) has attracted extensive attention due to its unique advantages, such as faster revisit time, lower payload costs, and all-weather operation. GNSS signal reflected at L-band also has significant advantages for SM estimation. Usually, SM is estimated based on the sensitivity of GNSS-R reflectivity to SM, but the noise in observations can significantly impact SM estimation results. A new SM retrieval method based on robust regression is proposed to address this issue in this work, and the effects of roughness and vegetation on the effective reflectivity of the Cyclone Global Navigation Satellite System (CyGNSS) are reconsidered. Ancillary data are provided by the SM Active Passive (SMAP) mission. The retrieved results from the training sets and test sets agree well with the referenced SMAP SM data. The correlation coefficient R is 0.93, the root mean square error (RMSE) is 0.058 cm³cm⁻³, the unbiased RMSE (ubRMSE) is 0.042 cm³cm⁻³, and the mean absolute error (MAE) is 0.040 cm³cm⁻³ in the training sets. For the test, the correlation coefficient is 0.91, the RMSE is 0.067 cm³cm⁻³, the ubRMSE is 0.051 cm³cm⁻³, and the MAE is 0.044 cm³cm⁻³. The proposed method has been evaluated using in situ measurements from the SMAP/in situ core validation site; in situ measurements and retrieval results exhibit good consistency with the ubRMSE value below 0.35 cm³cm⁻³. Moreover, the SM retrieval results using robust regression methods show better performance than CyGNSS official SM products that use linear regression. In addition, the land cover types significantly affect the accuracy of SM retrieval, and the incoherent scattering in densely vegetated areas (tropical forests) usually leads to more errors.

1. Introduction

The crucial impact of soil moisture (SM) on the climate system, hydrologic processes, and vegetation growth has long been recognized and was named one of the essential climate variables (ECVs) in 2010 [1,2]. At present, two L-band passive microwave radiometers provide global SM measurements: the SM Active Passive (SMAP) mission and the SM and Ocean Salinity (SMOS) mission, both of which have a coarse spatial resolution of about 40 km and a revisit period of about 2–3 days, and provide SM retrieval at a depth of 5 cm on the soil surface [3]. SM can also be measured using synthetic aperture radar (SAR) systems, such as Sentinel-1 and the upcoming NASA-ISRO SAR (NISAR) mission [4], which can provide SM data with high spatial resolution. However, the temporal resolution is limited, and both are severely affected by vegetation structure and rough surfaces.

In recent decades, Global Navigation Satellite System Reflectometry (GNSS-R) has demonstrated significant potential for retrieving geophysical parameters. Although the technique was originally designed to measure sea winds in tropical oceans [5], some recent studies and projects have demonstrated the sensitivity of reflected signals from the land surface to hydrological parameters, such as SM [6,7,8], mapping flood [9,10], wetlands [11,12], and vegetation [13,14]. Retrieving SM from spaceborne GNSS-R data has become a research hotspot due to its unique advantages (such as faster revisit time, all-weather operation, and lower payload costs). Existing SM estimation methods can be roughly divided into two categories: empirical models and machine learning (ML). ML methods mostly characterize the complex nonlinear relationship between GNSS-R observations (such as reflectivity and SNR) and reference SM through the inclusion of land surface characteristics, such as random forest [15,16], neural networks [17,18], and XGBoost approaches [19], and so on. ML requires a large number of GNSS-R observations and a great quality of ancillary data for a long period of training to obtain the inversion model, and it is difficult to explain the relationship between the involved variables. On the contrary, model-based SM estimation does not rely on a large amount of ancillary data and can clearly explain the relationship between GNSS-R observations and desired SM. Clarizia et al. [8] and Yan et al. [20] developed a trilinear regression model and multiple linear regression model to estimate SM, explaining the relationship between reflectivity, vegetation, and roughness. Based on the relationship between Cyclone GNSS (CyGNSS) reflectivity and SM using linear regression, Chew and Small [21] developed UCAR (University Corporation for Atmospheric Research) SM products (https://cmr.earthdata.nasa.gov/virtual-directory/collections/C2205122332-POCLOUD, accessed on 2 June 2023). However, the retrieval results using linear regression are affected by observation noise, such as the uncertainty in the antenna pattern correction (0.25 dB), GPS transmitter effective isotropic radiated power (EIRP) error (0.24 dB), noise floor uncertainty (~0.3 dB), and other noise [22]. Additionally, previous research on SM retrieval has not taken into account the fact that microwave-scale roughness (h) is insufficient to simulate the impact of roughness on reflectometry [23]. The vegetation optical depth (VOD) used for simulating vegetation attenuation effects needs to be reconsidered due to the wider observation angle of CyGNSS [24]. Al-Khaldi et al. presented a time-series model of SM obtained from the CyGNSS, and this method can offset vegetation attenuation by continuously measuring the ratio of CyGNSS DDM [25]. However, the roughness effect was solved using a mean square slope with a constant of 0.01 for all observations.

In this study, a new SM estimation method using robust regression is proposed to address the impact of any noise in CyGNSS observations on classical linear regression methods. Unlike studies [8,20] using a unified model to handle large-scale areas, the model in this work is parameterized with a pixel mode (36 × 36 km). In addition, this work has reconsidered the impact of surface roughness and vegetation attenuation on reflectometry. The remainder of this paper is organized as follows: Collected data are described in Section 2. The method and data process used in this study are shown in Section 3. Section 4 describes and discusses the experimental result. Section 5 discusses the results of the study, and Section 6 summarizes the work.

2. Datasets

In this section, the collection and quality control (QC) of CyGNSS data, as well as the use of referenced SMAP SM, in situ SM, and ancillary data are described.

2.1. CyGNSS Data

CyGNSS was launched on 15 December 2016. It is made up of eight microsatellites and is known for its nearly gapless earth coverage. An orbital inclination of about 35° from the equator can provide GNSS-R observations in the pantropical regions and allows CyGNSS to measure the reflected signals between the latitudes of about 38° N and 38° S, with a minimum spatial resolution of about 3.5 km × 0.5 km and a revisit time of about several hours [26,27].

The reflected GPS signals from the land surface are recorded by a delay Doppler mapping instrument (DDMI) at CyGNSS satellites, and DDMs are processed by cross-correlating with a replica of the receivers and reflected GPS signals. CyGNSS bistatic radar cross section (BRCS) is a 17 Delay × 11 Doppler array obtained by inverting the forward scattering model, and its peak value varies with SM, vegetation, roughness, etc. [15]. In addition to BRCS, the metadata related to CyGNSS DDM and the observational geometry are used here, such as incidence angles, signal-to-noise ratio (SNR), gain, geographic location of the specular point (SP), and the distance of the transmitter and receiver from the specular point. The CyGNSS Level 1 (L1) v3.0 datasets from 2019 to 2020 are used in this work (available at https://cmr.earthdata.nasa.gov/virtual-directory/collections/C2205618435-POCLOUD, accessed on 2 June 2023). For more variable descriptions of CyGNSS L1 V3.0 data, please see the CyGNSS User Guide [26] and Data Dictionary (https://archive.podaac.earthdata.nasa.gov/podaac-ops-cumulus-docs/cygnss/open/L1/docs/148-0346-8_L1_v3.0_netCDF_Data_Dictionary.xlsx, accessed on 2 June 2023). The data for 2019 are used to generate the roughness lookup table (LUT), and the data for 2020 are used to retrieve SM.

To filter low-quality DDM observations, similar procedures from previous studies [19,20,21] are referenced. CyGNSS data collected over land with SNR over 2 dB, antenna gain more than 0 dB (towards SP), and incidence angles lower than 65° are applied here. In addition, the quality flags of DDM are used, and the applied quality flags are listed in

Table 1. Applied quality flags in 2020 data processing.

Quality Flag Number	Quality Flag Name
2	S-Band Powered Up
4	Large Spacecraft Attitude Error
5	Blackbody DDM
6	DDMI Reconfigured
7	Space wire CRC Invalid
8	DDM is Test Patten
9	Channel Idle
16	Direct Signal in DDM
17	Low Confidence GPS EIRP Estimate
18	RFI Detected
22	GPS PVT sp3 error
23	SP Non-Existent Error
26	Blackbody Framing Error

. Figure 1 shows an example of grided mean reflectivity (January 2020) before and after QC. To avoid too many blanks on the map, the scope of maps was appropriately adjusted to longitude = [−130, 160] and latitude = [−38, 38]. The remaining maps are processed in the same manner. Note that to obtain the roughness information at all grids, we did not apply QC to 2019 data.

2.2. SMAP Data

The SM derived from CyGNSS will be referenced and compared using the SMAP L3 Radiometer Global Daily 36 km equal-area scalable earth grid (EASE-Grid) version 8 SM data (SMAP_L3_SM_P). The data are freely available through the National Snow and Ice Data Center (NSIDC) at https://nsidc.org/data/data-access-tool/SPL3SMP/versions/8/ (accessed on 2 June 2023). Although clay ratio (CR) and vegetation water content (VWC) from SMAP ancillary data were used (see Figure 2), CR and VWC data are independent of SMAP. Specifically, CR data (Soil texture) are assembled from an optimized combination of multiple global or regional soil databases, and VWC data are calculated based on the Normalized Difference Vegetation Index (NDVI) information provided by moderate-resolution imaging spectroradiometer (MODIS) [3]. The period of SMAP data used corresponds to the CyGNSS observations, i.e., 2019 to 2020. CyGNSS data are spatially averaged into the 36 km EASE-Grid used to facilitate the experimental evaluation in the following section.

2.3. In Situ Data

To investigate and evaluate the accuracy of CyGNSS SM retrievals on the temporal scale and avoid the issue of mismatch in resolution between retrievals and in situ, the SMAP/in situ core validation site land surface parameters match-up datasets were used here [28]. Please visit the SMAP cal/val activities website (https://smap.jpl.nasa.gov/science/validation/calvalpartners/, accessed on 30 June 2023) for more details. The dataset is produced by matching SMAP SM products with in situ SM estimates from core validation sites. These data have been used to evaluate the performance of SMAP SM products in different regions, such as arid/semi-arid regions [29] and forested sites [30]. Thus, this dataset is reliable in validating CyGNSS SM retrievals. SMAP cal/val in situ data include not only the surface SM (0–5 cm) and soil temperature but also the verifiable grid scale of the station (3 km, 9 km, and 36 km). Based on the CyGNSS observation range, available periods, and the verifiable grid scale of sites (36 km), 4 in situ sites were chosen as validation data. Refer to Figure 3 for the distribution of sites and to

Table 2. Details of in situ sites.

Site Name	Longitude, Latitude	IGBP Land Cover	Climate
Fort Cobb	35.36°N, 98.55°W	Grasslands	Temperate
Little Washita	34.97°N, 97.97°W	Grasslands	Temperate
Monte Buey	32.96°S, 62.52°W	Croplands	Arid
Yanco	34.8°S, 146.11°E	Croplands/Grasslands	Semi-arid

for detailed information on the sites. Note that soil temperature is an important dynamic auxiliary parameter because SM sensors do not provide reliable measurements when the soil temperature is below 4 degrees Celsius [31], so these data were removed during validation.

3. Methods

The section explains how to estimate and calibrate reflectivity from CyGNSS data, then goes into detail about the roughness LUT, and finally, describes a complete data processing flow.

3.1. SM Retrieval Method

The CyGNSS effective reflectivity is the observable used for SM retrieval. Assuming that the reflected signal over land is mostly determined by the coherent reflection [7,8,10], it is calculated as follows:

$$P_{coh}=\frac{{\lambda }^2{P}_t{G}_t{G}_r}{{{\left(4\pi \right)}^2\left(r_{st}+r_{sr}\right)}^2}{\Gamma }_{eff}\left(\theta \right),$$

(1)

where $\theta$ is the incidence angle; $P_{coh}$ is the peak value of the DDM of the analog scattered power; $\lambda$ is the wavelength of GPS, about 19 cm; $P_t$ is transmitter power; $G_t$ is transmitter gain, and $P_t{G}_t$ is the transmitter equivalent isotopically radiated power (EIRP); $G_r$ is CyGNSS antenna gain in the direction of SP; $r_{st}$ and $r_{sr}$ are the distances from SP to the transmitter and receiver, respectively.

The effective reflectivity can be modeled as

$${\Gamma }_{eff}\left(\theta \right)={\mathfrak{R}}_{RL}^2\left(ϵ,\theta \right){\gamma }^{2}exp\left(-h^2{\mathit{cos}}^2\theta \right)={{\Gamma }_{SM}\times \Gamma }_{Veg}\times {\Gamma }_{Rou},$$

(2)

where ${ \Gamma }_{SM}={\mathfrak{R}}_{RL}^2\left(ϵ,\theta \right)$, ${\Gamma }_{Veg}={\gamma }^2$, and ${\Gamma }_{Rou}=exp\left(-h{\mathit{cos}}^2\theta \right)$ describe the effect of SM, vegetation, and roughness, respectively. ${\mathfrak{R}}_{RL}$ is the surface’s Fresnel reflection coefficient, and $ϵ$ indicates the soil dielectric constant. Transmissivity $\gamma$ represents the signal attenuation as the signals propagate through the vegetation canopy, and is a function of vegetation optical depth (VOD) $\tau$,

$$\gamma =exp\left(-\tau \right)$$

(3)

VOD related to vegetation water content (VWC), vegetation attenuation parameter (b), and incidence angle,

$$\tau =b\times VWC/\cos \theta$$

(4)

A static global gridded map of b based on land cover types was successfully applied to the SMAP mission [32]. However, parameter b depends not only on the land cover types but also on the observation angle (GNSS-R satellites observed at a wider angle of about 0~75°, SMAP observed at a fixed angle of around 40°). Parameter b estimated by [24] and VWC data are used here to calculate vegetation effects. The exponential term in Equation (2) accounts for the surface roughness effects, $h=2ks$ denotes assumed linearly related to the root mean squared height (RMSH) $s$ of the land surface, $k$ is the wave number in free space. The influence of surface roughness on the reflectometer is much larger than that on passive microwave radiometry, and it is usually not appropriate to directly use the h parameter from SMAP for CyGNSS roughness modeling [23]. Here, a semi-empirical model is used to estimate the roughness LUT (${\Gamma }_{Rou}\left[\mathrm{d}\mathrm{B}\right]$), as described in Section 3.2. Taking the logarithm in Equation (2), it is as follows:

$${\Gamma }_{eff}={{\Gamma }_{SM}+\Gamma }_{Veg}+{\Gamma }_{Rou},\mathrm{unit}\mathrm{in}\mathrm{dB}.$$

(5)

By inverting Equation (5), the calibrated reflectivity can be obtained, that is, ${{\Gamma }_{SM}={\Gamma }_{eff}-\Gamma }_{Veg}-{\Gamma }_{Rou}$. The calibrated reflectivity is converted to SM using the concept proposed in [21]. Therefore, a linear model between the variations of reflectivity ($\mathsf{\Delta }{\Gamma }_{SM}$) and SM ($\mathsf{\Delta }SM$) is established and performed in each pixel manner. It can be expressed as

$$\mathsf{\Delta }SM=f\left(\mathsf{\Delta }{\Gamma }_{SM}\right)=\beta \times \mathsf{\Delta }{\Gamma }_{SM}+\alpha ,$$

(6)

$${SM}_{cyg}=\Delta SM+\overline{SM},$$

(7)

where $\beta$ and $\alpha$ are coefficients that need to be determined by robust regression, ${SM}_{cyg}$ is retrieved SM from CyGNSS, and $\overline{SM}$ is mean SMAP SM in each grid. Considering the sample size (one grid), regression speed, and the ability to handle outliers, the Random Sample Consensus (RANSAC) regression is selected for parameter estimation. RANSAC is a robust model-fitting algorithm that aims to find a solution that is resistant to outliers in the data by iteratively fitting models to randomly sampled subsets of the input data. The RANSAC Regressor, which is interfaced via the “scikit-learn” library within the Python framework [33], provides a convenient solution and easy-to-use environment, which was used in this work.

The flowchart of the proposed method is shown in Figure 4. The 2020 CyGNSS data and corresponding auxiliary data (SMAP VWC and SM) were used for SM retrieval. In order to weaken the impact of seasonal changes on retrieval results, the 2020 CyGNSS data were randomly divided into two parts: one part was used for the training phase (estimation of coefficients $\beta$ and $\alpha$), and the remaining for testing.

3.2. Estimation of Roughness

The h parameter (microwave-scale surface roughness) of passive microwave radiometry is too small for the modeling of reflectometry; it may not apply to SM retrieval from GNSS-R [23]. Previous research on SM retrieval did not take that into account; here, we use the semi-empirical model proposed by [24] to estimate the roughness effect, which includes the combined effects of microroughness and terrain. The semi-empirical model is expressed as

$${\Gamma }_{eff}\left(\theta \right)={\gamma }^2\left(\alpha +f_{s}S\right){\mathfrak{R}}_{RL}^2\left(ϵ,\theta \right)+{\gamma }_{v0}S,$$

(8)

where $\alpha =exp\left(-4k^2{s}^2{\mathit{cos}}^2\theta \right)$ represents the loss of coherent power due to microwave-scale roughness, and $f_{s}S$ accounts for the contributions of incoherent scattering, depending on topography. $S$ is the scale factor related to the distances from SP to the transmitter and receiver. It can be expressed as

$$S=A\frac{{\left(r_{st}+r_{sr}\right)}^2}{4\pi {r_{rs}}^2{r_{ts}}^2}.$$

(9)

where $A$ is the effective reflection area related to the Woodward ambiguity function (WAF). ${\gamma }_{v0}$ is an empirical parameter, and the correction used here is ${\gamma }_{v0}$ = −30.5 dB for VWC ≥ 5 kg/m² and ${\gamma }_{v0}$ = 0 for VWC < 5 kg/m².

Equation (8) can be inverted to

$$\left(\alpha +f_{s}S\right)=\frac{{\Gamma }_{eff}\left(\theta \right){-\gamma }_{v0}S}{{\gamma }^2{\mathfrak{R}}_{RL}^2\left(ϵ,\theta \right)},$$

(10)

and take the logarithm of its left and right sides to obtain the roughness LUT (${\Gamma }_{Rou}$), which is

$${\Gamma }_{Rou}=10log\left(\alpha +f_{s}S\right)=10log\left({\Gamma }_{eff}\left(\theta \right){-\gamma }_{v0}S\right)-20log\left(\gamma \right)-20log\left({\mathfrak{R}}_{RL}\left(ϵ,\theta \right)\right).$$

(11)

The flowchart for estimating roughness LUT is presented in Figure 5. The SM and CR from SMAP data in 2019 are used as the input of the Mironov dielectric model to calculate the Fresnel coefficient (${\mathfrak{R}}_{RL}$). Then, the SMAP VWC is input into the semi-empirical model to calculate the vegetation attenuation ($\gamma$) and scattering (${\gamma }_{v0}S$) effects. As a result, a roughness LUT with EASE-Grid is created, assuming that it does not change with time. It should be noted that the one-day CyGNSS observations and the mean values of three-day SMAP data match because of the SMAP revisit time (2–3 days).

3.3. Validation Metrics

We will evaluate the accuracy of CyGNSS SM retrievals ${SM}_{cyg}$ using reference SM ${SM}_{ref}$, including SMAP L3 SM (SMAP_L3_SM_P), in situ SM, and UCAR SM. The evaluation will involve four metrics: correlation coefficient (R), mean absolute error (MAE), root mean square error (RMSE), and unbiased root mean square error (ubRMSE). The correlation coefficient R indicates the relative accuracy between the retrieved SM and reference values. The calculation of correlation coefficient R is as follows:

$$R=\frac{1}{N-1}{{\displaystyle \sum }}_{i=1}^N\left[\left(\frac{{SM}_{cyg}-\overline{{SM}_{cyg}}}{{\sigma }_{cyg}}\right)\left(\frac{{SM}_{ref}-\overline{{SM}_{ref}}}{{\sigma }_{ref}}\right)\right],$$

(12)

where $N$ represents the total number of samples, $\overline{{SM}_{cyg}}$ and $\overline{{SM}_{ref}}$ are mean CyGNSS SM and mean reference SM, ${\sigma }_{cyg}$ and ${\sigma }_{ref}$ are the standard deviation of CyGNSS SM and reference SM, respectively.

The MAE, given by Equation (13), represents the average absolute difference between the CyGNSS SM and the reference data.

$$MAE=\frac{1}{N}{{\displaystyle \sum }}_{i=1}^N\left|{SM}_{cyg}-{SM}_{ref}\right|$$

(13)

The RMSE is a measure of random error and represents the accuracy between the retrieved SM and the reference SM. The calculation of RMSE is as follows:

$$RMSE=\sqrt{\frac{1}{N}{{\displaystyle \sum }}_{i=1}^N{\left({SM}_{cyg}-{SM}_{ref}\right)}^2}$$

(14)

But, RMSE essentially depends on the bias. To obtain a more reliable estimate of RMSE, the unbiased metric ubRMSE is used to evaluate the reliability of CyGNSS retrievals [34,35].

$$ubRMSE=\sqrt{{RMSE}^2-{\left(\overline{{SM}_{cyg}}-\overline{{SM}_{ref}}\right)}^2}$$

(15)

4. Results

The roughness LUT and coefficient estimation process are presented in this section, and then the CyGNSS SM retrievals are evaluated and validated for comparison with SMAP, in situ measurements, and UCAR SM products. The influence of land cover on CyGNSS SM retrieval is also discussed.

4.1. Calibration of Reflectivity

As explained in Section 3.1, the daily reflectivity obtained from CyGNSS is gridded using the approach of 36 km EASE-Grid, which means one sample presents one pixel with 36×36 km resolution. Figure 6a shows the annual mean effective reflectivity that was estimated from CyGNSS data in 2020; Figure 6b, roughness LUT; Figure 6c, vegetation effect; and Figure 6d, calibrated reflectivity. In other words, Figure 6 shows the product of the whole 2020 data processing, which is consistent with Equation (5), namely, Figure 6a = Figure 6b + Figure 6c + Figure 6d, (dB).

The Sahara Desert, the Arabian Desert, the Indian croplands, and most of Australia have higher reflectivity in Figure 6a, which ranges from −15 to −5 dB. These regions have limited vegetation and dry soil; thus, the primary source of reflectivity loss is surface roughness. In contrast, the reflectivity of CyGNSS is centered at −30 to −20 dB in the Amazon, Congo, and Southeast Asia. The vegetation is dense in these areas, and VWC is greater than 15 kg/m² (as shown in Figure 2c), indicating that the vegetation causes a loss of reflectivity. The roughness LUT is displayed in Figure 6b. The roughness effects depend on the position, whereby smoother surfaces lead to signals being closer to coherent reflection, while rougher surfaces cause stronger scattering. The high values (>−10 dB) are mostly located over deserts and croplands with lower surface roughness. The low values (<−10 dB) are mostly located over areas with complex topography, such as the Rocky Mountains, Andes Mountains, Tibetan plateau, etc. This demonstrates the influence of topography, that is, using only small-scale surface roughness (e.g., SMAP h-parameter) from a radiometer is insufficient to simulate surface scattering. Figure 6c shows the two-way attenuation of vegetation. Figure 6d displays the calibrated reflectivity (${\Gamma }_{SM}$) for the SM retrieval using robust regression in the next step.

Figure 7 illustrates the annual average differences in CyGNSS reflectivity before and after calibration. Smaller improvements originate from sparsely vegetated or bare soil regions, such as the Sahara Desert, Arabia, South Africa, and most parts of Australia, with the improvement centered around 0–10 dB. The primary contribution to these differences is the roughness calibration. Larger differences arise from densely vegetated areas (Amazon, Congo, and Southeast Asia) or complex terrain regions (eastern Australia, western Americas), with the improvements concentrated in the range of 10–20 dB. The primary contribution is vegetation calibration or the combined effect of vegetation and surface roughness.

4.2. Estimation of Coefficients

A well-known robust estimator of the slope is the RANSAC estimator, which was first proposed by Fischler and Bolles [36]. RANSAC iteratively computes the residuals of random subsets within each grid cell to classify all data as either signal or noise, and subsequently uses the signal to estimate the parameters of the model. Its advantages over linear regression include a greater capacity for dealing with noise and outliers in the data and a greater robustness to perturbations. Figure 8 shows an example to illustrate the advantages of robust regression. It can be found that robust regression is more consistent with the trend of data, whereas the results of linear regression are biased by outliers. This sets it apart from other similar studies [21]. Another distinction is that this method of parameter estimation is conducted within each grid cell, as opposed to employing a single parameter across a broad scale [8,20].

Robust regression was used to estimate the coefficients in each separate grid. It should be pointed out that when there were fewer than 10 collocated observations in a grid, results of robust regression were removed in this work. To confirm the contribution of reflectivity to SM retrieval and the effectiveness of the method used, we calculated the correlation between $\mathsf{\Delta }{\Gamma }_{SM}$ and $\mathsf{\Delta }\mathrm{S}\mathrm{M}$, as shown in Figure 9, from which strong positive correlations can be obtained for most regions. However, most wet areas have poor correlation, such as the Amazon, Congo, and Southeast Asia. Coefficients $\beta$ of the robust regression between $\mathsf{\Delta }{\Gamma }_{SM}$ and $\mathsf{\Delta }\mathrm{S}\mathrm{M}$ represent the sensitivity of CyGNSS to SM, as illustrated in Figure 10. Lower $\beta$ have been found in dry areas with almost constant SM, e.g., North Africa and Western Australia. Humid areas with a lower correlation also showed lower $\beta$. Although some data in these areas have been filtered out after applying QC (SNR > 2 dB), the remaining data may not be reliable for SM retrieval. The IGBP land cover type of each EASE-Grid cell is shown in Figure 3. The land classification map is based on the most dominant land types with more than 50% coverage in each EASE-Grid. Average coefficients $\beta$ over different land cover types, VWC, and SM are listed in

Table 3. Mean values of VWC, SM, coefficients $\beta ,$ and statistical error over different land types.

Land Cover Type	Mean VWC (kg/m2)	Mean SM (cm3/cm3)	$\mathbf{Mean} \mathit{\beta }$	RMSE (cm3cm−3)	ubRMSE (cm3cm−3)	MAE (cm3cm−3)
Barren/Sparsely Vegetated	0.01	0.0708	0.0010	0.021	0.016	0.014
Open Shrublands	0.44	0.1047	0.0045	0.037	0.031	0.026
Grasslands	1.01	0.1628	0.0121	0.050	0.043	0.040
Croplands	2.00	0.2478	0.0150	0.051	0.045	0.039
Savannas	2.77	0.1731	0.0113	0.054	0.048	0.045
Cropland/Natural Vegetation	3.57	0.2358	0.0118	0.056	0.051	0.046
Woody Savannas	4.28	0.2653	0.0117	0.059	0.050	0.049
Deciduous Broadleaf Forest	9.31	0.2472	0.0104	0.066	0.056	0.056
Mixed Forest	9.49	0.3809	0.0054	0.072	0.059	0.058
Evergreen Broadleaf Forest	15.99	0.4257	0.0024	0.079	0.062	0.066

. Coefficient $\beta$ increases with the increase in mean VWC and SM; however, the land type of forests exhibit inconsistency with coefficient $\beta$ decreasing as VWC and SM increase.

4.3. Evaluation of the Retrieved SM

Assessing the accuracy of SM retrieved from the CyGNSS satellite and understanding its spatial distribution of uncertainty is crucial for the development of retrieval algorithms. The CyGNSS data from 2020 were split into two random subsets to mitigate the impact of seasonal factors on parameter estimation. One subset was employed for training purposes, whereas the remaining subset was utilized for testing, and both the training and testing sets consist of approximately 12.82 million samples. This indicates that the training and testing datasets are independent in time. They are also spatially independent, owing to the pseudo-random distribution of CyGNSS SP. A good agreement between the retrieved SM and SMAP SM was obtained by examining the training and test data. The density diagram comparing the retrieved and SMAP SM is shown in Figure 11. For the training data (half of 2020 data), the correlation coefficient R reaches 0.93, the RMSE is 0.058 cm³cm⁻³, the ubRMSE is 0.042 cm³cm⁻³, and the MAE is 0.040 cm³cm⁻³. For the test (remaining data in 2020), the correlation coefficient is 0.91, the RMSE is 0.067 cm³cm⁻³, the ubRMSE is 0.051 cm³cm⁻³, and the MAE is 0.044 cm³cm⁻³. The generalizability of this strategy was demonstrated by very small accuracy loss between the test dataset and the training dataset.

We assessed the correlation and error between SM derived from CyGNSS and SMAP to further validate the capability of CyGNSS to retrieve SM in each EASE-Grid cell. Figure 12 displays the statistical maps that were computed using the daily CyGNSS SM and SMAP SM, (a) correlation, (b) RMSE, (c) ubRMSE, and (d) MAE. The majority of pantropical areas have a good correlation (>0.8), low RMSE (<0.06 cm³cm⁻³), ubRMSE (<0.06 cm³cm⁻³), and MAE (<0.06 cm³cm⁻³) overall. In contrast, the results in the Amazon, the Congo, and Southeast Asia are inconsistent, with low correlation (<0.6) and significant error (>0.08 cm³cm⁻³). The SMAP SM data in these regions are always masked as not recommended for retrieval, since these areas are covered by forests with a high VWC of 15 kg/m². These reference SMAP data were kept for the investigation, and the low-quality reference data may be the reason for the inconsistency [20].

In addition, retrieval errors for different land types were statistically analyzed, as listed in Table 3. It can be noted that mean values of VWC, SM, and retrieval errors are roughly positively correlated and that the RMSE, ubRMSE, and MAE of the retrieved results increase with the increase in SM and VWC. The error in densely forested areas, especially evergreen broadleaf forests, is usually greater than in areas with sparse vegetation, designated as Barren or Sparsely Vegetated or Shrublands. Furthermore, the correlation and errors rely on the land types when comparing Figure 10 and Figure 12 visually for resemblance. Broadleaf and mixed forests are primary locations with weak correlation and significant errors in CyGNSS daily SM retrievals.

4.4. Validation Using the In Situ Measurements

Due to the errors of CyGNSS and SMAP SM being expected to be correlated, the proposed method was evaluated with 4 in situ SM stations from the SMAP core validation/calibration sites. Figure 3 represents the sensor distribution. The EASE-Grid information and the geographical position of the ground sites are used to match the SM time-series data of the in situ sites, CyGNSS, and SMAP. Figure 13 shows the SM time series of Fort Cobb (USA), Little Washita (USA), Monte Buey (Argentina), and Yanco (Australia) from top to bottom. Statistical results have been listed in the title of Figure 13. It can be observed that the variations in the three datasets exhibit good consistency (ubRMSE is less than 0.35 cm³cm⁻³). It is important to note that under soil freezing conditions (the soil temperature is below 4 degrees Celsius), in situ SM records are not taken into account. Furthermore, we have included rainfall data from Global Precipitation Measurement (GPM, the spatial resolution is 0.1° × 0.1°) mission as a reference. Although the differences in the resolution of rainfall data and retrieved SM may lead to the inability to capture the spatial variability of rainfall in the study, overall, it shows a sharp increase in SM after rainfall events. The difference between CyGNSS SM and ground observations seems to be seasonally dependent. In the rainy season, the differences are relatively significant, whereas the disparities are minimal during the period of drought, which may be attributed to changes in surface conditions (e.g., vegetation).

4.5. Comparison of UCAR SM Products

Here, the proposed robust regression method is compared with UCAR SM Products. It has been selected because it is a classic method of using linear regression to retrieve SM. Figure 14 shows the SMs from (a) CyGNSS, (b) UCAR, and (c) SMAP on 1 January 2020, and the error of (d) CyGNSS and (e) UCAR obtained concerning SMAP. Similar to previous discussions, significant errors mainly focus on forest-covered areas. We calculated the correlation and error between UCAR and SMAP SM: R = 0.88, RMSE = 0.074 cm³cm⁻³, ubRMSE = 0.053 cm³cm⁻³, MAE = 0.052 cm³cm⁻³. Figure 15 shows the density map for the error of CyGNSS SM and UCAR SM compared with SMAP for the entire year of 2020, with a standard deviation of 0.064 cm³cm⁻³ for retrievals and a slightly larger standard deviation of 0.072 cm³cm⁻³ for UCAR products. This indicates that the retrieval results obtained using the present method are superior to those obtained using classical linear regression.

5. Discussions

Regarding the impact of observation noise on classical linear regression, a new method based on robust regression for retrieving SM from CyGNSS data is developed, and the impact of roughness and vegetation is reconsidered. Similar to most remote sensing technologies, the proposed retrieval algorithm has limitations: (1) It may be incorrect to assume that the sensitivity of reflectivity to SM does not change over time in certain areas, such as agricultural areas, although this possibility is currently ignored. (2) SMAP SM errors may propagate to CyGNSS, especially for SMAP data that are not recommended for retrieval. Using other methods or models may reduce the dependence on SMAP SM information.

The retrieval capacity of CyGNSS is limited in densely vegetated areas. First, the density of canopies governs the feasibility of GNSS signals reaching the ground and being received by CyGNSS, making a precise characterization of vegetation coverage and its impact on microwave signals indispensable for SM retrieval. Second, like most existing methods, retrieving SM using reflectometers is performed under the assumption of coherent reflection. However, the incoherent scattering is the main contribution to the reflected signal over the tropical forest. Several recent studies [18,37,38,39] on the use of reflectometers to estimate above-ground biomass (AGB) demonstrated that the sensitivity of reflectivity to forest canopies improves when the incoherent scattering term is dominant on the reflected signals and different tropical forests have different AGB saturation. Last, in these regions, the influence of seasonal floods on CyGNSS reflectivity may overwhelm the SM signals, resulting in a decline in retrieval capability [21].

The results using ground observations for validation are site-specific and depend on sensors’ locations, land cover, climate, and freeze-thaw conditions. Through the utilization of ground observations, CyGNSS showcases its ability to capture SM trends. It exhibits good performance during dry periods but shows a slight decrease during periods of increased rainfall.

As droughts and floods occur more frequently, the demand for water information is growing. In the future, GNSS-R SM products with high temporal resolution could be utilized as inputs for continental-scale hydrological models, such as the GEOGloWS ECMWF streamflow service [40] and the National Oceanic and Atmospheric Administration (NOAA) Water Model [41], enhancing streamflow prediction through SM data assimilation. Additionally, such products can play a crucial role in regions with often ineffective or limited hydrological services.

6. Conclusions

In this research, we present a robust regression-based approach for SM retrieval from CyGNSS reflectivity data, incorporating calibration for vegetation and roughness impacts on reflectivity. The proposed method’s efficacy is demonstrated by SMAP, in situ measurements, and UCAR SM products, and satisfactory consistency is attained.

Spatially, CyGNSS retrieval yields satisfactory results in most of pantropical regions. However, the accuracy of CyGNSS is decreased in densely vegetated regions. This feature may belong to the more complex structure of the canopy layer and surface characteristics. We should be careful when using SM data over dense forests. In terms of time-series analysis, the consistency between CyGNSS retrieval and ground observations has reached a reasonable level. In the future, investigating and quantifying the impact of additional vegetation parameters on SM retrieval may be of interest.