4.2. Multiple Polarization Roughness Estimates
The methodology from
Section 3.2 is implemented using the coherent and incoherent components from
Figure 1, and its corresponding Stokes parameters. The surface rms height (
) is estimated from Equation (24), and the slope rms (
s) is estimated from Equation (25). The results are presented in
Figure 2 for H, V, R, and L signals, respectively, for
, and in
Figure 3 for the slope rms (
s).
The surface roughness parameter shows slight differences depending on the polarization selected.
(rms height for H-pol) has an average value of 5.66 cm,
of 5.40 cm,
of 4.94 cm, and
of 5.20 cm. The standard deviation of each roughness is ~0.82 cm. Moreover, the ubRMSD and correlation coefficient between the different roughness values are summarized in
Table 2 and
Table 3. All four
present slight biases between them, but with a high correlation coefficient between the H, V, and L, with a small ubRMSD. However, this is not the case for the roughness in the RHCP polarization. As seen in the map from
Figure 2, very small roughness values are shown in vegetated areas.
Looking at the retrieved slope (
s) in
Figure 3, we see average values of 9.28, 6.75, 4.01, and 5.92 for H, V, R, L, and its standard deviations are 9.52, 5.92, 3.18, 5.75. In this case, it is surprising the very small bias and error between the surface slope at V polarization and L polarization, even larger than in the rms height (
) case.
The results show slight differences among the different polarizations. In terms of correlation between the products, PO and GO modeled roughness shows a very high correlation between H, V, and L, but a lower correlation with the R signal. The R signal SNR being lower than H, V, or L, as shown in [
53], may bury the RHCP component under the noise, making it impossible to measure and resulting in an increased bias in the retrieved roughness. As for H, V, or L, the ubRMSD is approximately 3% of the actual value, indicating that the assumption of a negligible cross-polarization component only produces a 3% error in the surface roughness retrieval, for the PO model. However, in the GO case, the ubRMSD is higher due to the logarithmic shape of the rms slope (
s), with respect to the average value. Still, this ubRMSD is two times lower than the slope rms standard deviation. This finding confirms that the assumption of negligible cross-polarization does not significantly bias the results, but it should be taken into account for very accurate retrievals. Note that, simulation studies on polarimetric roughness have studied the effects of scattering in HR, VR, LR, and RR, showing that the real and imaginary parts of the Fresnel reflection coefficient at h and v polarization are slightly affected by the surface roughness [
58].
Additionally, despite
and
s represent different physical phenomena, the correlation between them is noticeable. In this regard,
and
s are related by:
where
a and
b are two coefficients that can be estimated via least squares.
Table 6 summarizes the coefficients (
a and
b), the correlation coefficient, and the root mean square difference (RMSD) of the fit. Note that, the units of
are centimeters.
The results are highly consistent for the four different parameters, with a high correlation coefficient, and very similar parameters, a, and b, that link both magnitudes. This similarity between both methods is linked to the fact that the roughness effect, simulated via the coherent or incoherent component, is similar. The scattered wave coherently integrated for 30 ms, and modeled via the PO model produces a coefficient that is proportional to the incoherently averaged waveform modeled using the GO model. This implies that the signal at 30 ms coherent integration is neither coherent nor incoherent, but a mix of both since the GO model has a formulation to relate to the PO model at the selected integration time. It is worth noting that the RMSD between the fitted GO and the PO model is approximately 0.4 cm, equivalent to around 8% of the average rms height. This means that the difference between the PO and the GO models leads to an error of approximately 8% in the retrieved rms height.
To further analyze the similarity between the rms height and the rms slope, both retrieved magnitudes are compared to the actual DEMSTD and DEMSLPSTD described in
Table 1. Note that, DEMSTD is commonly known as the surface height rms, and DEMSLPSTD is the surface slope rms. Given its similarity with other roughness coefficients, we will only compute this for the L polarization. The scatter plot for both
and
s are shown in
Figure 4.
In order to relate the DEM rms height (e.g., DEMSTD) to the PO model, we would require a vertical and horizontal resolution of the DEM better than the signal wavelength ([
42], p. 423), which is not feasible. However, several studies dealing with different DEM resolutions conclude that a resolution change basically produced a scale on the rms height product [
55,
59] (e.g., if one wants an equivalent resolution of 1 cm from a 9 km product, the resultant rms height shall be scaled by the ratio of resolutions). In our case, the base resolution of the DEM used here is 30 m, and the standard deviation has been computed and upscaled to larger patches of 9 km.
As can be seen, the slope rms presents a slightly higher sensitivity to the DEM slope (DEMSLPSTD,
) and rms height (DEMSTD,
) variations. The correlation coefficient between
and
and
are 0.39 and 0.38, while for
s are 0.54 and 0.52, respectively. Different models are tested to find a relationship between both magnitudes and
and
s. It is found that the following linear relationship has a large correlation between DEMSTD and DEMSLPSTD and
s:
This linear fit presents a correlation coefficient of 0.54, with an RMSD of 4.84. A similar fit can be implemented for the
parameter, however, the correlation and RMSD are, R = 0.40, and RMSD of 0.80 cm for the equation:
Note that, the resolution of our is meters, ranging from 0 to 500 m, with a resolution of 9 km. By scaling the magnitude by 0.0036 the model is telling us that scale is ~277 times smaller than the 9 km resolution from the DEMSTD product used, which is ~30 m, 10 times the size of the GNSS chip at L2C.
4.3. Roughness Uncertainty on Soil Moisture Retrievals
Both
and
have been shown to be correlated to DEMSTD and DEMSLPSTD. However, the non-negligible RMSD of this parameter could prevent the implementation of Fresnel reflection coefficient inversion algorithms to estimate surface soil moisture [
21].
In this section, we will use the model with
, estimated via Equation (24) with an RMSD of 0.80 cm, to quantify the soil moisture estimation uncertainty. We now introduce the error metric due to the roughness uncertainty as:
where
ε is the error estimating the surface rms height parameter.
The impact of different values of
ε is shown in
Figure 5 assuming an average value of
= 5.2 cm, its average value for L-polarization, and
A surface roughness uncertainty of 0.8 cm (the standard deviation of the fit from Equation (28)) produces an error of
5.0 dB, producing an error in the soil moisture retrieval of ~0.12 cm
3/cm
3 for
, and even larger for other polarimetric components as the
or
, with errors higher than 0.2 cm
3/cm
3 for the LHCP component. These large differences in single-polarization approaches suggest that by only using single-polarization measurements, soil moisture retrievals might not be feasible for single-pass retrievals if the surface roughness is not accurately modeled. In other words, because the correlation coefficient is very low with respect to the actual GNSS-R reflectivity measurements, retrievals using the Tau-Omega model, using VOD, and estimating surface roughness from DEM models, will not produce an accurate soil moisture product. As indicated in ([
42], p. 423) an acceptable DEM base product to produce the small-scale surface roughness that is required in the coherent model needs to be comparable to
, which is not feasible nowadays for a global coverage approach.
4.4. Dual-Polarization Differential Roughness Estimates
Solving the roughness effect for a single polarization gives us the possibility for single-polarization soil moisture retrievals using any polarization combination. However, the algorithm is highly dependent on the accuracy of the roughness map. To overcome this issue, combined polarimetric retrievals can be used to mitigate the effect of roughness. This approach has been proposed by several pieces of research [
28,
42,
60,
61] to allow a higher quality soil moisture estimation without the need to accurately model the surface roughness, reducing the error. As in the single-polarization case, we define an error term by comparing the measured reflectivity ratio with the Fresnel reflection coefficient ratio:
Being
p and
q two orthogonal polarizations, and
Q polarization coupling factor [
28]. Note that, VOD and roughness parameters are neglected here, and both are simplified by the
Q parameter.
Results for
for
p = {H,R} and
q = {V,L} for both
and
are shown in
Figure 6. As can be seen, results are very similar for the coherent and incoherent cases. For the HV case, the correlation between the coherent and the incoherent is 0.88, with an RMSD of 0.61 dB. The average
QHV value is −2.2 dB, and the standard deviation is 1.3 dB. First, these results indicate that the SMAP-R captured H/V ratio presents a ~2.29 dB bias with respect to the theoretical one. The analysis of this bias was discussed in [
52] to be produced by the GPS L2 antenna axial ratio for blocks IIR or newer [
62,
63].
For QRL the average value is 2.72 dB with a standard deviation of 3.86 dB, and the correlation between the coherent and incoherent cases is 0.87 and 1.33 dB, respectively.
In this case, the
QHV seems uncorrelated with any surface parameter (e.g., DEM), but it does correlate with some parameters of the reflection, as the incoherent reflectivity from Equations (5)–(7), once calibrated by VOD:
With an RMSD of 0.97 dB and a correlation coefficient of 0.64. If VOD is not compensated, the correlation coefficient is lowered down to 0.55 and the RMSD increases up to 1.09 dB. Results for the coherent part show a lower correlation coefficient (0.4) and larger RMSD (1.38), which could indicate that this parameter is more sensitive to incoherent scattering rather than specular scattering.
On the contrary,
QRL is correlated with vegetation, notably in areas with large vegetation, e.g., Congo or Amazonian rainforests. VOD and
QRL are related via the linear relationship:
In this case, the correlation is 0.65, and the RMSD is 2.93 dB. The scatter-density plots of both fits are shown in
Figure 7.