**3. Methodology**

In this study, the procedure for the retrieval of soil moisture using CYGNSS based on proposed methods was illustrated in Figure 3. The first step involved the quality control and processing of data used, such as CYGNSS, SMAP and GSWE. The surface reflectivity derived from CYGNSS was corrected using an improved method of water removal and the normalization method of the Fresnel reflection coefficient. A linear regression equation of soil moisture was established by combining the resampled soil moisture product of SMAP with the corrected surface reflectivity. Finally, the results of soil moisture were obtained by averaging the retrieved soil moisture, and the accuracy was comparatively verified based on the measured data, SMAP products and CYGNSS products.

## *3.1. Removal of Water*

The current solution for observations influenced by water bodies is to exclude observations that carry information about water bodies [19,20]. A square grid of 7 × 7 km to exclude observations influenced by water bodies was designed by Chew et al. [19], based on the "Seasonality" product from GSWE data. According to the research of Chew et al. [19], an improved method for removing observations affected by water bodies in a 3 × 3 km square grid was proposed by analyzing the characteristics of CYGNSS data. The process was as follows:

Step 1: Based on the latitude and longitude of the specular reflection point from CYGNSS, its corresponding location in the "Seasonality" product was searched.

Step 2: A square grid of 3 × 3 km was created, with this corresponding location as the center.

Step 3: When there was a value marked as 1 (i.e., there is a water body) in this 3 × 3 km square grid, then the point was eliminated, i.e., the specular reflection point was eliminated.

Step 4: The above process was repeated to complete the removal of specular reflection points affected by water bodies.

The removal diagram of the observations affected by the water was shown in Figure 4:

**Figure 3.** Flowchart of this study.

**Figure 4.** Schematic diagram of the removal method for observations affected by water bodies.

#### *3.2. The Normalization of the Fresnel Reflection Coefficient*

According to the Kirchhoff approximation, the surface reflectivity Γ(θ) of CYGNSS can be further expressed as [29–31]:

$$\Gamma(\theta) = \left| R\_{lr}(\theta) \right|^2 \gamma \exp\left( -\left( 2kr\cos\theta \right)^2 \right) \tag{2}$$

*γ* is the vegetation attenuation term; exp(−(2*kσ* cos *θ*) 2 ) is the attenuation term of surface roughness; and *Rlr*(*θ*) is the Fresnel reflection coefficient, which is a function of the incident angle θ and the soil dielectric constant *ε* [32]. *ε* is calculated by the Dobson model [24], which is adapted to the frequency range of 0.3–1.3 Ghz and 1.4–18 Ghz and consists of microwave frequency, soil temperature, soil type composition and soil moisture:

$$\varepsilon\_{\rm soil}^a = \left[ 1 + \frac{p\_b}{p\_s} (\varepsilon\_s^a - 1) + m\_v^\beta \varepsilon\_{fw}^a - m\_v \right]^{\frac{1}{a}} \tag{3}$$

$$\beta = 1.2748 - 0.00519 P\_{sand} - 0.00152 P\_{clay} \tag{4}$$

where *Pb* and *Ps* are the bulk density of soil and the density of the solid medium in soil, respectively. α is generally 0.65, and *εfw* and *ε<sup>s</sup>* are the permittivity of free water and solid soil, respectively. *mv* is soil moisture. *Psand* and *Pclay* represent the sand and clay contents of soil (%), respectively.

According to Formulas (3) and (4), the response of these relevant variables to the Fresnel reflection coefficient was shown in Figure 5. Soil type parameters refer to the table of physical parameters published by the Dobson model [24] (Table 1).

**Table 1.** Physical parameters of typical soil types.


From Figure 5a–d, it can be seen that the Fresnel reflection coefficients obtained from different soil moisture values varied greatly under the condition of constant soil temperature and the same soil type. For a constant soil temperature and the same soil moisture, the Fresnel reflection coefficient corresponding to different soil types was also different. The above results indicate that differences in the soil type and soil moisture can lead to changes in the Fresnel reflection coefficient. Figure 5e–h show the response of the Fresnel reflection coefficient for a constant value of soil moisture and different soil temperatures. It can be observed that the changes in Fresnel reflection coefficients obtained from soil temperature differences were small relative to those caused by changes in soil moisture. Of course, differences in the Fresnel reflection coefficient between soil types are always present. Moreover, in Figure 5, the Fresnel reflection coefficient becomes smaller and smaller, with an increasing incident angle regardless of the differences in soil moisture, temperature and soil type, indicating that the incident angle plays a very important role.

In this study, a normalization method of the Fresnel reflection coefficient was proposed to reduce the influence of relevant parameters on the Fresnel reflection coefficient, and thus reduced the surface reflectivity error of CYGNSS caused by the Fresnel reflection coefficient and improved the accuracy of soil moisture retrieval. The process of establishing this method consisted of four steps in total.

**Figure 5.** Response of the Fresnel reflection coefficient under different conditions: sm is soil moisture, and st is soil temperature ((**a**–**d**) are response maps with soil temperature as a constant value, (**e**–**h**) are response maps with soil moisture as a constant value).

Step 1: A database of Fresnel reflection coefficients with incident angle as the independent variable was created by freely combining values in the range of soil moisture, soil temperature, soil type and incident angle. Soil moisture, soil temperature and incident angle were limited to [1, 100], [1, 60] and [1, 90], respectively. The increments for these three parameters were set as 1%, 1 ◦C and 1◦, respectively. Soil types referred to the physical parameters of the Dobson model (Table 1). Microwave frequency was set to 1.57542 GHz. Through the combination of the above variables, a total of 2,160,000 Fresnel reflection coefficient values in the range of incident angles from 1 to 90◦ were formed.

Step 2: Based on the combination of variables in Step 1, a database of a total of the values of 2,160,000 Fresnel reflection coefficients with an incident angle of 0◦ was additionally composed.

Step 3: Based on the database created in Step 1 and Step 2, the correction variable was obtained using the following equation:

$$\mathcal{R}\_{lr}(\theta)\_{car} = \mathcal{R}\_{lr}(\theta) / \mathcal{R}\_{lr}(0) \tag{5}$$

where *Rlr*(*θ*)*cor* is the corrected Fresnel reflection coefficient, *Rlr*(*θ*) is the Fresnel reflection coefficient obtained from Step 1 and *Rlr*(*0*) is the Fresnel reflection coefficient for an incident angle of 0◦ in Step 2.

Step 4: With incident angle as the independent variable and corrected Fresnel reflection coefficient values of 2,160,000 as the dependent variable, the functional relationship between corrected Fresnel reflection coefficient and incident angle (1–90◦) was established. A functional expression for the angle of incidence was as follows:

$$f(\theta) = a \cdot \exp(b \cdot \theta) + c \tag{6}$$

where *a*, *b* and *c* are all empirical parameters, which can be obtained by solving the parameters using the least square method.
