Microwave Vegetation Index from Multi-Angular Observations and Its Application in Vegetation Properties Retrieval: Theoretical Modelling

Monitoring global vegetation dynamics is of great importance for many environmental applications. The vegetation optical depth (VOD), derived from passive microwave observation, is sensitive to the water content in all aboveground vegetation and could serve as complementary information to optical observations for global vegetation monitoring. The microwave vegetation index (MVI), which is originally derived from the zero-order model, is a potential approach to derive VOD and vegetation water content (VWC), however, it has limited application at dense vegetation in the global scale. In this study, we preferred to use a more complex vegetation model, the Tor Vergata model, which takes into account multi-scattering effects inside the vegetation and between the vegetation and soil layer. Validation with ground-based measurements proved this model is an efficient tool to describe the microwave emissions of corn and wheat. The MVI has been derived through two methods: (i) polarization independent ( MVI B P ) and (ii) time invariant ( MVI B T ), based on model simulations at the L band. Results show that the MVI B T has a stronger sensitivity to vegetation properties compared with MVI B P . MVI B T is used to retrieve VOD and VWC, and the results were compared to physical VOD and measured VWC. Comparisons indicated that MVI B T has a great potential to retrieve VOD and VWC. By using L band time-series information, the performance of MVIs could be enhanced and its application in a global scale could be improved while paying attention to vegetation structure and saturation effects.


Introduction
Vegetation plays an important role in our ecosystems and its interactions with the Earth system. Vegetation properties such as biomass, coverage, and water content are widely studied. It is a challenge to measure vegetation properties in global scale using traditional in situ techniques. Remote-sensing techniques including various vegetation indices could serve as an important tool to derive the dynamic spatial-temporal distributions of vegetation. An optical index, such as the normalized difference vegetation index (NDVI) is related to the greenness of vegetation which depends on the leaf's adjacent angles of the L band is less influenced by partial penetrating through the canopy, as compared to multi-frequency MVI. Grant et al. [23] compared the Soil Moisture Observation Mission (SMOS) VOD to AMSR-E VOD, which indicates that AMSR-E VOD has a higher correlation with the optical indices than SMOS VOD. Studies have mentioned that lower frequencies of microwave spectrum are more suitable for identifying the canopy volume than higher frequencies [16,[24][25][26][27]. It can penetrate through the vegetation more than higher frequencies. Cui et al., [19] presented a multi-angular MVI approach and tested it with SMOS H-polarized multi-angular measurements. The retrieved VOD using the MVI approach was compared with above-ground biomass, which showed a correlation of R = 0.726, especially for shorter vegetation. The study considered a constant for vegetation single scattering albedo, which may introduce uncertainties in the retrieval of VOD. Again, the applicability of this approach on a global scale, especially validation over moderate and dense vegetation (>3 kg/m 2 ), lacks in evaluation from a theoretical viewpoint. Therefore, a comprehensive study on the deviation of MVI from multi-angular observations of L-band in dense vegetation, and its evaluation using a more rigorous theoretical modeling is necessary.
In this paper, MVI was calculated and evaluated based on the Tor Vergata model, which considers multiple scattering effects inside the vegetation and interactions between the vegetation layer and soil surface [28]. We simulated T B for corn and wheat as two typical crops with significant multiple scattering effects using the ground observation. To solve the MVI function, two assumptions were considered: the first assumption is considered as "no polarization dependent" (MVI P B ) and the second one as "no vegetation changes during a time window" (MVI T B ). The purpose of this study was to explore how multi-angular MVI at the L-band, as a zero-order model index, works in dense vegetation with multiple scattering, and to provide new insights for developing future satellite missions and algorithms. The following section describes the data, model and methods used for MVI derivations, and Section 3 presents the results of MVI application in VOD and VWC retrieval. Section 4 gives a further discussion of its advantages and limitations, and Section 5 summarizes the main conclusions.

Data
Data for wheat canopy were acquired in summer 1993 at the Institute National de Recherches Agronomiques (INRA) test site near Avignon, France, from day of year (DOY) 90 (shortly after seeding) to DOY 179 (shortly before harvest). Measurements were recorded for leaf width, length, and thickness; stalk diameter and length; stalk and leaf moisture; soil and plant temperature; canopy height; stalk length; leaf inclination angle (alpha, beta, and gamma); density; and volumetric soil moisture (SMC) at different depths. Fresh and dry biomass, leaf area index (LAI), and VWC were also measured. Detailed information about field data can be found in Ferrazzoli et al., [29]. The passive microwave sensor system used to measure brightness temperature (T B ) of wheat, was the Multi-Frequency Microwave Radiometer (PORTOS). It has six different frequency (1.4, 5.05, 10.65, 23.8, 36.5, and 90 GHz) antennae with dual-polarized (V/H) measuring ability. Regular calibration was performed over calm water surfaces. The radiometer absolute accuracy was 3 K. More detail about the instrument, calibration, and field data are provided in previous studies [1,7,8,29].
The data for the corn canopy were obtained in summer 2017 in Inner Mongolia, China, from DOY 155 (shortly after seeding) to DOY 195 (crop about to reach maximum height). A vehicle-mounted microwave radiometer that contains three frequency bands, L (1.4 GHz), C (6.925 GHz), and X (10.65 GHz), was used to measure the T B . Ground measurements included soil moisture, surface roughness, VWC, and LAI. Geometric and physical parameters for corn were measured on selected days. To represent the growth progress of corn, we used a crop growth model created at Tor Vergata University. In this growth model, canopy variables are computed as a function of crop height and crop type. This version of the growth model contains empirical relationships that were established mainly based on ground measurements over sunflower and corn fields during several experimental campaigns [30]. The model was found to match the measurements recorded in China. Table 1 summarizes the input parameters for wheat and corn.

Radiative Transfer Model (Tor Vergata Model)
The Tor Vergata model developed by Ferrazzoli et al. [31] ncludes volume scattering and interaction between vegetation volume and soil boundary [28]. The model describes the canopy as a group of discrete scatters and the soil as infinite half-space with the rough interface. The electromagnetic properties of the scatters are described by the absorbed cross-section and bistatic scattering cross-section. The Tor Vergata model is a discrete modeling approach. The descriptions of wheat and corn in the Tor Vergata model are shown in Figure 1. This uses simple geometric elements (discs, cylinders, and ellipsoids) to simulate the scattering matrix and extinction vector [31]. For wheat, the upper layer of small vertical cylinders represents ears and the lower layer with randomly oriented discs represents leaves. For corn, the upper layer with randomly oriented discs represent the leaves and the lower layer with cylinders is the main stems. As corn and wheat leaves have a strip shape, the Tor Vergata model treats them as several circular discs in an equivalent area. The model assumed that the disc orientation is distributed over the Euler angles between: 0 • < α < 360 • and 5 • < β < 85 • . To simulate the scattering matrix and the extinction of discs, the General Rayleigh-Gans (GRG) approximation is used when the frequency is under 2 GHz, and the physical optical (PHO) approximation is used when the frequency is above 2 GHz [32,33]. For small cylinders (secondary stems and petioles), the scatter matrices and the extinction vectors are computed using the Rayleigh-Gans (RG) approximation when the frequency is under 2 GHz, and infinite length (IL) approximation when the frequency is above 2 GHz. The main stem is another important component of corn with larger size and is mostly vertically distribution. The infinite length cylinder approximation is adopted to simulate the stalk [32,33].  In the Tor Vergata model, the vegetation layer containing different types of scattering is divided into n number of thin elementary sub-layers (Δzn) that are assumed to be symmetrical concerning the azimuth (here, n=10). In each sub-layer, the incident and scattering angles are divided into several small intervals for considering directions as many as possible. For each incident angle, the scattering In the Tor Vergata model, the vegetation layer containing different types of scattering is divided into n number of thin elementary sub-layers (∆z n ) that are assumed to be symmetrical concerning the azimuth (here, n = 10). In each sub-layer, the incident and scattering angles are divided into several small intervals for considering directions as many as possible. For each incident angle, the scattering matrix and transmission matrix at the nearby sub-layer can be obtained by radiative transfer solution [34]. A combination is performed between ∆z n and ∆z n+1 to form a new sub-layer ∆z , so on and so forth, till the combination reaches n = 10; finally, the total microwave scattering of the wheat and corn layer can be extracted. For the soil surface emission, the advanced integral equation model (AIEM) [35] was adopted to account for roughness effects. Then, the emissivity of the total vegetation layer was finally computed.

Multi-Angular Microwave Vegetation Index (MVI)
In the MVI technique, T B from the τ-ω model is rearranged and linearly linked to the soil emissivity by two vegetation components, transmission (V t (θ)) as slope and emission (V e (θ)) as intercept at specific incident angle (θ), and their relationship is explained as: where, T BP (θ) is the total T B simulated and E s p (θ) is the soil surface emission at a specific incident angle (θ) and polarization (p): where E v p is the vegetation emission, L p (θ) is the one-way attenuation in the specified incident angle and polarization, F v is the vegetation fraction, and T v and T s are vegetation and soil temperature, respectively. To minimize the effect of the soil emission signal, the bare soil emission signals at different angles are evaluated through numerical simulations from the AIEM model. The bare soil surface emissivity at two adjacent angles are highly correlated and the linear function can be expressed as: where the coefficients b(θ 11 , θ 2 ) and a(θ 1 , θ 2 ) are constant, only dependent on the pair of angles to be used [7]. Here, for 40 • and 50 • incident angles a is 0.087 and b is 1.035. By using Equations (1) and (4) to eliminate the soil emissivity, total T B observations at two adjacent angles can be described as a linear function as shown in Equation (5). The intercept, A p (θ 1 , θ 2 ), and slope, B p (θ 1 , θ 2 ), of this linear function are the microwave vegetation indices as shown in Equations (6) and (7). This technique minimizes the surface emission signal and maximizes the vegetation signal [13] More information about the MVI technique can be found in Shi et al. [5].
The slope and intercept of the linear relationship of T B (Equation (5)) in two adjacent angles are MVI B and MVI A , which is represented as B p (θ 1 , θ 2 ) or MVI B and A p (θ 1 , θ 2 ) or MVI A , respectively, from Equations (6) and (7).

Derivation of Multi-Angular MVI
In this study MVI was derived through two different assumptions: At the passive microwave footprint scale, there are numerous different types of vegetation canopies with different scatter sizes, shapes, and orientations. Under these circumstances, it can be assumed that there is no considerable impact of the polarization dependence [5,36,37]. By this assumption, the slope (B) of the linear relationship of T B (Equation (5)) in two adjacent angles, MVI P B , can be calculated from the ratio of polarization difference in two adjacent angles, as shown in Equation (8). Detailed information of this method can be found in Shi et al. [5]. The second method to drive MVI was based on Equation (5). MVI is independent of underlying soil surface signals and dependent only on vegetation properties [10,19,38]. By assuming that vegetation is not changing in a short time window, the soil moisture changes will generate a series of different T B s. Figure 2 presents the T B simulation results with linear fit in vertical polarization. As noted, the slope and the intercept of each fitted line are MVI T B and MVI T A , respectively. As the canopy height increases, the slope (MVI T B ) increases and the intercept (MVI T A ) decreases.
The slope and intercept of the linear relationship of TB [Eq. 5] in two adjacent angles are MVIB and MVIA, which is represented as ( , ) or MVIB and ( , ) or MVIA, respectively, from equations (6) and (7).

Derivation of Multi-Angular MVI
In this study MVI was derived through two different assumptions: At the passive microwave footprint scale, there are numerous different types of vegetation canopies with different scatter sizes, shapes, and orientations. Under these circumstances, it can be assumed that there is no considerable impact of the polarization dependence [5,36,37]. By this assumption, the slope (B) of the linear relationship of TB (Equation (5)) in two adjacent angles, MVI , can be calculated from the ratio of polarization difference in two adjacent angles, as shown in Equation (8). Detailed information of this method can be found in Shi et al. [5]. (

2) No vegetation changes in a certain time window (MVI )
The second method to drive MVI was based on Equation (5). MVI is independent of underlying soil surface signals and dependent only on vegetation properties [10,19,38]. By assuming that vegetation is not changing in a short time window, the soil moisture changes will generate a series of different TBs. Figure 2 presents the TB simulation results with linear fit in vertical polarization. As noted, the slope and the intercept of each fitted line are MVI and MVI , respectively. As the canopy height increases, the slope (MVI ) increases and the intercept ( MVI ) decreases.

Optical Depth
The definition of optical depth is the ratio of the natural logarithm of the incident to the transmitted radiant power through a material. Absorption, reflection, scattering, and other physical processes in the medium affect the value of optical depth, which is dimensionless. The VOD is used to describe the attenuation properties of the vegetation cover [10,39]. VOD is a function of vegetation dielectric properties, responding strongly to the vegetation water content. Its value depends on

Optical Depth
The definition of optical depth is the ratio of the natural logarithm of the incident to the transmitted radiant power through a material. Absorption, reflection, scattering, and other physical processes in the medium affect the value of optical depth, which is dimensionless. The VOD is used to describe the attenuation properties of the vegetation cover [10,39]. VOD is a function of vegetation dielectric properties, responding strongly to the vegetation water content. Its value depends on vegetation type and the wavelength used. Optical depth at lower frequencies, which can penetrate more in vegetation, can provide unique information about vegetation biomass and water content [40].

Optical Depth Simulation
For the forward scattering theorem of vegetation transmissivity (λ p (θ) = e −τ p secθ ) function, τ is vegetation optical depth, which is given by (VOD p (θ) = k ep (θ)d), where θ is the observation angle from the nadir and d is canopy layer height. The average vegetation extinction coefficient is defined by  [41], where ρ α is the number density of the scatter type α and K 0 is wave number. Here, K 0 = 2π/λ 0 and λ is the wavelength. The angular bracket in this function defines the ensemble average over the angularity and size of particles. For leaves, the average size of a circular disc is based on orientation angles. No averaging is performed for trunks because they are vertical and have a typical size. f (α) f pp is the forward scattering amplitude of the αth group of scatters [42]. The optical depth obtained by this theoretical function is hereafter called VOD S in this paper.

Optical Depth and Vegetation Water Content (VWC) retrieval from MVI
According to the MVI function in Equation (6) and the vegetation transmission in Equation (3), we have: To derive optical depth from MVI, we assumed the emissivity of vegetation was approximately equal at two adjacent incident angles, given by . By rearranging Equation (9), we acquired a simple formulation for the optical depth: where b(θ 1 , θ 2 ) is calculated using Equation (4). The soil surface emissivity is extracted from the AIEM model. VOD T and VOD P can be obtained through MVI T B and MVI P B for wheat and corn. According to the VOD and VWC relationship (VOD = b·VWC), VWC can be estimated from VOD T and VOD P and compared with measured VWC. Here b is a parameter depending on vegetation structure, microwave frequency, and polarization and it is obtained from literature [18,43].

The Validity of the Tor Vergata Model and MVI Technique
We simulated T B using the Tor Vergata model under different soil moisture and roughness conditions. Ranges of volumetric soil moisture (0.05-0.35 m 3 /m 3 , intervals 0.05 m 3 /m 3 ) and roughness parameters (root mean square height 0.5-1.5 cm, intervals 0.5 and correlation length 5-15 cm, intervals of 5) were used. Based on field observations and the growth model, we completed T B simulations from the lowest to highest canopies. The Tor Vergata model simulation has been validated several times using field data for corn and wheat [29,31,[44][45][46]. Figure 3 indicates the comparison results of measured T B from radiometer and simulated T B from Tor Vergata model at L band at incident angle of 40 • for wheat and corn. Correlation coefficients between measured and simulated T B for wheat is 0.93 in both polarizations and 0.78 and 0.83 for corn at V and H polarizations, respectively. The results provide a satisfactory level of accuracy for the Tor Vergata model to be accepted as the theoretical basis for this study.
One of the theoretical bases of simulated MVI by the τ-ω model is the linear relationship of T B with soil emissivity. This linear relationship was evaluated using the Tor Vergata model simulations, as shown in Figure 4. Only simulations for a constant canopy height (corn at 150 cm and wheat at 70 cm) are shown. We found the linear relationships between T B and bare soil emissivity were valid in all angles. The coefficients of determination were high (more than 0.95) over all incident angles. The sensitivity of T B to soil emissivity decreases with an increase in the incidence angle, as more vegetation is visible over a longer penetration path. This angular effect is not so significant for wheat at H-pol, as wheat is vertical structure dominated vegetation. cm) are shown. We found the linear relationships between TB and bare soil emissivity were valid in all angles. The coefficients of determination were high (more than 0.95) over all incident angles. The sensitivity of TB to soil emissivity decreases with an increase in the incidence angle, as more vegetation is visible over a longer penetration path. This angular effect is not so significant for wheat at H-pol, as wheat is vertical structure dominated vegetation.

Multi-Angular MVI for Corn and Wheat
Multi-angular MVI at the L-band was derived using the two methods based on the Tor Vergata model simulations. In the first method, assuming no polarization dependence on vegetation signal, vegetation emission components were eliminated while calculating MVI P B using the ratio of T B polarization difference at two adjacent angles. The result shows that MVI P B increases with an increase in canopy height (θ 1 being < θ 2 ).
The second method to derive MVI is based on the linear function of T B at two adjacent angles, such that the slope and intercept of these linear relationships are MVI T B and MVI T A , respectively. MVI T B and MVI T A can be derived by simulating T B in various soil parameters in two angles. By increasing canopy height, MVI T B value is increased (θ 1 < θ 2 ). A high coefficient of variation (CV) for MVIs denotes a higher sensitivity to vegetation change, as vegetation changes significantly from sowing to harvesting according to experimental data. Table 2 indicates the CV values for two MVI types for a different pair of angles of wheat and corn. The CV of MVI T B is greater than that of MVI P B . In deriving MVI P B , we assumed there was no polarization dependence on the vegetation signal and the effect of the vegetation emission component was eliminated. This may have caused a lower CV value in MVI P B compared to MVI T B . We also found that the CV of MVI T B in V-pol was more than that in H-pol. As Ferrazzoli et al. [29] indicated, L-band emissivity at V-pol is more sensitive to vegetation than at H-pol. In fact, MVI T B in V-pol could better display vegetation changes than others and it is a superior indicator for monitoring vegetation properties.  [40,47] are considered in this study because the effect of vegetation at higher incident angles is more significant compared to lower incident angles. Figure 5a shows a promising increase in MVI T B with the growth of wheat, while MVI P B has fewer variations during the growth of wheat. Figure 5d represents similar response of MVI T B only during the growth stage of corn. Canopy height and VWC were also compared for wheat (b) and corn (e). Since MVI is based on total brightness temperature, the trend in T B is also represented for wheat (c) and corn (f).

Application for Vegetation Optical Depth (VOD) and VWC Retrieval
Microwave signals are sensitive to vegetation properties, including above-ground biomass, VOD, and VWC, which are liked with each other [7,8,19]. As mentioned in Section 2.4.2., VOD can be retrieved through its relationship with MVI , which may serve as a good indicator of global vegetation properties monitoring. In this section, we further explore the use of MVI for VOD and VWC retrievals.

Application for Vegetation Optical Depth (VOD) and VWC Retrieval
Microwave signals are sensitive to vegetation properties, including above-ground biomass, VOD, and VWC, which are liked with each other [7,8,19]. As mentioned in Section 2.4.2., VOD can be retrieved through its relationship with MVI B , which may serve as a good indicator of global vegetation properties monitoring. In this section, we further explore the use of MVI T B for VOD and VWC retrievals. Figure 6 provides the VOD S of corn and wheat at different incident angles in V-and H-pol. In general, optical depth increased with incident angle and canopy height. With increasing incident angle and canopy height, the extinction of vegetation and optical depth also increased [42]. In other words, when vegetation was growing, vegetation attenuation increased, especially for denser vegetation [31,44]. Vegetation canopies with a structure dominated by vertical stalks, like corn and wheat, were shown to be strongly polarization-dependent [17,[48][49][50]. The figure show generally the optical depth values at H-pol were less than those at V-pol [29].
Remote Sens. 2019, 11, x FOR PEER REVIEW 12 of 20 Figure 6 provides the VOD of corn and wheat at different incident angles in V-and H-pol. In general, optical depth increased with incident angle and canopy height. With increasing incident angle and canopy height, the extinction of vegetation and optical depth also increased [42]. In other words, when vegetation was growing, vegetation attenuation increased, especially for denser vegetation [31,44]. Vegetation canopies with a structure dominated by vertical stalks, like corn and wheat, were shown to be strongly polarization-dependent [17,[48][49][50]. The figure show generally the optical depth values at H-pol were less than those at V-pol [29]. To validate and explore the applicability of MVI to retrieve VOD, VOD was calculated at the L-band and compared with VOD , as shown in Figure 7. The correlation coefficients of VOD and VOD for corn and wheat were both as high as 0.99. But RMSE value of wheat figure was less than RMSE value of corn figure. The color bar shows the single leaf area of wheat and corn (cm 2 ). It is found that the retrieval accuracy decreases as leaf area increases, especially for corn. VOD is underestimated comparing to VOD for corn. It is believed that the assumption, the emissivity of vegetation was approximately equal at two adjacent incident angles, may introduce underestimation in VOD retrieval. To validate and explore the applicability of MVI to retrieve VOD, VOD T was calculated at the L-band and compared with VOD S , as shown in Figure 7. The correlation coefficients of VOD S and VOD T for corn and wheat were both as high as 0.99. But RMSE value of wheat figure was less than RMSE value of corn figure. The color bar shows the single leaf area of wheat and corn (cm 2 ). It is found that the retrieval accuracy decreases as leaf area increases, especially for corn. VOD T is underestimated comparing to VOD S for corn. It is believed that the assumption, the emissivity of vegetation was approximately equal at two adjacent incident angles, may introduce underestimation in VOD T retrieval. Remote Sens. 2019, 11, x FOR PEER REVIEW 13 of 20 The VOD measured by passive microwave signals is an indicator of both VWC and structural effects. Estimating VWC is also critical for many vegetation applications [14,[51][52][53]. As ground measurements of wheat and corn included fresh and dry biomass (g/m 2 ), LAI, and VWC, here, we simulated VWC from VOD and compared with measurement one. Figure 8 illustrates the comparison of estimated VWC from VOD and measured VWC for wheat and corn at V-pol. The Pearson correlation coefficients of measured and simulated VWC for corn and wheat were both high, 0.99 and 0.98 at V-and H-pol respectively. But similarly, MVI retrieved VWC was underestimated.

4.Discussion
In this study, the MVI was derived at the L-band from multi-angular observations. The original derivation of MVI was based on the zero-order model, and its validity over dense vegetation was not clear. Previous studies [10] have shown that the MVI from the zero-order model is underestimated due to ignoring the vegetation volume scattering. Here, we continued to evaluate its validity and applicability in VOD and VWC retrievals based on the Tor Vergata model, which includes multiple scattering effects and is closer to reality compared with the zero-order model. In this section hereafter, we discuss the assumptions of MVI and its sensitivity on vegetation properties. The VOD measured by passive microwave signals is an indicator of both VWC and structural effects. Estimating VWC is also critical for many vegetation applications [14,[51][52][53]. As ground measurements of wheat and corn included fresh and dry biomass (g/m 2 ), LAI, and VWC, here, we simulated VWC from VOD T and compared with measurement one. Figure 8 illustrates the comparison of estimated VWC from VOD T and measured VWC for wheat and corn at V-pol. The Pearson correlation coefficients of measured and simulated VWC for corn and wheat were both high, 0.99 and 0.98 at V-and H-pol respectively. But similarly, MVI retrieved VWC was underestimated. The VOD measured by passive microwave signals is an indicator of both VWC and structural effects. Estimating VWC is also critical for many vegetation applications [14,[51][52][53]. As ground measurements of wheat and corn included fresh and dry biomass (g/m 2 ), LAI, and VWC, here, we simulated VWC from VOD and compared with measurement one. Figure 8 illustrates the comparison of estimated VWC from VOD and measured VWC for wheat and corn at V-pol. The Pearson correlation coefficients of measured and simulated VWC for corn and wheat were both high, 0.99 and 0.98 at V-and H-pol respectively. But similarly, MVI retrieved VWC was underestimated.

4.Discussion
In this study, the MVI was derived at the L-band from multi-angular observations. The original derivation of MVI was based on the zero-order model, and its validity over dense vegetation was not clear. Previous studies [10] have shown that the MVI from the zero-order model is underestimated due to ignoring the vegetation volume scattering. Here, we continued to evaluate its validity and applicability in VOD and VWC retrievals based on the Tor Vergata model, which includes multiple scattering effects and is closer to reality compared with the zero-order model. In this section hereafter, we discuss the assumptions of MVI and its sensitivity on vegetation properties.

Discussion
In this study, the MVI was derived at the L-band from multi-angular observations. The original derivation of MVI was based on the zero-order model, and its validity over dense vegetation was not clear. Previous studies [10] have shown that the MVI from the zero-order model is underestimated due to ignoring the vegetation volume scattering. Here, we continued to evaluate its validity and applicability in VOD and VWC retrievals based on the Tor Vergata model, which includes multiple scattering effects and is closer to reality compared with the zero-order model. In this section hereafter, we discuss the assumptions of MVI and its sensitivity on vegetation properties.

Assumptions
We made two assumptions to solve the MVI function: no polarization dependence on vegetation signal (MVI P B ) and no vegetation changes during MVI derivation (MVI T B ). The first method assumes that microwave signals are polarization-independent as various types of vegetation contain scatters with different sizes, shapes, and orientations. So, in the MVI function, the ratio of polarization difference at two adjacent angles can eliminate the soil contribution. However, this assumption could introduce errors for the stalk domain canopy [54]. For vegetation with a preferred orientation structure such as corn, the no polarization dependence assumption is not reasonable. The vertical structure of the stalks leads to a strong polarization dependence, as predicted by the model and supported by experimental observations [5,55]. By using this method to derive MVI P B , the effects of the vegetation emission component are eliminated, which may cause error because emission component is polarization-dependent for vegetation with significant multi-scattering effects. This method of MVI derivation may work better for real microwave satellite data (coarse resolution) rather than the model simulations, as there can be various vegetation scatters within a satellite microwave footprint.
In the second method, a series of T B observations with different soil conditions (multi-temporal information) was used to calculate MVI. The slope and intercept of this linear function were MVI T B and MVI T A , respectively. Based on the results, MVI T B is a better indicator for monitoring vegetation properties. Applying the second method of MVI derivation to satellite images is possible by assuming no vegetation changes during a certain time window (2-4 days). However, the method also has limitations, as solving the function will be difficult if soil moisture does not change much. The accuracy of MVI T B calculation is compromised over areas having less soil moisture content. Thus, use of this method in areas with limited soil moisture content is restricted. Also, a constant temperature is an important parameter during MVI derivation in a specific time window.
There is another method for retrieving vegetation properties by Konings et al. [56], using multi-temporal information, which is referred to as the multi-temporal dual channel algorithm (MT-DCA). By using dual-polarized observations at L-band Soil Moisture Active Passive (SMAP), simultaneous retrievals of vegetation optical depth, effective single scattering albedo, and soil dielectric constant could be achieved. The method is also applicable to SMOS sensor. However, the ground parameters are retrieved based on the iteration algorithm, which may have multiple solutions without prior knowledge. The main difference or advantage of our method is that it is an analytical solution to retrieve vegetation properties. However, we need further requirement of significant soil moisture changes (no soil moisture changes will also lead to noisy retrievals for MT-DCA method), and our method is only applicable to multi-angular observations, such as SMOS. Both methods assume that no vegetation changes during a certain time window and artifacts may happen when sudden changes in vegetation happen, such as fire or harvest.

Sensitivity of MVI to Vegetation Properties
It is expected that vegetation optical depth can be as an indicator of vegetation growth. Chaparro et al. [16] analyzed SMAP MT-DCA VOD for one year and compared with in-suit crop yield data. They indicated the minimum value of VOD accrued during crop emergence, and when the crops were mature the maximum value of VOD occurred. Patton and Hornbuckle [25] found optical depth changes of the crop in the growing season is related to crop yield. They indicated the SMOS optical depth increased in the late spring and early summer and decreased while the crop slowly dried out. Similarly, in Figure 9, the MVI derived VOD corresponds to literatures in terms of the trend and the time of maximum and minimum value of VOD T , which is correlated to measured VWC and LAI. In the case of corn, unfortunately measured VOD and LAI did not span through the entire crop season and the figures are represented from sowing to maximum canopy height. It should be noted that results ( Figure 5) indicated the maximum value of MVI for wheat was more than that of corn. This issue can be explained by paying attention to how wheat and corn have been described in the Tor Vergata model. In the wheat model, upper layers of small vertical cylinders represent ears, while in the corn model the upper layers of randomly oriented disks represent leaves ( Figure 1). As Ferrazzoli et al. [35,37,44] confirmed (not only by model but also by radiometer data), discs generate a significant scattering directed toward the upper half of the space and reduces the overall emissivity of corn. Wheat cylinders that represent ears are mainly absorbers and can increase the emissivity. Wheat acts like an absorption layer, which is different from wide leaf crops. For corn, the contribution of the lower vertical cylinder is strongly attenuated by the upper layers. In fact, MVI is strongly dependent on vegetation structure and leaf shape. The distribution of dielectric discs and cylinders significantly affect the absorption and scattering characteristics of vegetation components. The dependency of MVI on the vegetation structure requires more research.
Another point that should be noted is saturation effects that can happen in a microwave signal. Vegetation indices suffer saturation, especially for multi-layer dense vegetation like corn with a lot of leaves on top. Many studies have been done on microwave responses to corn canopies [57][58][59]. They indicated crop canopies can saturate at higher LAI or biomass and the exact point of saturation is dependent on crop types and frequency. It has been reported that the saturation level of L-band was found to vary between 40 and 150 ton ha For corn. [60,61] Steele-Dunne et al. [57] and Zhao et al. [58] showed that microwave brightness temperature becomes saturated when LAI exceeds 3. We confirmed this in our results that saturation started from LAI more than 3 with height more than 1 m. As it is clear from the figure 10 for corn, the brightness temperature is increasing as LAI increases, but brightness temperature started saturation while corn became dense and LAI exceeded around 3. This is because of the strong scattering effects when leaf size is comparable to the wavelength. This could strongly affect corn emissivity and reduce the sensitivity. In fact, vegetation like corn with many large leaves on top anticipate saturation [44].
It is interesting to note that not all vegetation types had reported saturation. Prevot et al. [62] indicated winter wheat backscatter continue to be sensitive to crop development throughout the season. Figure 10 (left, wheat) also indicated brightness temperature was increasing by increasing LAI and no saturation was observed during the wheat growth period. Thus, a saturation event in the canopy should depend on frequency and vegetation structure. It should be noted that results ( Figure 5) indicated the maximum value of MVI T B for wheat was more than that of corn. This issue can be explained by paying attention to how wheat and corn have been described in the Tor Vergata model. In the wheat model, upper layers of small vertical cylinders represent ears, while in the corn model the upper layers of randomly oriented disks represent leaves ( Figure 1). As Ferrazzoli et al. [35,37,44] confirmed (not only by model but also by radiometer data), discs generate a significant scattering directed toward the upper half of the space and reduces the overall emissivity of corn. Wheat cylinders that represent ears are mainly absorbers and can increase the emissivity. Wheat acts like an absorption layer, which is different from wide leaf crops. For corn, the contribution of the lower vertical cylinder is strongly attenuated by the upper layers. In fact, MVI is strongly dependent on vegetation structure and leaf shape. The distribution of dielectric discs and cylinders significantly affect the absorption and scattering characteristics of vegetation components. The dependency of MVI on the vegetation structure requires more research.
Another point that should be noted is saturation effects that can happen in a microwave signal. Vegetation indices suffer saturation, especially for multi-layer dense vegetation like corn with a lot of leaves on top. Many studies have been done on microwave responses to corn canopies [57][58][59]. They indicated crop canopies can saturate at higher LAI or biomass and the exact point of saturation is dependent on crop types and frequency. It has been reported that the saturation level of L-band was found to vary between 40 and 150 ton ha For corn. [60,61] Steele-Dunne et al. [57] and Zhao et al. [58] showed that microwave brightness temperature becomes saturated when LAI exceeds 3. We confirmed this in our results that saturation started from LAI more than 3 with height more than 1 m. As it is clear from the Figure 10 for corn, the brightness temperature is increasing as LAI increases, but brightness temperature started saturation while corn became dense and LAI exceeded around 3. This is because of the strong scattering effects when leaf size is comparable to the wavelength. This could strongly affect corn emissivity and reduce the sensitivity. In fact, vegetation like corn with many large leaves on top anticipate saturation [44].
It is interesting to note that not all vegetation types had reported saturation. Prevot et al. [62] indicated winter wheat backscatter continue to be sensitive to crop development throughout the season. Figure 10 (left, wheat) also indicated brightness temperature was increasing by increasing LAI and no saturation was observed during the wheat growth period. Thus, a saturation event in the canopy should depend on frequency and vegetation structure. Further investigation on comparing the result with satellite L-band VOD estimates is necessary and future work for improving MVI-derived VOD estimation should be done. However, it should be very difficult to find a pure corn footprint from satellite observations and is out of the scope of this theoretical investigation.

Conclusions
Passive microwave-derived VOD is a useful indicator of vegetation properties, especially at lower frequencies. VOD increases during the growing stage when plants grow and water content rises and decreases during maturity. Hence for an agricultural crop that changes basically from planting to harvest, VOD has great potential in monitoring vegetation grow stage, phenology and water stress. Therefore, in this paper, we conducted a theoretical study on how to improve and use the MVI technique to retrieve VOD, which is correlated to vegetation properties. In this study, we simulated TB based on the Tor Vergata model for corn and wheat at the L-band. The validity of the Tor Vergata model and MVI technique was evaluated based on ground-based observations. It is found that the Tor Vergata model had satisfactory accuracy and should be an effective tool for simulation microwave emission from those two kinds of crops. It is also found that the MVI technique is robust under the descriptions of the Tor Vergata model, that the overall brightness temperature is linearly correlated with soil emissivity with coefficients depending on vegetation properties. Results indicated that the MVI should work better at V-pol due to its sensitivity to vegetation.
We calculated MVI under two different assumptions of (1) no polarization dependence: MVI , and (2) no vegetation changes in a certain time window: MVI . The first method of MVI derivation is suitable without a preferred orientation for vegetation. The second method is a new development compared with previous studies. MVI from both of those two methods were found to be correlated with vegetation water content, and the MVI , which is derived based on multi-temporal information, could achieve larger variation with vegetation changes and has a great potential to be used to retrieve vegetation properties.
By comparing theoretical VOD with VOD derived from MVI , it has been found that the retrieved VOD has a very high correlation with theoretical ones. Good results were obtained by using the MVI to retrieve VOD and VWC for wheat. However, this also indicated that VOD is an underestimation of VOD for corn. The percent of leaves and cylinders in vegetation canopy, structure and wavelength can be effective parameters in VOD response. The same result was obtained when comparing retrieved VWC from VOD with the measurements. Regardless of the method that we used to derive VOD, vegetation structure and saturation effects may change the vegetation emission characteristics and affect the MVI.
Obtaining the theoretical relationship between MVI and VOD, make MVI a promising index for global vegetation monitoring and soil moisture retrievals. The suitability of the MVI is dependent Further investigation on comparing the result with satellite L-band VOD estimates is necessary and future work for improving MVI-derived VOD estimation should be done. However, it should be very difficult to find a pure corn footprint from satellite observations and is out of the scope of this theoretical investigation.

Conclusions
Passive microwave-derived VOD is a useful indicator of vegetation properties, especially at lower frequencies. VOD increases during the growing stage when plants grow and water content rises and decreases during maturity. Hence for an agricultural crop that changes basically from planting to harvest, VOD has great potential in monitoring vegetation grow stage, phenology and water stress. Therefore, in this paper, we conducted a theoretical study on how to improve and use the MVI technique to retrieve VOD, which is correlated to vegetation properties. In this study, we simulated T B based on the Tor Vergata model for corn and wheat at the L-band. The validity of the Tor Vergata model and MVI technique was evaluated based on ground-based observations. It is found that the Tor Vergata model had satisfactory accuracy and should be an effective tool for simulation microwave emission from those two kinds of crops. It is also found that the MVI technique is robust under the descriptions of the Tor Vergata model, that the overall brightness temperature is linearly correlated with soil emissivity with coefficients depending on vegetation properties. Results indicated that the MVI should work better at V-pol due to its sensitivity to vegetation.
We calculated MVI under two different assumptions of (1) no polarization dependence: MVI P B , and (2) no vegetation changes in a certain time window: MVI T B . The first method of MVI derivation is suitable without a preferred orientation for vegetation. The second method is a new development compared with previous studies. MVI from both of those two methods were found to be correlated with vegetation water content, and the MVI T B , which is derived based on multi-temporal information, could achieve larger variation with vegetation changes and has a great potential to be used to retrieve vegetation properties.
By comparing theoretical VOD S with VOD derived from MVI T B , it has been found that the retrieved VOD has a very high correlation with theoretical ones. Good results were obtained by using the MVI T B to retrieve VOD and VWC for wheat. However, this also indicated that VOD T is an underestimation of VOD S for corn. The percent of leaves and cylinders in vegetation canopy, structure and wavelength can be effective parameters in VOD response. The same result was obtained when comparing retrieved VWC from VOD T with the measurements. Regardless of the method that we used to derive VOD, vegetation structure and saturation effects may change the vegetation emission characteristics and affect the MVI.
Obtaining the theoretical relationship between MVI and VOD, make MVI a promising index for global vegetation monitoring and soil moisture retrievals. The suitability of the MVI T B is dependent on vegetation structure and soil moisture changes. The accuracy of MVI T B retrieval is compromised, over areas having less soil moisture content. Thus, use of this method in areas with limited soil moisture content is restricted. Hence, more investigation needs to be done to resolve those issues in real satellite data applications.
Author Contributions: J.S. and T.Z. conceived and designed the work; S.T. analyzed the data and wrote the manuscript, T.Z. modified the manuscript; P.F. provided the model and useful suggestion; J.-P.W. provided some experimental data; M.Z. reviewed and helped in programming and P.P. revised and edited the English language of the manuscript.