3. Polarimetric decomposition for soil moisture estimation

Depending whether the sensor generates the microwave by itself, the microwave remote sensing can be categorized into the active and passive, which are reviewed separately. Polarimetric SAR is a coherent active microwave remote sensing system, providing backscattering signals in quad-polarization states with fine spatial resolution. Unlike the optical remote sensing, the SAR system monitors the earth using a side-look geometry, resulting in the issues of overlap, shadow, and forth short. Furthermore, at the microwave bands, the signals are sensitive to the permittivity and the structure of the targets. Thus, the interpretation and modeling of the SAR data differ from those of optical domain. The SAR system generates the microwave, so that it operates regardless the light and day/night and clear/cloudy conditions. This is particularly interesting for monitoring the soil moisture over the area frequently covered by the cloud.

### 3.1. Decomposition theories

#### 3.1.1. Polarimetric SAR data expression

The microwave scattering process over the ground can be formulated ES ¼ ½ � S EI, where the Sinclair matrix [S] relates the incident wave EI to the scattering wave ES. Thus, the polarimetric SAR data extracted as [S] includes the target dielectric and structural properties:

$$\mathbf{[S]} = \begin{bmatrix} \mathbf{S}\_{HH} & \mathbf{S}\_{HV} \\ \mathbf{S}\_{VH} & \mathbf{S}\_{VV} \end{bmatrix} \tag{15}$$

½ �¼ C3

3.1.2. Eigen-based decomposition

<sup>H</sup> <sup>¼</sup> <sup>X</sup> 3

3.1.3. Model-based decomposition

H > 0.7:

i¼1

�pi log <sup>3</sup> pi

metric parameters to the soil characteristics at C-band.

C<sup>11</sup> C<sup>12</sup> C<sup>13</sup>

<sup>12</sup> C<sup>22</sup> C<sup>23</sup>

<sup>23</sup> C<sup>33</sup>

j j SHH <sup>2</sup> D E ffiffiffi

ffiffiffi 2 <sup>p</sup> <sup>S</sup><sup>∗</sup>

S∗

2 <sup>p</sup> SHHS<sup>∗</sup>

2 <sup>p</sup> <sup>S</sup><sup>∗</sup>

HHSHV � � 2 Sj j HV

HHSVV � � ffiffiffi

The polarimetric decompositions are often done on the coherency matrix [T3] and the covariance matrix [C3], which can be converted between each other via unitary transformation. However, the elements of the [T3] matrix are physically convenient. For instance, the T11, T22, and T33 can be used to approximate the surface, dihedral, and volume scattering powers.

Both [T3] and [C3] matrices are characterized by nonnegative eigenvalues and orthogonal eigenvector. The classical decomposition approach proposed by Cloude and Pottier relies on the eigenanalysis on the [T3] matrix. The scattering mechanism and the corresponding relative

From the eigenvalues and eigenvectors, the entropy H and α angle are defined to characterize

In addition, the scattering anisotropy A is introduced to discriminate the ambiguous case of

These polarimetric parameters are used to describe the scattering mechanisms under a variety of scenarios. However, in Baghdadi et al. [18], the sensitivity of entropy and α angle to soil moisture and surface roughness is analyzed, indicating insignificant response of these polari-

Under the assumption of reflection symmetry (zero correlation between the co- and crosspolarization channels), the Freeman-Durden decomposition models the covariance matrix [C3] as the incoherent summation of the surface, dihedral, and volume scattering components.

power were quantified by the eigenvector (Ti) and eigenvalues (λi), respectively:

the randomness of the scattering scene and the dominant scattering mechanism:

i¼1

, <sup>α</sup> <sup>¼</sup> <sup>X</sup> 3

HV � � SHHS<sup>∗</sup>

Soil Moisture Retrieval from Microwave Remote Sensing Observations

HVSVV � � j j <sup>S</sup>VV

<sup>2</sup> D E ffiffiffi

½ �¼ T3 λ1T<sup>1</sup> þ λ2T<sup>2</sup> þ λ3T<sup>3</sup> (19)

A ¼ ð Þ λ<sup>2</sup> � λ<sup>3</sup> =ð Þ λ<sup>2</sup> þ λ<sup>3</sup> (21)

X 3

λ<sup>i</sup> (20)

i¼1

pi acosð Þ j j ei<sup>1</sup> and pi ¼ λi=

VV � �

http://dx.doi.org/10.5772/intechopen.81476

39

<sup>2</sup> D E

2 <sup>p</sup> SHVS<sup>∗</sup> VV � �

C∗

C∗ <sup>13</sup> C<sup>∗</sup>

where SHH and SVV are the co-polarized scatterings, and SHV and SVH represent the crosspolarized scattering power. They are all complex numbers. For the monostatic case of backscattering, the satisfied reciprocity results in SHV = SVH. This format of [S] matrix is considered as single-look data suffering from the speckle effect, as no averaging process is performed.

However, the natural targets dynamically vary with time, requiring a statistical description such as the second-order moment approach. In order to extract more polarimetric information such as the correlation between different polarimetric channels, the Pauli and Lexicographic vectors are constructed from the [S] matrix, respectively:

$$k = \begin{bmatrix} \mathbf{S}\_{HH} + \mathbf{S}\_{VV}, & \mathbf{S}\_{HH} - \mathbf{S}\_{VV}, & \mathbf{2}\mathbf{S}\_{HV} \end{bmatrix}^{T} \tag{16}$$

$$\mathbf{Q} = \begin{bmatrix} \mathbf{S}\_{HH}, & \sqrt{2}\mathbf{S}\_{HV}, & \mathbf{S}\_{VV} \end{bmatrix}^T \tag{17}$$

From the Pauli and Lexicographic vectors, the coherency matrix [T] and the covariance matrix [C] are obtained by T½ �¼ k � k <sup>∗</sup><sup>T</sup> � � and ½ �¼ <sup>C</sup> <sup>Ω</sup> � <sup>Ω</sup><sup>∗</sup><sup>T</sup> � �, where the symbol hi means the temporal or spatial averaging to reduce the randomness of the polarimetric images. In the monostatic condition (transmitter and receiver in the same location), they are expressed as

$$\begin{aligned} \text{(T3)} &= \begin{bmatrix} T\_{11} & T\_{12} & T\_{13} \\ T\_{12} & T\_{22} & T\_{23} \\ T\_{13} & T\_{23} & T\_{33} \end{bmatrix} \\ &= \mathbf{0.5} \begin{bmatrix} \left< \mathbf{S\_{HH}} + \mathbf{S\_{VV}} \right>^{2} & \left< (\mathbf{S\_{HH}} + \mathbf{S\_{VV}})(\mathbf{S\_{HH}} - \mathbf{S\_{VV}})^{\*} \right> & 2\left< (\mathbf{S\_{HH}} + \mathbf{S\_{VV}})\mathbf{S\_{HV}^{\*}} \right> \\ \left< (\mathbf{S\_{HH}} + \mathbf{S\_{VV}})^{\*}(\mathbf{S\_{HH}} - \mathbf{S\_{VV}}) \right> & \left< \mathbf{S\_{HH}} - \mathbf{S\_{VV}} \right>^{2} & 2\left< (\mathbf{S\_{HH}} - \mathbf{S\_{VV}})\mathbf{S\_{HV}^{\*}} \right> \\ & 2\left< (\mathbf{S\_{HH}} - \mathbf{S\_{VV}})^{\*}\mathbf{S\_{HV}} \right> & 2\left< (\mathbf{S\_{HH}} - \mathbf{S\_{VV}})^{\*}\mathbf{S\_{HV}} \right> \end{bmatrix} \end{aligned} \tag{18}$$

Soil Moisture Retrieval from Microwave Remote Sensing Observations http://dx.doi.org/10.5772/intechopen.81476 39

$$\begin{bmatrix} \mathbf{C} \mathbf{3} \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} & \mathbf{C}\_{13} \\\\ \mathbf{C}\_{12}^\* & \mathbf{C}\_{22} & \mathbf{C}\_{23} \\\\ \mathbf{C}\_{13}^\* & \mathbf{C}\_{23}^\* & \mathbf{C}\_{33} \end{bmatrix} = \begin{bmatrix} \begin{Bmatrix} |\mathbf{S}\_{HH}|^2 \end{Bmatrix} & \sqrt{2} \begin{Bmatrix} \mathbf{S}\_{HH}\mathbf{S}\_{HV}^\* \end{Bmatrix} & \begin{Bmatrix} \mathbf{S}\_{HH}\mathbf{S}\_{VV}^\* \end{Bmatrix} \\\\ \sqrt{2} \begin{Bmatrix} \mathbf{S}\_{HH}^\*\mathbf{S}\_{HV} \end{Bmatrix} & 2 \begin{Bmatrix} |\mathbf{S}\_{HV}|^2 \end{Bmatrix} & \sqrt{2} \begin{Bmatrix} \mathbf{S}\_{HV}\mathbf{S}\_{VV}^\* \end{Bmatrix} \\\\ \begin{Bmatrix} \mathbf{S}\_{HH}^\*\mathbf{S}\_{VV} \end{Bmatrix} & \sqrt{2} \begin{Bmatrix} \mathbf{S}\_{HV}^\*\mathbf{S}\_{VV} \end{Bmatrix} & \begin{Bmatrix} |\mathbf{S}\_{HV}|^2 \end{Bmatrix} \end{bmatrix}$$

The polarimetric decompositions are often done on the coherency matrix [T3] and the covariance matrix [C3], which can be converted between each other via unitary transformation. However, the elements of the [T3] matrix are physically convenient. For instance, the T11, T22, and T33 can be used to approximate the surface, dihedral, and volume scattering powers.

#### 3.1.2. Eigen-based decomposition

This is particularly interesting for monitoring the soil moisture over the area frequently cov-

The microwave scattering process over the ground can be formulated ES ¼ ½ � S EI, where the Sinclair matrix [S] relates the incident wave EI to the scattering wave ES. Thus, the polarimetric

where SHH and SVV are the co-polarized scatterings, and SHV and SVH represent the crosspolarized scattering power. They are all complex numbers. For the monostatic case of backscattering, the satisfied reciprocity results in SHV = SVH. This format of [S] matrix is considered as single-look data suffering from the speckle effect, as no averaging process is performed.

However, the natural targets dynamically vary with time, requiring a statistical description such as the second-order moment approach. In order to extract more polarimetric information such as the correlation between different polarimetric channels, the Pauli and Lexicographic

k ¼ ½ � SHH þ SVV; SHH � SVV; 2SHV

2 <sup>p</sup> SHV; SVV h i<sup>T</sup>

From the Pauli and Lexicographic vectors, the coherency matrix [T] and the covariance matrix

ral or spatial averaging to reduce the randomness of the polarimetric images. In the monostatic

<sup>∗</sup><sup>T</sup> � � and ½ �¼ <sup>C</sup> <sup>Ω</sup> � <sup>Ω</sup><sup>∗</sup><sup>T</sup> � �, where the symbol hi means the tempo-

ð Þ <sup>S</sup>HH <sup>þ</sup> SVV ð Þ <sup>S</sup>HH � SVV <sup>∗</sup> h i 2 Sð Þ HH <sup>þ</sup> SVV <sup>S</sup><sup>∗</sup>

<sup>2</sup> D E

2 Sð Þ HH <sup>þ</sup> SVV <sup>∗</sup> h i <sup>S</sup>HV 2 Sð Þ HH � SVV <sup>∗</sup> h i <sup>S</sup>HV 4 Sj j HV

<sup>Ω</sup> <sup>¼</sup> SHH; ffiffiffi

condition (transmitter and receiver in the same location), they are expressed as

ð Þ <sup>S</sup>HH <sup>þ</sup> SVV <sup>∗</sup> h i ð Þ <sup>S</sup>HH � SVV j j <sup>S</sup>HH � SVV

SHH SHV SVH SVV � �

(15)

(17)

<sup>T</sup> (16)

HV

(18)

HV

� �

� �

<sup>2</sup> D E

2 Sð Þ HH � SVV <sup>S</sup><sup>∗</sup>

SAR data extracted as [S] includes the target dielectric and structural properties:

½ �¼ S

vectors are constructed from the [S] matrix, respectively:

ered by the cloud.

38 Soil Moisture

3.1. Decomposition theories

[C] are obtained by T½ �¼ k � k

T<sup>11</sup> T<sup>12</sup> T<sup>13</sup>

<sup>12</sup> T<sup>22</sup> T<sup>23</sup>

<sup>23</sup> T<sup>33</sup>

j j SHH þ SVV <sup>2</sup> D E

T∗

¼ 0:5

T∗ <sup>13</sup> T<sup>∗</sup>

½ �¼ T3

3.1.1. Polarimetric SAR data expression

Both [T3] and [C3] matrices are characterized by nonnegative eigenvalues and orthogonal eigenvector. The classical decomposition approach proposed by Cloude and Pottier relies on the eigenanalysis on the [T3] matrix. The scattering mechanism and the corresponding relative power were quantified by the eigenvector (Ti) and eigenvalues (λi), respectively:

$$\mathbf{T} \begin{bmatrix} \mathbf{T} \mathbf{3} \end{bmatrix} = \lambda\_1 T\_1 + \lambda\_2 T\_2 + \lambda\_3 T\_3 \tag{19}$$

From the eigenvalues and eigenvectors, the entropy H and α angle are defined to characterize the randomness of the scattering scene and the dominant scattering mechanism:

$$H = \sum\_{i=1}^{3} -p\_i \log\_3 p\_{i^\*} \quad \alpha = \sum\_{i=1}^{3} p\_i \arccos\left(|e\_{i1}|\right) \quad \text{and} \quad p\_i = \lambda\_i / \sum\_{i=1}^{3} \lambda\_i \tag{20}$$

In addition, the scattering anisotropy A is introduced to discriminate the ambiguous case of H > 0.7:

$$A = (\lambda\_2 - \lambda\_3)/(\lambda\_2 + \lambda\_3) \tag{21}$$

These polarimetric parameters are used to describe the scattering mechanisms under a variety of scenarios. However, in Baghdadi et al. [18], the sensitivity of entropy and α angle to soil moisture and surface roughness is analyzed, indicating insignificant response of these polarimetric parameters to the soil characteristics at C-band.

#### 3.1.3. Model-based decomposition

Under the assumption of reflection symmetry (zero correlation between the co- and crosspolarization channels), the Freeman-Durden decomposition models the covariance matrix [C3] as the incoherent summation of the surface, dihedral, and volume scattering components. In order to be consistent with previous eigen-based approach, we express the Freeman-Durden decomposition based on [T3] matrix [19]:

$$\begin{aligned} \mathbf{[T3]} = \begin{bmatrix} T\_{11} & T\_{12} & 0 \\ T\_{12}^\* & T\_{22} & 0 \\ 0 & 0 & T\_{33} \end{bmatrix} = f\_s \begin{bmatrix} 1 & \beta^\* & 0 \\ \beta & |\beta|^2 & 0 \\ 0 & 0 & 0 \end{bmatrix} + f\_d \begin{bmatrix} |a|^2 & a & 0 \\ a^\* & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} + f\_v \begin{bmatrix} V\_{11} & 0 & 0 \\ 0 & V\_{22} & 0 \\ 0 & 0 & V\_{33} \end{bmatrix} \end{aligned} \tag{22}$$

The surface component is modeled using the simple Bragg model. The polarimetric parameter <sup>β</sup> <sup>¼</sup> RH�RV RHþRV and <sup>f</sup> <sup>s</sup> <sup>¼</sup> <sup>0</sup>:5j j RH <sup>þ</sup> RV <sup>2</sup> are constructed from the Bragg scattering coefficients:

$$R\_H = \frac{\cos\theta - \sqrt{\varepsilon - \sin^2\theta}}{\cos\theta + \sqrt{\varepsilon - \sin^2\theta}},\\ R\_V = \frac{(\varepsilon - 1)\left(\sin^2\theta - \varepsilon\left(1 + \sin^2\theta\right)\right)}{\left(\varepsilon\cos\theta + \sqrt{\varepsilon - \sin^2\theta}\right)^2} \tag{23}$$

decompositions were mainly applied to the image classification, target detection by analyzing the scattering mechanisms. Hajnsek et al. [19] proposed to estimate the soil moisture from the L-band polarimetric decomposition. In their approach, after removing the volume component from the full signature, the soil moisture is retrieved from the surface and dihedral scattering

Soil Moisture Retrieval from Microwave Remote Sensing Observations

http://dx.doi.org/10.5772/intechopen.81476

41

For the surface scattering component, the polarimetric parameter β is related to the soil moisture and incidence angle (Figure 3). Unlike the traditional radar backscattering coefficients which are more sensitive to soil moisture at low incidence angle condition, the polarimetric parameter β is more sensitive to the soil moisture at high incidence angle. Thus, depending on the incidence angle ranges of the radar data, the traditional direct backscattering approach or the advanced polarimetric approach is preferable. In Hajnsek et al. [19], the surface scattering component is adapted by replacing the Bragg model with the X-Bragg

In contrary to the surface scattering component, the dihedral scattering component is influenced by both the soil and vegetation dielectric constants. Thus, two equations were required to decouple the soil and vegetation contributions on the dihedral component, in order to extract the soil moisture from it. In the literature [19, 20], the parameter α and fd are used to construct an equation system, from which the soil and vegetation dielectric constants are solved. Nevertheless, the vegetation dielectric constant is not furthermore considered, as the main purpose of this

model in order to take the surface roughness effect into account.

Figure 3. Sensitivity of surface scattering parameter β to soil moisture.

chapter is to estimate the soil moisture from microwave remote sensing data.

component, respectively.

The dihedral component is developed from the Fresnel coefficients of the orthogonal dielectric planes between the plant stalks and the underlying soils. The scattering amplitude <sup>f</sup> <sup>d</sup> <sup>¼</sup> <sup>0</sup>:<sup>5</sup> RSHRTH <sup>þ</sup> RSVRTVe<sup>i</sup><sup>ψ</sup> � � � � <sup>2</sup> and polarization ratio <sup>a</sup> <sup>¼</sup> RSHRTH�RSVRTV ei<sup>ψ</sup> RSHRTHþRSVRTV ei<sup>ψ</sup> are related to the Fresnel coefficients of soil and plant:

$$R\_{\rm jH} = \frac{\cos \theta\_{\rm j} - \sqrt{\varepsilon\_i - \sin^2 \theta\_{\rm j}}}{\cos \theta\_{\rm j} + \sqrt{\varepsilon\_i - \sin^2 \theta\_{\rm j}}} \text{ and } R\_{\rm jV} = \frac{\varepsilon\_i \cos \theta\_{\rm j} - \sqrt{\varepsilon\_i - \sin^2 \theta\_{\rm j}}}{\varepsilon\_i \cos \theta\_{\rm j} + \sqrt{\varepsilon\_i - \sin^2 \theta\_{\rm j}}} \tag{24}$$

where j represents soil (S) or plant (T). In the dihedral geometric configuration, the incidence angle over soil <sup>θ</sup><sup>S</sup> and over the plant <sup>θ</sup><sup>T</sup> is supplementary (θ<sup>S</sup> <sup>þ</sup> <sup>θ</sup><sup>T</sup> <sup>¼</sup> <sup>π</sup> 2).

The vegetation volume is simulated by the dipole with a uniform statistical distribution. Consequently, the volume component is derived as

$$[V] = fv \begin{bmatrix} 0.5 & 0 & 0 \\ 0 & 0.25 & 0 \\ 0 & 0 & 0.25 \end{bmatrix} \tag{25}$$

The Freeman-Durden model is firstly fitted to the forest scenario, and it is reported to be effective to discriminate the forest and deforest areas.

#### 3.2. Soil moisture retrieval using polarimetric decomposition techniques

#### 3.2.1. Model-based decomposition

The polarimetric soil moisture retrieval can be conducted based on the model-based decomposition, in which the soil dielectric constant is related to the surface scattering component through the Bragg scattering model and to the dihedral component through the combined Fresnel scattering model. Nevertheless, in the past decades, the model-based polarimetric decompositions were mainly applied to the image classification, target detection by analyzing the scattering mechanisms. Hajnsek et al. [19] proposed to estimate the soil moisture from the L-band polarimetric decomposition. In their approach, after removing the volume component from the full signature, the soil moisture is retrieved from the surface and dihedral scattering component, respectively.

In order to be consistent with previous eigen-based approach, we express the Freeman-Durden

The surface component is modeled using the simple Bragg model. The polarimetric parameter

The dihedral component is developed from the Fresnel coefficients of the orthogonal dielectric planes between the plant stalks and the underlying soils. The scattering amplitude

where j represents soil (S) or plant (T). In the dihedral geometric configuration, the incidence

The vegetation volume is simulated by the dipole with a uniform statistical distribution.

The Freeman-Durden model is firstly fitted to the forest scenario, and it is reported to be

The polarimetric soil moisture retrieval can be conducted based on the model-based decomposition, in which the soil dielectric constant is related to the surface scattering component through the Bragg scattering model and to the dihedral component through the combined Fresnel scattering model. Nevertheless, in the past decades, the model-based polarimetric

0:50 0 0 0:25 0 0 00:25 3 7

2 6 4 j j <sup>α</sup> <sup>2</sup> <sup>α</sup> <sup>0</sup> α<sup>∗</sup> 1 0 0 00

<sup>2</sup> are constructed from the Bragg scattering coefficients:

<sup>ε</sup> cos <sup>θ</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ε � sin <sup>2</sup>θ � � <sup>p</sup> <sup>2</sup> (23)

p <sup>ε</sup><sup>i</sup> cos <sup>θ</sup><sup>j</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>ε</sup> � sin <sup>2</sup><sup>θ</sup> <sup>p</sup> , RV <sup>¼</sup> ð Þ <sup>ε</sup> � <sup>1</sup> sin<sup>2</sup><sup>θ</sup> � <sup>ε</sup> <sup>1</sup> <sup>þ</sup> sin<sup>2</sup><sup>θ</sup> � � � �

<sup>2</sup> and polarization ratio <sup>a</sup> <sup>¼</sup> RSHRTH�RSVRTV ei<sup>ψ</sup>

<sup>p</sup> andRjV <sup>¼</sup> <sup>ε</sup><sup>i</sup> cos <sup>θ</sup><sup>j</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3 7 <sup>5</sup> <sup>þ</sup> <sup>f</sup> <sup>v</sup> 2 6 4

V<sup>11</sup> 0 0 0 V<sup>22</sup> 0 0 0 V<sup>33</sup>

RSHRTHþRSVRTV ei<sup>ψ</sup> are related to the

<sup>5</sup> (25)

<sup>p</sup> (24)

ε<sup>i</sup> � sin <sup>2</sup>θ<sup>j</sup>

ε<sup>i</sup> � sin <sup>2</sup>θ<sup>j</sup>

2).

3 7 <sup>5</sup> <sup>þ</sup> <sup>f</sup> <sup>d</sup>

1 β<sup>∗</sup> 0 β β� � � � <sup>2</sup> 0

000

<sup>ε</sup> � sin <sup>2</sup><sup>θ</sup> <sup>p</sup>

ε<sup>i</sup> � sin <sup>2</sup>θ<sup>j</sup>

ε<sup>i</sup> � sin <sup>2</sup>θ<sup>j</sup>

angle over soil <sup>θ</sup><sup>S</sup> and over the plant <sup>θ</sup><sup>T</sup> is supplementary (θ<sup>S</sup> <sup>þ</sup> <sup>θ</sup><sup>T</sup> <sup>¼</sup> <sup>π</sup>

½ �¼ V fv

3.2. Soil moisture retrieval using polarimetric decomposition techniques

2 6 4

decomposition based on [T3] matrix [19]:

3 7 <sup>5</sup> <sup>¼</sup> <sup>f</sup> <sup>s</sup>

RH <sup>¼</sup> cos <sup>θ</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�

RjH <sup>¼</sup> cos <sup>θ</sup><sup>j</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Consequently, the volume component is derived as

effective to discriminate the forest and deforest areas.

p cos <sup>θ</sup><sup>j</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cos <sup>θ</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 6 4

T<sup>11</sup> T<sup>12</sup> 0

<sup>12</sup> T<sup>22</sup> 0 0 0 T<sup>33</sup>

RHþRV and <sup>f</sup> <sup>s</sup> <sup>¼</sup> <sup>0</sup>:5j j RH <sup>þ</sup> RV

<sup>f</sup> <sup>d</sup> <sup>¼</sup> <sup>0</sup>:<sup>5</sup> RSHRTH <sup>þ</sup> RSVRTVe<sup>i</sup><sup>ψ</sup> � � �

3.2.1. Model-based decomposition

Fresnel coefficients of soil and plant:

T∗

2 6 4

½ �¼ T3

40 Soil Moisture

<sup>β</sup> <sup>¼</sup> RH�RV

For the surface scattering component, the polarimetric parameter β is related to the soil moisture and incidence angle (Figure 3). Unlike the traditional radar backscattering coefficients which are more sensitive to soil moisture at low incidence angle condition, the polarimetric parameter β is more sensitive to the soil moisture at high incidence angle. Thus, depending on the incidence angle ranges of the radar data, the traditional direct backscattering approach or the advanced polarimetric approach is preferable. In Hajnsek et al. [19], the surface scattering component is adapted by replacing the Bragg model with the X-Bragg model in order to take the surface roughness effect into account.

In contrary to the surface scattering component, the dihedral scattering component is influenced by both the soil and vegetation dielectric constants. Thus, two equations were required to decouple the soil and vegetation contributions on the dihedral component, in order to extract the soil moisture from it. In the literature [19, 20], the parameter α and fd are used to construct an equation system, from which the soil and vegetation dielectric constants are solved. Nevertheless, the vegetation dielectric constant is not furthermore considered, as the main purpose of this chapter is to estimate the soil moisture from microwave remote sensing data.

Figure 3. Sensitivity of surface scattering parameter β to soil moisture.

Figures 4, 5 plot the α and fd in terms of soil and vegetation dielectric constants:

• Parameter α is more sensitive to soil moisture when the incidence angle is less than 45; otherwise, it is more sensitive to vegetation dielectric constants.

It is in the consensus that the most challenging issue is the modeling of the volume scattering component. With the crop growth, the shape and crop structures vary dynamically, which makes the unique volume coherency matrix fail to capture the high complexity of the crop growth. In order to analyze this issue, Hajnsek et al. [19] compared several volume scattering formulations. One is the flexible volume model in Yamaguchi et al. [22], where the crops are described in vertical, random, and horizontal orientations. The volume coherency matrix was derived considering the dipoles with different orientation angle distribution widths. The parameter Pr = 10 log10(VV)/10 log10(HH) is used to determine the dominant orientation:

> 0:50 0 0 0:25 0 0 00:25

Another volume coherency matrix is proposed by narrowing the dipole orientation angle around radar line of sight. However, for all the volume models in [19], the corresponding soil moisture retrieval results indicate an underestimation for the wheat and corn fields, while an over-/underestimation for the rape fields. So far, there is no universal volume coherency matrix which performs well for all the crop types and the whole phenological development stages.

Furthermore, Jagdhuber et al. [20] developed an L-band polarimetric decomposition for the multiangular soil moisture retrieval over the agricultural fields covered by low vegetation. In the study, the multiangular observation was conducted by three flight lines over the same area. The effects of microwave extinction and phase shift on the surface and dihedral scattering component were accounted. For each pixel, multiple β, α, and fd were obtained for a joint retrieval process. The soil moisture retrieval obtained an RMSE ranging from 0.06 to 0.08 m3

Recently, the hybrid decomposition which combines the model-based and eigen-based decompositions is used for the soil moisture retrieval [1]. After extracting the volume scattering component using the model-based approach, the remaining ground scattering component is decomposed again using the eigen-based approach in order to better discriminate the surface and dihedral scattering mechanisms, taking advantages of the orthogonality of the eigenvector. This avoids the assumption of the dominant scattering mechanism in the ground compo-

In addition, the deorientation process is accounted before conducting the polarimetric decomposition, to reduce the fluctuation due to the random orientation angle of each pixel. This was done by minimizing the cross-polarization power [24]. After the deorientation process, the pixel with different orientation angles will result in the same decomposition results. Wang et al. [25] studied effectivity of the deorientation on the polarimetric soil moisture, indicating that the surface scattering component is significantly enhanced, as a result of the deorientation process. The increase in the surface scattering power is assumed to benefit the soil moisture retrieval. This is understandable, as the surface component is a function of the soil characteristics, while the dihedral component is complicated due to the coupling between the soil and

, TH <sup>¼</sup> fv 30

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(26)

43

/m3 .

TV <sup>¼</sup> fv 30

, TR ¼ fv

nent, in the original Freeman-Durden decomposition approach [23].

• For the fd parameter, the sensitivity to soil moisture is the same between the low and high incidence angles, while the absolute value of fd is different.

The dihedral scattering component is complementary to the surface component, increasing the overall retrieval rate. The surface scattering component which is the function of only soil dielectric constant is generally easier for the soil moisture retrieval than the dihedral component which is the function of both soil and vegetation dielectric constants. However, for some crop types such as canola and wheat, the significant dihedral scattering power at the early phenological stages contributes largely to the soil moisture [21]. There is a limitation in the dihedral component at incidence angle around 45, when the soil and vegetation dielectric constants are not possible to be decoupled from each other.

Figure 4. Sensitivity of dihedral parameter alpha to soil and vegetation dielectric constants under low and high incidence angles.

Figure 5. Sensitivity of alpha parameter fd to soil and vegetation dielectric constants under low and high incidence angles.

It is in the consensus that the most challenging issue is the modeling of the volume scattering component. With the crop growth, the shape and crop structures vary dynamically, which makes the unique volume coherency matrix fail to capture the high complexity of the crop growth. In order to analyze this issue, Hajnsek et al. [19] compared several volume scattering formulations. One is the flexible volume model in Yamaguchi et al. [22], where the crops are described in vertical, random, and horizontal orientations. The volume coherency matrix was derived considering the dipoles with different orientation angle distribution widths. The parameter Pr = 10 log10(VV)/10 log10(HH) is used to determine the dominant orientation:

Figures 4, 5 plot the α and fd in terms of soil and vegetation dielectric constants:

otherwise, it is more sensitive to vegetation dielectric constants.

incidence angles, while the absolute value of fd is different.

constants are not possible to be decoupled from each other.

angles.

42 Soil Moisture

• Parameter α is more sensitive to soil moisture when the incidence angle is less than 45;

• For the fd parameter, the sensitivity to soil moisture is the same between the low and high

The dihedral scattering component is complementary to the surface component, increasing the overall retrieval rate. The surface scattering component which is the function of only soil dielectric constant is generally easier for the soil moisture retrieval than the dihedral component which is the function of both soil and vegetation dielectric constants. However, for some crop types such as canola and wheat, the significant dihedral scattering power at the early phenological stages contributes largely to the soil moisture [21]. There is a limitation in the dihedral component at incidence angle around 45, when the soil and vegetation dielectric

Figure 5. Sensitivity of alpha parameter fd to soil and vegetation dielectric constants under low and high incidence angles.

Figure 4. Sensitivity of dihedral parameter alpha to soil and vegetation dielectric constants under low and high incidence

$$T\_V = \frac{fv}{30} \begin{bmatrix} 15 & 5 & 0 \\ 5 & 7 & 0 \\ 0 & 0 & 8 \end{bmatrix}, T\_R = fv \begin{bmatrix} 0.5 & 0 & 0 \\ 0 & 0.25 & 0 \\ 0 & 0 & 0.25 \end{bmatrix}, T\_H = \frac{fv}{30} \begin{bmatrix} 15 & -5 & 0 \\ -5 & 7 & 0 \\ 0 & 0 & 8 \end{bmatrix} \tag{26}$$

Another volume coherency matrix is proposed by narrowing the dipole orientation angle around radar line of sight. However, for all the volume models in [19], the corresponding soil moisture retrieval results indicate an underestimation for the wheat and corn fields, while an over-/underestimation for the rape fields. So far, there is no universal volume coherency matrix which performs well for all the crop types and the whole phenological development stages.

Furthermore, Jagdhuber et al. [20] developed an L-band polarimetric decomposition for the multiangular soil moisture retrieval over the agricultural fields covered by low vegetation. In the study, the multiangular observation was conducted by three flight lines over the same area. The effects of microwave extinction and phase shift on the surface and dihedral scattering component were accounted. For each pixel, multiple β, α, and fd were obtained for a joint retrieval process. The soil moisture retrieval obtained an RMSE ranging from 0.06 to 0.08 m3 /m3 .

Recently, the hybrid decomposition which combines the model-based and eigen-based decompositions is used for the soil moisture retrieval [1]. After extracting the volume scattering component using the model-based approach, the remaining ground scattering component is decomposed again using the eigen-based approach in order to better discriminate the surface and dihedral scattering mechanisms, taking advantages of the orthogonality of the eigenvector. This avoids the assumption of the dominant scattering mechanism in the ground component, in the original Freeman-Durden decomposition approach [23].

In addition, the deorientation process is accounted before conducting the polarimetric decomposition, to reduce the fluctuation due to the random orientation angle of each pixel. This was done by minimizing the cross-polarization power [24]. After the deorientation process, the pixel with different orientation angles will result in the same decomposition results. Wang et al. [25] studied effectivity of the deorientation on the polarimetric soil moisture, indicating that the surface scattering component is significantly enhanced, as a result of the deorientation process. The increase in the surface scattering power is assumed to benefit the soil moisture retrieval. This is understandable, as the surface component is a function of the soil characteristics, while the dihedral component is complicated due to the coupling between the soil and vegetation dielectric constants. Three different polarimetric decompositions (Freeman-Durden, Hajnsek, and An) were compared for the soil moisture retrieval. However, the performances depend on the crop types and phenological stages, and none of them can perform well for all the crop types and the whole growth stages. The Hajnsek decomposition is better for the early growth stage, while the An decomposition is overperformed for the crop's later development season. The Freeman decomposition obtained better results on the corn fields with sparse planting density. Furthermore, the incidence angle normalization is conducted on the polarimetric parameters (β, α, and fd) to reduce the incidence angle effect on the soil moisture retrieval.

where the sign nos denotes no order in size. The corresponding analytical eigenvectors can be

VV

VV

conditions, three polarimetries SERD, DERD, and SDERD are defined to characterize the difference among three scattering mechanisms (single bounce, double bounce, and multiple

SERD <sup>¼</sup> <sup>λ</sup><sup>s</sup> � <sup>λ</sup><sup>m</sup>

DERD <sup>¼</sup> <sup>λ</sup><sup>d</sup> � <sup>λ</sup><sup>m</sup>

SDERD <sup>¼</sup> <sup>λ</sup><sup>s</sup> � <sup>λ</sup><sup>d</sup>

where λ<sup>s</sup> ¼ λ1nos and λ<sup>s</sup> ¼ λ2nos if a<sup>1</sup> < a2. In contrary, λ<sup>s</sup> ¼ λ2nos and λ<sup>s</sup> ¼ λ1nos if a<sup>1</sup> > a2. The λ<sup>m</sup> ¼ λ3nos holds on in all cases. It is reported [10] that SERD is suitable to characterize vegetation, while DERD is appropriate to quantify the surface roughness. SDERD can be

In order to find a polarimetric parameter which is sensitivity to soil moisture, the α<sup>1</sup> from the

λ<sup>s</sup> þ λ<sup>m</sup>

λ<sup>d</sup> þ λ<sup>m</sup>

λ<sup>s</sup> þ λ<sup>d</sup>

2SHHS<sup>∗</sup>

2SHHS<sup>∗</sup>

2SHHS<sup>∗</sup>

2SHHS<sup>∗</sup>

VV þ j j SHH

Soil Moisture Retrieval from Microwave Remote Sensing Observations

VV � j j SHH

VV þ j j SHH

VV � j j SHH

. Based on the eigenvalues in reflection symmetry

<sup>2</sup> � j j <sup>S</sup>VV

http://dx.doi.org/10.5772/intechopen.81476

<sup>2</sup> � j j <sup>S</sup>VV

<sup>2</sup> � j j <sup>S</sup>VV

<sup>2</sup> � j j <sup>S</sup>VV

� � p

� � p

0

0

� � p

� � p

<sup>2</sup> <sup>þ</sup> ffiffiffi Δ

45

(28)

(29)

<sup>2</sup> <sup>þ</sup> ffiffiffi Δ

<sup>2</sup> � ffiffiffi Δ

<sup>2</sup> � ffiffiffi Δ

<sup>þ</sup> 4 SHHS<sup>∗</sup>

<sup>þ</sup> 4 SHHS<sup>∗</sup>

� � � � 2

� � � � 2

<sup>e</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>e</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>þ</sup> ffiffiffi Δ

<sup>2</sup> � ffiffiffi Δ

<sup>þ</sup> 4 SHHS<sup>∗</sup>

� � � � 2

applied to discriminate between bare and sight vegetation soils.

VV

s � �

s � �

<sup>2</sup> � j j <sup>S</sup>VV

<sup>2</sup> � j j <sup>S</sup>VV

<sup>2</sup> � j j <sup>S</sup>VV <sup>2</sup> � �<sup>2</sup>

first eigenvector is derived as

� � p <sup>2</sup>

� � p <sup>2</sup>

derived as

e<sup>3</sup> ¼

2 Sj j HH

2 Sj j HH

0

0

1

with Δ ¼ j j SHH

scattering):

Similar to the idea of X-Bragg model which rotates the Bragg surface around radar line of sight, the extended Fresnel model was developed for the dihedral scattering component [26]. It is achieved by rotating the soil plane of the dihedral component around the radar line of sight, to introduce the surface roughness effect on the dihedral component. Unlike the introduction of the surface roughness in the dihedral component in Hajnsek et al. [19], which did not change the matrix rank, the dihedral coherency matrix obtained in the extended Fresnel model increases the matrix rank from 1 to 3. Thus, both the amplitude and phase of the dihedral component have been changed.

#### 3.2.2. Eigen-based decomposition

The eigenvalues and eigenvectors of [T] matrix were computed to construct the polarimetric parameters for characterization of the scattering mechanisms. However, the currently eigenbased decomposition is mainly limited for soil moisture retrieval over the bare soils. The first one is the X-Bragg model [27], introducing the surface roughness effect into the Bragg model by rotating the soil plane around the radar light of sight. In order to estimate the soil moisture, the X-Bragg model relates the entropy H and α angle to the soil dielectric constant. A lookup table is established to determine the soil dielectric constant from the data-derived entropy and α angle. In addition, the surface roughness is derived from the polarimetric anisotropy parameter.

Furthermore, under the assumption of the reflection symmetry, the polarimetric parameters which are dominated by only the soil moisture or the surface roughness were constructed from the eigenvalue and eigenvector of the coherency matrix. According to Allain [28], the analytical eigenvalues is derived as

$$\begin{aligned} \lambda\_{1\text{ms}} &= 0.5 \left( < |\mathbf{S}\_{HH}|^2 > + < |\mathbf{S}\_{VV}|^2 > + \sqrt{\left( < |\mathbf{S}\_{HH}|^2 > - < |\mathbf{S}\_{VV}|^2 > \right)^2 + 4 < |\mathbf{S}\_{HH}\mathbf{S}\_{VV}^\*|^2} > \right) \\ \lambda\_{2\text{ms}} &= 0.5 \left( < |\mathbf{S}\_{HH}|^2 > + < |\mathbf{S}\_{VV}|^2 > - \sqrt{\left( < |\mathbf{S}\_{HH}|^2 > - < |\mathbf{S}\_{VV}|^2 > \right)^2 + 4 < |\mathbf{S}\_{HH}\mathbf{S}\_{VV}^\*|^2} > \right) \\ \lambda\_{3\text{ms}} &= 2 < |\mathbf{S}\_{HV}|^2 > \end{aligned} \tag{27}$$

where the sign nos denotes no order in size. The corresponding analytical eigenvectors can be derived as

vegetation dielectric constants. Three different polarimetric decompositions (Freeman-Durden, Hajnsek, and An) were compared for the soil moisture retrieval. However, the performances depend on the crop types and phenological stages, and none of them can perform well for all the crop types and the whole growth stages. The Hajnsek decomposition is better for the early growth stage, while the An decomposition is overperformed for the crop's later development season. The Freeman decomposition obtained better results on the corn fields with sparse planting density. Furthermore, the incidence angle normalization is conducted on the polarimetric parameters (β, α, and fd) to reduce the incidence angle effect on the soil moisture

Similar to the idea of X-Bragg model which rotates the Bragg surface around radar line of sight, the extended Fresnel model was developed for the dihedral scattering component [26]. It is achieved by rotating the soil plane of the dihedral component around the radar line of sight, to introduce the surface roughness effect on the dihedral component. Unlike the introduction of the surface roughness in the dihedral component in Hajnsek et al. [19], which did not change the matrix rank, the dihedral coherency matrix obtained in the extended Fresnel model increases the matrix rank from 1 to 3. Thus, both the amplitude and phase of the dihedral

The eigenvalues and eigenvectors of [T] matrix were computed to construct the polarimetric parameters for characterization of the scattering mechanisms. However, the currently eigenbased decomposition is mainly limited for soil moisture retrieval over the bare soils. The first one is the X-Bragg model [27], introducing the surface roughness effect into the Bragg model by rotating the soil plane around the radar light of sight. In order to estimate the soil moisture, the X-Bragg model relates the entropy H and α angle to the soil dielectric constant. A lookup table is established to determine the soil dielectric constant from the data-derived entropy and α angle. In addition, the surface roughness is derived from the polarimetric anisotropy parameter. Furthermore, under the assumption of the reflection symmetry, the polarimetric parameters which are dominated by only the soil moisture or the surface roughness were constructed from the eigenvalue and eigenvector of the coherency matrix. According to Allain [28], the analyti-

< j j SHH

< j j SHH

r !

r !

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> >

<sup>2</sup> >

<sup>þ</sup> <sup>4</sup> <sup>&</sup>lt; <sup>S</sup>HHS<sup>∗</sup>

<sup>þ</sup> <sup>4</sup> <sup>&</sup>lt; <sup>S</sup>HHS<sup>∗</sup>

� � � � 2 >

� � � � 2 >

VV

VV

(27)

<sup>2</sup> <sup>&</sup>gt; � <sup>&</sup>lt; j j <sup>S</sup>VV

<sup>2</sup> <sup>&</sup>gt; � <sup>&</sup>lt; j j <sup>S</sup>VV

� �<sup>2</sup>

� �<sup>2</sup>

retrieval.

44 Soil Moisture

component have been changed.

3.2.2. Eigen-based decomposition

cal eigenvalues is derived as

<sup>2</sup> <sup>&</sup>gt; <sup>þ</sup> <sup>&</sup>lt; j j <sup>S</sup>VV

<sup>2</sup> <sup>&</sup>gt; <sup>þ</sup> <sup>&</sup>lt; j j <sup>S</sup>VV

<sup>2</sup> >

<sup>2</sup> <sup>&</sup>gt; <sup>þ</sup>

<sup>2</sup> <sup>&</sup>gt; �

λ1nos ¼ 0:5 < j j SHH

λ2nos ¼ 0:5 < j j SHH

λ3nos ¼ 2 < j j SHV

$$\begin{aligned} \left\lVert e\_{1} = \frac{1}{\sqrt{2\left[\left(\left|\mathbf{S}\_{HH}\right|^{2} - \left|\mathbf{S}\_{VV}\right|^{2} + \sqrt{\Delta}\right)^{2} + 4\left|\mathbf{S}\_{HH}\mathbf{S}\_{VV}^{\*}\right|^{2}\right]}} \\ \frac{1}{\sqrt{2\left[\left(\left|\mathbf{S}\_{HH}\right|^{2} - \left|\mathbf{S}\_{VV}\right|^{2} + \sqrt{\Delta}\right)^{2} + 4\left|\mathbf{S}\_{HH}\mathbf{S}\_{VV}^{\*}\right|^{2}\right]}} \end{aligned} \tag{28}$$

$$\begin{aligned} e\_{2} = \frac{1}{\sqrt{2\left[\left(\left|\mathbf{S}\_{HH}\right|^{2} - \left|\mathbf{S}\_{VV}\right|^{2} - \sqrt{\Delta}\right)^{2} + 4\left|\mathbf{S}\_{HH}\mathbf{S}\_{VV}^{\*}\right|^{2}\right]}} \begin{bmatrix} 2S\_{HH}\mathbf{S}\_{VV}^{\*} + \left(\left|\mathbf{S}\_{HH}\right|^{2} - \left|\mathbf{S}\_{VV}\right|^{2} - \sqrt{\Delta}\right) \\ 2S\_{HH}\mathbf{S}\_{VV}^{\*} - \left(\left|\mathbf{S}\_{HH}\right|^{2} - \left|\mathbf{S}\_{VV}\right|^{2} - \sqrt{\Delta}\right) \\ 2S\_{HH}\mathbf{S}\_{VV}^{\*} - \left(\left|\mathbf{S}\_{HH}\right|^{2} - \left|\mathbf{S}\_{VV}\right|^{2} - \sqrt{\Delta}\right) \\ 0 \end{bmatrix} \end{aligned} \tag{28}$$

with Δ ¼ j j SHH <sup>2</sup> � j j <sup>S</sup>VV <sup>2</sup> � �<sup>2</sup> <sup>þ</sup> 4 SHHS<sup>∗</sup> VV � � � � 2 . Based on the eigenvalues in reflection symmetry conditions, three polarimetries SERD, DERD, and SDERD are defined to characterize the difference among three scattering mechanisms (single bounce, double bounce, and multiple scattering):

$$\begin{aligned} \text{SERD} &= \frac{\lambda\_s - \lambda\_m}{\lambda\_s + \lambda\_m} \\ \text{DERD} &= \frac{\lambda\_d - \lambda\_m}{\lambda\_d + \lambda\_m} \\ \text{SDERD} &= \frac{\lambda\_s - \lambda\_d}{\lambda\_s + \lambda\_d} \end{aligned} \tag{29}$$

where λ<sup>s</sup> ¼ λ1nos and λ<sup>s</sup> ¼ λ2nos if a<sup>1</sup> < a2. In contrary, λ<sup>s</sup> ¼ λ2nos and λ<sup>s</sup> ¼ λ1nos if a<sup>1</sup> > a2. The λ<sup>m</sup> ¼ λ3nos holds on in all cases. It is reported [10] that SERD is suitable to characterize vegetation, while DERD is appropriate to quantify the surface roughness. SDERD can be applied to discriminate between bare and sight vegetation soils.

In order to find a polarimetric parameter which is sensitivity to soil moisture, the α<sup>1</sup> from the first eigenvector is derived as

$$a\_{1} = \arctan\left(\frac{2\sigma\_{HHVV} - \left(\sigma\_{VVV} - \sigma\_{HHHH} + \sqrt{\left(\sigma\_{VVV} - \sigma\_{HHHH}\right)^2 + 4\left|\sigma\_{HHVV}\right|^2}\right)}{2\sigma\_{HHVV} + \left(\sigma\_{VVV} - \sigma\_{HHHH} + \sqrt{\left(\sigma\_{VVV} - \sigma\_{HHHH}\right)^2 + 4\left|\sigma\_{HHVV}\right|^2}\right)}\right) \tag{30}$$

<sup>V</sup><sup>12</sup> <sup>¼</sup> <sup>A</sup><sup>2</sup>

4.4 m<sup>3</sup>

/m<sup>3</sup>

parameters.

4.1. IEM model

<sup>p</sup> � 1 <sup>2</sup>

<sup>V</sup><sup>22</sup> <sup>¼</sup> <sup>0</sup>:5 A<sup>p</sup> � <sup>1</sup> <sup>2</sup>

<sup>V</sup><sup>33</sup> <sup>¼</sup> <sup>0</sup>:5 A<sup>p</sup> � <sup>1</sup> <sup>2</sup>

functions for the vertical and horizontal orientations. Finally, a RMSE of 6.12 m3

for the soil moisture retrieval using the C-band RADARSAT-2 dataset.

and the water cloud model (WCM) over the vegetated condition.

4. Radiative scattering model

sinc 2ð Þ Δφ

. In addition, for the covariance matrix, a generalized volume scattering model is

The parameters Ap and Δϕ were used to characterize the vegetation shape and its distribution width, respectively. With the dynamic volume model and hyper-decomposition approach, the soil moisture estimation obtained an inversion rate higher than 95% and RMSE from 4.0 to

proposed in [29] to quantify the vegetation scattering using the cosine-square distribution. Although the formulation varies from one to another study, the main idea relies on the characterization of the vegetation shape and orientation using the minimum number of

However, the model-based polarimetric decomposition for the soil moisture retrieval is mainly valid at L-band. When it comes to C-band, the surface roughness condition is beyond the valid range of Bragg (ks < 0.3) or X-Bragg model (ks < 1). In order to overcome this limitation, Huang et al. [2] first proposed a C-band polarimetric decomposition for the slight vegetation condition. In their approach, the surface scattering component is simulated using the IEM model, while the volume scattering component is formulated using the first-order sine and cosine

The soil moisture retrieval is performed using either physical or empirical models. We introduced below the application of integral equation model (IEM) and Oh model over the bare soil

The IEM model can be used to simulate the backscattering coefficients from incidence angle θ and soil parameters (surface roughness ks, correlation length kl, and soil moisture mv). Two surface roughness conditions (Gaussian or exponential) are considered to compute the corresponding backscattering coefficients. Regarding the applicability of IEM model, some studies show reasonable agreements between measurements and the model [30, 31]. However, the disagreements between measurements and model predictions are frequently observed [32–36], because the IEM model backscattering behavior depends on the autocorrelation function (ACF). Furthermore, the measurement of correlation length l is difficult to be accurate enough,

ð Þ 1 þ sinc 4ð Þ Δφ

Soil Moisture Retrieval from Microwave Remote Sensing Observations

http://dx.doi.org/10.5772/intechopen.81476

47

ð Þ 1 � sinc 4ð Þ Δφ

/m<sup>3</sup> is obtained

In Allain [28], the IEM model is used to simulate the backscattering coefficients. It is found that α<sup>1</sup> tends to be invariable with respect to the radar frequency higher than 8 GHz. At such high frequency, the α<sup>1</sup> is approximated using the IEM model as

$$\lim\_{\begin{subarray}{c}f\rightarrow\text{light}\\\text{frequency}\end{subarray}}\alpha\_{1}=\arctan\left(\frac{2f\_{hh}f\_{vv}^{\*}-\left(\left|f\_{vv}\right|^{2}-\left|f\_{hh}\right|^{2}+\sqrt{\left(\left|f\_{vv}\right|^{2}-\left|f\_{hh}\right|^{2}\right)^{2}+4\left|f\_{hh}f\_{vv}^{\*}\right|^{2}}\right)}{2f\_{hh}f\_{vv}^{\*}+\left(\left|f\_{vv}\right|^{2}-\left|f\_{hh}\right|^{2}+\sqrt{\left(\left|f\_{vv}\right|^{2}-\left|f\_{hh}\right|^{2}\right)^{2}+4\left|f\_{hh}f\_{vv}^{\*}\right|^{2}}\right)}\right)\tag{31}$$

where the fhh and fhh are the parameters in the IEM model. In this case, the α<sup>1</sup> is independent of surface roughness and mainly depends on the soil dielectric constant. The potential of α<sup>1</sup> for the soil moisture retrieval is investigated in Baghdadi et al. [18] using the C-band RADARSAT-2 data, indicating it is possible to discriminate two soil moisture levels or provide necessary a priori information to enhance the accuracy of soil moisture retrieval.

#### 3.2.3. Hybrid decomposition

The eigen-based decomposition is more empirically used for soil moisture retrieval, as it is inherently a mathematical approach. In contrast, the model-based decomposition based on the Bragg and Fresnel scattering models is more physically used. Recently, the combination between the model-based and eigen-based decompositions results in the hyper-decomposition [1]. Firstly, the volume scattering component is removed using the model-based decomposition. Then, the remaining ground scattering is decomposed using the eigen-based decomposition. This process overcomes the requirement of assumption on the dominant surface or dihedral scattering mechanism in the ground component (in that case, we need to assume the β or α to be constant in order to solve the undetermined equation system).

Furthermore, as the vegetation shape and structure vary with the phenological growth, the limited volume scattering model is not sufficient to capture this complex variability. Thus, the dynamic volume scattering is developed [1], which is suitable for the entire crop phenological cycle:

$$\begin{aligned} \text{[Tv]} &= \frac{f\_v}{2 + 2A\_p^2} \begin{bmatrix} V\_{11} & V\_{12} & 0 \\ V\_{12}^\* & V\_{22} & 0 \\ 0 & 0 & V\_{33} \end{bmatrix} \\ V\_{11} &= \begin{pmatrix} \mathbf{A}\_p + \mathbf{1} \end{pmatrix}^2 \end{aligned} \tag{32}$$

$$\begin{aligned} V\_{12} &= \left(\mathbf{A}\_p^2 - 1\right)^2 \text{sinc}(2\Delta\varphi) \\\\ V\_{22} &= 0.5 \left(\mathbf{A}\_p - 1\right)^2 (1 + \text{sinc}(4\Delta\varphi)) \\\\ V\_{33} &= 0.5 \left(\mathbf{A}\_p - 1\right)^2 (1 - \text{sinc}(4\Delta\varphi)) \end{aligned}$$

The parameters Ap and Δϕ were used to characterize the vegetation shape and its distribution width, respectively. With the dynamic volume model and hyper-decomposition approach, the soil moisture estimation obtained an inversion rate higher than 95% and RMSE from 4.0 to 4.4 m<sup>3</sup> /m<sup>3</sup> . In addition, for the covariance matrix, a generalized volume scattering model is proposed in [29] to quantify the vegetation scattering using the cosine-square distribution. Although the formulation varies from one to another study, the main idea relies on the characterization of the vegetation shape and orientation using the minimum number of parameters.

However, the model-based polarimetric decomposition for the soil moisture retrieval is mainly valid at L-band. When it comes to C-band, the surface roughness condition is beyond the valid range of Bragg (ks < 0.3) or X-Bragg model (ks < 1). In order to overcome this limitation, Huang et al. [2] first proposed a C-band polarimetric decomposition for the slight vegetation condition. In their approach, the surface scattering component is simulated using the IEM model, while the volume scattering component is formulated using the first-order sine and cosine functions for the vertical and horizontal orientations. Finally, a RMSE of 6.12 m3 /m<sup>3</sup> is obtained for the soil moisture retrieval using the C-band RADARSAT-2 dataset.
