2.2.1. Fractal description

The fractal geometry theory was introduced in [13] to describe the complicated surface roughness structure, especially for the irregular and fragmented soil structures. This surface roughness description approach is proved to be suitable for natural soil because of its self-similarity, no matter what the surface scale is. In addition, many basic natural physical processes generate fractal surface; thus fractal structure is quite common in natural environment.

The fractal models describe the local structure of the soil surface using one parameter, the fractal dimension D, ranged from 1 to 2. The higher the fractal dimension, the rougher the surface. One of the frequently used fractal approaches is the Brownian model [14, 15] for a limited fractal profile. In this model, the surface profile height h(x) at location x is considered to be a fractional Brownian function: For any x and Δx>0, the increase of surface height h(x + Δx) � h(x) follows a Gaussian distribution with mean value zero and variance AΔx2H. Consequently, the expected value of the surface elevation increase is derived as

$$E[\mathbf{h}(\mathbf{x} + \Delta \mathbf{x}) - \mathbf{h}(\mathbf{x})] = 2 \left[ \frac{\mathbf{u}}{\sqrt{2\pi A} \Delta \mathbf{x}^H} \exp\left(\frac{-\mathbf{u}^2}{2A \Delta \mathbf{x}^{2H}}\right) \mathbf{d}\mathbf{u} = \Delta \mathbf{x}^H E[\mathbf{h}(\mathbf{x} + \mathbf{1}) - \mathbf{h}(\mathbf{x})] \tag{5}$$

where A is the variance of the normal distribution h(x + 1) � h(x) and H is the Hurst exponent constant ranged from 0 to 1. The equation is rewritten in logarithm format in order to resolve H:

$$\log\left[\mathbf{h}(\mathbf{x} + \Delta \mathbf{x}) - \mathbf{h}(\mathbf{x})\right] = \mathbf{H} \log(\Delta \mathbf{x}) + \log\left[\mathbf{h}(\mathbf{x} + 1) - \mathbf{h}(\mathbf{x})\right] \tag{6}$$

In this equation, the parameter H equals to the slope of log h x ½ � ð Þ� þ Δx h xð Þ in terms of log ð Þ Δx . It is calculated by using minimum RMSE method [16]. Consequently, the fractal dimension D can be obtained directly from H by the relationship D = 2 � H.

#### 2.2.2. Statistical description

The second approach to describe the surface roughness is from the statistical point of view. There are two parameters to describe the statistical variations of the surface height relative to a reference surface: the standard deviation of the surface height s is to quantify the vertical roughness, while the correlation length l (with autocorrelation function) is to characterize the horizontal roughness [9, 17].

Suppose a surface in the x-y plane and the height of point (x, y) are assumed to be z(x, y) above the x-y plane. A representative surface with dimensions Lx and Ly is segmented statistically, which is centered at the original point.

The average height of the surface is given by

$$\overline{z} = \frac{1}{L\_x L\_y} \int\_{-L\_x/2}^{L\_x/2} \int\_{-L\_{y/2}}^{L\_{y/2}} z(\mathbf{x}, \mathbf{y}) \, \mathbf{dx} \, \mathbf{dy} \tag{7}$$

and the second moment is given by

2.2. Surface roughness

2.2.1. Fractal description

description.

34 Soil Moisture

derived as

to resolve H:

2.2.2. Statistical description

E½h xð Þ� þ Δx h xð Þ� ¼ 2

ð ∞

u ffiffiffiffiffiffiffiffiffi

dimension D can be obtained directly from H by the relationship D = 2 � H.

<sup>2</sup>π<sup>A</sup> <sup>p</sup> <sup>Δ</sup>xH exp �u<sup>2</sup>

where A is the variance of the normal distribution h(x + 1) � h(x) and H is the Hurst exponent constant ranged from 0 to 1. The equation is rewritten in logarithm format in order

In this equation, the parameter H equals to the slope of log h x ½ � ð Þ� þ Δx h xð Þ in terms of log ð Þ Δx . It is calculated by using minimum RMSE method [16]. Consequently, the fractal

The second approach to describe the surface roughness is from the statistical point of view. There are two parameters to describe the statistical variations of the surface height relative to a reference surface: the standard deviation of the surface height s is to quantify the vertical

2AΔx<sup>2</sup><sup>H</sup> � �

log h x ½ ð Þ� þ Δx h xð Þ� ¼ Hlogð Þþ Δx log h x ½ � ð Þ� þ 1 h xð Þ (6)

du <sup>¼</sup> <sup>Δ</sup>xHE½ � h xð Þ� <sup>þ</sup> <sup>1</sup> h xð Þ (5)

0

Besides the soil moisture, the surface roughness is another important factor that affects the backscattering SAR signature, because it determines how the incidence wave interacts with the surface. There exist several ways to describe the natural surface roughness, and two frequently used methods are mentioned here: the fractal geometry theory and the statistical

The fractal geometry theory was introduced in [13] to describe the complicated surface roughness structure, especially for the irregular and fragmented soil structures. This surface roughness description approach is proved to be suitable for natural soil because of its self-similarity, no matter what the surface scale is. In addition, many basic natural physical processes generate

The fractal models describe the local structure of the soil surface using one parameter, the fractal dimension D, ranged from 1 to 2. The higher the fractal dimension, the rougher the surface. One of the frequently used fractal approaches is the Brownian model [14, 15] for a limited fractal profile. In this model, the surface profile height h(x) at location x is considered to be a fractional Brownian function: For any x and Δx>0, the increase of surface height h(x + Δx) � h(x) follows a Gaussian distribution with mean value zero and variance AΔx2H. Consequently, the expected value of the surface elevation increase is

fractal surface; thus fractal structure is quite common in natural environment.

$$\overline{z}^2 = \frac{1}{L\_x L\_y} \int\_{-L\_x/2}^{L\_x/2} \int\_{-L\_y/2}^{L\_{y/2}} z^2(\mathbf{x}, \mathbf{y}) \, \mathbf{dx} \, \mathbf{dy} \tag{8}$$

Consequently, the standard deviation of the surface height within the area Lx X Ly is defined as

$$s = \sqrt{(z^2) - \overline{z}^2} \tag{9}$$

The formulation above can be reduced to a discrete condition. The surface profiles are digitized into discrete values zi(xi) at spacing rate Δx which is satisfied the criterion Δx < 0.1λ as described in [3]. The standard deviation s for discrete condition is formulated as

$$s = \sqrt{\frac{\sum\_{i=1}^{N} \left(\mathbf{z}\_i\right)^2 - N\left(\hat{\mathbf{z}}\right)^2}{N-1}}\tag{10}$$

where ^z ¼ P<sup>N</sup> <sup>i</sup>¼<sup>1</sup> zi <sup>N</sup> is the mean surface height and N is the number of samples.

For the horizontal surface roughness description, the surface autocorrelation function (ACF) has to be determined. The autocorrelation function r characterizes the independence of two points at a distance ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>ξ</sup><sup>2</sup> <sup>þ</sup> <sup>ζ</sup><sup>2</sup> <sup>p</sup> :

$$\rho(\xi,\zeta) = \frac{\int\_{-\mathrm{l}x/2}^{\mathrm{l}x/2} \int\_{-\mathrm{l}y/2}^{\mathrm{l}y/2} z\,(\mathbf{x},\mathbf{y})z\,(\mathbf{x}+\xi,\mathbf{y}+\zeta)\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{y}}{\int\_{-\mathrm{l}x/2}^{\mathrm{l}x/2} \int\_{-\mathrm{l}y/2}^{\mathrm{l}y/2} z^2\,(\mathbf{x},\mathbf{y})\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{y}}\tag{11}$$

In the discrete case, the autocorrelation function for a spatial displacement xi = (j � 1)Δx is defined as

$$\rho(\xi) = \frac{\sum\_{i=1}^{N+1-j} z\_{j} z\_{j+i-1}}{\sum\_{i=1}^{N} z\_{i}^{2}} \tag{12}$$

2.2.4. Bragg phenomenon

the signals will add in phase.

2.3. Vegetation

analysis.

Except the surface roughness and soil moisture, the row direction also influences the backscattering SAR wave from the bare agricultural soils, because it induces the Bragg phenomenon. Bragg resonance is a type of coherent scattering, which is present in some agricultural fields due to the plowing or other row structures' tillage. The resonance occurs in case that the distance between radar and each of the periodic structures has an additional phase difference of λ/2 in the slant-range direction. Under this condition, the additional phase shift is 2π, and

Vegetation has two effects on the radar signal: (1) attenuate the backscattering from the underlying soils and (2) produce the volume scattering adding to the radar signal. These two effects increase the complexity of soil moisture retrieval from microwave signal. The vegetation attenuation and scattering effects were parameterized by the vegetation scattering albedo

and optical depth, which are related to the vegetation water content or leaf area index.

can be obtained from the LAI through a linear relationship:

The b1 and b2 are assumed to be dependent on the vegetation type.

3. Polarimetric decomposition for soil moisture estimation

Α. Vegetation optical depth τ is linked to the vegetation water content through b parameter:

The b parameter depends on the crop type, structure, and growth stage and microwave polarization. The vegetation water content is often obtained from the NDVI. Alternatively, τ

B. Vegetation scattering albedo ω is set to be zero or a low value in the passive radiometer

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.

τ ¼ bVWC (13)

Soil Moisture Retrieval from Microwave Remote Sensing Observations

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

37

τ ¼ b1LAI þ b<sup>2</sup> (14)

where zj+i � <sup>1</sup> is a point with the spatial displacement xi from the point zi. The surface correlation length l is then defined as the displacement xi, under which the r(xi) between the two points equals 1/e. The correlation length characterizes the statistical independence of two points. In case that the distance between two points is larger than l, their heights can be considered statistically independent from each other. For very smooth surface, the correlation length is toward infinity.

#### 2.2.3. Wave interaction with the surface roughness

Furthermore, the effective surface roughness observed by SAR system depends on microwave wavelength. For instance, a given surface that appears smooth in L-band may seem rough in C-band. The relative surface roughness status (compared with wavelength) affects the surface scattering behaviors:


Thus, in the electromagnetic models, the effective vertical and horizontal surface roughness is given in terms of the production with EM wave number (k=2 π f/c): ks and kl. It is obvious that ks and kl are decreasing with increasing wavelength. Consequently, as shown in Figure 2, the surface roughness is one of the important factors that determine the electromagnetic wave response from bare soil.

Figure 2. Scattering patterns determined by surface roughness.
