# Soil Moisture

## Lecture 4: Incidence Angle Normalisation

Created by: Vienna University of Technology

Funded by:

### Backscatter from natural land surfaces

• "Scattering": the random distortion of coherent electromagnetic (EM) waves by surface and structures similar to/larger in size than the radar wavelength

• any interface separating two media with different dielectric properties affects an incident EM wave

• perfectly flat interface ⇒ resulting field: two plane waves (refracted and reflected wave)

### Fresnel's law

• upper medium: reflected wave, same angle $\theta$ as incident angle

• lower medium: refracted wave, angle $\theta'$

• Fresnel's law:
$\theta'\;=\;\arcsin{\frac{\sqrt{\varepsilon_2}\sin{\theta}}{\sqrt{\varepsilon_1}}}$
$\varepsilon_{1,\;2}$...dielectric constant of medium 1, 2

### Surface Scattering I

• occurs if lower medium is uniform and homogenous

• scattering only occurs at the surface interface between the upper and the lower medium

• dependent on two surface roughness parameters:

• standard deviation of the surface height variations $s$
• surface correlation length $l$

• Fraunhofer criterion: a surface is considered smooth if $s\;\lt\;\frac{\lambda}{128\cos(\theta_0)}$

### Surface Scattering II

• Specular reflection: if the surface interface between two media is smooth compared to the wavelength of the incident wave, the interface acts like a mirror (see Fig. a)

• Diffuse reflection: for rougher surface, less energy is reflected in specular direction, and more energy is scattered diffusely (Fig. b-d)

Interesting for us: radiation in sensor direction (= backscatter)

A larger incidence angle automatically means more specular reflection, and the other way round!

### Surface roughness

Whether a surface appears to be smooth or rough depends on the wavelength of the incident wave.

Rayleigh-criterion: a surface is classified as rough, if the root mean square height $h$ > $\frac{\lambda}{8}*\cos{\theta}$
$\lambda$...wavelength, $\theta$...incidence angle

### ACF and correlation length

• normalised autocorrelation function (ACF) $\rho(x)$

• measure of the similarity between the heights at two points

• surface correlation length $l$: the displacement $x'$ for which $\rho(x)$ is equal to $\frac{1}{e}$ (see next slide)

• $l$ is a measure for the statistical independence of two points: if they are separated by a horizontal distance greater than $l$, their heights may be considered to be approximately stastistically independent

• perfectly smooth (specular) surface: $l\;=\;\infty$

### Volume Scattering I

• In case of an inhomogeneous medium, scattering can occur within the medium itself.

• Dielectric discontinuities within the medium cause absorption and scattering in all directions ⇒ energy loss of the propagating wave

• Size, shape and distribution of the individual dielectric discontinuities is more critical than the roughness of the surface boundary

### Volume Scattering II

Such inhomogeneities can be leafs, twigs etc. of a vegetation canopy, snow flakes of a dry snow pack, or air bubbles in ice

### Relationship Incidence angle - backscatter coefficient I

The backscatter coefficient $\sigma^0$ is strongly related to the incidence angle $\theta$:
Increasing incidence angle ⇒ rapidly decreasing backscatter values

Therefore, to compare measurements taken at different incidence angles, an incidence angle normalisation needs to be applied first!

### Relationship Incidence angle - backscatter coefficient II

Main influence factors:

• surface roughness
• amount of biomass

The figure shows the backscatter $\sigma^0$ as a function of the incidence angle for three different biomes.

The slope of the curves is indicative for the scattering mechanism taking place at the observed target.

### Backscatter incidence angle behaviour

The backscatter incidence angle behaviour can be sufficiently modelled using a second order polynomial:

$$\begin{split} \sigma^0(\theta,\;t)\;=\;&\sigma^0(40°,\;t)\;+\;\sigma'(40°,\;t)(\theta\;-\;40°)\;\\ &+\;\frac{1}{2}\sigma''(40°,\;t)(\theta\;-\;40°)^2 \end{split}$$

• slope $\sigma'(40°,\;t)$ - first derivative of $\sigma^0(\theta)$
• curvature $\sigma''(40°,\;t)$ - second derivative of $\sigma^0(\theta)$
• both parameters are a function of time $t$ and incidence angle $40°$

### Soil moisture change

Assumption: slope $\sigma'(40°,\;t)$ and curvature $\sigma''(40°,\;t)$ are unaffected by soil moisture variations

⇒ changes in soil moisture are only reflected in the magnitude of the backscatter coefficient $\sigma^0(40°,\;t)$!

### Changes of vegetation or surface roughness

Changes in the slope or curvature show variations in the vegetation phenology or changes in the surface roughness.

### slope and curvature estimation I

• ESCAT: 3 beams (fore-, mid-, aft-beam)

• estimation of the local slope $\sigma'_L$

• each backscatter triplet yields two local slope estimates:

$$\sigma'_L(\frac{\theta_m\;+\;\theta_{f,a}}{2})\;=\;\frac{\sigma^0_m(\theta_m)\;-\;\sigma^0_{f,a}(\theta_{f,a})}{\theta_m\;-\;\theta_{f,a}}$$

### slope and curvature estimation II

• regression line fitted to the computed local slopes $\sigma'_L$ aquired during a particular period:

$$\begin{split} \sigma'(\theta,t)\;=&\;\sigma'(40°,t)\;\\ &+\;\sigma''(40°,t)(\theta\;-\;40°) \end{split}$$
• slope and curvature can be obtained from the regression line

### slope and curvature estimation III

$\sigma^0$ depends on the incidence angle $\theta$. This dependency is not constant over the year but changes with soil moisture and vegetation phenology.

⇒ 12 values for slope and curvature are computed for each month of the year

Values for each day: interpolation of the monthly values

### Global monthly slope

Compare the slope variability in rainforest areas to that in temperate zones (e.g. Europe or North America)!

### Incidence angle normalisation

The incidence angle normalisation can be performed using a full complement of 366 slope $\sigma'(40°,t)$ and curvature $\sigma''(40°,t)$ values.
Finally, the following equation is used to derive backscatter coefficients normalised to 40° incidence angle $\sigma^0(40°,t)$:

$$\begin{split} \sigma_i^0(40°,t) =\; & \sigma_i^0(\theta,t)\;-\;\sigma'(40°,t)(\theta\;-\;40°) \\ & -\;\frac{1}{2}\sigma''(40°,t)(\theta\;-\;40°)^2 \end{split}$$