# Soil Moisture

## Lecture 7: Soil Moisture Uncertainty Estimation

Created by: Vienna University of Technology

Funded by:

### Introduction

Parallel to the soil moisture calculation, an estimate of the uncertainty of the soil moisture retrieval is determined by rules of error propagation.

### Error Propagation I

• Determination of the noise level of the backscatter measurements

• Assumptions:

• all three beams observe the same region
• the fore- and aft-beam have the same incidence angle and are thus comparable (as long as there are no azimuthal effects)

### Error Propagation II

• The difference $\delta$ should be 0:
$\delta\;=\;\sigma^0_{fore}\;-\;\sigma^0_{aft}$

• Its variance should be twice the variance of one of the beams (assuming independent measurements)
$var[\delta]\;=\;2\;*\;var[\sigma^0]$

• Estimate of the standard deviation of the backscatter $\sigma^0$ (the so-called ESD):
$ESD\;=\;std[\sigma^0]\;=\;\frac{std[\delta]}{\sqrt{2}}$

### Error Propagation III

A noise estimate of the normalised backscatter $\sigma^0(40°,t)$ is obtained by:

$var[\sigma^0(40°,t)]\;=\;\frac{1}{9}\sum_{i=1}^3{\left(\begin{array}{l l}ESD^2\;+\;var[\sigma'(40°,t)]\;*\;(\theta_i\;-\;40°)^2\\ +\;\frac{1}{4}var[\sigma''(40°,t)]\;*\;(\theta_i\;-\;40°)^4\end{array}\right)}$

$i$...the individual fore-, mid- and aft-beam measurements of the backscatter triplet

### Uncertainty estimate of the soil moisture retrieval I

• no simple general expression for the noise estimate of the dry and wet reference

• noise estimate of wet reference is simpler because the wet crossover angle and the reference incidence angle are both equal to 40 degrees

### Uncertainty estimate of the soil moisture retrieval II

• Final equation for an uncertainty estimate of the soil moisture retrieval:

$$\begin{split} var[sm(t)]\;=\;&\frac{var[\sigma^0(40,t)]}{\sigma^0_{dry}(40,t)\;-\;\sigma^0_{wet}(40,t)}\\ &+\;var[\sigma^0_{dry}(40,t)]\left(\frac{\sigma^0(40,t)\;-\;\sigma^0_{wet}(40,t)}{\sigma^0_{dry}(40,t)\;-\;\sigma^0_{wet}(40,t)}\right)^2\\ &+\;var[\sigma^0_{wet}(40,t)]\left(\frac{\sigma^0(40,t)\;-\;\sigma^0_{dry}(40,t)}{\sigma^0_{dry}(40,t)\;-\;\sigma^0_{wet}(40,t)}\right)^2 \end{split}$$