In: Wagner W., Székely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
lea [cm] 
60 
50 
40 
30 
20 
10 
0 
t 
feb 27 mar 06 
t 
mar 23 mar 30 apr 02 
Date 
Probability 
Figure 4: Effective correlation lengths (Z e fr) obtained with the Figure 6: Example of a probability distribution for modeled cor- 
IEM for the different acquisition dates in 2003 relation length (¿ mo( i) 
left [cm] 
Uod=-5.12-ffS -8.16 
V • 
• 
X 
• • 
[dB] 
with n the number of observations, e* the difference between 
the fth observed and modeled values of ¿ e fr, cr° h the normalized 
backscatter coefficient for which the regression model is applied, 
cr^j the ¿th observed normalized backscatter coefficient and <7° 
the mean of all observed normalized backscatter coefficients. As 
an example, Figure 6 shows the t-distribution for an arbitrary 
value of from which it can be seen that this distribution is 
symmetric around the mean value. The obtained distribution for 
¿mod is propagated through the inversion of the IEM by means 
of a Monte Carlo method. To this end, 1000 values of Z mod are 
randomly sampled from the distribution and further used as in 
put to the IEM. This results in 1000 corresponding soil moisture 
contents, representing the histogram of soil moisture. 
Figure 5: Dependence of the effective correlation length (¿ e ff) on 
the normalized backscatter coefficient (<7°) 3 RESULTS AND DISCUSSION 
et al. (2010) observed that for a large number of different (s,l)- 
combinations a very small soil moisture retrieval error is obtained. 
They furthermore concluded that a fixed value for s is best used 
on which basis the corresponding value for ¿ e ff is determined. 
Therefore s is fixed at 1.0 cm and l ranges from 1.0 cm to 120 cm 
in the inversion of the IEM. The value resulting in the lowest ob 
servation error, ¿cff, is retained. Figure 4 shows the effective corre 
lation lengths corresponding to the observed normalized backscat 
ter coefficients on every acquisition date. 
A comparison of Figures 3 and 4 shows that the behaviour of the 
field average effective correlation lengths is strongly related to 
the normalized backscatter coefficients. A plot of the values of 
¿efr versus <7°, as presented in Figure 5, reveals this relationship 
can be modeled by a linear regression model: 
¿mod = CL • Un + b + e, (2) 
with ¿mod the modeled correlation length, a and b regression pa 
rameters and e a random error term, usually considered to be 
normally distributed. The values of parameters a and b are also 
shown in Figure 5. Lievens et al. (2010) performed an exten 
sive cross-validation, indicating the robustness of this regression 
model, however, latter exercise will not be discussed in this work. 
3.1 Soil moisture histogram 
Figure 7 shows the histogram of the obtained soil moisture val 
ues for the arbitrary example. The histograms are cut off at a 
minimum soil moisture content of 3 vol% (residual soil moisture 
content) and a maximum of 45 vol% (saturated soil moisture con 
tent), which can influence the mean value. Furthermore, it should 
be noticed that this example histogram is asymmetric, skewed to 
wards higher soil moisture values and therefore not normal. This 
was confirmed using a Lillifors normality test (Lilliefors, 1967) 
for about 70% of the obtained histograms. The remaining his 
tograms were found to be normal, which mostly occurred at low 
soil moisture values (< 25 vol%). Consequently, using the mean 
and standard deviation in further applications as representatives 
for the obtained non-normal soil moisture histograms, may lead 
to a distorted view of the underlying distributions. Furthermore, 
the mean value and standard deviation of the normal histograms 
may be influenced by the fact that the histograms are cut off at 
the residual and saturated soil moisture content. Therefore it is 
recommended to use the median value and the interquartile range 
divided by 1.35 (converted IQR), which are insensitive to the val 
ues of residual and saturated soil moisture content. 
3.2 Retrieved median soil moisture content 
The linear regression model can then be further used to estimate 
the uncertainty around the predicted value. This uncertainty is de 
scribed by a t-distribution with variance a 2 , calculated as follows 
(Neter et al., 1996): 
T n e 2 
, , 1 , Kh-*n°) 2 
. * EÏU«-*- 0 ) 2 
(3) 
Figure 8 shows a scatterplot of the retrieved versus the observed 
soil moisture values, where the error bars represent the converted 
IQR of the resulting histograms. It can be seen that low soil mois 
ture contents (< 20vol%) are slightly overestimated. Overall, 
a root mean square error (RMSE) of 3.51 vol% is obtained be 
tween the measured soil moisture content and the median of the