Summary
To separate potentially spurious geomorphic change that could be due to survey noise a method is needed to create a threshold of noise on a cell by cell basis. This section covers the basic equations and concepts used in probabilistic thresholding which filter out potentially spurious geomorphic change due to uncertainty.
Concepts
Probabilistic Thresholding & Confidence Limits
Due
to the lack of direct verification of elevation measurements resulting from
surveying sub-aqueous surfaces, statistical estimation of uncertainty is
required. This statistical estimation is governed by the laws of probability
and is represented by the following Equations 1, 2, and 3.
Equation 1 - Propagated Uncertainty
Equation 2 - T-Score
Equation 3 - Critical Uncertainty Threshold
The combination of these equations
is used to model the distribution of uncertainty for each cell within a geomorphic
change detection study. The propagated
uncertainty calculated in Equation 1 is an estimation of the standard deviation
of the distribution of uncertainty for each cell. By assuming the uncertainty
in each raster cell is random and independent, the standard deviation can be
used to create a normal distribution centered on zero. An example of using
these equations to compute the standard deviation to model a normal
distribution for uncertainty is shown in Figure
1.
Figure 1 - Example of Probabilistic Thresholding
Figure 1 - An example of computing the probabilistic threshold at 95% for a single cell.
In GCD computing the 95% probabilistic threshold level is equivalent to
computing the threshold value to which 95% of the uncertainty values would be
contained. To compute a 95% probabilistic threshold (Equation 3) the standard
deviation of uncertainty (result from Equation 1) is multiplied by 1.96 the
standard value for computing the threshold for which 95% of values would lie
within the mean. This distribution and 95% threshold is shown in Figure
2. To compute a different probabilistic threshold
the Standard T-Distribution Critical Values Table is referred to. The comparison of different critical t-values
and their effect on the critical threshold error are shown in Figure
2. By
viewing these values it can be understood that lowering the probabilistic
threshold level is synonymous with lowering the critical threshold error. By viewing the differing critical t-values it
can be understood that the sensitivity of changing the probabilistic threshold
level is a simple linear relationship that increases linearly with increasing
propagated uncertainty. For example the difference between the critical
threshold error for a propagated uncertainty of .2496 ft. at 95% and 80%
threshold levels is 0.07875 ft. While when a higher propagated uncertainty of
2.5 ft. is used the difference between 95% and 80% threshold level is 0.7875
ft.
Figure 2 - Probabilistic Thresholding and Different Confidence Levels
Figure 2 - Example of the normal distribution computed at a 95% probabilistic threshold for a propagated uncertainty value of 0.2496 ft. The table within this displays the different critical t-values, which are independent of propagated uncertainty, and their effect on the critical threshold error.
