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.
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.
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.
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.