### Probabilistic Thresholding

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.

#### ET-AL Recommendation to IPC

ET-AL recommends using an 80% confidence interval for sediment budgeting purposes.

### 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 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.