Process Error, Q
In the early versions of cBathy, the phase 3 Kalman filter processing was key to obtaining reliable, robust output based on noisy individual collection results that can sometimes be partly or completely unusable. Version 2.0 produces much better estimates from single runs, but Kalman filtering will always improve the output. New Kalman results are averaged into the running average result according to a pixel-by-pixel Kalman gain that expresses the believability of the new result compared to that of the running average (section 2.3 of HPH13). Since the running average bathymetry already reflects an average of many individual results, it is likely that new estimates will only slowly nudge the average, i.e. Kalman gains will typically be small.
If bathymetry were unchanging, then the estimated error variance of the running average bathymetry should monotonically reduce with additional estimates, much as predicted by the central limit theorem. However, bathymetry and nearshore morphology change in time due to sediment transport processes, so that confidence in a prior running average estimate should degrade with time since that estimate. For example, you would believe that a survey from yesterday would be a much better representation of today’s bathymetry than a survey from last month or last year. This temporal degradation in confidence in a prior estimate (or temporal increase in error variance) is represented by the process error, Q. There is little available data from which estimates of the magnitude and environmental dependences of Q can be made. HPH13 discuss one very extensive set of daily bathymetry surveys which were used to develop the following form:
Bathymetric variability was found to increase linearly with time, Δt, and with the square of the offshore wave height, and is centered as a location xo with a Gaussian form with standard deviation σx.
The 38-‐day bathymetric data set used to establish this form showed the largest variability to be in a narrow spatial band around the bar location at that time. However, we know that bars move in the cross-‐shore over time so have used a user-‐ selected broader set of values for the spatial Gaussian form (x0 = 150, σx = 100 for Duck, NC).
It is recommended that users applying cBathy to new sites use their best judgment to establish the coefficients for their new site in the routine findProcessError, choosing spatial parameters that center on locations where the maximum variability is expected and choosing a spread, σx, that allows reasonable variability over the region of expected climatological bathymetric variability (a reasonable sand bar envelope region). Criteria for selecting an appropriate value of CQ for a new site are not well known. A value that is too large will lead to running average bathymetries that respond quickly to changes and presumably to noise. A value that is too small means that bathymetric results will vary more slowly than truth and will lag behind natural changes at the site. Development of an improved understanding of appropriate values is an ongoing research topic at the Coastal Imaging Lab.
Examples of Q for different sites are contained in the routine findProcessError.