Uncertainty in depth grids

The Norwegian territorial waters and continental shelf have been mapped by The Norwegian Mapping Authority, The Norwegian Defence Research Establishment (FFI), and other actors over a number of years. The highest resolution data within the territorial limit are classified due to their military value. A project to make depth data available as high-resolution raster maps (distributed as GeoPackage) was established in 2020. The depth in these raster maps will be the average within each grid cell. The users also want an uncertainty attribute connected to each grid cell. The project is a cooperation between the Mapping Authority, The Norwegian Armed Forces, and FFI. The Mapping Authority will deliver depth data and produce the depth grids. FFI will deliver depth data and represent possible users of the depth grids. As users, FFI was asked for a definition or description of uncertainty This work was performed within FFI project 1600 “Further support for the procurement phase for new submarines”. The goal of this report is to find a good estimate for the standard deviation of a grid cell, with good meaning that the difference between the depth in a random position and the average value can be modeled as a normal distribution with a standard deviation equal to this estimate. The estimate must be based on the measurements used to compute the grid depth. Three possible estimators were examined, along with the effect of the number of measurements/cell. The three estimators are: the empirical standard deviation (the usual estimator for uncertainty), the extreme deviation (the largest deviation between average and single measurements), and the span (largest minus smallest measurement). The latter two must be scaled to work as estimators. Two areas were examined: The side of a fjord and an open, uneven terrain off the coast. The investigation has shown that number of measurements/cell is an important parameter as it is difficult to find a good uncertainty estimate when there are few measurements. In this report, ‘few’ are five or less, but it might be worthwhile to investigate this in other areas as well. The empirical standard deviation is an adequate estimator as long as there are enough measurements. The extreme deviation and the span can also provide good estimators, but it has not been possible to find scale factors that fits all the grids in this investigation. The extreme deviation can, when compared with the standard deviation, be used to see if the distribution is Gaussian. It is also important to note that there will be cases where the depth difference simply cannot be modeled as normally distributed. In such cases, the difference between the average and an independent measurement can be much larger than the standard deviation indicates. FFI recommends that each grid cell have the following attributes: average depth, standard deviation, extreme deviation, and number of measurements. The average standard deviation/cell and extreme deviation/cell should be added as meta information for the grid, if possible.