\noindent\textbf{Nearshore bathymetry needs} Knowledge of nearshore bathymetry is crucial for every aspect of the blue economy. Currently, it is expensive to obtain nearshore bathymetry at regional scales, and therefore this data is not available for many coasts. The data scarcity occurs especially in the global south and big ocean states that are at the highest risk from climate change. Currently the only global bathymetric dataset, GEBCO, provides depth data at approximately a 500m resolution. This data is of limited accuracy in the nearshore zone because the source data is sparse in nearshore areas. Recent research has found that NASA's ICESat-2, launched in 2018 to study the cryosphere, can incidentally capture bathymetric data. Some studies have been doing using small sites, but the potential for a global product has not yet been investigated. This thesis proposes a method of using GEBCO data as a starting point and incorporating ICESat-2 data via a Kalman filter. This results in a product with an downscaled spatial resolution and improved the accuracy without requiring any in-situ data. If other local bathymetry data is available, it can be added as input to the Kalman filter or used for validation of the method.

\noindent\textbf{Data assimilation via Kalman Updating} To downscale the data using a Kalman filter, first global data from GEBCO is clipped to the area of interest, and then resampled bilinearly to 50m resolution. Then, the ICESat-2 photon data for the area is processed to generate point measurements of bathymetric depth. To fill in the gaps between these point measurements, the bathymetric points are subsampled and interpolated to the same resolution as the GEBCO data using a universal kriging interpolator. This interpolator results in a raster of the estimated depth, and a raster of the estimated uncertainty. To update the interpolated GEBCO grid, the Kalman gain is calculated for each raster cell in and using the Kalman state equation a new bathymetry grid is produced. If other data is available, the process can be applied recursively with other depth and uncertainty grids, allowing the Bayesian combination of any number of bathymetry datasets for the site.

\noindent\textbf{Outlook} The results of this research could allow easier methods of characterizing nearshore bathymetry in remote areas. It could also be extended to include temporal variation – as more bathymetric data becomes available, it could be used to measure the dynamic changes of coastal systems. Data with this temporal dimension could provide valuable validation data for coastal dynamics models. However, the method is limited by strict water clarity requirements and missing data due to atmospheric conditions. Despite these limitations, it could potentially be downscale to a global product for clear water areas.

The largest practical limitation to the Bayesian updating method is the requirement to interpolate the data. Kriging interpolaters are ideal because they are robust to outliers and provide an uncertainty estimate as well as a depth estimate. The downside of the kriging interpolator is both the computational complexity and the requirement that points are not too close together. If points are too close to one another, the kriging matrix is not soluble, and the algorithm has a complexity of $\mathcal{O}(n^4)$ where $n$ is the number of points being interpolated \parencite{}. This means that there is a relatively strict practical limitation to the number of points that can be used as input to the interpolator. The upper limit with a laptop with 32GB RAM was found to be approximately 2000 points without exceeding the available memory. One way to deal with this is to use a tiling strategy, and repeat the process using tiles which contain 2000 bathymetric points or less. This would need to be done adaptively since the bathymetric points are not evenly distributed. Using a larger number of smaller sub-sites would take longer to process, but it would at least be feasible using a consumer-grade computer.

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