# Elevation model (Water Overlay)

The elevation model is :

- Rasterization of the height sectors
- Piecewise linear reconstruction of the bottom

### Piecewise linear reconstruction of the bottom

The implementation of the Water Module is based on second-order semi-discrete central-upwind scheme by Kurganov and Petrova (2007)^{[1]}. The surface elevation, also named bottom in the paper, is slightly adjusted to support the scheme to become *well balanced and positivity preserving*. The process of adjusting the original surface elevation is called *piecewise linear reconstruction of the bottom*.

The first requirement the scheme to become *well balanced and positivity preserving* is to ensure that each grid cell has a constant linear slope in both the x- and y- direction. Secondly the end points of the slope should meet in the center of the cell's edges. This ensures that the bottom is continuous along cells in the x- and y- direction. Thirdly, the linear slope in the x- and y-direction within a cell should meet in a single center point.

To fulfill these requirements, the following steps are taken:

- Pick or calculate the height points for the 4 corners of the cell.
- Form a rectangle with the 4 corners and calculate the centers of these edges. (These are the points that have to meet for continuity)
- Calculate a new center point based on the 4 edge center points.

Given that the adjacent cells share the same corner points, and thus share an edge center point, the bottom will be continuous in the x and y direction. Furthermore, the cell has an linear slope in both the x- and y-direction. The only downside is that the new center point might have been placed heigher or lower in a situation where the terrain's slope was originally not linear within the cell.

### Notes

- The smaller the grid size, the closer the bottom reconstruction will approximate the original surface elevation.
- The resulting elevation model can be inspected in a project using the SURFACE_ELEVATION result type.

## References

- ↑
^{1.0}^{1.1}Kurganov A, Petrova G (2007) ∙ A Second-Order Well-Balanced Positivity Preserving Central-Upwind Scheme for the Saint-Venant System ∙ found at: http://www.math.tamu.edu/~gpetrova/KPSV.pdf (last visited 2018-06-29) - ↑
^{2.0}^{2.1}Zsolt Horváth, Jürgen Waser, Rui A. P. Perdigão, Artem Konev and Günter Blöschl (2014) ∙ A two-dimensional numerical scheme of dry/wet fronts for the Saint-Venant system of shallow water equations ∙ found at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.700.7977&rep=rep1&type=pdf ∙ http://visdom.at/media/pdf/publications/Poster.pdf ∙ (last visited 2018-06-29)