Author(s): Alastair Barnett
Linked Author(s): Alastair G. Barnett
Keywords: 3D Scalar solution Free surface solution Flood mapping
Abstract: Mapping of a fixed 3D ground surface has long been achieved by the conventions of contour mapping on a 2D horizontal plan view. Here the third (height) dimension is represented by colour gradations, with a key provided to define the chosen contour ranges. Established packages such as Surfer © Copyright Golden Software offer advanced facilities such as kriging which usually offer the best detection of incised channel bed shapes. However, the free surface of a 3D water body is usually low in gradient because of the tendency of water to flow down surface gradients, eliminating all height discrepancies in primary longitudinal flow, except those which are small enough to be maintained by resistance from the channel bed and sides. Lateral gradients are even smaller because the channel walls inhibit secondary cross-channel flows, although superelevation at the outside of channel bends may locally create a lateral slope comparable with the longitudinal (downstream) slope. Contour mapping is not a suitable technology for directly displaying such low surface gradients. 3D scalar computational hydraulic solutions through a channel reach are based on a segmented channel s axis along which scalar longitudinal distances are measured. A series of n cross-sections is surveyed, each judged to be approximately perpendicular to the axis at the intersection with the cross-section. Both ends of each cross-section are fixed to a suitable 2D map grid by surveyed markers, which then become the vertices of a closed polygon inside which the height of the free surface is available by interpolation of the solution. This can be done first by noting that the solution provides the water surface level at each cross-section, which with this construction extends along the whole straight line joining the two relevant end markers. Computational cell boundaries are defined by two adjacent cross-sections, so in the X-Y (map grid) plane the cell is defined by a quadrilateral joining the four relevant vertices, two for each cross-section. Each of those vertices has a water level available from the solution, so the 3D water surface for the whole cell can be interpolated by fitting a warped (hyperbolic paraboloid) surface between those four vertices. This interpolated solution is still difficult to present directly on a contour map, but it now provides a water level at every X, Y point that the bed level is available, allowing a solution to be derived for flow depth by subtraction. This will of course produce negative depths where the bed level is above the water level, but it is easy to discard such results in a plotting package. This procedure is very much more efficient than attempting to work directly with computational solutions for depth because the bed surface and the free surface have very different computational requirements. The bed surface requires very high spatial resolution in places such as narrow channels, where the width and depth are required with resolution no larger than a 1m grid for stormwater channels, but bed levels do not usually require high temporal resolution with changes happening slowly if at all. The water surface is exactly the opposite, usually requiring low spatial resolution without warped surface interpolation while at times (especially during flood peaks) changing almost discontinuously in a single time step. Attempting to compute depths by subtraction with bed kriging at every spatial grid point and warped free surface interpolation at every time step will take orders of magnitude more computing time and storage space compared with completing that final precise subtraction step only at times and places (such as during flood peaks) where graphical displays of the solution are of real interest, and so worth the effort of assembling a high precision presentation.
Year: 2025