Improving flood models using LiDAR: progress and problems

Airborne scanning laser altimetry (LiDAR) is an important new data source for river flood modelling. LiDAR can give dense and accurate DTMs of floodplains for use as model bathymetry. Spatial resolutions of 0.5m or less are possible, with a height accuracy of 0.15m. LiDAR gives a Digital Surface Model (DSM), so vegetation removal software (e.g. TERRASCAN) must be used to obtain a DTM. An example used to illustrate the current state of the art will be the LiDAR data provided by the EA, which has been processed by their in-house software to convert the raw data to a ground DTM and separate vegetation height map. Their method distinguishes trees from buildings on the basis of object size. EA data products include the DTM with or without buildings removed, a vegetation height map, a DTM with bridges removed, etc. Most vegetation removal software ignores short vegetation less than say 1m high. We have attempted to extend vegetation height measurement to short vegetation using local height texture. Typically most of a floodplain may be covered in such vegetation. The idea is to assign friction coefficients depending on local vegetation height, so that friction is spatially varying. This obviates the need to calibrate a global floodplain friction coefficient. It’s not clear at present if the method is useful, but it’s worth testing further. The LiDAR DTM is usually determined by looking for local minima in the raw data, then interpolating between these to form a space-filling height surface. This is a low pass filtering operation, in which objects of high spatial frequency such as buildings, river embankments and walls may be incorrectly classed as vegetation. The problem is particularly acute in urban areas. A solution may be to apply pattern recognition techniques to LiDAR height data fused with other data types such as LiDAR intensity or multispectral CASI data. We are attempting to use digital map data (Mastermap structured topography data) to help to distinguish buildings from trees, and roads from areas of short vegetation. The problems involved in doing this will be discussed. A related problem of how best to merge historic river cross-section data with a LiDAR DTM will also be considered. LiDAR data may also be used to help generate a finite element mesh. In rural area we have decomposed a floodplain mesh according to taller vegetation features such as hedges and trees, so that e.g. hedge elements can be assigned higher friction coefficients than those in adjacent fields. We are attempting to extend this approach to urban area, so that the mesh is decomposed in the vicinity of buildings, roads, etc as well as trees and hedges. A dominant points algorithm is used to identify points of high curvature on a building or road, which act as initial nodes in the meshing process. A difficulty is that the resulting mesh may contain a very large number of nodes. However, the mesh generated may be useful to allow a high resolution FE model to act as a benchmark for a more practical lower resolution model. A further problem discussed will be how best to exploit data redundancy due to the high resolution of the LiDAR compared to that of a typical flood model. Problems occur if features have dimensions smaller than the model cell size e.g. for a 5m-wide embankment within a raster grid model with 15m cell size, the maximum height of the embankment locally could be assigned to each cell covering the embankment. But how could a 5m-wide ditch be represented? Again, this redundancy has been exploited to improve wetting/drying algorithms using the sub-grid-scale LiDAR heights within finite elements at the waterline.

Boundary value problems for the N-wave interaction equations

We consider boundary value problems for the N-wave interaction equations in one and two space dimensions, posed for x [greater-or-equal, slanted] 0 and x,y [greater-or-equal, slanted] 0, respectively. Following the recent work of Fokas, we develop an inverse scattering formalism to solve these problems by considering the simultaneous spectral analysis of the two ordinary differential equations in the associated Lax pair. The solution of the boundary value problems is obtained through the solution of a local Riemann–Hilbert problem in the one-dimensional case, and a nonlocal Riemann–Hilbert problem in the two-dimensional case.

Well-posed boundary value problems for integrable evolution equations on a finite interval

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n–N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient n in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.

A Nyström method for a boundary value problem arising in unsteady water wave problems

This paper is concerned with solving numerically the Dirichlet boundary value problem for Laplace’s equation in a nonlocally perturbed half-plane. This problem arises in the simulation of classical unsteady water wave problems. The starting point for the numerical scheme is the boundary integral equation reformulation of this problem as an integral equation of the second kind on the real line in Preston et al. (2008, J. Int. Equ. Appl., 20, 121–152). We present a Nystr¨om method for numerical solution of this integral equation and show stability and convergence, and we present and analyse a numerical scheme for computing the Dirichlet-to-Neumann map, i.e., for deducing the instantaneous fluid surface velocity from the velocity potential on the surface, a key computational step in unsteady water wave simulations. In particular, we show that our numerical schemes are superalgebraically convergent if the fluid surface is infinitely smooth. The theoretical results are illustrated by numerical experiments.