A numerical study of the probe method

The goal of this work is the numerical realization of the probe method suggested by Ikehata for the detection of an obstacle D in inverse scattering. The main idea of the method is to use probes in the form of point source (., z) with source point z to define an indicator function (I) over cap (z) which can be reconstructed from Cauchy data or far. eld data. The indicator function boolean AND (I) over cap (z) can be shown to blow off when the source point z tends to the boundary aD, and this behavior can be used to find D. To study the feasibility of the probe method we will use two equivalent formulations of the indicator function. We will carry out the numerical realization of the functional and show reconstructions of a sound-soft obstacle.

A study on orthogonality sampling.

The goal of this paper is to study and further develop the orthogonality sampling or stationary waves algorithm for the detection of the location and shape of objects from the far field pattern of scattered waves in electromagnetics or acoustics. Orthogonality sampling can be seen as a special beam forming algorithm with some links to the point source method and to the linear sampling method. The basic idea of orthogonality sampling is to sample the space under consideration by calculating scalar products of the measured far field pattern , with a test function for all y in a subset Q of the space , m = 2, 3. The way in which this is carried out is important to extract the information which the scattered fields contain. The theoretical foundation of orthogonality sampling is only partly resolved, and the goal of this work is to initiate further research by numerical demonstration of the high potential of the approach. We implement the method for a two-dimensional setting for the Helmholtz equation, which represents electromagnetic scattering when the setup is independent of the third coordinate. We show reconstructions of the location and shape of objects from measurements of the scattered field for one or several directions of incidence and one or many frequencies or wave numbers, respectively. In particular, we visualize the indicator function both with the Dirichlet and Neumann boundary condition and for complicated inhomogeneous media.

An iterative contractive framework for probe methods: LASSO

We present a new iterative approach called Line Adaptation for the Singular Sources Objective (LASSO) to object or shape reconstruction based on the singular sources method (or probe method) for the reconstruction of scatterers from the far-field pattern of scattered acoustic or electromagnetic waves. The scheme is based on the construction of an indicator function given by the scattered field for incident point sources in its source point from the given far-field patterns for plane waves. The indicator function is then used to drive the contraction of a surface which surrounds the unknown scatterers. A stopping criterion for those parts of the surfaces that touch the unknown scatterers is formulated. A splitting approach for the contracting surfaces is formulated, such that scatterers consisting of several separate components can be reconstructed. Convergence of the scheme is shown, and its feasibility is demonstrated using a numerical study with several examples.

An objective, niche-based approach to indicator species selection

1. Species-based indices are frequently employed as surrogates for wider biodiversity health and measures of environmental condition. Species selection is crucial in determining an indicators metric value and hence the validity of the interpretation of ecosystem condition and function it provides, yet an objective process to identify appropriate indicator species is frequently lacking. 2. An effective indicator needs to (i) be representative, reflecting the status of wider biodiversity; (ii) be reactive, acting as early-warning systems for detrimental changes in environmental conditions; (iii) respond to change in a predictable way. We present an objective, niche-based approach for species' selection, founded on a coarse categorisation of species' niche space and key resource requirements, which ensures the resultant indicator has these key attributes. 3. We use UK farmland birds as a case study to demonstrate this approach, identifying an optimal indicator set containing 12 species. In contrast to the 19 species included in the farmland bird index (FBI), a key UK biodiversity indicator that contributes to one of the UK Government's headline indicators of sustainability, the niche space occupied by these species fully encompasses that occupied by the wider community of 62 species. 4. We demonstrate that the response of these 12 species to land-use change is a strong correlate to that of the wider farmland bird community. Furthermore, the temporal dynamics of the index based on their population trends closely matches the population dynamics of the wider community. However, in both analyses, the magnitude of the change in our indicator was significantly greater, allowing this indicator to act as an early-warning system. 5. Ecological indicators are embedded in environmental management, sustainable development and biodiversity conservation policy and practice where they act as metrics against which progress towards national, regional and global targets can be measured. Adopting this niche-based approach for objective selection of indicator species will facilitate the development of sensitive and representative indices for a range of taxonomic groups, habitats and spatial scales.