Topological approach to remote sensing

Abstract

The topological problem underlying remote sensing is analysed by determining the geometric singularities that an unknown surface or structure generically impresses on a sensing wavefield. It is shown that the analytical singularities observed in scattering amplitudes and echograms are produced by the topological singularities of the scattering system. Imposing the principle of structural stability on the inverse scattering problem, the singularities that generically occur in recorded signals, travel-time curves, surface contour maps and Fresnel-zone topographies can, together with the associated high-intensity diffraction patterns, be classified into a few universal standard forms described by catastrophe polynomials. As the source-receiver positions vary, the patterns change their morphologies in terms of specific bifurcation sets. By applying singularity and bifurcation theory to allow the effects of caustics (both in ray and wave theory) to be incorporated into three-dimensional techniques for reconstructing surfaces and subsurface structures from their echoes, the interpretation process is considerably simplified and permits an on-site 3D survey. Universal power laws for singularity-dominated echo amplitude variations with the source frequency are deduced. The shape of a scattering surface is reconstructed using the high-frequency regime alone. Discontinuities in the surface, edges and faults, are discussed in terms of constraint catastrophes and the patterns they produce in echograms are classified. on a sensing wave system because remote sensing is a structurally stable process. This view of the inverse scattering problem generalizes classical S-matrix theory in the sense that the observed analytical singularities in scattering amplitudes are linked with the topological singularities of the scatterer. Since the latter can be classified, so can the former, and a new structural approach to remote sensing emerges which considerably simplifies the interpretation process.Briefly the genesis of the theory is this. In order that the reconstruction of surfaces or subsurface structures from backscattered waves be physically feasible, the scattering process has to enjoy an inherent stability property, namely to preserve its quality under slight deformations of the scattering object. Otherwise we could hardly think about or describe it, and today's experiment would not reproduce yesterday's result. Imposing this principle of structural stability (Thom 1975) on the inverse scattering problem has two important consequences. First, it permits us to identify the significant patterns that can generically occur in travel-time curves, contour maps and Fresnel-zone topographies, and to classify them together with the associated diffraction patterns into a set of canonical normal forms or singularities. As the source-receiver positions vary, the patterns change their morphologies according to universal bifurcation sets (caustics). These bifurcation sets and the diffraction catastrophe amplitudes around them are produced by the...

Singularities of dispersion relations

“…The standard approach to the bifurcation equation (13) in the framework of singularity theory is to choose a coordinate system in which the solutions of (1.3) can be easily determined. According to imperfect bifurcation theory [5], the following coordinate changes are allowed. DEFINITION…”

“…Consider first the ideal I: = (x5 , x 2 A, A 2 ) with dim E l+l /I = 7. If r 1 (0) = 0, we find Gel, Kd k GeI andM 1+1 (d x G)d,whence TG + I = U{d x G, d x G} +1, implying cod G g 5.…”

Geometrical inversion for a scattering curve

“…Assuming a homogeneous isotropic wave medium, the travel time(s) to a scatterer measured at some point are proportional to the perpendicular distance(s) to the scatters, from which only specular (geometrical) reflections are assumed to be measured (see e.g. Dangelmayr and Giittinger 1982 for more details). The number of specular reflections observed at a point depends on the position of the point; it changes by two on crossing a sheet of the caustic (envelope of normals) of the scatterer.…”

Remarks on the singularity indices of Arnol'd and Berry

“…For accounts of this theory and references to the literature see Guillemin and Sternberg [18], Duistermaat [15], Hdrmander [19], Arnol'd [1,3], and for applications to phenomena in statistical optics see Berry [6], Berry and Upstill [8]. Applications in other areas of physics are explored by Dangelmayr and co-workers [11][12][13][14] and by Berry [7].…”