3-D acoustic modelling of edge diffractions — revisited

An accurate, fast, and simple algorithm for 3-D acoustic modelling of seismic edge diffractions, originally developed in the 1980s, is revisited in this paper. The main objective is to reintroduce this simple approach to edge-diffraction modelling and for the first time give the details of the theory in the open literature. The method is based on a combination of Kirchhoff theory and uniform asymptotic techniques developed within a high-frequency assumption. The diffraction contributions are then computed at stationary edge points only, by analogy with the geometrical ray contributions associated with internal stationary points or specular points. To be able to handle sampling inaccuracies of the critical edge points, a modified algorithm is proposed. Its robustness is verified in case of scattering from a circular edge. Also the extension from rigid or free boundary conditions to the case of edges defined by two penetrable surfaces is discussed in this paper. Both experimental and synthetic 3-D data are presented to demonstrate the potential of this edge-diffraction modelling technique. Since all parameters needed in the computations are obtained from dynamic ray tracing, the algorithm can readily be incorporated in existing software packages for 3-D seismic ray modelling.     

Diffraction‐limited imaging and beyond – the concept of super resolution‡

It is well‐known that experimental or numerical backpropagation of waves generated by a point‐source/‐scatterer will refocus on a diffraction‐limited spot with a size not smaller than half the wavelength. More recently, however, super‐resolution techniques have been introduced that apparently can overcome this fundamental physical limit. This paper provides a framework of understanding and analysing both diffraction‐limited imaging as well as super resolution. The resolution analysis presented in the first part of this paper unifies the different ideas of backpropagation and resolution known from the literature and provides an improved platform to understand the cause of diffraction‐limited imaging. It is demonstrated that the monochromatic resolution function consists of both causal and non‐causal parts even for ideal acquisition geometries. This is caused by the inherent properties of backpropagation not including the evanescent field contributions. As a consequence, only a diffraction‐limited focus can be obtained unless there are ideal acquisition surfaces and an infinite source‐frequency band. In the literature various attempts have been made to obtain images resolved beyond the classical diffraction limit, e.g., super resolution. The main direction of research has been to exploit the evanescent field components. However, this approach is not practical in case of seismic imaging in general since the evanescent waves are so weak – because of attenuation, they are masked by the noise. Alternatively, improvement of the image resolution of point like targets beyond the diffraction limit can apparently be obtained employing concepts adapted from conventional statistical multiple signal classification (MUSIC). The basis of this approach is the decomposition of the measurements into two orthogonal domains: signal and noise (nil) spaces. On comparison with Kirchhoff prestack migration this technique is showed to give superior results for monochromatic data. However, in case of random noise the super‐ resolution power breaks down when employing monochromatic data and a limited acquisition aperture. For such cases it also seems that when the source‐receiver lay out is less correlated, the use of a frequency band may restore the super‐resolution capability of the method.     

EM target location1

We consider the problem of computing the most probable location of a target based on radar measurements of the subsurface. Our algorithm makes use of the maximum likelihood estimator (MLE), which represents a correlation between the measured data and synthetic data generated for the object of interest at different locations. Previous studies assume a plane-wave acquisition geometry and target object(s) embedded in a uniform background. In this paper, a generalization of the MLE method is presented which is valid for discrete point sources (and receivers) and a 2D model (i.e. a 2.5D acquisition geometry). Within this formulation the treatment of a non-uniform background model is also possible. We concentrate on geotechnical ground investigations and assume that the characteristic dimensions of the target object are in the range 1–2λ, (λ being the wavelength). The potential of the method is demonstrated employing cross-hole radar data acquired in a controlled field experiment. The MLE result is also compared with the image obtained employing a full reconstruction method such as diffraction tomography.