Diffraction-limited imaging and beyond - the concept of super resolution‡

Abstract

A B S T R A C TIt 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 weakbecause 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.

Time-reversal multiple signal classification in case of noise: A phase-coherent approach

“…It is assumed that the rank of the matrix K is D < min(N s , N r ), with D representing the total number of scatterers. 7,11 In the same way, the quantities R \ , U \ , and V \ represent the counterparts of R sig , U sig , and V sig , associated to the noise subspace. We finally have the orthonormality properties, namely,…”

“…(9a) and (9b), an SVD of K(x) results in signal subspace singular functions that are normalized versions of the Green's function vectors associated with the scatterers. 5,11 Thus, mathematically the left-and right-singular matrices, U sig (x) and V sig (x), in Eq. (4) take the forms…”

“…The singular vectors output from an SVD analysis of a complexvalued matrix system will be non-unique by an arbitrary phase. 11,14 The phase angles in Eqs. (10) represent symbolically this non-uniqueness.…”

“…12 The corresponding effect on the pseudo-spectrum of time-reversal MUSIC has also been addressed for smaller amounts of noise. 11 In this paper the term super-resolution is used to characterize any method with a resolving power beyond the diffraction limit. For a finite aperture, this limit is quantitatively described by the Rayleigh criteria, which for an ideal system approach the classical half wavelength limit.…”

“…5,9 In the case of noisy data, time-reversal MUSIC will get easily distorted unless a rather idealized acquisition geometry is applied. 11 An analytical description has been introduced discussing the effect of noise on the singular values of the array-response matrix. 12 The corresponding effect on the pseudo-spectrum of time-reversal MUSIC has also been addressed for smaller amounts of noise.…”