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Linear Radon transform

Images - Appendix D (a) Linear events in the CMP gather and (b) their slant-stack transforms. Theoretically, an event with linear moveout in the time-offset domain can be mapped to a point with the slant-stack transform, and a hyperbolic event, such as a primary or a multiple event, can be mapped to an ellipse in the τ-p domain (Treitel et al, 1982).

The Radon transform is a widely mathematical technique in seismic data processing and image analysis. Three types of Radon transforms used in seismic data processing: the slant-stack or τ-p (or linear Radon transform) transform; the hyperbolic Radon transform; and the parabolic Radon transform. Events with linear moveout in the time-offset domain are mapped to a point with the slant-stack transform as we can see here.

For many modern seismic applications, like multiple-elimination in the (τ-p)-domain, it is important to find a fast, digital Radon transform for sampled seismic data. Over the last twenty years, in seismic as well as other disciplines, attention has
been given to this problem. Mersereau and Oppenheim (1974) introduced a non-Cartesian grid in the 2-D Fourier plane, called the concentric squares grid. Recently, Averbuch et al. (2003) have proposed a discrete Radon transform that is rapidly computible and invertible by means of FFTs. Its basis is the concentric squares grid, which they call the pseudo-polar grid. In this appendix, we take advantage of the idea of the concentric squares grid, or the pseudo-polar grid, as introduced by these authors, to transform data to the 2-D Fourier space. For most seismic applications, it is sufficient to transform data to a triangle subdomain of the concentric squares grid. Therefore, we choose to call the transform the triangle-Fourier transform.

Multiple suppression by predictive deconvolution builds on the periodicity of multiples. However, on time-distance gathers, like common shot gathers, common midpoint gathers, and common receiver gathers, multiples are not periodic in time for non-zero offsets. Taner (1980) first recognized that multiples in layered media
are periodic along radial traces (fixed p). The time separation is different from one radial trace to another. Therefore, a predictive deconvolution operator can be designed from the autocorrelogram of eace p-trace and applied to attenuate multiples in the (τ,p)-domain, in which the primary and subsequent multipes are ellipses.

Parabolic Radon transform

Images - Appendix D Here is a variation of the Radon transform known the parabolic Radon transform. The other well known variations of the Radon transform is the hyperbolic Radon transform. As described in the previous figure, the slant-stack transform maps a linear event in the x-t domain to a focused point, and a hyperbolic event to an ellipse in the transformed domain.

The variations of Radon transform, such as the parabolic, hyperbolic and Radon transform, ideally map events with corresponding patterns to focused points in the Radon domain as we can see here. Events with paraobolic moveout in the time-space domain are mapped to a focused point in the model domain by the hyperbolic Radon transform.

In practical terms, Radon Transforms are not not reversible. Most invere Radon transform are solve linear inverse problem using typical least-square techniques.

Hyperbolic Radon transform

Images - Appendix D An illustration of the hyperbolic Radon transform. (a) Hyperbolic events in the CMP domain are mapped to (b) focused points in the Radon domain by the hyperbolic Radon transform.

NMO corrected CMP gathers

Images - Appendix D NMO corrected CMP gathers.

CMP gathers after Radon demultiple

Images - Appendix D NMO corrected CMP gathers after Radon demultiple.

Time-space data have NT=32 and NX=16 samples (Cartesian)

Images - Appendix D Time-space data have NT=32 and NX=16 samples. The red triangles show the related Cartesian kx-ω-grid with nkx=-NX/2+1,...,NX/2 along the horizontal axis and nω=0,NT/2 along the vertical axis.

The grid in this is not suitable for the discrete Radon transform. The Fourier slice theorem shows that the Fourier transform of the Radon transform with respect to the intercept variable τ is equal to the 2-D Fourier transform of u(x,t) evaluated on the line kx=pω. In numerical computations, however, since the 2-D FFT of u(nx,nt)gives data on a regular grid as a function of frequency ω and wavenumber kx, that is, $U(nkx,nω), interpolation on the ω-kx-grid to the lines kx=pω (for varying p) is generally required.

On the other hand, the use of the so-called pseudo-polar, or triangle, Fourier transform eliminates the interpolation problem as illustrated in the next figure. We call this grid, a subset of the pseudo-polar grid, the triangle grid. Mersereau and Oppenheim (1974) are the pioneers of this type of grid, which enables fast Fourier computations.

Time-space data have NT=32 and NX=16 samples (pseudo-polar)

Images - Appendix D Time-space data have NT=32 and NX=16 samples. The blue triangles show the related triangle (pseudo-polar) grid with nkx=-NX/2+1,...,NX/2 along the horizontal axis and nω=0,NT/2 along the vertical axis.

Cartesian (red) and triangle (blue) grids

Images - Appendix D The Cartesian (red) and triangle (blue) grids in Figures D-1 and D-2, respectively, plotted on top of each other.