Multidimensional seismic data processing forms a major component of seismic profiling, a technique used in geophysical exploration. The technique itself has various applications, including mapping ocean floors, determining the structure of sediments, mapping subsurface currents and hydrocarbon exploration. Since geophysical data obtained in such techniques is a function of both space and time, multidimensional signal processing techniques may be better suited for processing such data.
Data acquisition
There are a number of data acquisition techniques used to generate seismic profiles, all of which involve measuring acoustic waves by means of a source and receivers. These techniques may be further classified into various categories, depending on the configuration and type of sources and receivers used. For example, zero-offset vertical seismic profiling (ZVSP), walk-away VSP etc. The source (which is typically on the surface) produces a wave travelling downwards. The receivers are positioned in an appropriate configuration at known depths. For example, in case of vertical seismic profiling, the receivers are aligned vertically, spaced approximately 15 meters apart. The vertical travel time of the wave to each of the receivers is measured and each such measurement is referred to as a “check-shot” record. Multiple sources may be added or a single source may be moved along predetermined paths, generating seismic waves periodically in order to sample different points in the sub-surface. The result is a series of check-shot records, where each check-shot is typically a two or three-dimensional array representing a spatial dimension (the source-receiver offset) and a temporal dimension (the vertical travel time).
Data processing The acquired data has to be rearranged and processed to generate a meaningful seismic profile: a two-dimensional picture of the cross section along a vertical plane passing through the source and receivers. This consists of a series of processes: filtering, deconvolution, stacking and migration.
Multichannel filtering
Multichannel filters may be applied to each individual record or to the final seismic profile. This may be done to separate different types of waves and to improve the signal-to-noise ratio. There are two well-known methods of designing velocity filters for seismic data processing applications.
Two-dimensional Fourier transform design The two-dimensional Fourier transform is defined as:
F ( k _ , ω ) = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x _ , t ) e − j ( ω t − k _ x _ ) d x _ d t {\displaystyle F({\underline {k}},\omega )=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f({\underline {x}},t)e^{-j(\omega t-{\underline {k}}{\underline {x}})}d{\underline {x}}dt}
where k _ {\displaystyle {\underline {k}}} is the spatial frequency (also known as wavenumber) and ω {\displaystyle \omega } is the temporal frequency. The two-dimensional equivalent of the frequency domain is also referred to as the k _ − ω {\displaystyle {\underline {k}}-\omega } domain. There are various techniques to design two-dimensional filters based on the Fourier transform, such as the minimax design method and design by transformation. One disadvantage of Fourier transform design is its global nature; it may filter out some desired components as well.
τ-p transform design The τ-p transform is a special case of the Radon transform, and is simpler to apply than the Fourier transform. It allows one to study different wave modes as a function of their slowness values, p {\displaystyle p} . Application of this transform involves summing (stacking) all traces in a record along a slope (slant), which results in a single trace (called the p value, slowness or the ray parameter). It transforms the input data from the space-time domain to intercept time-slowness domain.
p = 1 v = d t d x {\displaystyle p={\frac {1}{v}}={\frac {dt}{dx}}}
Each value on the trace p is the sum of all the samples along the line
t = τ + p x {\displaystyle t=\tau +px}
The transform is defined by:
F ( p , τ ) = ∫ − ∞ ∞ f ( x , τ + p x ) d x = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , t ) δ ( t − τ − p x ) d x d t {\displaystyle F(p,\tau )=\int _{-\infty }^{\infty }f(x,\tau +px)dx=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,t)\delta (t-\tau -px)dxdt}
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