Coherent linear events in the t − x domain can be separated in the f − k domain by their dips. This allows us to eliminate certain types of unwanted energy from the data. In particular, coherent linear noise in the form of ground roll, guided waves, and side-scattered energy commonly obscure primary reflections in recorded data. These types of noise usually are isolated from the reflection energy in the f − k domain. From the field record in Figure 6.2-1a, note how ground-roll energy can dominate the data. Ground roll is a type of dispersive waveform that propagates along the surface and is low-frequency, large-amplitude in character. Typically, ground roll is suppressed in the field by using a suitable receiver array.
Figure 6.2-1b is the 2-D amplitude spectrum of the field record in Figure 6.2-1a. Here, various types of energy are well isolated from one another. Ground roll A, its backscattered component B, and guided waves C, are identifiable. Reflections D are situated around the frequency axis. As shown in Figure 6.2-1c, a fan is imposed on this spectrum within which the undesired energy is rejected. This is followed by inverse mapping back to the t − x domain. The resulting filtered record in Figure 6.2-1d is largely free of ground-roll energy, except for the backscattered component. Defining a reject fan in the f − k domain is one implementation of the process known as f − k dip filtering.
Note that dip filtering is but one type of f − k filtering. The reject zone in the f − k domain may be specified not just as a fan but also as a shape suitable for the objective in mind. For instance, the reject zone may be defined as one entire quadrant of the f − k plane in the case of multiple attenuation. It may only be one half of one quadrant in the case of a spatial antialiasing filter (the 2-D Fourier transform). The following are the steps involved in f − k filtering:
- Starting with a common-shot or a CMP gather, or a CMP-stacked section, apply 2-D Fourier transform.
- Define a 2-D reject zone in the f − k domain by setting the 2-D amplitude spectrum of the f − k filter to zero within that zone and set its phase spectrum to zero.
- Apply the 2-D f − k filter by multiplying its amplitude spectrum with that of the input data set.
- Apply 2-D inverse Fourier transform of the filtered data.
Practical issues associated with the 2-D Fourier transform and specifying a fan reject zone are outlined below:
- Conventional implementations of the Fourier transform itself produce wraparound noise. This is apparent in Figure 6.2-1d, location F. To avoid this problem, the data must be extended beyond the ranges of the spatial and temporal axes by padding with zeroes. The size of the input gather typically is increased by a factor of 4, which is equivalent to doubling the length in t and x. This increases the cost but removes the wraparound effects.
- The fan width must not be too narrow. This follows from previous observations of the 1-D Fourier analysis of frequency filters (the 1-D Fourier transform). If the bandwidth of the reject zone were narrow, then the t − x response of the dip filter would have a large array of nonzero elements. Fortunately, coherent noise with large stepouts, such as ground roll, often is isolated in the f − k domain from the zone that includes the reflection signal. This is demonstrated by the example in Figure 6.2-1b. In such cases, ground-roll energy A is attenuated without damaging the reflection signal by using a large fan (Figure 6.2-1c).
- As for the 1-D frequency filters (the 1-D Fourier transform), the amplitude spectrum of the f − k filter must not have sharp boundaries. There must be a smooth transition from the reject zone to the pass zone. This is accomplished by tapering the fan edges, which is analogous to using slopes in frequency filtering. The amount of tapering must be large enough to be effective. On the other hand, it must not be so wide that it suppresses signal in the pass zone. Extra precaution is taken at low frequencies. As the fan thins to a zero width at the origin of the f − k domain, as in a wedge, the actual reject zone may invade the low-frequency components of the pass zone. This invasion occurs because the fan cannot get too narrow. It may be desirable to stop the reject zone just short of the low frequencies. This effectively excludes the low frequencies from the f − k dip filtering action.
- Spatial aliasing often causes poor f − k filter performance. The fan reject zone must be extended to the spatially aliased frequency components. A practical approach to this problem is to apply linear moveout correction to the data before f − k filtering so that the unwanted signal appears at lower dips, thus eliminating the spatial aliasing effects. The linear moveout then is removed after f − k filtering. Unfortunately, this may not always work, since events that are not spatially aliased before may be spatially aliased after linear moveout.
- Introduction to noise and multiple attenuation
- Multiple attenuation in the CMP domain
- The slant-stack transform
- The radon transform
- Linear uncorrelated noise attenuation
- Multichannel filtering techniques for noise and multiple attenuation