Cascaded Dipole Filters: Extending the Limits of Seismic Resolution

Introduction

In the challenging world of mapping thin stratigraphic targets, a geophysicist may choose to try a tactic for increasing resolution, particularly if it is simple, quick and has a known effect. The dipole filters described below, with either even or odd iterations (zero or 90 degree phase shift), can be used to "see" the details of a thinning layer through the fog of the lower frequency portion of a signal. In particular, a horizon with a doublet character may become split into separate events which can be "auto picked" more easily on a workstation. A catchy phrase to describe this method to landmen and engineers might be "first we catch the wave... then we ride the ripples".

The (1, -1) Dipole Filter

The dipole filter featured in this article (1, -1) consists of two time samples of equal amplitude with opposite signs. This filter has an approximately linear 6dB/octave slope, and a constant 90 degree phase shift (Anstey 1970). Successive convolutions re-apply the amplitude spectrum ramp and the 90 degree phase shift.

The dipole and even numbered cascades of two to six times with itself are shown in Figure 1. The cascaded filter becomes progressively longer and adds an extra lobe with each cascade. The amplitude spectrum is progressively enhanced at the high end and diminished at the low end.

Derivatives

Both Anstey (1970) and Claerbout (1976) showed that this dipole is a digital approximation for differentiation of seismic data or wavelets. The subsequent cascaded convolution of the (1, -1) filter with itself, when convolved with the wavelet, approximates the second and higher order derivatives of the wavelet.

Resolution of Wavelets

Resolution of thin beds by a zero-phase wavelet is determined by the dominant frequency and the shape of its amplitude spectrum. The dominant period T, is the trough-to-trough time of the central side lobes. The corresponding frequency F=1/T is the dominant frequency. As a simple rule of thumb, the minimum seismic time separation for which the top and bottom of a layer can be clearly resolved as two separate time events is half the dominant period of the wavelet or T/2. (Widess, 1973; Kallweit and Wood, 1982). In this article the Raleigh limit R for temporal resolution is thus R=T/2.

Extending Resolution

The motivation for the use of these dipoles may be analogous to the motivation for the use of non-linear sweeps in Vibroseis recording, the use of higher low cut filters and higher natural frequency geophones, and several edge or fault detection strategies for image enhancement. The geophysicist is trying to extend the resolution of seismic layers.

What happens if we successively convolve or cascade this dipole with a seismic wavelet and with our seismic data? Will the amplification of the higher frequencies by the dipole enhance the resolution of stratigraphic events? Will this allow picking of isolated events that were previously merged in doublets or thin-layer tuning?

The illustrations that follow may generate discussion and experimentation with practical applications of these dipoles to assist in detailed interpretation.

Examples of Dipoles on Wavelets

The resolution effects of differentiation with cascaded dipoles are shown for a Ricker, an Ormsby, and a Butterworth filter in Figure 2 through Figure 4. The labels T_n and R_n will be used to refer to the period and resolution after n cascades of the dipole. Note that in each case the shape of the amplitude spectrum is multiplied by the ramp of each additional dipole. The low frequencies are attenuated and the higher frequencies are amplified. The dominant frequency increases as the dominant or central trough-to-trough period is reduced and additional lobes appear in the wavelet.

Ricker Wavelet

The Ricker wavelet in Figure 2 has an initial dominant frequency of 25 Hz (corresponding to T₀=40 ms, R₀=20ms). The dominant frequency increases from 25 Hz to approximately 31 Hz (T₂=32 ms, R₂=16ms) and 43 Hz (T₆=23 ms, R₆=12ms) for the second (2x cascaded dipole) and sixth (6x cascaded dipole) derivatives, respectively. The bandwidth and time envelope stay approximately the same while extra side-lobes appear. The resolution is finer as the dominant frequency increases.

Ormsby Wavelet

Figure 3 shows the dipole effect on an Ormsby type filter defined as 8/12-48/96 Hz.

This filter has a bandwidth of 36 Hz equivalent to two octaves in width and a one octave high-end taper. The frequencies above 96 Hz are removed and cannot be recovered. The dominant frequency moves from 53 Hz (T₀= 19ms, R₀= 9ms) to 62 Hz (T₂ = 16ms, R₂=8ms) and 71 Hz (T₆=14ms, R₆=7ms) for the second and sixth derivatives, respectively. The initial side-lobes are exaggerated and the time envelope is lengthened. The ringy waveform is characteristic of the steep high-frequency cut-off of the resulting narrow bandwidth (see Appendix for further discussion).

Butterworth Wavelet

Figure 4 shows the dipole effect on a Butterworth wavelet (8 Hz@ 18 dB/octave-30 Hz@24 dB/octave). This filter was selected to illustrate the possibilities for enhancing an initially low frequency waveform provided the high frequencies, although weak, are present. The initial Butterworth filter has a low frequency side-lobe character with few visible side-lobes created after the series of cascaded dipoles. The dominant frequency moves from approximately 25 Hz (T₀=40 ms, R₀=20ms) to 40 Hz (T₂=25 ms, R₂= 12ms) and to 200 Hz (T₆=5 ms, R₆=2ms) for the second and sixth derivatives, respectively. While this improvement is beyond that expected in practice, this Butterworth wavelet will be used in 2-D stratigraphic examples in Figure 5 and Figure 6 to illustrate the potentially dramatic enhancement in resolution by cascaded dipoles.

Resolution and Interpretation Goals

The convolution of these short filters with the interpreter's zero-phase broadband data may raise the effective dominant frequency. Effectively the low frequency portions of the signal are reduced, and we can see the ripples on the waveform and refine our picking of events.

Resolution and a Wedge Model

The effect of extending the limits of resolution are visually demonstrated in Figure 5 with a simple wedge model. The time separation between the equal amplitude positive spikes of this wedge increases by I ms/trace. Thus the trace number corresponds directly to the time separation in ms. Figure 5 shows the wedge model before and after six cascades of the dipole on the Ricker wavelet from Figure 2 and the Butterworth wavelet from Figure 4, respectively.

Figure 5a and 5b show the before and after response of the wedge model with an initial 25 Hz Ricker wavelet corresponding to a dominant period T₀ of 40 ms, for which the Raleigh temporal resolution is R₀=T₀/2=20ms. Figure 5a demonstrates this expectation with trace 20 being the first trace where the two events are separate. Figure 5b show the results after the application of the sixth derivative filter (i.e., the 6x cascaded dipole). A clear separation of two events can be seen on trace 12. This matches the expected resolution for the 6x dipole on this Ricker wavelet where T₆=23ms giving R₆=12ms.

Figures 5c and 5d show a more dramatic extension of the resolution limit for the same wedge model using the Butterworth wavelet from Figure 4a. The initial 8 Hz@18 dB/octave-30 Hz@24 dB/octave Butterworth wavelet has T₀=40 ms with expected resolution R₀=20 ms or less. Figure 5c shows the initial separation for the layers clearly identifiable at trace 16 (i.e., 16 ms resolution). After applying the 6x cascaded dipole we expect a resolution of R₆=2-3 ms. The result in Figure 5d shows that the wedge is resolved at trace 2.

Resolution and a 2-D Synthetic Model

Figure 6 shows an example applicable to resolution issues in NW Alberta Slave Point patch reefs and illustrates the potential for these short filters to extend the resolution of thin layers below the initial apparent temporal resolution limit. Figure 6a shows the interval velocity model corresponding to a 20ms thick patch reef with basinal infill or porosity development. In Figure 6b the reflection coefficients are filtered with the rather low-frequency Butterworth filter from Figure 4. The effect of the sixth derivative cascaded dipole filter on this Butterworth synthetic is shown in Figure 6c.

Initially the character details of the reef are hidden as undulations in the shape of the waveform between the strong peaks on the synthetic. By taking the derivatives of Figure 6b to get Figure 6c we have separated the finer local events. The high-frequency "ripples" can be seen and can be more easily identified and interpreted.

Implications for Interpretation

These examples suggest that one might try the dipoles on seismic data until one gets the desired resolution or until the noise at higher frequencies becomes too strong to see coherent events. In practice, laterally incoherent noise may be reduced with FX deconvolution methods between multiple applications of the dipole.

On the seismic section the resulting improvement in resolution will move the onset of thin-bed tuning to a thinner bed thickness. Depending on the target thickness and the achieved wavelet resolution the interpreter may not need to switch between mapping strategies for time-based resolution to amplitude-based resolution of thin beds. Doublet events which could not be autopicked on a workstation may be split into separately pickable events.

Interpretation Strategy

The geophysicist attempts to retain a wide range of frequencies throughout the processing flow. The resulting final migrated section usually has a broadband, possibly flat spectrum of at least two octaves with good signal to noise ratio. The post-deconvolution interpretation wavelet is approximately zerophase, or made so by whatever method preferred by the geophysicist.

The selection of the final bandwidth and spectrum shape affect the limits and strategy used for resolution and interpretation of layers (see Appendix for further discussion). At any stage in detailed mapping, one might want to further refine the wavelet resolution. The dipole filters offer a method to enhance the effective higher frequencies and extend the limits of resolution of layer boundaries.

Future Work

We want to demonstrate the practical limitations of the dipole effects on several types of data and with different types of targets. We hope to show examples of the effects of these dipoles on the effective resolution of seismic data and the resulting refinement in the mapping of exploration or development targets.