The Inconvenient Truth About High Frequencies
Practitioners in the marine seismic industry have focused upon the benefits of enhanced low-frequency amplitudes in recent years: benefits to improved velocity model building with FWI (full waveform inversion), enhanced quantitative prediction of subsurface rock and fluid properties, and cleaner seismic event character for seismic interpretation.
In some sense, the benefits of enhanced high-frequency amplitudes has less prominence, partly because the inescapable effects of absorption and dispersion are seen as insurmountable. I revisit this topic and show some encouraging solutions to enhance high-frequency signal content in seismic images.
Temporal Frequency Loss: Attenuation Effects
The introductory figure above is an unfortunate reminder that the resolution of seismic images is extremely coarse. At exploration targets in the range of 3 or 4 kilometers below the surface, the dominant seismic wavelength will be several tens of meters, so using a well-known rule-of-thumb that the visible resolution is about one-quarter of the dominant wavelength, this translates to a resolution of 10 or more meters.
At the physical outcrop scale, such as the exposed cliff face above, we can see and touch thin geological strata only millimeters of centimeters in thickness. However, seismic reflections from subsurface geology (superimposed to scale on the photo) are only sensitive to gross contrasts in the 'acoustic impedance' (impedance = seismic velocity x rock density) from packages of geological strata. How can we improve the sensitivity of seismic imaging to thinner sequences of geological strata?
Any effort to recover a larger range of frequencies in seismic images from seismic data is confronted by many phenomena, as schematically illustrated below. I will assume some knowledge of fundamental seismic wavefield propagation terminology.
Although the combined effects of the source-side and receiver-side ghosts penalize the low and high frequency amplitudes quite dramatically, other factors are also relevant, including the band-limited nature of the recording system, attenuation, spatial aliasing associated with multi-channel processing and imaging algorithms, introduced processing artifacts, mechanical noise modes, and so on.
If you are interested in more detail, I wrote a 5-page document titled "High Resolution Marine Seismic: A Short Webinar Series" that accompanies a related four-part series of short webinars.
Although the ultra-low frequency range of 0-10 Hz has received quite significant industry attention in recent years, being of particular relevance to quantitatively accurate seismic inversion, the high frequency end has been rather neglected in the so-called 'broadband seismic revolution'.
Whilst denser acquisition geometry with smaller bin dimensions in processing can enable the unaliased preservation of somewhat higher frequencies from the start to the end of the processing and imaging workflow, there is no escape from the fact that various attenuation and scattering phenomena in the earth rapidly remove higher frequency amplitudes with increasing depth below the seafloor.
Indeed, it is not uncommon for marine broadband seismic data after deghosting and low frequency shaping (often referred to as 'designature') to have maximum amplitude in the 10-20 Hz range at target level – before any effort is made to recover attenuated higher frequency amplitudes.
Conventional marine seismic image including all ghost effects (upper) vs. fully deghosted broadband seismic (lower). The multisensor GeoStreamer platform has noticeably removed the various ghost events associated with each impedance contrast in the lower image, improving the seismic representation of many small features. 3 m vertical resolution is achieved at the wedge pinchout annotated by yellow arrows. The frequency range for this shallow window is 2-200 Hz, but this decays rapidly with increasing depth due to the inescapable effects of attenuation.
It may be popular in the 'broadband seismic' community to use examples such as that above to showcase the often dramatic improvements in image clarity and resolution after deghosting, but it should be noted that these images are from a window reasonably close to the seafloor and therefore attenuation has had little effect upon the high frequency content.
Claims of '2-200 Hz' or almost eight octaves of bandwidth are not representative of the (much) smaller recoverable frequency bandwidth at typical target depths.
Unfortunately, there exists no acquisition-based solution for injecting and recovering extremely high frequencies from deep targets. The only solution, albeit rather limited, is to recover higher frequency amplitudes in processing and imaging—a challenge confronted by the fact that the signal-to-noise ratio (SNR) tends to decay rapidly towards high frequencies and high frequency wavefronts are easily distorted during propagation through the earth.
Improving Temporal Resolution by Q Compensation
A common rule of thumb for vertical or temporal resolution is:
Provided at least two octaves of frequency bandwidth are available in the seismic data, the maximum temporal resolution is proportional to the highest 'recoverable' frequency—ultimately determined by the SNR
Whilst strong ultra-low frequency amplitudes with stable phase behavior (often overlooked) will suppress the side lobes on zero phase reflection wavelets, thereby minimizing interference/tuning between thin layers, the highest recoverable frequency determines the minimum resolvable bed thickness.
The most common approach to spectral recovery from attenuation is 'Q compensation' wherein Q, the 'quality factor', is a term that describes the frequency-dependent rate of attenuation.
According to a given Q model, either simple deterministic or based upon some kind of tomographic solution, amplitudes and phase are corrected in a frequency- and time-dependent manner.
The phase correction is required as attenuation creates frequency-dependent time shifts that degrade event resolution and focusing. As SNR inevitably decreases with increasing frequency it is usually observed that Q compensation can tend to 'blow up' higher frequency noise, and a remedial filter is applied to suppress the high frequency content of the data after Q compensation.
Industry best practice in the past has consequently applied the phase component of Q compensation before (pre-stack) migration and the amplitude component after migration as a pragmatic way to minimize high frequency noise being exaggerated.
The most correct platform for Q compensation in acoustic media is 'viscoacoustic migration' wherein the frequency-dependent amplitude and phase corrections are applied within the migration operator.
Horizontal slice through a 3D wavefront being propagated in a constant velocity model. In contrast to acoustic modeling (upper left), incorporation of the amplitude term related to attenuation (upper right) reduces amplitude with increasing propagation distance, incorporation of the phase term related to attenuation (lower left) affects phase and therefore travel time with increasing propagation distance, and viscoacoustic modeling includes both the amplitude and phase terms. Correspondingly, viscoacoustic migration accounts for both (frequency-dependent) amplitude and phase changes within the migration operator.
In the case of viscoelastic Kirchhoff migration this means the travel time tables are complex valued, and viscoelastic wavefield extrapolation solutions (i.e., finite difference-based migration solutions) apply the Q term within the dispersion relation.
In addition to the migration operator being a powerful (random) noise filter when the amplitude component of Q compensation is applied this way, 3D wavefield propagation effects are naturally accounted for. The figure above illustrates the respective amplitude and phase components of wavefield propagation.
The figure below provides an example of a stack before and after viscoelastic Kirchhoff migration (sometimes generically referred to as 'Q migration').
Viscoelastic TTI anisotropic Kirchhoff pre-stack time migration (PSTM) stacks with no attenuation compensation (upper) and with Q compensation (lower). Note the improved resolution at depth after stable Q compensation within the migration operator.
Regards the "TTI" (Tilted Transverse Isotropy ) terminology, refer to my article titled "Anisotropic Seismic Imaging".
The example above also makes the point that the benefits of Q compensation increasingly become obvious with increasing depth below the seafloor (there are no attenuation effects within the water column so attenuation does not really affect shallow events).
Going Beyond Simple Temporal Frequency Compensation
To be more explicit, Q compensation compensates for degraded spatial wavenumbers, but does not account for acquisition geometry or account for illumination variations in the subsurface.
As I discuss in my article titled "Least Squares Migration: 1 of 2 Articles", Least Squares Migration (LSM), which comes in many forms, will compensate for imperfect acquisition geometry, resolve shadow zones on seismic images, and enhance the spatial wavenumber content--resulting in sharper images of discontinuities (e.g., faults and fractures) and higher resolution imaging of dipping features.
By comparison to traditional migration (upper), which often fails to recover the true earth reflectivity in areas affected by subsurface illumination variations and imperfect acquisition geometry, LSM (middle) recovers the reflectivity much closer to the true model (lower). If implemented correctly, local image focusing should be much sharper (e.g. the point diffractors embedded into the synthetic data model), and spatial resolution should be better due to the improvement in high wavenumber content (e.g. local event terminations and fault plane imaging).
Overall, we can see that both Q compensation and LSM are broadening the (temporal and spatial) frequency content of seismic images: Q compensation on the high temporal frequency end and LSM on the high spatial frequency end.
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