High-Density 3D (HD3D) EAGE Workshop 092004 Andrew Long
The relationships between pre-stack migration
resolution and 3D spatial sampling with the
High Density 3D method
PGS Marine Geophysical/Geophysical Support/ 1060 Hay Street, West Perth, WA, 6005, Australia
High Density 3D (HD3D) acquisition pursues a straightforward strategy: The use of dense trace
acquisition and tight 3D spatial sampling to provide an optimal platform for subsequent processing
operations. In recent years, several HD3D surveys, both land and marine, have provided substantial
improvements over the existing data. The following discussion attempts to isolate the significance of
certain key elements in the HD3D story, notably the issue of resolution after migration. We acknowledge
that several issues contribute to improved temporal and spatial resolution, improved imaging quality, and
improved data repeatability. When observing an improvement in data quality, both acquisition and
processing technology must be considered, and it is unfair in any circumstance to attribute all benefits to a
specific technology or parameter. For brevity, various acquisition components are discussed in the
context of improved “resolution”, before focus is given to the issue of spatial sampling vs. pre-stack
Spatial Sampling vs. Migration Resolution
Seismic resolution requires maximization of the spectral bandwidth after migration. The link between
acquisition parameters and resolution is often dictated by simple geometrical relationships. For example,
the dip spectrum of the geological structure is mostly limited by the spatial extent of the acquisition
geometry, and the reflection angle range is limited by the maximum offset where pre-critical reflections
are recorded. Vermeer (1999) demonstrates that spatial resolution is proportional to maximum horizontal
wavenumber (kx), and the minimum resolvable distance is inversely proportional to maximum
wavenumber. The limited temporal bandwidth of the seismic signal mostly determines vertical
resolution. However, vertical resolution of dipping and wide reflection angle components is linked to the
temporal bandwidth and the spatial sampling rate in less than obvious ways.
For stationary plane waves, the well-known expression for the Nyquist frequency is (Equation 1):
f Nyquist = (1)
4.∆x. sin ϑ
where ∆x is the spatial sampling interval, and ϑ is the reflector dip for the constant velocity, zero-offset
case. Note that in a more realistic application, ϑ is the emergence angle of the upgoing (plane) wave, and
consequently, will vary by a trivial amount for updip vs. downdip shooting. Furthermore, velocity
increasing with depth means that the zero offset emergence angle is less than the target dip, so the
Nyquist frequency increases with respect to the above expression. As source-receiver offset increases,
Nyquist frequency decreases.
The generalisation of plane wave anti-aliasing approximations to multi-dimensional migration operators
with curved surfaces becomes inadequate when the summation trajectory has such strong curvature that it
EAGE Fall Research Workshop on Advances in Seismic Acquisition Technology
Rhodes, Greece, 19 - 23 September 2004
cannot be considered planar across a Fresnel zone of the data. In the discussions here, only plane wave
effects are considered.
As described by Biondi (2001), there are three types of aliasing in the context of migration processing:
Data, operator, and image space aliasing. Data space aliasing is the well-known aliasing addressed by
standard Nyquist considerations. Operator space aliasing is generally specific to Kirchhoff algorithms,
and is a function of the local time derivatives of the operator summation surface (often called operator
dip), and of the frequency and dip bandwidth of the data. Artifacts are generated when data that are not
aligned with the summation path stack coherently into the image. This phenomenon is often described as
operator aliasing, and the artifacts are called aliasing noise. Yilmaz (1987) illustrates pictorially how
operator aliasing “blows up” the migration image, rapidly destroying data quality and resolution. As
discussed later, low-pass “anti-alias” (AA) filtering is used to avoid operator aliasing in practice. Image
aliasing occurs when the spatial sampling of the image is too coarse to adequately represent the steeply
dipping reflectors that the imaging operator attempts to build in the imaging process. The only remedy to
data aliasing is to sample better in the field. Most noise is coherent, and 3D in nature.
Figure 1 illustrates that as the spatial sampling interval increases, the spatial bandwidth decreases.
Dipping events “wrap around” in FK space, contaminating low-dip data. The solution is to increasingly
band-limit the frequencies for increasingly steep data. Note that the availability of a larger spatial
bandwidth does not improve the frequency bandwidth (or temporal resolution) of wavenumbers that are
already unaliased. However, the frequency bandwidth that is unaffected by aliased energy is increased
for those wavenumbers. Again, if anti-aliasing processing is perfectly successful, contamination by
aliased dipping events will not be problematic. In that case, dipping events simply benefit from an
increased spatial and temporal bandwidth, as spatial sampling interval decreases.
Migration will have three main effects: 1. Increased dip for dipping events, 2. Focussing of diffractions,
and 3. Smearing of noise. The first two effects are explicitly linked to Nyquist considerations. We desire
a large frequency bandwidth for the reconstruction of primary events, and we desire crisp resolution of
event truncations and discontinuities. The third effect is a function of the residual noise types persisting
after pre-migration processing, and any noise artifacts arising from the migration processing itself.
In terms of resolution, the “dispersion relation” (Claerbout, 1985) links the (x, y, z) spatial frequencies,
the temporal frequency (ω), and velocity (V) for frequency domain migration as follows (Equation 2):
k x2 + k y + k z2 − ( ) 2 = 0
This can be recast to provide the frequency after migration as (Equation 3):
ω out = ω in − V 2 (k x2 + k y )
For zero kx and ky (flat dips), the input frequency will be unchanged at output. As the (apparent) dip
increases, and kx and ky increase, then the output frequency decreases (refer to Figure 1). This fits the
previous Nyquist discussions, and demonstrates how frequency domain migration algorithms
“automatically” avoid operator aliasing. Obviously, if the unaliased (data) frequency going into
migration is larger (tighter 3D spatial sampling), then the output frequency will also be larger. As
described in Robein (2003), a large range of specular (obeying the laws of reflection at the subsurface
image point) reflection angles are available for all subsurface image locations, complemented by tight
spatial sampling being available for all source-receiver azimuths. The former pursuit is essentially a 3D
illumination issue, and is addressed by shooting strategies such as sail line overlap (Hoffmann et al.,
2002) or multi-azimuth shooting (Hegna and Gaus, 2003). Note that steeper dips are typically illuminated
less consistently, and with a smaller range of specular reflection angles, than flat dips. This will
compound the degraded imaging and resolution of steep dips, which have already been “penalised” by a
smaller available spatial and temporal bandwidth going into migration.
When spatial sampling improves, aliasing becomes less of a problem (low-dip events are relatively
unaffected), and steeper-dip events benefit in terms of both spatial and temporal resolution. Imaging
quality and S/N ratio must also be considered (below). Note that conventional 3D seismic acquisition
samples the shot and receiver domains quite densely in the inline (shooting) direction, but the cross-line
receiver direction is more coarsely sampled by a comparative factor of between 8 and 12:1, and the cross-
line shot direction is more coarsely sampled by a comparative factor of between 3 and 30:1 (typically
about 10:1). The HD3D (marine) method typically reduces the comparative cross-line receiver sampling
to a factor of 3 or 4:1. Shooting strategies such as sail line overlap reduce the comparative cross-line shot
sampling by a factor appropriate to the degree of overlap. The point made here is that HD3D acquisition
can provide something close to true “3D symmetry”, where tight inline spatial sampling in the common
midpoint (CMP) domain will not be degraded or compromised by coarse cross-line spatial sampling, as is
the case for “standard” 3D acquisition.
Figure 1. FK spectra of the wavenumber and frequency range available for migrated data with
different spatial sampling frequency. Modified from Yilmaz (1987).
Finally, we acknowledge that dense trace acquisition benefits the post-migration S/N ratio, in the manner
described by Krey (1987). If the 3D spatial sampling improves, then the S/N ratio for any given velocity
or frequency will increase. Improved velocity analysis contributes to better temporal and spatial
resolution, Q compensation for higher frequency recovery benefits from higher S/N ratio, and data is
naturally more reliably interpreted. Overall, it is difficult to quantify the comparative imaging quality and
resolution of data that has different spatial sampling, as there are competing contributions in terms of
improvements in the relief of coherent events (both “signal” and “noise”) vs. improvements in spatial
bandwidth (and, as we saw, improvements in temporal bandwidth for dipping events). Figure 2 provides
a multi-streamer case example, where 13.33 x 26.66 m vs. 6.25 x 12.5 m CMP sampling is compared.
There are many issues contributing to the difference in data quality. Detailed analysis (Long, 2003)
demonstrates that a significant benefit to resolution and S/N ratio arises from the HD3D approach.
How dense should we go in acquisition? Modern acquisition technologies (multi-streamer, OBC, and
land) allow symmetric inline and cross-line CMP sampling, at the scale of typical inline receiver
sampling, without incurring much additional survey time. Every effort should be made to avoid aliasing
primary events and notably, noise, which is typically far more prone to aliasing, and cannot be removed
without proper 3D spatial sampling. Given the marginal cost penalty vs. the data quality benefits, HD3D
acquisition at the operational limits should always be considered during pre-survey planning.
EAGE Fall Research Workshop on Advances in Seismic Acquisition Technology
Rhodes, Greece, 19 - 23 September 2004
Figure 2. 13.33 x 26.66 m CMP bins (left: 95,000 traces/km2) vs. 6.25 x 12.5 m CMP bins (right:
691,000 traces/km2. High-density (HD3D) data acquisition (right) provided many improvements.
Theoretically, the relationships between spatial sampling and migration resolution are straightforward,
although the isolation and quantification of the improvements observed in migration quality with denser
trace acquisition are less straightforward. This is because denser 3D trace sampling manifests itself in
overlapping and intertwined ways: Improved S/N ratio is both a function of less (Kirchhoff) migration
operator aliasing and the “stacking” power of the migration process. Increased spatial bandwidth also
translates to increased (unaliased) temporal bandwidth for dipping events, and we cannot forget that all
other multi-channel processing operations preceding migration will have benefited in various manners
from better sampling too. It is always critical that noise be unaliased and fully described in 3D if
processing can successfully remove all noise types. Otherwise, image quality and resolution is destroyed.
Frustratingly, an estimate of the likely temporal and spatial bandwidths during any pre-survey planning
exercise is not necessarily a quantifier of data resolution and interpretability. However, as there are
roughly linear relationships between bandwidth and resolution, analysis of any existing data should be
useful in appreciating what comparative benefits can be achieved by different trace densities during new
acquisition. Symmetric sampling has become a realistic pursuit in the CMP domain (typically 12.5 x 12.5
m, for both land and marine), although the shot domain remains coarsely sampled in the cross-line.
Biondi, B., 2001. Kirchhoff imaging beyond aliasing: Geophysics, 66, 654–666.
Clærbout, J.F., 1985. Imaging the Earth's Interior: Blackwell Publications.
Hegna, S., & Gaus, D., 2003. Improved imaging by prestack depth migration of multi-azimuth towed
streamer seismic data: Annual Meeting Abstracts, EAGE, Paper C-02.
Hoffmann, J., Rekdal, T., & Hegna, S., 2002. Improving the data quality in marine streamer seismic by
increased cross-line sampling: Annual Meeting Abstracts, SEG, Session ACQ 3.7.
Krey, T. C., 1987. Attenuation of random noise by Two-D and Three-D CDP stacking and Kirchhoff
migration: Geophysical Prospecting, 35, 135-147.
Long, A., 2003, Marine acquisition: Moving away from the S/N ratio?, First Break, 22, 12.
Robein, E., 2003. Velocities, Time-imaging and depth-imaging: Principles and methods: EAGE
Vermeer, G.J.O., 1999. Factors affecting spatial resolution: Geophysics, 64, 942-953.
Yilmaz, O., 1987. Seismic data processing: Society of Exploration Geophysicists, Tulsa.