# Introduction to velocity analysis and statics corrections

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

A sonic log represents *direct measurement* of the velocity with which seismic waves travel in the earth as a function of depth. Seismic data, on the other hand, provide an *indirect measurement* of velocity. Based on these two types of information, the exploration seismologist derives a large number of different types of velocity — interval, apparent, average, root-mean-square (rms), instantaneous, phase, group, normal moveout (NMO), stacking, and migration velocities. However, the velocity that can be derived reliably from seismic data is the velocity that yields the best stack.

Assuming a layered media, stacking velocity is related to normal-moveout velocity. This, in turn, is related to the root-mean-squared (rms) velocity, from which the average and interval velocities are derived. Interval velocity is the average velocity in an interval between two reflectors.

Several factors influence interval velocity within a rock unit with a certain lithologic composition:

- Pore shape,
- Pore pressure,
- Pore fluid saturation,
- Confining pressure, and
- Temperature.

These factors have been investigated extensively under laboratory conditions. Figure 3.0-1 shows laboratory measurements of velocity as a function of the confining pressure in a Bedford limestone sample with pores in the form of microcracks. The experiment was conducted using enclosed samples to control the pore fluid pressure independent of the confining pressure.

From Figure 3.0-1, we make the following observations:

- Both compressional (
*P*) and shear (*S*) wave velocities increase with increasing confining pressure. More specifically, velocity generally increases rapidly with confining pressure at small confining pressures, then gradually levels off at high confining pressures. The reason for this is that as the confining pressure increases, pores close. However, at a high confining pressure, not much deformable pore space is left. Therefore, any further increase in the confining pressure will not cause a significant increase in velocity. - Note that, regardless of confining pressure,
*P*-wave velocity is greater than*S*-wave velocity. This is true for any rock type. - The saturated rock sample has a higher
*P*-wave velocity than the dry sample at low confining pressure. At high confining pressures,*P*-wave velocity in the dry sample approaches the magnitude of the*P*-wave velocity in the saturated sample. - Note also that the
*P*-wave velocity in the saturated sample does not change as rapidly as in the dry sample. This is because the fluid is almost as incompressible as the rock. Whether the pores are filled with fluid or not has little effect on*S*-wave velocity, since fluids cannot support shear-wave propagation.

We now examine velocity as a function of confining pressure for an enclosed sample of Berea sandstone with rounded pores (Figure 3.0-2). Again, note the increase in velocity with increasing confining pressure. The important difference between this sample and the one in Figure 3.0-1 is the range of magnitude of the velocity. The rock with microcracks has a higher velocity than the rock with rounded pores at any given confining pressure. The reason for this is that it is easier to close the pores formed as microcracks than it is to close those that are round.

The most prominent factor influencing velocity in a rock of given lithology and porosity probably is confining pressure. This type of pressure arises from the overburden and increases with depth. It is generally true that velocity increases with depth. However, because of factors such as pore pressure, there may be inversion in the velocity within a layer.

Figure 3.0-3 shows the variation of velocity with depth for various types of lithology. We make the following observations:

- Tertiary clastics, which usually are less indurated than other rocks, occupy the low-velocity end of the graph. They generally start out with a velocity that ranges from 1.5 to 2.5 km/s at or near the surface, then gradually increase to from 4.5 to 5.5 km/s at depths greater than 5 km.
- Carbonates with high porosity occupy the central portion of the graph, starting at about 3 km/s and increasing to nearly 6 km/s.
- Carbonates with low porosity, on the other hand, have a smaller range of variation in velocity. If there is not much pore space to close, then the confining pressure cannot cause much of an increase in velocity.

This chapter discusses ways to estimate velocities from seismic data. Velocity estimation requires the data recorded at nonzero offsets provided by common-midpoint (CMP) recording. With estimated velocities, we can correct reflection traveltimes for nonzero offset and compress the recorded data volume (in midpoint-offset-time coordinates) to a stacked section (Figure 1.5-1).

For a single constant-velocity horizontal layer, the reflection traveltime curve as a function of offset is a hyperbola (Normal moveout). The time difference between traveltime at a given offset and at zero offset is called *normal moveout* (NMO). The velocity required to correct for normal moveout is called the *normal moveout velocity*. In the case of an earth model with a single horizontal reflector, the NMO velocity is equal to the velocity of the medium above the reflector. In the case of an earth model with a single dipping reflector, the NMO velocity is equal to the medium velocity divided by the cosine of the dip angle. When the dipping reflector is viewed in three dimensions, then the azimuth angle (between the dip direction and the profiling direction) becomes an additional factor. Traveltime as a function of offset for a series of horizontal isovelocity layers is approximated by a hyperbola. This approximation is better at small offsets than large offsets. For short offsets, the NMO velocity for a horizontally layered earth model is equal to the rms velocity down to the layer boundary under consideration. In a medium composed of layers with arbitrary dips, the traveltime equation gets complicated. However, in practice, as long as dips are gentle and the spread is small (less than reflector depth), the hyperbolic assumption still can be made. For layer boundaries with arbitrary shapes, the hyperbolic assumption breaks down.

There is a difference between the NMO and stacking velocities that often is ignored in practice. The NMO velocity is based on the small-spread hyperbolic traveltime ^{[3]}; ^{[4]}, while stacking velocity is based on the hyperbola that best fits data over the entire spread length. Nevertheless, stacking velocity and NMO correction velocity generally are considered equivalent.

Conventional velocity analysis is based on the hyperbolic assumption. Various methods for velocity analysis are discussed in Velocity analysis. The hyperbolic traveltime equation is linear in the *t*^{2} − *x*^{2} plane, where *t* is the two-way traveltime and *x* is the source-receiver offset. Zero-offset time and stacking velocity for a given reflector can be estimated from the line that best fits the traveltime picks plotted on the *t*^{2} − *x*^{2} plane. Another way to estimate the NMO velocity is to apply different NMO corrections to a CMP gather using a range of constant velocity values, then display them side by side. The velocity that best flattens each event as a function of offset is picked as its NMO velocity. Alternatively, a small portion of a line can be stacked with a range of constant velocity values. These constant-velocity stacks (CVS) can be plotted in the form of a panel. Stacking velocities that yield the desired stack then can be picked from the CVS panel.

Another commonly used velocity analysis technique is based on computing the *velocity spectrum* ^{[3]}. The idea is to display some measure of signal coherency on a graph of velocity versus two-way zero-offset time. The underlying principle is to compute the signal coherency on the CMP gather in small time gates that follow a trajectory in offset. Stacking velocities are interpreted from velocity spectra by choosing the velocity function that produces the highest coherency at times with significant event amplitudes.

Occasionally the stacking velocity variation needs to be determined in detail along a particular reflector. Horizon-consistent velocity analysis provides the stacking velocity variation in the lateral direction along a particular horizon of interest.

Reflection traveltimes are not always hyperbolic in horizontally layered media. One reason that traveltime often deviates from a perfect hyperbola is the presence of static time shifts caused by near-surface velocity variations. Statics can severely distort the reflection hyperbola when there are large surface elevation changes or when the weathering layer varies horizontally. Shown in Figure 3.0-4 is a stacked section that exhibits severe distortions of reflection traveltimes in the susburface, which is known to have generally flat layers, caused by the complexity in the near-surface that is composed of glacial tills with irregular shapes. The traveltime distortions caused by the near-surface also are observed in the CMP gather shown in Figure 3.0-5a. The velocity spectrum derived from the CMP gather that has not been corrected for the near-surface does not exhibit a reliable stacking velocity trend (Figure 3.0-6a). By estimating a model for the near-surface and correcting for its effects on the reflection traveltimes in the subsurface using the refracted arrivals from the near-surface, the resulting CMP gather (Figure 3.0-5b) yields a more accurate estimate of stacking velocities (Figure 3.0-6b). The CMP stack derived from the CMP gathers with statics corrections exhibits reflection traveltimes free of the near-surface distortions (Figure 3.0-7). A close-up of portions of the CMP stack without (Figure 3.0-4) and with (Figure 3.0-7) statics corrections clearly demonstrates the improvement achieved by the statics corrections as shown in Figure 3.0-8.

Residual statics variations usually remain in the data even after initial corrections for estimated weathering layer variations and elevation changes (field statics) (Refraction statics corrections). Corrections for residual statics normally must be estimated and applied to CMP gathers before stacking. Estimation is done after a preliminary NMO correction using either a regional velocity function or information from a series of preliminary velocity analyses along the line. Following the residual statics corrections, velocity analyses usually are repeated to revise the velocity picks for stacking. Various aspects of residual statics corrections are discussed in residual statics corrections and refraction statics corrections.

As a final note, velocities required by stacking and migration are not necessarily the same. In fact, for data collected parallel to the dip direction of a single dipping reflector, stacking velocity is the velocity of the medium above the reflector divided by the cosine of the dip angle, while migration velocity is the velocity of the medium itself. In other words, stacking velocity is dip-dependent, while migration velocity is not. Migration velocity estimation is discussed in migration velocity analysis.

## See also

- Normal moveout
- Velocity analysis
- Residual statics corrections
- Refraction statics corrections
- Exercises
- Topics in moveout and statics corrections

## References

- ↑
^{1.0}^{1.1}Nur, 1981, Nur, A., 1981, Physical properties of rocks: Soc. Expl. Geophys. Continuing Education Course Notes. - ↑ Sheriff, 1976, Sheriff, R. E., 1976, Inferring stratigraphy from seismic data: Am. Assn. Petr. Geol. Bull., 60, 528–542.
- ↑
^{3.0}^{3.1}Taner and Koehler, 1969, Taner, M. T. and Koehler, F., 1969, Velocity spectra — digital computer derivation and applications of velocity functions: Geophysics, 39, 859–881. - ↑ Al-Chalabi, 1973, Al-Chalabi, M., 1973, Series approximations in velocity and traveltime computations: Geophys. Prosp., 21, 783–795.