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Seismic Migration

After collecting seismic data and processing to obtain a stacked section, the data, still, may not represent the layers of earth since the actual reflection points are unknown. At this stage, the wave propagation effects should also be taken into account to determine the correct reflection point of the seismic waves (Fig. 1). This is done using a process known as Seismic Migration. Migration is the process of assigning the correct surface locations in terms of vertical depth and time to their corresponding reflection events. Migration corrects the effect of dipping reflectors by which the seismic data are misreported.

The basis of time migration is Huygens’ Principle (Mousa and Al-shuhail 2009). For applying this principle to reflection seismology, each point on a reflector (geological interface) is considered as secondary source in response to the incident wave field. This is known as the exploding reflector model which is the basis for all travel time correction techniques. Three most important methods for travel time corrections in seismic processing are: normal move out (NMO), dip move out (DMO) and migration (Liner 2004) (Fig. 2). NMO reduces time so that the reflection point would have been recorded at the midpoint of source and receiver by removing the travel time effect of offset. DMO adjusts times to normal incident path which goes through the original reflection point. On the other hand, Migration assigns data to the vertical travel path through the reflection point.

Time Migration

When doing migration, the obtained seismic cross sections should be migrated in which the seismic line is normal to the strike so that the dip move out indicates true dip. Time migration is done by assuming the constant vertical velocity to the reflector and drawing an arc with radius equal to the half of arrival time multiplied by the average velocity (Fig. 3) (Telford, 1990).

Migration can be performed in time or depth domain before and after stacking (Liner 2004). Migration of seismic section in time domain is called time migration which gives the accurate measure of reflection points in constant velocity. Time migration is capable to solve simple problems in which the velocity is constant or only depends on depth (Fig. 4). For more complex problems in which lateral change in velocity exits, depth migration is required. Time migration is a fast and easy way to analyze velocity but it does not give accurate measure in the area where there is complex geology and great variation in lateral velocity (Liner 2004). Since seismic velocity increases with travel time and the lateral variations in geology are gradual in most of the sedimentary basins, time migration is applicable everywhere.

Time Migration Techniques

Many migration techniques have been introduced in literature. However, the most important time migration techniques which also dominate seismic processing in industry are Kirchhoff, Fourier and Downward Continuation. These techniques render quick and robust velocity analysis. Two widely used Fourier transform time migration methods are known as Stolt and Gazdag. The simplest forms of these techniques are used for time migration of constant velocity models; however these methods can be extended for migration of models in which velocity only changes with depth.

Stolt technique or F-K migration

Stolt technique (also known as F-K migration and phase shift migration) is the fastest of all migration methods. This method is applicable to both poststack and prestack data. Since Stolt migration is based on Fourier transform of time axis and time sampling, it requires that the data have uniform trace spacing which is commonly accomplished by data interpolation (Liner 2004). This method makes use of the 2-D Fourier transform to convert the input section into the 2-D Fourier domain where it is migrated with a simple algorithm. The inverse transform provides the migrated structure (Fig. 5). The F denotes the Fourier transform of time, and K denoted the Fourier transform of space or distance (Bancroft 2007).

Consider a 2D constant velocity wave equation (Liner 2004):

(∂^2 p)/(∂x^2 )+(∂^2 p)/(∂z^2 )-1/v^2  (∂^2 p)/(∂t^2 )=0            (1)

In which p(t,z,x) is a time-dependent wave field in 2D (z,x) domain. Taking the Fourier transform of this equation with respect to all three coordinates renders the mapping of the equation from physical space (t,x,z) into Fourier space (ω,k_z,k_x ) in which (k_z,k_x ) are vertical and horizontal wavenumbers and the result is:

(-k_x^2-k_z^2+ω^2/v^2 )p(ω,k_z,k_x )=0                              (2)

In this equation since p(ω,k_z,k_x )=0 is trivial and means that the wave field is zero, the only possibility would be: -k_x^2-k_z^2+ω^2/v^2 =0                                             (3) Equation (3) gives the relationship between transform variables of (t,z,x) and indicates that for data that adhere to wave equation, (w,k_z,k_x ) are not independent variables, hence vertical wavenumber can be expressed by:

k_z=ω/v √(1-((vk_x)/ω)^2 )

If we have a zero offset section or CMP stack data as a function of time and CMP coordinate, p(t,x); taking the 2D Fourier transform maps data into frequency-wavenumber space p(ω,k_x ). Now if assume a velocity for the model, then equation (4) gives the k_z value at every point. An inverse Fourier transform of the calculated p(k_z,k_x ) will give the migrated data p(z,x).

Kirchhoff method

The Kirchhoff method has descended from one of the oldest methods of migration, the diffraction stack (Bancroft 2007). It is considered by many to be the best. Its implementation may vary from a very simple algorithm to one that is complex. Kirchhoff migration can be used for both time and depth migration. This method is capable of handling irregular data (not uniform trace spacing) and is used in 3D seismology. Kirchhoff can be applied to both prestack and poststack data (Liner 2004). Schneider in 1978 showed the diffraction sum method could be an exact solution to the wave equation if scaling and filtering were included. His method was based on the Kirchhoff integral solution used in optics. The term Kirchhoff migration has been used since then for all algorithms that used the summation method.

In Kirchhoff method, for every migrated sample, energy is summed along the diffraction, or hyperbolic paths, on the input section. The summed value becomes the amplitude value at the output location (Fig. 6). Additional scaling and filtering may be required for the data to represent its true location in time or space.

Kirchhoff time migration is used when the velocities of a structure vary smoothly or depth migration is not really required. It can be used as a starting point in defining a structured model for depth migration or wherever the structure is too complex for depth migration.