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Time Migration

Seismic processing techniques started to evolve around 1920s. At that time, scientists expected that the signal would travel straight down the subsurface and reflect of the subsurface structures straight back up. There was no expectation of off-vertical reflections. Nevertheless, we currently know that after collecting seismic data and applying various processing techniques; the data, still, may not represent the layers of earth because seismic data do not directly stand for the location of reflector surface. At this stage, the wave propagation effects and acquisition geometry should also be taken into account to determine the correct reflection point of the seismic waves.

Description


Migration is the process of assigning the correct subsurface locations in terms of vertical depth and time to their corresponding reflection events (Fig. 1).

The basis of time migration is Huygens’ Principle. According to this principle, each point on a reflector (geological interface) is considered as a secondary source in response to the incident wave field. This secondary source is known as the exploding reflector model which is the basis for all travel time correction techniques. In its simplest form, 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. 2).





Time Migration Methods
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.

Fourier Transform Migration
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


Stolt technique (also known as F-K migration or phase shift migration) is the fastest of all migration methods. Stolt migration basically uses one forward and one backward 2-D Fast Fourier Transform (FFT) computations. 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. 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.

Consider a 2D constant velocity wave equation.


 * $$\frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial z^2} - \frac{1}{v^2} \frac{\partial^2 p}{\partial t^2} = 0. \qquad \qquad (1)$$

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 $$(t,z,x)$$ into Fourier space $$(\omega,k_z,k_x )$$ in which $$(k_z,k_x)$$ are vertical and horizontal wavenumbers and the result is:

$$ (-k_x^2-k_z^2+\frac{\omega^2}{v^2})*p(\omega,k_z,k_x)=0 \qquad \qquad (2)$$

in this equation since $$p(\omega,k_z,k_x )=0$$ means that the wave field is zero and is a trivial solution, the only possibility would be:

$$ -k_x^2-k_z^2+\frac{\omega^2}{v^2}=0 \qquad \qquad (3) $$

Equation (3) gives the relationship between transform variables of $$(t,z,x)$$ and indicates that for data that adhere to wave equation, $$(\omega,k_z,k_x )$$ are not independent variables, hence vertical wavenumber can be expressed by:

$$ k_z=\frac{\omega}{v} \sqrt{1-(\frac{v k_x}{\omega})^2} \qquad \qquad (4) $$

If we have a zero offset section or common midpoint stack data as a function of time coordinate, $$ p(t,x) $$; taking the 2D Fourier transform maps data into frequency-wavenumber space $$ p(\omega,k_x ). $$ Now if we 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 $$ p(z,x). $$

Kirochhoff Approach


The Kirchhoff method descends from one of the oldest methods of migration, which is known as diffraction stack. Kirchhoff migration can be seen as a general form of Hagedoorn's graphical time migration. Many geophysicists consider Kirchhoff method to be the best migration method. The algorithm of Kirchhoff method may vary from a very simple algorithm to one that is complex. . Kirchhoff migration can be used for both time and depth migration. Kirchhoff time migration is used when the velocities of a structure vary smoothly or depth migration is not really required. 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.

Kirchhoff migration starts with the works of Schneider in 1978, who showed that diffraction sum can approach the solution of wave equation if some other processing methods (e.i. scaling and filtering) were included. Schneider implemented Kirchhoff integral solution for his works. Since then, any migration algorithm that uses this summation is classified as Kirchhoff migration.

In the Kirchhoff method, first a final migration point (output location) is selected. Then a diffraction curve, or hyperbolic path is defined in the output location and the energy within that diffraction shape is summed and assigned to the amplitude value at the output location. Additional scaling and filtering may be required for the data to represent their true location in time or space. This procedure is repeated for each migrated output sample.

==Hagedoorn's graphical time migration method ==

The Hagedoorn’s graphical migration is based on sketching all probable reflection points from which seismic arrivals could have been triggered. This method draws a circle centered at time t=0 of a given seismic trace and a seismic arrival. These circles represent the incident field. We can use this sketch to locate the place where the incident field interacts with the reflector surface. If we repeat Hagedoorn’s method for every arrival on every trace, we get a collection of circles whose envelope depicts the reflector. Clearly, in case a reflector does not exist, these circles will add up and cancel each other.

Time Migration versus Normal moveout and Depth moveout


Three most important methods for travel time corrections in seismic processing are: normal moveout (NMO), dip move out (DMO) and migration. 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 versus depth migration
Migration can be performed in time or depth before and after stacking. Migration of seismic section in time domain is 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. For more complex problems in which lateral change in velocity exits, time migration does not give reliable results. 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. 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 to sedimentary basins.

Using Seismic Unix for time migration
Seismic Unix has various routines for investigating time migration. Stolt migration method is available in Seismic Unix for stacked data or common-offset gathers. We can access the help of Seismic Unix for more information about Stolt migration by typing sustolt:



For investigating the application of sustolt routine, we choose a simple dataset which is sorted into common-offset gathers. We illustrate the content of this dataset using following codes:

We can access the header information of this seismic dataset using surange. The data contain 80 traces. The number of samples per each trace is 501 and the time interval between samples is 4000 microseconds.

For applying time migration using sustolt keyword, we choose the rms migration velocity to be 2000 m/s and distance between adjacent cdp bins as 40 m.

Then, we plot the result of applying time migeration correction.