Seismic Migration in geophysics
It is obvious that after the Normal Move Out (NMO) correction and stacking, only a zero-offset section is synthesized, and the fact that the offset dependence of the receiver position with respect to the source has been removed.
Migration deals with a further removal of wave phenomena via focussing in order to arrive at a section which is a true representation of the subsurface.
Migration is the focussing process which results in a true image of the subsurface from primary-reflection data, assuming the velocity model is correct.
Migration is the process of reconstructing a seismic section so that reflection events are repositioned under their correct surface location and at a correct vertical reflection travel time.
Equivalently, this mean that migration obtains the true image in (x, y, z) from seismic data that are obtained in (x, y, t), where x, y and z stand for the two horizontal and vertical coordinate, respectively; again, under the assumption the velocity model is correct.
BASIC PRINCIPLES OF MIGRATION
The principles of migration are shown in the figure below. A single point diffractor will generate a hyperbolic trajectory in a travel time section. Migration collapses the hyperbolic event back to its apex.
Let us discuss three typical cases regarding Migration which are commonly observed from field practices, these are point diffractor, dipping reflector and a syncline. However we have to note that usually much more complicated structures may be considered in complex field observation.
Let us start with the simple example of a point diffractor in the subsurface. A point diffractor is like a ”ball” in the subsurface: when wave energy impinges on it, it scatters (reflects) energy back in all directions. When the source and receiver are at the same point at the surface, the receiver will only receive the ray that is scattered back as shown in figure. So notice that even not right above the diffractor, we will receive energy.
The (zero-offset) time section for a diffractor at position (xd, zd) is described by:
T² = [2R/c]² = Td² + 4(xs - xd)²/c²
This time section is (again) a hyperbola. As may be clear now, a zero-offset section is not a good representation of the subsurface, since that should be the left picture in figure . The process that converts the right picture (hyperbola) into the left picture (ball) is called seismic migration.
Figure: A diffractor (left) and its seismic response (right), a hyperbola in the zero offset section
Figure: 4 point diffractors and its corresponding seismic response.
For the case of dipping reflector, when we consider any reflector, then by putting all point diffractors on that reflector while keeping the spacing between the point diffractors infinitely small, the responses become identical.
Let us now look at a full dipping reflector. Of course, it has some of the characteristics as we saw with the previous point diffractors above, only with a full reflector we no longer see the separate hyperbolae. Actually, we will only see the apparent dip. As we saw with the point diffractors, we need to bring the reflection energy back to where they came from, namely the apex of each hyperbola. When connecting all the apexes of the hyperbolae, we get the real dip as shown in figure below.
Figure: Relation between the reflection points in depth (a) and the traveltimes in the zero offset section (b) for a dipping reflector (from Yilmaz, 1987).
Then our last commonly typical observed case is the so-called ”bow-tie” shaped zero offset
response, which is due to synclinal structures in the earth. This is shown in figure, where it can be observed how in the middle above the syncline multi-valued arrivals are present.
Figure: A syncline reflector (left) yields ”bow-tie” shape in zero offset section (right).
CONCEPT OF DIFFRACTION STACK
So far, we haven’t described how to migrate a full dataset like the one shown in figure. The simplest case of a migration is adding (stacking) the data along hyperbolae. In that case, each point of the section (each time and space point!) is seen as a diffractor. As we saw in the 4-point-diffractor case compared with the dipping-reflector case, any reflector can be synthesized by point diffractors (although infinite).
In case no real diffractor is present, the energy along the hyperbola at that point does not add up constructively and therefore the output signal (=migrated) will be small. This procedure is called a diffraction stack. In the early days of computers the diffraction stack was used to apply the migration.
Pzo(xd, td) = (Sum)Xs Pzo ( xs, t = √[td² + 4(xs - xd)²/c²])
where Pzo stands for zero-offset data and c is the stacking velocity.
What is lacking in the approach of the diffraction stack is the basis on deeper physical principles than (kinematic) ray theory alone.
∂²P/∂t² = V[∂²P/ ∂x² + ∂²P/ ∂z²]
Where P(x, z, t) is the seismic amplitude as a function of reflection time t at any position (x, z) in the subsurface, and
Migration involves a running of the wave equation backward in time, starting with the measured waves at the earth's surface P(x, z = 0, t), in effect pushing the waves backward and downward to their reflecting locations.
All current computer-based approaches to migration involve this backward solution to the wave equation.
1. Kirchhoff Summation Migration
The earliest form, at which in essence, recorded amplitudes on CMP traces are summed along the diffraction trajectories dictated by the assumed subsurface velocity distribution, and the sums are placed at the apexes of the curves, one curve for each sample point in the output migrated section.
2. Finite Difference Migration
This numerically integrates the wave equation by the method of finite differences to push seismic waves backward into the subsurface.
3. f-k Migration
This is a migration approach which operates via Fourier transforms in the frequency wavenumber (f-k) domain. In general, Fourier transform methods provide elegant means of solving partial differential equations. When applied to migration, this elegance is often complemented by high computational efficiency.
Each of the different approaches has specific advantages such as computational efficiency, accuracy for imaging steep reflectors, and accuracy in the presence of spatial variation of velocity. Also each can produce undesirable processing artifacts related to some limitation in data quality such as poor signal to noise ratio.
From above explanation, let us jump into the facts that the effects of migration are to: Collapse a diffraction hyperbola back to a point, Make dipping structures appear with correct dip angle and remove the bow ties from synclines and shorten anticlines.
To your success!
References
Gardner, G. H. F., ed., 1985, Migration of Seismic Data: Tulsa, OK, Society of Exploration Geophysicists Monograph Series, 462 p.
That means, we have a section as if we did a seismic survey with source and receiver at the same place, thus zero-offset.
Migration deals with a further removal of wave phenomena via focussing in order to arrive at a section which is a true representation of the subsurface.
Migration is the focussing process which results in a true image of the subsurface from primary-reflection data, assuming the velocity model is correct.
Migration is the process of reconstructing a seismic section so that reflection events are repositioned under their correct surface location and at a correct vertical reflection travel time.
Equivalently, this mean that migration obtains the true image in (x, y, z) from seismic data that are obtained in (x, y, t), where x, y and z stand for the two horizontal and vertical coordinate, respectively; again, under the assumption the velocity model is correct.
BASIC PRINCIPLES OF MIGRATION
The principles of migration are shown in the figure below. A single point diffractor will generate a hyperbolic trajectory in a travel time section. Migration collapses the hyperbolic event back to its apex.
Let us discuss three typical cases regarding Migration which are commonly observed from field practices, these are point diffractor, dipping reflector and a syncline. However we have to note that usually much more complicated structures may be considered in complex field observation.
Let us start with the simple example of a point diffractor in the subsurface. A point diffractor is like a ”ball” in the subsurface: when wave energy impinges on it, it scatters (reflects) energy back in all directions. When the source and receiver are at the same point at the surface, the receiver will only receive the ray that is scattered back as shown in figure. So notice that even not right above the diffractor, we will receive energy.
The (zero-offset) time section for a diffractor at position (xd, zd) is described by:
T² = [2R/c]² = Td² + 4(xs - xd)²/c²
where:
R being the distance in a homogeneous medium with velocity c,
Td being the time,
R being the distance in a homogeneous medium with velocity c,
Td being the time,
2zd/c and xs being the surface position of source/receiver.
This time section is (again) a hyperbola. As may be clear now, a zero-offset section is not a good representation of the subsurface, since that should be the left picture in figure . The process that converts the right picture (hyperbola) into the left picture (ball) is called seismic migration.
For the case of dipping reflector, when we consider any reflector, then by putting all point diffractors on that reflector while keeping the spacing between the point diffractors infinitely small, the responses become identical.
This concept agrees with Huygens’ principle as the number of point diffractors the separate diffractors are hardly observable any more, and the response also looks more like a dipping reflector (with some end-point effects) see figure below.
Figure: 32 point diffractors and its corresponding seismic response look like a dipping reflector.
Let us now look at a full dipping reflector. Of course, it has some of the characteristics as we saw with the previous point diffractors above, only with a full reflector we no longer see the separate hyperbolae. Actually, we will only see the apparent dip. As we saw with the point diffractors, we need to bring the reflection energy back to where they came from, namely the apex of each hyperbola. When connecting all the apexes of the hyperbolae, we get the real dip as shown in figure below.
Then our last commonly typical observed case is the so-called ”bow-tie” shaped zero offset
response, which is due to synclinal structures in the earth. This is shown in figure, where it can be observed how in the middle above the syncline multi-valued arrivals are present.
This behaviour can be predicted by considering small portions of the reflected signal, and increasing the dip of each portion of the reflected signal.
CONCEPT OF DIFFRACTION STACK
So far, we haven’t described how to migrate a full dataset like the one shown in figure. The simplest case of a migration is adding (stacking) the data along hyperbolae. In that case, each point of the section (each time and space point!) is seen as a diffractor. As we saw in the 4-point-diffractor case compared with the dipping-reflector case, any reflector can be synthesized by point diffractors (although infinite).
So if each point of the zero-offset time section is seen as a point diffractor (and the velocity is known), we can add data along the particular hyperbola for that point. In case a real hyperbola is present in the observed time section, due to a real point diffractor, energy will be added up constructively to give a relatively large output signal (=migrated) at that point.
In case no real diffractor is present, the energy along the hyperbola at that point does not add up constructively and therefore the output signal (=migrated) will be small. This procedure is called a diffraction stack. In the early days of computers the diffraction stack was used to apply the migration.
In formula form, the diffraction stack is given by (assumed to have a discrete number of x’s, being the traces in a zero offset section):
Pzo(xd, td) = (Sum)Xs Pzo ( xs, t = √[td² + 4(xs - xd)²/c²])
where Pzo stands for zero-offset data and c is the stacking velocity.
From the formula it may be obvious that data are added along hyperbolae for each output point (xd, td), being the apex of the hyperbola for point (xd, td). What we do when stacking along hyperbolae, is actually removing the wave propagation effect from a point diffractor to the source/receiver positions.
A very nice feature about the diffraction stack is that it visualizes our intuitive idea of migration, and is very useful in a conceptual sense. Of course, for this procedure to be effective we need to know the stacking velocity.
What is lacking in the approach of the diffraction stack is the basis on deeper physical principles than (kinematic) ray theory alone.
The final migrated result may be correct in position (if the diffraction responses can be assumed to have a hyperbolic shape, i.e. if the subsurface exhibits moderate variations in velocity), but not in amplitude.
As the diffraction stack described above example below shows an unmigrated stacked section (left) and its corresponding time migrated section (right).
Figure: Stacked section (a) and its time migrated version (b) (from Yilmaz, 1987).
TYPES OF MIGRATION APPROACHES
All of the many methods of doing migration are founded on solutions to the scalar wave equation, a partial differential equation that models how waves propagate in the earth. A simple form of the wave equation is as follows:
As the diffraction stack described above example below shows an unmigrated stacked section (left) and its corresponding time migrated section (right).
TYPES OF MIGRATION APPROACHES
All of the many methods of doing migration are founded on solutions to the scalar wave equation, a partial differential equation that models how waves propagate in the earth. A simple form of the wave equation is as follows:
∂²P/∂t² = V[∂²P/ ∂x² + ∂²P/ ∂z²]
Where P(x, z, t) is the seismic amplitude as a function of reflection time t at any position (x, z) in the subsurface, and
V is the seismic wave velocity in the subsurface, a function of both x and z.
Disturbances initiated by a seismic energy source are assumed to propagate in accordance with solutions to the wave equation.
Migration involves a running of the wave equation backward in time, starting with the measured waves at the earth's surface P(x, z = 0, t), in effect pushing the waves backward and downward to their reflecting locations.
All current computer-based approaches to migration involve this backward solution to the wave equation.
1. Kirchhoff Summation Migration
The earliest form, at which in essence, recorded amplitudes on CMP traces are summed along the diffraction trajectories dictated by the assumed subsurface velocity distribution, and the sums are placed at the apexes of the curves, one curve for each sample point in the output migrated section.
2. Finite Difference Migration
This numerically integrates the wave equation by the method of finite differences to push seismic waves backward into the subsurface.
3. f-k Migration
This is a migration approach which operates via Fourier transforms in the frequency wavenumber (f-k) domain. In general, Fourier transform methods provide elegant means of solving partial differential equations. When applied to migration, this elegance is often complemented by high computational efficiency.
Each of the different approaches has specific advantages such as computational efficiency, accuracy for imaging steep reflectors, and accuracy in the presence of spatial variation of velocity. Also each can produce undesirable processing artifacts related to some limitation in data quality such as poor signal to noise ratio.
From above explanation, let us jump into the facts that the effects of migration are to: Collapse a diffraction hyperbola back to a point, Make dipping structures appear with correct dip angle and remove the bow ties from synclines and shorten anticlines.
To your success!
References
Gardner, G. H. F., ed., 1985, Migration of Seismic Data: Tulsa, OK, Society of Exploration Geophysicists Monograph Series, 462 p.
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