User:Esmailansari

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

File:Simple migration.png

Figure 1. This illustration shows why migration is needed for correcting data. In this figure, true location of reflector is unknown (black line) and needs to be calculated using seismic traces. S/R (Source/Reciever) shows that the offset for each trace is zero. If migration doesn't apply to the data, each normal incident wave will indicate a point with the travel time distance below the subsurface (red line). This will result in calculating a smaller dip for the reflector. See figure (2).

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).

File:Simple migration2.png

Figure 2. Correction of arrival time for each reflector by drawing an arc with radius equal to the half of arrival time multiplied by the average velocity. The effect of migration on the slope line segment in the data is to make it steeper, shorter and move it up-dip. see Hagedoorn's graphical time migration.

File:Complex migration.png

Figure 3. Classical "bowtie" feature seen over a syncline. The red digram shows the seismic data. This diagram does not represent the image of subsurface but it contains information about the subsurface reflector characteristics (e.g. dip). The blcak diagram represents correction of arrival time for each reflector by drawing an arc with radius equal to the half of arrival time multiplied by the average velocity. See Hagedoorn's graphical time migration.

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

File:Stolt method.png

Sketches illustrating Stolt (F-K) migration. In this figure FFT represents Fast Fourier transform. In Stolt migration methods, first, the data are mapped into fourier domain. Then the migraion of data takes place in this domain with a simple algorithm. Finally, an inverse transform maps the data back to seismic data's initial domain. .

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:

$k_{z}={\frac {\omega }{v}}{\sqrt {1-({\frac {vk_{x}}{\omega }})^{2}}}\qquad \qquad (4)$

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

File:Kirchhoff migration.png

Sketch illustrating Kirchhoff migration. In the Kirchhoff method, first a final migration point (output location) is selected. Then a diffraction curve (black hyperbola) is defined in the input and the energy within that diffraction shape is summed. Afterwards, the summed value is inserted at the migration position. This procedure is repeated for each migrated output sample.

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

File:Hagedoorn method.png

Hagedoorn's method corrects data which is acquired from synthetic seismogram by sketching a circle, centered at t=0 and passing through the maximum amplitudes on each arrival on each trace. The red dashed line shows the interpreted location of the reflector which is the envelope of the circles.

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

Travel time corrections to seismic data. Path 1 shows raw travel time from the datum plane which is corrected by Normal Move Out (path 2), Depth Move Out (path 3) and Migration (path 4)

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:

$ sustolt

Simple seismic data before time migration. The data are sorted into common-offset gathers. Before applying the migration, the data don't represent the true location of the reflectors.

The simple seismic data after applying time migration. Without employing time migration, the complex structure of the subsurface is not clear.

 SUSTOLT - Stolt migration for stacked data or common-offset gathers	
									
 sustolt <stdin >stdout cdpmin= cdpmax= dxcdp= noffmix= [...]		
									
 Required Parameters:							
 cdpmin=		  minimum cdp (integer number) in dataset	
 cdpmax=		  maximum cdp (integer number) in dataset	
 dxcdp=		  distance between adjacent cdp bins (m)	
									
 Optional Parameters:							
 noffmix=1		number of offsets to mix (for unstacked data only)
 tmig=0.0		times corresponding to rms velocities in vmig (s)
 vmig=1500.0		rms velocities corresponding to times in tmig (m/s)
 smig=1.0		stretch factor (0.6 typical if vrms increasing)
 vscale=1.0		scale factor to apply to velocities		
 fmax=Nyquist		maximum frequency in input data (Hz)		
 lstaper=0		length of side tapers (# of traces)		
 lbtaper=0		length of bottom taper (# of samples)		
 verbose=0		=1 for diagnostic print				
 tmpdir=		if non-empty, use the value as a directory path	
			prefix for storing temporary files; else if the	
			the CWP_TMPDIR environment variable is set use	
			its value for the path; else use tmpfile()	
									
 Notes:								
 If unstacked traces are input, they should be NMO-corrected and sorted
 into common-offset  gathers.  One common-offset gather ends and another
 begins when the offset field of the trace headers changes. If both	
 NMO and DMO are applied, then this is equivalent to prestack time 	
 migration (though the velocity profile is assumed v(t), only).	
									
 The cdp field of the input trace headers must be the cdp bin NUMBER, NOT
 the cdp location expressed in units of meters or feet.		
									
 The number of offsets to mix (noffmix) should be specified for	
 unstacked data only.	noffmix should typically equal the ratio of the	
 shotpoint spacing to the cdp spacing.	 This choice ensures that every	
 cdp will be represented in each offset mix.  Traces in each mix will	
 contribute through migration to other traces in adjacent cdps within	
 that mix.								
									
 The tmig and vmig arrays specify a velocity function of time that is	
 used to implement Stolts stretch for depth-variable velocity.  The	
 stretch factor smig is often referred to as the "W" factor.		
 The times in tmig must be monotonically increasing.

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:

$ suxwigb < simple.su xcur=3 title="Simple" label1="Time (s)" label2="Trace number" &

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.

$ surange < simple.su

80 traces:
tracl    1 80 (1 - 80)
cdp      1 80 (1 - 80)
trid     1
ns 501 dt 4000

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.

$ sustolt < simple.su cdpmin=1 cdpmax=80 dxcdp=40 vmig=2000 tmig=0.0 > stolt.simple.su

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

$ suxwigb < stolt.simple.su xcur=3 title="Stolt migration of simple data"  label1="Time (s)" label2="Trace number" &

See also

• Huygens–Fresnel principle
• Reflection seismology
• Seismic Unix, open source software for processing of seismic reflection data
• Seismic waves

Further Reading

• Yilmaz, Öz (2001). Seismic data analysis. Society of Exploration Geophysicists. ISBN 1-56080-094-1.

Category:Geophysics