P51 2-D Kirchhoff Prestack Time Migration of Seismic Synthetic ...

P51 

2-D Kirchhoff Prestack Time Migration of Seismic 

Synthetic Records by MATLAB 

P. Hadian* (Amirkabir University of Technology) & A. Javaherian 

(University of Tehran) 

SUMMARY 

In this paper, a MATLAB software is presented that synthesizes 2-D shot records with up to four flat 

reflectors with different velocities and then migrate them using kirchhoff prestack time migration method. 

Migration algorithm needs two matrices as inputs which they are shot records and the velocity model. Shot 

records are either synthesized from the data entered in the graphic interface or received as an input file. 

Prestack Kirchhoff time migration is applied via software written with MATLAB. A GUI (Graphical User 

Interface) was designed to work with codes for convenience of operator. Same model was migrated with 

the industrial software. Results were compared and proved good agreement between them. 

Shiraz 2009 - First International Petroleum Conference & Exhibition 

Shiraz, Iran, 4 - 6 May 2009

1. Introduction 

Time migration which produces a migrated time section is appropriate as long as lateral 

velocity variations are mild to moderate. Dipping events on a stacked section call for time 

migration. Conflicting dips with different stacking velocities is one case in which a 

conventional stacked section differs from a zero-offset section. Thus, poststack time 

migration which assumes that the stacked section is equivalent to a zero-offset is not valid to 

handle the case of conflicting dips. Instead, one needs to do prestack time migration (Yilmaz, 

2001). 

Kirchhoff migration methods are based on diffraction summation technique, which sums 

the seismic amplitudes along a diffraction hyperbola whose curvature is governed by the 

medium velocity, and maps the result to apex of the hyperbola. The Kirchhoff summation 

technique (Schneider, 1978) applies amplitude and phase corrections to the data before 

summation. These corrections make the summation consistent with the wave equation in that 

they account for spherical spreading, the obliquity factor (angle-dependency of amplitudes), 

and the phase shift inherent in Huygens' secondary sources. Since Kirchhoff migration is 

based on summing the amplitudes along the hyperbolic trajectory, as long as the diffraction 

curves are known, it can be adapted for any domain. The velocity function used in diffraction 

curve equation for prestack migration is vrms. 

2. Data input 

The seismic response of a complex geological structure can be viewed as superposition of the 

responses of many point diffractors. Data can be either inputted as in a *.mat file or be 

generated. 

2.1. Making a Synthetic Shot Record 

This part of program generates data containing hyperbolic events correspondent to flat 

reflectors according to the velocity model. The GUI (Graphical User Interface) nature of 

program sets the operator free of working with codes. Each reflector is defined by its intercept 

time, velocity, and amplitude. Entering negative amplitude makes a reflector with reverse 

polarity. A random noise is added to seismogram and signal to noise ratio is determined in the 

S/N ratio. A Ricker wavelet with changeable central frequency is convolved with the 

reflectivity time series. Finally, the synthetic data is plotted using wiggle-trace variable-area 

display. 

2.2. Entering a .mat file 

This software is able to accept MATLAB's mat files as an input. The mat file should contain 

following matrices: 

shot: matrix containing the shot record: one trace per column. 

t: time coordinate vector for shot. 

x: space coordinate vector for shot. 

velmod: velocity model. This is a matrix of vrms as a function of lateral position and time. 

tv: time coordinate vector for velmod. 

xv: space coordinate vector for velmod. 

3. Prestack Time Migration 

Kirchhoff prestack time migration sums through the input space along hyperbolic paths to 

compute each point in the output space. For variable velocity, the hyperbola is replaced by 

more general shape. Amplitudes change under migration. Velocity model is a matrix of 

interval velocity at each sampling point. Velocity model is generated using reflector 

velocities. Traveltime from a source to a scatterpoint (i.e the image point) is approximated by 

a Dix equation using the vrms from the model at the lateral position halfway between the 

source and the receiver at the vertical traveltime of the scatterpoint. Similarly, from the 

scatterpoint to a receiver, a Dix equation using the vrms at halfway between the scatterpoint 



and the receiver is used. The source and all receivers are assumed to be on the same 

horizontal plane. 

4. Models 

The first model is a shot record with 10 m and 1500 m of minimum and maximum offsets, 

respectively. The sampling interval of 4 ms and the record length of 2.2 s are used. Signal to 

noise ratio equals 10. There are four reflectors in this shot record at intercepts of 0.1, 0.2, 0.9 

and 2 s and velocities of 500, 1500, 2000 and 3000 m/s respectively. Figure 2 shows this shot 

record. The prestack time migrated is shown in Figure 3. 

The second model is a trapezoid model which is synthesized externally and inputted to 

migration algorithm part. The traveltimes are calculated via rayracing with respect to a shot in 

the center of the model. The synthetic shot record and its velocity model are shown in Figures 

6 and 7, respectively. First shot record was migrated with the interval velocity model and the 

result is depicted in Figure 8. Since prestack migration is very sensitive to velocity model, an 

rms velocity model was used. Using rms velocity field (Figure 9) improved the migrated 

section (Figure 10). An important factor in Kirchhoff migration is migration aperture. 

Reducing aperture reduces the maximum dip to migrate. Figure 11 illustrates the effect of 

migration aperture, using a 1000 trace aperture eliminates steeply dipping events from the 

migrated section. 

5. Conclusions 

We have written a software for migrating prestack 2-D seismic data by Kirchhoff method. 

Another part of software was developed for synthesizing shot records as an input for 

migration. The main advantage of such approach is that it does not need data previously 

prepared for it, shot records and the velocity model are generated from the information 

entered in the graphic interface. Migration were done successfully on shot records using the 

velocity model generated and the results were displayed for unmigrated and migrated 

sections. 

References 

Bancroft, J.C., Geiger, H.D., 1994, Equivalent offset CRP gathers: SEG Expanded Abstracts 

13, 672-675. 

Fowler, P.J., 1997, A comparative overview of prestack time migration methods: SEG 

Expanded Abstracts, 16, 67 th annual meeting- Dallas, 1571-1574. 

Gardner, G.H.F., Wang, S. Y., Pan, N.D., and Zhang, Z., 1986, Dip moveout and prestack 

imaging: 18 th Offshore Tech. Conf., 2, 75-84. 

Margrave, G.F., 2001, Numerical methods of exploration seismology with algorithms in 

MATLAB, The University of Calgary. 

Schneider, W.A., 1978, Integral formulation for migration in two and three dimensions: 

Geophysics, 43, 49–76. 

Yilmaz, O., 2001, Seismic data analysis: Society of Exploration Geophysicists, Tulsa, 

Oklahoma. 

Yilmaz, O., and Claerbout, J. F., 1980, Prestack partial migration: Geophysics, 45, 1753-1779 



Figure 1 Software main interface for 

synthesizing a shot and migration. 

Figure 2 A shot record with four reflectors 

of velocities 500, 1500, 2000 and 3000 

m/s. 

Figure 3 Migrated section of the first model 

by MATLAB. 

Figure 4 First model regenerated by 

industrial software. 

Figure 5 Migrated section of the first model 

by industrial software. 

Figure 6 Synthesized shot record of the 

trapezoid model. 



Figure 7 Raytracing of the trapezoid model. 

Figure 8 Migrated section of trapezoid 

model with interval velocity field. 


model with a 1000 trace migration 

aperture width. 

Figure 9 rms velocity field. 


model with rms velocity field.

P51 2-D Kirchhoff Prestack Time Migration of Seismic Synthetic ...

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