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# Lab 6: Phase shift downward and upward continuation ... · PDF file BEI - Lab 6 2 Phase shift...

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• Due Date: Wednesday, November 14, 2012 TA: Mandy Wong ([email protected])

Lab 6: Phase shift downward and upward

continuation (Fortran version)

Sven Treitel1

ABSTRACT

In this exercise you are provided with a modified version of the phase shift migra- tion and modeling programs found in Basic Earth Imaging (BEI). The program has been arranged so that it outputs snapshots of a wavefield as it is upward continued and downward continued. This week’s programming task will be to add depth-variable velocity to the phase shift program.

INTRODUCTION

Phase shift modeling/migration is described in Chapter 7 of BEI. The forward model- ing process works by starting deep in the Earth and “upward continuing” the wavefield one step at a time until we reach the surface, adding in exploding reflectors as we go up. Migration works by “downward continuing” the wavefield that is recorded at the surface, until the exploding reflector sources are imaged.

Wavefield upward/downward continuation is done in the frequency domain by multiplying by an extrapolation operator. For migration, this downward extrapola- tion operator for a single time step, ∆τ , can be expressed as

C = exp

( −iω∆τ

√ 1 − v

2k2x 4ω2

)

The upward continuation operator is simply the complex conjugate of C.

Phase-shift migration is more subtle than the brute force hyperbola summations of Kirchhoff. Hopefully by watching movies of wavefields being upward and downward continued you will gain a better physical understanding of what is going on, and why the two methods give similar (but not identical) results.

1e-mail: [email protected]

• BEI - Lab 6 2 Phase shift continuation

EXERCISE

Figure 1 shows the results of applying a phase-shift forward modeling operator, and then a phase-shift migration operator to focus the model again.

The source code files that generate the figures in this document are Phase.f90, phaseshift, phasevofz1. The programs use ft1axis() and ft2axis() (BEI Chap- ter 6) to perform Fourier transforms. Upward/downward continuation is then carried out by multiplying the wavefield by a complex exponential. After several continua- tion steps the wavefield is Fourier transformed back into time-space domain and a snapshot of it is saved as a frame in a movie file. The movies can be seen by building Fig/movie-up.v3 and Fig/movie-down.v3.

Take a look at these two movies, and change the values in up.P and down.P to pick your favorite frames to include in this paper. Remember to burn and build the figures to make sure your favorite frame is included.

Figure 1: Initial model (right), phase-shift modeling (center) and phase-shift migra- tion of modeled data (right). Constant velocity.

The next part of this Lab is to modify the code to simulate a depth-variable velocity function. To do this, edit phasevofz1. You do not need to convert the code to handle a full v(z) function. Make a simple hack, so that the lower half of the model, has velocity twice that of the top half: vtop = 2000 m/s, vbottom = 4000 m/s. Currently, the velocity is constant and is set to 2000 m/s in the Makefile. Burn, build,

• BEI - Lab 6 3 Phase shift continuation

Figure 2: Movie of wavefield being upward continued.

Figure 3: Movie of wavefield being downward continued.

and view vofz to see your updated Figure 4.

A problem with frequency domain operations is that the Fourier Transform as- sumes periodic boundary conditions. This causes artifacts that can contaminate the image, the final task is to eliminate the wrap-around artifacts in Figure 4. To do this, you need to edit Makefile to do some padding of the input before running migration. Use the SEPlib command Pad and add in the appropriate lines in your Makefile (Hint: You do not need to modify any Fortran code for this.)

QUESTIONS

Question 1

Imagine that you have a seismic survey which corresponds to the geometry and prop- agation velocity of Figure 1. What is the maximum angle of propagation that can be observed from a reflection point at two way travel time depth τ = 0.15s and hor- izontal position x = 160m? How about travel time depth τ = 0.3s and horizontal position x = 320m?

• BEI - Lab 6 4 Phase shift continuation

Figure 4: Phase shift modeling (left) and phase shift migration (right) of a simple synthetic. Change phasevofz1() so that this Figure shows the result of migration and modeling with depth variable velocity.

Question 2

Why are the first couple of frames of the upward continuation movie blank?

Question 3

Look at the downward continuation movie and compare frames 0 and 14. Notice any similarity? Explain the relationship between these two frames and why frame 14 looks the way it does.

Question 4

Mark where the imaging condition is invoked in phaseshift.

Question 5

What effect do you think the imaging condition has on the output amplitudes?

• BEI - Lab 6 5 Phase shift continuation

Question 6

Question 7

If you programmed v(z) correctly in phasevofz1(), you should notice that the deep- est diffractor in Figure 4 looks different than the shallower diffractors. Explain how it looks different and why it looks different.

Question 8

Can this method be applied to media with lateral velocity variations (v(x, z))?

Question 9

If trying to apply this method to v(x, z) media, how would you extend the current code (don’t do it, just tell me)? What velocity would you use?

Question 10

If the media is v(x, z) and you use the v(z) code, what artifacts would you expect to see?

Phase-shift vs. Kirchhoff

Type make compmig.view to see a movie of two zero-offset migration results with Kirchhoff and Phase-shift with the data of Lab4. I encourage you to look at the pro- gram that generates the Phase-shift result to answer some of the following questions.

• What result do you prefer and why?

• List at least two differences between the two results and try to explain where they come from.

• Do you see wrap-around effects in the Phase-shift migration result? If yes, mark them in Figure 5.

• Compare the computing cost of both methods. Why is one method much slower than the other?

• BEI - Lab 6 6 Phase shift continuation

• Limit the angle of propagation in the Kirchhoff method (see Lab4). Include a figure that best compares with the Phase-shift result. (If you normally complete the Matlab version of the labs, you may simply include your ”favorite” result from Lab 4.) Explain your result. Is Phase-shift still better? Why?

Figure 5: Left:Kirchhoff migration of the Gulf of Mexico data. Right: Phase-shift migration.

Question 11

What are the advantages and disadvantages of phase-shift migration compared with Kirchhoff migration?

Reducing boundary artifacts

Type make compbnd.view to see a movie of two zero-offset migration results with Phase-shift with/without boundary conditions with the data of Lab4. Type make compbndnew.view to see a movie of two zero-offset migration results with a subset of the same data. Apply the same padding technique you’ve learned from the last section, implement the padding boundary condition. You should be able to see less artifacts in the results using migration with boundary conditions, this is more obvious in the second comparison. Answer the following questions.

• BEI - Lab 6 7 Phase shift continuation

Figure 6: Left:Phase-shift migration of the Gulf of Mexico data. Right: Same migra- tion with less boundary artifacts.

• BEI - Lab 6 8 Phase shift continuation

Figure 7: Left:Phase-shift migration of the Gulf of Mexico data. Right: Same migra- tion with less boundary artifacts.

• BEI - Lab 6 9 Phase shift continuation

Question 12

Why there are different amount of artifacts in the original migration(left panel) be- tween figure6 and figure 7

Question 13

Explain what factors affect your choice of boundary region size, dont just list com- putational factors.

HAND IN

Once your figures and answers are all in place, and you have checked that they are all OK, type scons to compile a pdf of this lab’s paper, and email it to your TA.

PROGRAMS

Extra Credit: Who was Sven Treitel, what was he/she famous for?

Extra Credit: With a green pen mark all spelling and gramar error’s in this Lab. Also comment on confusing statements in the Lab or other broken and outdated features/questions.

module phasevofz1 mod

use sep use f o u r i e r

imp l i c i t none in t ege r , p r i va t e : : nz , nt , nx , movie j r ea l , p r i va t e : : dz , dt , dx , v l o g i c a l , p r i va t e : : wantmovie complex , dimension ( : , : ) , a l l o c a t ab l e , p r i va t e : : cmodl , cdata , ctemp

conta in s

subrout ine pha s e v o f z 1 i n i t ( nz in , nt in , nx in , dz in , dt in , dx in , v in , wantmovie in , mov i e j i n ) i n t e g e r : : nz in , nt in , nx in , mov i e j i n r e a l : : dz in , dt in , dx in , v in l o g i c a l : : wantmovie in nz = nz in nt = nt i n nx = nx in dz = dz in dt = dt i n dx = dx in v = v in

• BEI - Lab 6 10 Phase shift continuation

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