Download - Lab 6: Phase shift downward and upward continuation ...sep · BEI - Lab 6 2 Phase shift continuation EXERCISE Download the Lab 6 tar ﬁle from the website, and log onto sad. Type

Due Date: Wednesday, November 14, 2012TA: 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 programhas been arranged so that it outputs snapshots of a wavefield as it is upwardcontinued and downward continued. This week’s programming task will be toadd 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 wavefieldone step at a time until we reach the surface, adding in exploding reflectors as we goup. Migration works by “downward continuing” the wavefield that is recorded at thesurface, until the exploding reflector sources are imaged.

Wavefield upward/downward continuation is done in the frequency domain bymultiplying 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 − v2k2

x

4ω2

)

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

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

1e-mail: [email protected]

BEI - Lab 6 2 Phase shift continuation

EXERCISE

Download the Lab 6 tar file from the website, and log onto sad. Type tar -xf

Lab6.tar to obtain your directory.

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 carriedout by multiplying the wavefield by a complex exponential. After several continua-tion steps the wavefield is Fourier transformed back into time-space domain and asnapshot of it is saved as a frame in a movie file. The movies can be seen by buildingFig/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 topick your favorite frames to include in this paper. Remember to burn and build thefigures 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-variablevelocity function. To do this, edit phasevofz1. You do not need to convert the codeto handle a full v(z) function. Make a simple hack, so that the lower half of themodel, 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,


Figure 2: Movie of wavefield beingupward continued.

Figure 3: Movie of wavefield beingdownward 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 theimage, 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 canbe 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 horizontalposition x = 320m?


Figure 4: Phase shift modeling (left) and phase shift migration (right) of a simplesynthetic. Change phasevofz1() so that this Figure shows the result of migrationand 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. Noticeany similarity? Explain the relationship between these two frames and why frame 14looks 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?


Question 6

Do you think Phase.x adj=0 and Phase.x adj=1 are true adjoint processes? Howwould you check?

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 howit 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 currentcode (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 tosee?

Phase-shift vs. Kirchhoff

Type make compmig.view to see a movie of two zero-offset migration results withKirchhoff 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 wherethey come from.

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

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


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

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

Question 11

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

Reducing boundary artifacts

Type make compbnd.view to see a movie of two zero-offset migration results withPhase-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 subsetof the same data. Apply the same padding technique you’ve learned from the lastsection, implement the padding boundary condition. You should be able to see lessartifacts in the results using migration with boundary conditions, this is more obviousin the second comparison. Answer the following questions.


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


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


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 areall 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 thisLab. Also comment on confusing statements in the Lab or other broken and outdatedfeatures/questions.

module phasevofz1 mod

use sepuse f o u r i e r

imp l i c i t nonein t ege r , p r i va t e : : nz , nt , nx , movie jr ea l , p r i va t e : : dz , dt , dx , vl o g i c a l , p r i va t e : : wantmoviecomplex , 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 nr e a l : : dz in , dt in , dx in , v inl o g i c a l : : wantmovie innz = nz innt = nt i nnx = nx indz = dz indt = dt i ndx = dx inv = v in


wantmovie = wantmovie inmovie j = mov i e j i na l l o c a t e ( cmodl ( nz , nx ) , cdata ( nt , nx ) , ctemp ( nt , nx ) )cmodl = 0 .cdata = 0 .ctemp = 0 .c a l l f t i n i t ( nt , nx )

end subrout ine

!===============================================================

subrout ine phasevofz1 ( adj , add , modl , data )l o g i c a l , i n t en t ( in ) : : adj , addrea l , dimension ( : , : ) : : modl , datai n t e g e r : : i z , nw, iw , ikxr e a l : : pi , s i gnt , s ignx , rv2r e a l : : w0 , dw, w, kx0 , dkx , kx , kz2complex : : phase

p i = 4 .∗ atan ( 1 . )s i g n t =1.s ignx =−1.rv2 =1./(v∗v )

nw = ntw0 = −pi /dtdw = 2.∗ pi /( nt∗dt )kx0 = −pi /dxdkx= 2 .∗ pi /(nx∗dx )

i f ( . not . add ) theni f ( adj ) then

modl = 0 .else

data = 0 .end i f

e nd i f

i f ( adj ) then ! DOWNWARD CONTINUATION ( migrat ion )cdata = datac a l l f t 1 a x i s ( . f a l s e . , s i gnt , cdata ) ! FFT over tc a l l f t 2 a x i s ( . f a l s e . , s ignx , cdata ) ! FFT over xdo i z =1,nz

do ikx=2,nxkx = kx0 + ( ikx −1)∗dkxdo iw=2,nw

w = w0 + ( iw−1)∗dwkz2 = w∗w∗ rv2 − kx∗kxi f ( kz2 >0) then

phase = cexp ( cmplx ( 0 . , signum (w)∗ dz∗ s q r t ( kz2 ) ) )cdata ( iw , ikx ) = cdata ( iw , ikx )∗ phase

end i fcmodl ( i z , ikx ) = cmodl ( i z , ikx ) + cdata ( iw , ikx )


enddoenddoi f ( wantmovie . and . mod( iz , movie j ) . eq . 1) then ! take snapshot for t h i s depth

ctemp = cdatac a l l f t 1 a x i s ( . t rue . , s i gnt , ctemp ) ! i nv e r s e FFT in tc a l l f t 2 a x i s ( . t rue . , s ignx , ctemp ) ! i nv e r s e FFT in xc a l l s ep wr i t e ( r e a l ( ctemp ) , ”movie” )

end i fenddoc a l l f t 2 a x i s ( . t rue . , s ignx , cmodl ) ! i n v e r s e FFT in xmodl = modl + r e a l ( cmodl )/ nt

else ! UPWARD CONTINUATION ( modeling )cmodl = modlc a l l f t 2 a x i s ( . f a l s e . , s ignx , cmodl ) ! FFT in xdo i z=nz ,1 ,−1

do ikx=2,nxkx = kx0 + ( ikx −1)∗dkxdo iw = 2 ,nw

w = w0 + ( iw−1)∗dwkz2 = w∗w∗ rv2 − kx∗kxi f ( kz2>0) then

phase = cexp ( cmplx ( 0 . , −signum (w)∗ dz∗ s q r t ( kz2 ) ) )else

phase = 0 .end i fcdata ( iw , ikx ) = cdata ( iw , ikx )∗ phase + cmodl ( i z , ikx )

enddoenddoi f ( wantmovie . and . mod( iz , movie j ) . eq . 1) then ! take snapshot for t h i s depth

ctemp = cdatac a l l f t 1 a x i s ( . t rue . , s i gnt , ctemp ) ! i nv e r s e FFT in tc a l l f t 2 a x i s ( . t rue . , s ignx , ctemp ) ! i nv e r s e FFT in xc a l l s ep wr i t e ( r e a l ( ctemp ) , ”movie” )

end i fenddoc a l l f t 2 a x i s ( . t rue . , s ignx , cdata ) ! i n v e r s e FFT in xc a l l f t 1 a x i s ( . t rue . , s i gnt , cdata ) ! i n v e r s e FFT in tdata = data + r e a l ( cdata )

end i f

end subrout ine

!===============================================================

r e a l func t i on signum (x )r e a l : : xi f ( x . eq . 0 . ) then

signum = 0 .else

signum = x/abs (x )end i f

end func t i on

end module


### Makef i l e for GP210 Homework 6

inc lude $ (SEPINC)/SEP. top

SRCDIR = ./ SrcOBJDIR = ./ ObjMODDIR = ./ModBINDIR = ./ BinRESDIR = ./ FigPARDIR = ./ ParDATDIR = ./ Dat

S = ${SRCDIR}O = ${OBJDIR}M = ${MODDIR}B = ${BINDIR}R = ${RESDIR}P = ${PARDIR}D = ${DATDIR}

UF90INCLUDES = −I /opt/FFTW/ inc ludeUF90LIBS = ${GEELIB} −L/opt/FFTW/ l i b −L./ Lib − l f f tw 3 − l f f t w 3 fUF90FLAGS = −C −module ${M}

################################################################################

# Proce s s ingsp ike .H:

Spike n1=64 n2=64 nsp=3 k1=15 ,30 ,45 k2=16 ,32 ,48 d1=10 d2=10 > sp ike .H

modeled .H up .H: ${B}/Phase . x sp ike .H${B}/Phase . x < sp ike .H nt=64 dt=0.01 adj=0 v=2000. movie=up .H > modeled .H

imaged .H down .H: modeled .H ${B}/Phase . x${B}/Phase . x < modeled .H nz=64 dz=10 adj=1 v=2000. movie=down .H > imaged .H

modvofz .H up−vo fz .H: ${B}/ Phasevofz . x sp ike .H${B}/ Phasevofz . x < sp ike .H nt=64 dt=0.01 adj=0 v=2000. movie=up−vo fz .H

> modvofz .H

imgvofz .H down−vo fz .H: ${B}/ Phasevofz . x modvofz .H${B}/ Phasevofz . x < modvofz .H nz=64 dz=10 adj=1 v=2000. movie=down−vo fz .H > imgvofz .H

################################################################################

# Figures

grey = Grey > /dev/ nu l l ga inpane l=a l l

${R}/%.v : %.H$ ( grey ) < $ ∗ .H p c l i p =100 t i t l e=$∗ out=$@

${R}/ vconst . v : ${R}/ sp ike . v ${R}/modeled . v ${R}/ imaged . v


vp SideBySideAniso ${R}/ sp ike . v ${R}/modeled . v ${R}/ imaged . v > $@

${R}/ vo fz . v : ${R}/ sp ike . v ${R}/modvofz . v ${R}/ imgvofz . vvp SideBySideAniso ${R}/ sp ike . v ${R}/modvofz . v ${R}/ imgvofz . v >$@

${R}/movie−%.v ${R}/movie−%.v3 : %.H ${P}/%.pWindow < $ ∗ .H n3=1 par=${P}/$ ∗ . p | $ ( grey ) t i t l e=$∗ out=${R}/movie−$ ∗ . v$ ( grey ) < $ ∗ .H t i t l e=$∗ out=${R}/movie−$ ∗ . v3

########################################################## GULF OF MEXICO ##########################################################

# pr ep ro c e s s i nggu l f l owb .H: ${D}/ Gulfnear .HH

Window f2=199 n2=400 n1=512 < ${D}/ Gulfnear .HH | \Bandpass f l o =10. f h i =50. | Tpow tpow=0.5 > $@

# bu i ld rms v e l o c i t y func t i on from parametersgul fvrms .H: gu l f lowb .H ${B}/Vel . x ${P}/ ve lpar . p

${B}/Vel . x < $< par=${P}/ ve lpar . p > $@

# bu i ld i n t e r v a l v e l o c i t y func t i on from rmsgu l f v i n t .H: gul fvrms .H

Vconvert < $< in type=2 outtype=1 inpar=1 outpar=1 inve l op t=1 outve lopt=1 > $@

# Kirchho f f migrat ionGulfmig .H: gu l f l owb .H ${B}/Kapp . x gul fvrms .H

${B}/Kapp . x < $< adj=1 i s t a r t=5 iend=5 i ang l e=10 ve l=gul fvrms .H > $@

# Downward cont inuat i on migrat ionGulfmig−PS .H: gu l f l owb .H ${B}/ Phasevofz2 . x g u l f v i n t .H

${B}/ Phasevofz2 . x < $< adj=1 ve l=gu l f v i n t .H > $@

Gulfmig−PS2 .H: gu l f l owb .H ${B}/ Phasevofz2 . x g u l f v i n t .HWindow < $< n2=250 > datnew .H${B}/ Phasevofz2 . x < datnew .H adj=1 ve l=gu l f v i n t .H > $@

####################################################################################### Downward cont inuat i on migrat ion with boundary cond i t i on# modify the make r u l e s here to run the migrat ion with reduced boundary a r t i f a c t s by padding the input######################################################################################

Gulfmig−PSbnd .H: gu l f l owb .H ${B}/ Phasevofz2 . x g u l f v i n t .H<gu l f l owb .H ${B}/ Phasevofz2 . x adj=1 ve l=gu l f v i n t .H > $@

Gulfmig−PSbnd2 .H: gu l f l owb .H ${B}/ Phasevofz2 . x g u l f v i n t .HWindow < $< n2=250 > datnew .H${B}/ Phasevofz2 . x < datnew .H adj=1 ve l=gu l f v i n t .H > $@


############################################################################################################################################################################

# Figure

grey1 = Grey l ab e l 1=”time ( s ) ” l a b e l 2=”x (m) ” t i t l e=” ” > /dev/ nu l l

${R}/compmig . v : Gulfmig .H Gulfmig−PS .H${ grey1 } < Gulfmig .H t i t l e=” Kirchho f f migrat ion ” out=${R}/ l e f t . v${ grey1 } < Gulfmig−PS .H t i t l e=”Downward cont inuat i on ” out=${R}/ r i gh t . vvp SideBySideAniso ${R}/ l e f t . v ${R}/ r i gh t . v > $@cat ${R}/ l e f t . v ${R}/ r i gh t . v > $@3

${R}/compbnd . v : Gulfmig−PS .H Gulfmig−PSbnd .H${ grey1 } < Gulfmig−PS .H t i t l e=”Downward migrat ion ” out=${R}/ l e f t . v${ grey1 } < Gulfmig−PSbnd .H t i t l e=”Downward cont inuat i on with l e s s boundary a r t i f a c t s ” out=${R}/ r i gh t . vvp SideBySideAniso ${R}/ l e f t . v ${R}/ r i gh t . v > $@cat ${R}/ l e f t . v ${R}/ r i gh t . v > $@3

${R}/compbndnew . v : Gulfmig−PS2 .H Gulfmig−PSbnd2 .H${ grey1 } < Gulfmig−PS2 .H t i t l e=”Downward migrat ion 2” out=${R}/ l e f t . v${ grey1 } < Gulfmig−PSbnd2 .H t i t l e=”Downward cont inuat i on with l e s s boundary a r t i f a c t s 2” out=${R}/ r i gh t . vvp SideBySideAniso ${R}/ l e f t . v ${R}/ r i gh t . v > $@cat ${R}/ l e f t . v ${R}/ r i gh t . v > $@3

################################################################################

# c l ean and Burn

c l ean : j c l e a n

burn : c l eanrm −f ${B}/∗rm −f ${O}/∗rm −f ${M}/∗rm −f ${R}/∗ . pd frm −f ${R}/∗ . psrm −f ${R}/∗ . v∗

################################################################################

de f a u l t : ${R}/ vcons t . pd f ${R}/ vo f z . pd f ${R}/movie−up . pdf ${R}/movie−down . pdf ${R}/compmig . pdf ${R}/compbnd . pdf ${R}/compbndnew . pdf

i n c l ude $ (SEPINC)/SEP. bottom