Lab 4: Kirchhoff migration (Matlab version)

Transcription

1 Due Date: Oktober 29th, 2012 TA: Mandy Wong Lab 4: Kirchhoff migration (Matlab version) Robert U. Terwilliger 1 ABSTRACT In this computer exercise you will modify the Kirchhoff migration and modeling subroutine that was presented in class. First you will limit the propagation angles in the subroutine, and then you will modify the routine to handle v(z). INTRODUCTION The subroutine kirchfast performs migration by summing input data along a hyperbolic trajectory, and modeling by spraying out along a hyperbolic trajectory. Later on in this course you will study the phenomena of operator aliasing. This occurs when operator moveout across adjacent traces exceeds the time sampling rate. Cycle skips occur and the operator is aliased. For a moveout curve with slope dt/dx, and data with a spatial Nyquist frequency of k n, temporal frequencies above are aliased. ω = k n dt/dx Later in the course you will learn about more sophisticated ways of dealing with operator aliasing. In this exercise you will observe the phenomena and deal with it by limiting the slope of the moveout curve. The criteria for limiting the curve will be based on the maximum angle of propagation that you will allow to be migrated. In Chapter 7 of Basic Earth Imaging (BEI) you learned that the ratio of first arrival time on a hyperbola to any other arrival time gives the cosine of the angle of propagation. After coding up the angle-limited migration and testing it on a simple synthetic, you will modify the subroutine to handle depth variable velocity. Your new program will be used to migrate a near-offset section from the Gulf of Mexico. 1 robert@terwilliger.com

2 BEI - Lab 4 2 Kirchhoff EXERCISE This lab is, once again, available online. First download the source code for this lab, lab4 matlab.tar from the BEI class webpage, and save it in the appropriate directory. Then type tar -xvf lab4 matlab.tar to create your Lab4 matlab directory. From your workstation, log onto ssh Xusername@cees tool.stanford.edu and continue to Part 1. To build the first copy of paper.pdf, type make default first and then type make paper. Edit the file paper.tex to answer questions. Part 1 Angle-limited migration The subroutine kirchfast is a version of the fast Kirchhoff modeling and migration routine presented in Chapter 5 of BEI. The routine has been modified to include a variable called amax which is passed from the main program Kapp.m. You should not have to make any modifications to Kapp.m. Edit kirchfast.m so that only the energy which is propagating at angles less than amax is migrated or modeled. On the matlab prompt, type run Kapp to rebuild the figure. Save the figure as Fig/migmodel.pdf to include in this paper. Sharp operator truncations can also cause artifacts in a final image. The Gibbs effect (ringing) is an example of this problem caused by truncation in the Fourier domain. If you are feeling ambitious (or are a SEP student) you can further improve the operator. Instead of truncating the operator at amax, apply a taper, so its amplitude decreases to zero over a small range. Part 2 Field data and v(z) You will now migrate the near-offset section of the Gulf of Mexico data. Figure 5 is an angle-limited migration with a depth variable velocity function. Your objective in this section is to make Figure 4 look better than Figure 5. Don t worry about the fact that you are migrating a near-offset section as if it were a zero-offset section. The approximation is good enough for our purposes. On your Matlab prompt, type run kirmiggom to see a movie of migration as a function of maximum propagation angle (amax). When you watch the movie, notice how the migrated image changes. As angle increases you should see the flanks of the salt dome getting resolved better. As angle increases you will also see more aliasing artifacts. These artifacts appear as systematic dipping noise and as precursors above the first arrival. Before you choose the best frame to include as your final answer, you need to improve the migration velocity function. The current velocity is shown in Figure 3. In kirmiggom.m, choose values for the parameters vwater, alpha, beta, and twater. Use what you have learned in Chapters 3 and 4 about typical Gulf of Mexico v(z)

3 BEI - Lab 4 3 Kirchhoff sections. In order to include the new velocity function, you will have to save the velocity figure with the name Fig/velgulf.pdf Finally edit the parameter slice in kirmiggom.m to choose which frame to include in the report you hand in. Save the best migrated figure with the name Fig/Gulfmig.pdf. QUESTIONS You should edit paper.tex file to answer the following questions. Don t forget to add your name. Question 1 Imagine that you have a seismic survey which corresponds to the geometry and propagation velocity of Figures 1 and 2. What is the maximum angle of propagation that can be observed from a reflection point at two way travel time depth τ = 0.3s and horizontal position x = 640m? How about travel time depth τ = 0.6s and horizontal position x = 640m? Question 2 Watch the migration movie of the Gulf data and describe your observations How does the image change as the angle changes? What kind of changes do you see in different parts of the section? How did adding v(z) change the image? What criteria did you use to select the frame for Figure 4. Question 3 Aperture limitation is sometimes used to avoid aliasing in Kirchhoff migration. What would you change in kirchfast() to limit the spatial size of the aperture and not the angular size. (Don t do it, just tell me what you would change)? What would be the advantage and/or disadvantage of aperture limitation compared to angle limitation?

4 BEI - Lab 4 4 Kirchhoff Question 4 Kirchhoff operators can be implemented as either push operators or pull operators. The forward operator for kirchfast models data by looping over its input space and pushing data along hyperbolas. The adjoint operation, migration, is done by looping over the output space and pulling in data from hyperbolas in the input space. 1. A different subroutine could be written that does Kirchhoff migration by looping over the dataspace and pushing it out along semi-circles (ellipses for v(z)) above it. What would the adjoint operation of this be? 2. Although mathematically equivalent, what would you think are the implementation advantages of push operators vs pull operators? (HINT: Think irregular data) 3. Can the forward and adjoint parts of an operator both be push operators or both be pull operators? 4. 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. HAND IN When you are all done, type make paper at the shell prompt to rebuild a pdf version of this lab. Print out a copy of your homework and hand it in to your TA s office or it to the TA. filt subroutine in matlab PROGRAMS

5 BEI - Lab 4 5 Kirchhoff % K i r c h h o f f migration and d i f f r a c t i o n % adj = 0, forward operation, data = data + L modl, L i s the K i r c h h o f f % modeling o p e r a t o r % adj = 1, a d j o i n t operation, modl = modl + L data, L i s the K i r c h h o f f % migration o p e r a t or % amax = maximum propagation angle in d e g r e e s f u n c t i o n [ modl, data ]= k i r c h f a s t ( adj, add, v e l h a l f, amax, t0, dt, dx, nt, nx, modl, data ) i f ( add==0) i f ( adj==0) data = z e r o s ( nt, nx ) ; else modl = z e r o s ( nt, nx ) ; pi = 2 acos ( 0. 0 ) ; r2d = 180./ pi ; % loop over o f f s e t for ih= nx : nx h = dx ih ; % h = o f f s e t % loop over t r a v e l time depth for i z =2: nt z = t0 + dt ( iz 1); % z= t r a v e l time depth %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % make chnages to handle v ( z ) media %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% t = s q r t ( z ˆ2 + ( h/ v e l h a l f ( 1 ) ) ˆ 2 ) ; % c a l c u l a t e the corresponding two way t r a v e l i t = f l o o r ( ( t t0 )/ dt ) ; % f i n d the index o f the t r a v e l time in the d i f ( i t > nt ) % i f out o f bounds, break break ; amp = ( z / t ) s q r t ( nt dt / t ) ; % weighting f u n c t i o n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % add your l i n e s here, you w i l l have to modify the % weighting f u n c t i o n to make i t as a f u n c t i o n o f maximum % migration angle : amax %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x s t a r t = max( f l o o r (1 ih ), 1 ) ; % x s t a r t and x d e f i n e the migration / modeling x = min ( f l o o r ( nx ih ), nx ) ; i f ( adj == 0) % i f modeling, we g e nerate the data for i x=x s t a r t : x data ( i t, i x+ih)=data ( i t, i x+ih)+modl ( iz, i x ) amp ; else % i f migration, we produce the model

6 BEI - Lab 4 6 Kirchhoff for i x=x s t a r t : x modl ( iz, i x )=modl ( iz, i x )+data ( i t, i x+ih ) amp ; return

7 BEI - Lab 4 7 Kirchhoff 0 Migration 0 Diffraction time [s] 0.6 time [s] x [m] x [m] Figure 1: After you have modified kirchfast() your migration and modeling result should look like Figure 2.

8 BEI - Lab 4 8 Kirchhoff 0 Migration 0 Diffraction time [s] 0.6 time [s] x [m] x [m] Figure 2: The answer: Migration and modeling limited to 45 propagation angle. Your task is to make Figure 1 look like this.

9 BEI - Lab 4 9 Kirchhoff Figure 3: Velocity function used for migrations.

10 BEI - Lab 4 10 Kirchhoff Figure 4: Migration of the Gulf of Mexico data.

11 BEI - Lab 4 11 Kirchhoff Figure 5: What you have to beat: Limited angle migration with preliminary depth variable velocity function.