MATLAB FACILITIES FOR SIMPLE MODELLING AND RAYTRACE MIGRATION 103

4.3. MATLAB FACILITIES FOR SIMPLE MODELLING AND RAYTRACE MIGRATION 103

meters meters 0 0 500

Figure 4.27: The response of five reflectors with Figure 4.28: The same dataset as Figure 4.28 is dips of 0 ◦ , 20 ◦ , 40 ◦ , 60 ◦ , and 80 ◦ .

shown with a strongly clipped display to reveal the underlying hyperbolae.

is migrated, then the migrated reflector will be found between the specified coordinates.

A more complex seismic section is shown in Figure 4.27. The geologic model is a fan of five dipping reflectors that originate at the point (x, z) = (250m, 200m) and extend to the right until x = 1500m. In Figure 4.28 the same seismic section is shown with a strongly clipped display to reveal the underlying hyperbolae.

Code Snippet 4.3.3. This code illustrates the use of event pwlinh to create a simple model of a reef. The result is shown in Figure 4.29.

1 v=2000;dx=10;dt=.004;%basic model parameters

2 x=0:dx:3000;%x axis

3 t=0:dt:1.5;%t axis

4 zreef=600;hwreef=200;hreef=50;%depth, half-width, and height of reef

5 xcntr=max(x)/2;

6 xpoly=[xcntr-hwreef xcntr-.8*hwreef xcntr+.8*hwreef xcntr+hwreef];

7 zpoly=[zreef zreef-hreef zreef-hreef zreef];

8 seis4=zeros(length(t),length(x));%allocate seismic matrix

9 seis4=event_diph(seis4,t,x,v,0,xcntr-hwreef,zreef,0,.1);%left of reef

10 seis4=event_diph(seis4,t,x,v,xcntr+hwreef,max(x),zreef,0,.1);%right of reef

11 seis4=event_diph(seis4,t,x,v,xcntr-hwreef,xcntr+hwreef,zreef,0,.2);%base of reef

12 seis4=event_pwlinh(seis4,t,x,v,xpoly,zpoly,-.1*ones(size(zpoly)));%reef

13 [w,tw]=ricker(dt,40,.2);%make ricker wavelet

14 seis4=sectconv(seis4,t,w,tw);%apply wavelet

End Code Code Snippet 4.3.3 illustrates the use of these hyperbolic summation tools to create a simple

model of a reef. The resulting seismic response is shown in Figures 4.29 and 4.30. The reef is a simple trapezoidal structure, 400 m wide and 50 m high, on top of a flat reflector 600 m below the recording plane. The reflection coefficient of the reef is modelled as -.1 (the acoustic impedance within the reef is assumed to be lower than the surrounding material), +.1 on the base reflector, and +.2 beneath the reef.

104 CHAPTER 4. ELEMENTARY MIGRATION METHODS

meters meters 0 0 500

Figure 4.29: The seismic response of a simple Figure 4.30: The same dataset as Figure 4.29 is reef model. This was created with Code Snippet

shown with a strongly clipped display to reveal

4.3.3 the underlying hyperbolae.

Code Snippet 4.3.4. This code uses event diph2 to construct the response of a trapezoidal struc- ture. The parameter ndelx (line 5) controls the spacing of the hyperbolae. Figures 4.31, 4.32, 4.33 and 4.34 correspond to values of ndelx of 15, 10, 5, and 1 respectively.

1 v=2000;dx=5;dt=.004;%basic model parameters

2 x=0:dx:3000;%x axis

3 t=0:dt:1.5;%t axis

4 xcntr=max(x)/2;

5 ndelx=30;

6 seis5=zeros(length(t),length(x));%allocate seismic matrix

7 seis5=event_diph2(seis5,t,x,v,0,500,1000,ndelx,0,.1);

8 seis5=event_diph2(seis5,t,x,v,500,xcntr-500,1000,ndelx,-45,.1);

9 seis5=event_diph2(seis5,t,x,v,xcntr-500,xcntr+500,500,ndelx,0,.1);

10 seis5=event_diph2(seis5,t,x,v,xcntr+500,max(x)-500,500,ndelx,45,.1);

11 seis5=event_diph2(seis5,t,x,v,max(x)-500,max(x),1000,ndelx,0,.1);

12 [w,tw]=ricker(dt,40,.2);%make ricker wavelet

13 seis5=sectconv(seis5,t,w,tw);%apply wavelet

End Code

The function event diph2 is similar to event diph except that it allows control over the spacing of hyperbolae through the input parameter ndelx. In event diph the hyperbolae spacing is never greater than the grid spacing while in event diph2 the spacing is ndelx times greater than in event diph . Code Snippet 4.3.4 illustrates the use of this function to create a series of figures illustrating the gradual formation of the seismic response of a trapezoidal structure. The sequence of Figures 4.31, 4.32, 4.33 and 4.34 shows the complete seismic response gradually forming as the density of hyperbolae increases.

Also present in the synsections toolbox are two functions that make “standard” synthetics called makestdsyn and makestdsynh . These differ in that one uses hyperbolic superposition for linear events and the other does not. Their use can be illustrated in the script makesections that can be run as a demonstration.