Modelling by hyperbolic superposition

4.3.1 Modelling by hyperbolic superposition

Section 4.2.5 describes how the seismic response of a complex geological structure can be viewed as the superposition of the responses of many point diffractors. Conceptually, infinitely many point diffractors are required and the seismic response is found in the limit as the number of diffractors becomes infinite and their spacing infinitesimal. In a practical computer implementation, only a finite number of diffractors can be simulated and their spacing can be no finer than the underlying computation grid.

There are six basic routines that support modelling by superposition of hyperbolae. These are collected in the synsections toolbox and assume that a constant-velocity, zero-offset synthetic is desired. The fundamental paradigm is that each routine inserts a single event in a matrix that represents the seismic section. These six basic commands are

event spike Inserts an isolated noise spike. event hyp Inserts a hyperbolic event. event dip Inserts a dipping (linear) event. event diph Builds a dipping event by superimposing hyperbolae. event diph2 Similar to event diph plus it allows control over the hyperbolae spacing. event pwlinh Superimposes hyperbolae along a piecewise linear track.

These all operate similarly and insert a spike, hyperbola, linear event, or piecewise linear event in

a matrix. The matrix must be created externally and assigned vertical (time) and horizontal (dis- tance) coordinates. Then the inserted event is described by its position in (x, t) or (x, z). Some of the commands require the specification of velocity and this should be the physical velocity. Within the functions, the velocity is halved to produce an exploding reflector synthetic. The final three commands create events by the superposition of hyperbolae along linear or piecewise linear struc- tures. By calling these functions repeatedly, the seismic responses from quite complex geometric shapes can be synthesized.

Code Snippet 4.3.1. This example creates the synthetic zero offset section shown in Figure 4.25.

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

2 x=0:dx:2000;%x axis

3 t=0:dt:2;%t axis

4 seis=zeros(length(t),length(x));%allocate seismic matrix

5 seis=event_hyp(seis,t,x,.4,700,v,1,3);%hyperbolic event

6 seis=event_dip(seis,t,x,[.75 1.23],[700 1500],1);%linear event

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

8 seis=sectconv(seis,t,w,tw);%apply wavelet

End Code

Code Snippet 4.3.2. This is similar to Code Snippet 4.3.1 but differs byusing event diph to create the dipping event. The result is shown in Figure 4.26.

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

2 x=0:dx:2000;%x axis

3 t=0:dt:2;%t axis

4 seis=zeros(length(t),length(x));%allocate seismic matrix

5 seis=event_hyp(seis,t,x,.4,700,v,1,3);%hyperbolic event

6 seis=event_dip(seis,t,x,[.75 1.23],[700 1500],1);%linear event

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

8 seis=sectconv(seis,t,w,tw);%apply wavelet

102 CHAPTER 4. ELEMENTARY MIGRATION METHODS

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Figure 4.25: The hyperbolic response of a point Figure 4.26: Similar to Figure 4.25 except that diffractor and a simple linear event are shown.

the linear event was created by the superposition The display is slightly clipped to make the

of many hyperbolae. The display is clipped to diffraction more visible. This was created by

match that of Figure 4.25. See Code Snippet Code Snippet 4.3.1.

End Code

Code Snippet 4.3.1 illustrates the use of event hyp and event dip to create the model section shown in Figure 4.25. The first three lines establish the basic model geometry and define the coordinate axes; then, line 4 initializes the seismic matrix to zero. On line 5, event hyp is invoked to make the hyperbolic diffraction response shown in the top left of Figure 4.25. The seismic matrix is input to event hyp and is replaced by the output. The coordinate vectors (second and third input parameters) are required to defined the geometry. The fourth and fifth input parameters are the (x, t) coordinates of the apex of the diffraction and the sixth parameter is the velocity. The seventh parameter sets the amplitude at the apex of the hyperbola and the eighth is a flag that determines how amplitude decays down the limbs of the hyperbola. There are four possibilities: (1) no amplitude

decay, (2) decay given by a(x) = a(0)t 0 /t x (where a(x) is the amplitude at offset x, t 0 is the zero- offset traveltime, and t x is the traveltime at offset x), (3) decay given by a(x) = a(0)(t 0 /t x ) 3/2 , and (4) decay given by a(x) = a(0)(t 0 /t x ) 2 . Line 6 invokes event dip to create the linear dipping event seen in the center of Figure 4.25. The fourth input parameter specifies the time of the event at its beginning and end while the fifth parameter gives the corresponding lateral positions. The last parameter gives the event amplitude. Once the events are created, the final two lines make a Ricker wavelet (dominant frequency of 40 Hz) and convolve it with the seismic section.

The linear event shown in Figure 4.25 is physically unrealistic because it lacks diffractions from its endpoints. Real wavefields are never discontinuous. A much more realistic event can by created using event diph . This function synthesizes a linear event by hyperbolic superposition exactly as illustrated in Figure 4.20. The result is the seismic section shown in Figure 4.26. In order to do so, the input parameters must describe the dipping event in (x, z) rather than (x, t). Code Snippet 4.3.2 shows how this is done and differs from Code Snippet 4.3.1 only on line 6. The fourth input parameter for event diph is the velocity while the next four parameters are respectively: the starting and ending x coordinates, the starting z coordinate, and the dip in degrees. The lateral extent of the resulting event is generally much greater than the prescribed coordinates. If the section