CHAPTER 3. WAVE PROPAGATION

74 CHAPTER 3. WAVE PROPAGATION

Figure 3.2: The velocity model created in Code Snippet 3.3.1.

algorithm used by afd vmodel is that points that lie exactly on the boundary of the polygon are considered outside the polygon and so do not acquire the new velocity. The variable dx2, defined on line 7 as half of the grid spacing, is used on line 8 to define a rectangle that is half a grid spacing above depths 0 and z2 and half a grid spacing outside the minimum and maximum x coordinates. This ensures that the rectangle extends to the limits of the velocity matrix. Line 9 fills this rectangle with the velocity of 2000 m/s and then lines 10 and 11 repeat this process for the next layer. The resulting velocity model is shown in Figure 3.2. This plot was made using plotimage(vmodel-3000,z,x).

A constant is subtracted from the velocity model so that the resulting matrix has both positive and negative values as expected by plotimage . The raypaths shown in this figure correspond to traveltimes shown in Figures 3.3 and 3.4.

Code Snippet 3.3.1 creates two seismograms, the first (line 21) uses a second-order Laplacian (equation 3.19) and the second (line 23) uses a fourth-order Laplacian (equation 3.20). The prepa- ration for the seismograms defines the time step (line 13), temporal sample rate (line 14), maximum time (line 14), the receiver locations (lines 15-16) and the source strength and geometry (line 19). The time step is generally chosen to be small enough to satisfy the stability requirement (equation

3.23) and the temporal sample rate is usually much more coarse. afd shotrec internally calculates the seismogram at the sample rate of the time step and then resamples it to the desired temporal sample rate. This is sensible because it is a well-known property of finite-difference methods that the higher frequencies are physically inaccurate. In this case, the Nyquist frequency for a sample rate of .001 seconds is 1000 Hz while for .004 seconds it is 125 Hz. Considering that the antialias filter for resampling to .004 seconds will attenuate frequencies above about half of Nyquist, this ex- ample anticipates using only frequencies below about 60 Hz and that is less than 10% of the original Nyquist.

The variables snap1 and snap2 created on lines 17-19 represent the the wavefield ψ(x, z, t = −∆t) and ψ(x, z, t = 0) respectively. They are created to be the same size as the velocity model and snap1 is a matrix of zeros. The source is specified by placing appropriate nonzero samples in snap2. In this case, a point source in the center of the x axis at z = 0 is simulated.

At this stage, all of the input parameters to afd shotrec have been described except for the final three. These specify the filter (or wavelet) to be convolved with the seismogram, the phase of the filter, and the order of the Laplacian. In this case, Ormsby filter specifications are given as [5 10 30 40] and this means that the filter pass band begins at 5 Hz., reaches full ampltude at 10 Hz., begins to

3.3. FINITE DIFFERENCE MODELLING WITH THE ACOUSTIC WAVE EQUATION

Figure 3.3: The second-order seismogram cre- Figure 3.4: The fourth-order seismogram cre- ated on line 21 of Code Snippet 3.3.1. Superim-

ated on line 23 of Code Snippet 3.3.1. See Figure posed on the right-hand side are raytraced trav-

3.3 for a description of the superimposed travel- eltimes for (from top to bottom) the first pri-

times.

mary reflection, the first-order multiple in the top layer, and the second primary reflection. The corresponding raypaths are shown in Fig- ure 3.2.

ramp down at 30 Hz., and rejects frequencies above 40 Hz. The penultimate parameter specifies the phase of the filter that is, in this case, zero (a value of 1 gives minimum phase). The last parameter is either 1, indicating a second-order Laplacian, or 2, meaning a fourth-order approximation.

The seismograms are shown in Figures 3.3 and 3.4 respectively. Both of these figures are plotted with a highly clipped display, otherwise, the direct wave near the source would be the only visible arrival. Though similar at first glance, these two seismograms are significantly different. The three hyperbolic reflections on each figure are (from the top) the primary reflection off the bottom of the first layer, the first-order multiple reflecting between the bottom of the first layer and the surface, and the primary reflection off the bottom of the second layer. The traveltimes for these events are superimposed on the right-hand sides of the figures and their raypaths are shown in Figure

3.2. The reflections for these events lag significantly behind the raytraced traveltimes but more so on the second-order solution. This time lag occurs because the finite difference approximations to the derivative in the wave equation have a frequency-dependent performance. Essentially, for wavelengths that are large compared to the computation grid, the approximate derivatives are acceptable but, as the wavelength approaches the grid spacing, the approximation becomes very poor. The result is a phenomenon known as grid dispersion, meaning that the actual propagation speed of waves on the grid is not simply v ins (x, z) but is a complex function of wavelength (or, equivalently, frequency). Grid dispersion makes the reflections appear to lag behind the corresponding raytrace traveltimes. It also makes the apparent wavelet appear more elongated (dispersed). Comparison of the first reflection at far offsets on both seismograms shows that the second-order result has a more dispersed waveform. Also, the fourth-order result has a smaller lag behind the raytrace traveltimes than the second-order for all reflections.

Careful inspection of both Figures 3.3 and 3.4 shows a set of events that originate where each

76 CHAPTER 3. WAVE PROPAGATION reflection meets the sides of the seismogram and then cross in the center of the figures. These are

typical artifacts known as edge effects. They arise because a computer simulation of wave propagation must always operate in a finite domain with definite boundaries. When a wave encounters such a boundary, it behaves as though it has met a perfect reflector. These boundary reflections would be much worse if afd shotrec did not incorporate absorbing boundaries (Clayton and Engquist, 1977). Obviously, absorbing boundaries are not perfect but they do help. The boundary related artifacts can be seen to be generally quite steeply dipping in (x, t). This is because the absorbing boundary conditions are optimized for a small range of wavefront incidence angles around normal incidence. That is, a wavefront that encounters the boundary at normal incidence is completely absorbed while one making an angle of, say, 30 ◦ is partially reflected.

Chapter 4