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Since velocity increases with depth, at wide enough offset a deep enough path will arrive sooner than a shallow path. In other words, traveltime curves for shallow events must cut across the curves of deeper events. Where traveltime curves cross, NMO is not a one-to-one transformation. To see what happens to the stacking process I prepared Figures 2-4 using a typical marine recording geometry (although for clarity I used larger $(\Delta t,\Delta x)$) and we will use a typical Texas gulf coast average velocity, $v(z)=1.5+\alpha z$ where $\alpha=.5$.

First we repeat the calculation of Figure 1 with constant velocity $\alpha=0$ and more reflectors. We see in Figure 2 that the stack reconstructs the model except for two details: (1) the amplitude diminishes with time, and (2) the early waveforms have become rounded.

Figure 2
Synthetic CMP gather for constant velocity earth and reconstruction.

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Then we repeat the calculation with the Gulf coast typical velocity gradient $\alpha=1/2$.The polarity reversal on the first arrival of the wide offset trace in Figure 3 is evidence that in practice traveltime curves do cross. Comparing Figure 2 to Figure 3 we see that an effect of the velocity gradient is to degrade the stack's reconstruction of the model. The velocity gradient has ruined the waveform on the shallowest event, at about 400ms. If the plot were made on a finer mesh with higher frequencies, we could expect ruined waveforms a little deeper too.

Figure 3
Synthetic CMP gather for velocity linearly increasing with depth (typical of Gulf of Mexico) and reconstruction. Notice that the Stack signal is not as sharp as in Figure 2.

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In designing NMO and stack subroutines we gave no thought to signal amplitudes (although results showed an interesting AVO effect in Figure 1.) We could redesign the programs so that the modeling operator has the most realistic amplitude that we can devise. Alternately, we could design the amplitudes to get the best approximation to $\bold S'\bold S\approx \bold I$which should result in the ``Stack'' signal being a good approximation to the ``Model'' signal. I experimented with various weighting functions until I came up with subroutine nmo1() [*] which embodies the weighting function $(\tau/t)(1/\sqrt{t})$and which produces the result in Figure 4.

Figure 4
Synthetic CMP gather for velocity linearly increasing with depth and reconstruction with weighting functions in subroutine nmo1(). Notice that the Stack signal is much sharper than in Figure 3. (Should make this figure interactive).

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This result is very pleasing. Not only is the amplitude as a function of time better preserved, more importantly, the shallow wavelets are less smeared and have recovered their rectangular shape. The reason the reconstruction is much better is the cosine weighting implicit in $\tau/t$.It has muted away much of the energy in the shallow asymptote. I think this energy near the asymptote is harmful because the waveform stretch is so large there. Perhaps a similar good result could be found by experimenting with muting programs. However, subroutine nmo1() [*] differs from muting in two significant respects: (1) nmo1() is based on a theoretical concept whereas muting requires observational parameters and (2) muting applies a weighting in the coordinates of the (t,x) input space, while nmo1() does that but also includes the coordinate $\tau$ of the the output space. With nmo1() events from different $\tau$ depths see different mutes which is good where a shallow event asympote crosses a deeper event far from its own asymptote.

subroutine nmo1( adj, add, slow,    x, t0, dt, n,zz,  tt )
integer  it, iz, adj, add,                        n
real  xs, t , z,           slow(n), x, t0, dt, zz(n), tt(n), wt
call adjnull(    adj, add,                     zz,n,  tt,n)
do iz= 1, n {	z = t0 + dt*(iz-1)
		xs = x * slow(iz)
		t = sqrt ( z * z + xs * xs) + 1.e-20
		wt = z/t * (1./sqrt(t)) 		#  weighting function
		it = 1 + .5 + (t - t0) / dt
		if( it <= n )
			if( adj == 0 )
				tt(it) = tt(it) + zz(iz) * wt
				zz(iz) = zz(iz) + tt(it) * wt
return; end

The most physically correct amplitudes are not likely to be those for which $\bold S'\bold S\approx \bold I$.Physically accurate amplitudes involve many theoretical and experimental issues not covered here. It is easy to include some effects (spherical divergence based on velocity depth variation) and harder to include others (surface ghosts and arrays). I will not set up the ``inverse problem'' here. It has many complicating pitfalls that you are welcome to work through.

previous up next print clean
Next: Ideal weighting functions for Up: Claerbout: Stacking fundamentals Previous: MOVEOUT AND STACK
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