Limiting flattening model space

The two-stage approach of the last section is only applicable when the $\boldsymbol \tau$ estimation is able to fully describe the moveout in the gather. When it cannot, another approach must be found. The left panel of Figure  is a synthetic CMP gather created by bandpassing random numbers, and then spraying them out with adjoint of NMO. The right panel of Figure  shows the result of estimating time shifts (7 Gauss-Newton steps) and then applying those time shifts to flatten the data. Note that the flattening approach has failed in several areas.

 

syn
Figure 6 The left panel is a synthetic CMP gather. The right panel shows the result of flattening the CMP gather using the standard approach. Note the waviness of several reflectors due to the the non-linear nature of the flattening technique.

Estimation of the time shifts is problematic because the problem is inherently non-linear. One successful strategy is to try to start with an initial guess that is as close as possible to the correct solution. Another is to limit the model space to feasible candidates. In this simple case we know that the moveout is governed by the NMO equation. We can linearize the NMO equation that relates time shifts $\tau$, zero offset time t₀, offset h, and slowness s through

$$\tau=\sqrt{t_0^2 +h^2s^2}-t_0$$ (12)

around our initial slowness $\bf s_0$. We obtain an equation,

$$\Delta \tau = \frac{h^2s_0}{\sqrt{t_0^2+h^2s_0^2}}h^2s_0\Delta s,$$ (13)

that relates $\boldsymbol \Delta \bf \tau$to $\boldsymbol \tau \bf s$. The implied operator $\bf H$ then helps to form the linearized optimization equation,

$$Q(\boldsymbol \Delta s) = \vert\bf W_\epsilon \nabla \bf H \Delta s -\boldsymbol \Delta \bf p\vert^2.$$ (14)

In practice we need to add an additional weighting operator $\bf W_0$ which accounts for areas effected wavelet stretch and for reflections that exist at zero offset, but not at larger offsets. As a result we must use a space-domain conjugate gradient scheme  

$$Q(\boldsymbol \Delta s) = \vert\bf W_0 \bf W_\epsilon \nabla \bf H \Delta s -\boldsymbol \Delta \bf p\vert^2 .$$ (15)

Figure  shows the flattened CMP gather using the slowness model space description. While not perfect, the result is significantly flatter than the alternative approach (right panel of Figure ).

 

cmp Figure 7 The flattened CMP gather using a Gauss-Newton scheme with the model space limited to hyperbolic moveout. The result is much flatter than the standard parameterization scheme show in the right panel of Figure .

Figure  shows the result of applying both techniques to a CMP gather from the same North Sea dataset used in the previous section. The left panel is the raw gather, the center panel uses the conventional technique, and the right panel limits the moveout description to a single hyperbolic parameter. Note how both approaches fail at early times but the hyperbolic description provides noticably better result.

 

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Figure 8 A CMP gather from the North Sea. The left panel shows the original gather. The center panel shows the result of using a standard parameterization for flattening. The right panel is the result using a hperbolic parameterization.