DETECTION OF PLANE WAVES

In his book, Claerbout 1994 shows that filters of the form  

$$\frac{\partial }{\partial x} - p_x \frac{\partial }{\partial t}$$ (1)

destroy plane waves u(x,t) = u(t + p_x x). Letting $<\cdots\gt$ denote averaging over time and $[\cdots]$ over space, the value,  

$$p_x = \frac{<\frac{\partial [u]}{\partial x}\times\frac{\par... ...rtial [u]}{\partial t}\times\frac{\partial [u]}{\partial t}\gt}$$ (2)

represents a smoothed least squares estimate of local dip. In the 3-D setting one can calculate both, p_x, and,  

$$p_y = \frac{<\frac{\partial [u]}{\partial y}\times\frac{\par... ...rtial [u]}{\partial t}\times\frac{\partial [u]}{\partial t}\gt}$$ (3)

to obtain estimates in the line and cross-line directions so that the vector

 

p = (p_x,p_y) (4)

defines a local plane at each point in the 3-D volume. Its magnitude,  

$$\parallel p \parallel = \sqrt{(p_x^2 + p_y^2)}$$ (5)

is a scalar which responds in a somewhat dramatic fashion to relatively minor deviations from local plane wave assumptions. The reasons for this response may not be completely clear. For time-migrated 3-D volumes, $p_x = \frac{\delta \tau}{\delta x}$ and $p_y = \frac{\delta \tau}{\delta y}$ are estimates of the post-migration dip. For a given velocity, they provide the basis for computing the zero offset point from which the local primary-reflection horizon migrated. In this case, the dip-magnitude is in some sense porportional to the distance to this zero-offset location. For large dipping events, such as faults, this distance can be quite large.

Abrupt changes in amplitude or wavelet phase can also result in major changes in dip-magnitude. The quantities p_x and p_y are estimates of differential change and so will be significant whenever discontinuities are present in the data volume.