Snell wave coordinates

A Snell wave has three intrinsic planes, which suggests a coordinate system. First are the layer planes of constant z, which include the earth's surface. Second is the plane of rays. Third is the moving plane of the wavefront. The planes become curved when velocity varies with depth.

The following equations define Snell wave coordinates:

$$\begin{eqnarray} z' (z,x,t) \ \ \ &=&\ \ \ z \ { \cos \, \theta \over v } \\ x'... ... \, \theta \over v }\ \ -\ \ x \ {\sin\,\theta \over v }\ \ +\ \ t\end{eqnarray}$$ (30) (31) (32)

Equation (30) simply defines a travel-time depth using the vertical phase velocity seen in a borehole. Interfaces within the earth are just planes of constant z'.

Setting x' as defined by equation (31) equal to a constant, say, x₀, gives the equation of a ray, namely, $(x - x_0 )/z = - \tan \theta$.Different values of x₀ are different rays.

Setting t' as defined by equation (32) equal to a constant gives the equation for a moving wavefront. To see this, set t' = t₀ and note that at constant x you see the borehole speed, and at constant z you see the airplane speed.

Mathematically, one equation in three unknowns defines a plane. So, setting the left side of any of the equations (4a,b,c) to a constant gives an equation defining a plane in (z,x,t)-space. To get some practice, we will look at the intersection of two planes. Staying on a wavefront requires dt' = 0. Using equation (32) gives  

$$dt' \ \ =\ \ 0 \ \ =\ \ {\cos\, \theta \over v }\ dz \ -\ {\sin\,\theta \over v }\ dx \ +\ dt$$ (33)

Combining the constant wavefront equation dt' = 0 with the constant depth equation dz' = dz = 0 gives the familiar relationship  

$${dt \over dx }\ \eq \ p$$ (34)

When coordinate planes are nonorthogonal, the coordinate system is said to be affine. With affine coordinates, such as (30), (31) and (32), we have no problem with computational tractability, but we often do have a problem with our own confusion. For example, when we display movies of marine field data, we see a sequence of (h,t)-planes. Successive planes are successive shot points. So the data is displayed in (s,h) when we tend to think in the orthogonal coordinates (y,h) or (s,g). With affine coordinates I find it easiest to forget about the coordinate axis, and think instead about the perpendicular plane. The shot axis ${\bf s}$ can be thought of as a plane of constant geophone, say, cg. So I think of the marine-data movie as being in (cs,ch,ct)-space. In this movie, another plane, really a family of planes, the planes of constant midpoints cy, sweep across the screen, along with the ``texture'' of the data.

To define Snell coordinates when the velocity is depth-variable, it is only necessary to interpret (30), (31) and (32) carefully. First, all angles must be expressed in terms of p by the Snell substitution $\sin\theta = pv(z)$.Then z must everywhere be replaced by the integral with respect to z.