Snell’s law

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
Series Geophysical References Series
Title Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author Enders A. Robinson and Sven Treitel
Chapter 2
DOI http://dx.doi.org/10.1190/1.9781560801610
ISBN 9781560801481
Store SEG Online Store

Suppose we have two homogeneous isotropic media separated by a straight horizontal interface (Figure 10). We plot the depth axis y in the upward direction. Let . The low-velocity surface layer (layer 1, with constant slowness ) is in the range . The high-velocity deep layer (layer 2, with constant slowness ) is in the range . Ray EFG starts in a vertical direction. It strikes the horizontal interface at right angles. A particle on the ray speeds up when it passes from the low-velocity layer 1 into the high-velocity layer 2. The ray does not bend at the interface and travels in layer 2 in the same vertical direction. Ray ABC starts in a direction oblique to the interface. A particle on ray ABC speeds up when it passes from the low-velocity layer 1 into the high-velocity layer 2, just as the particle did for ray EFG. The particles on ray EFG and on ray ABC experience exactly the same contrast in velocity. However, ray ABC bends at the interface. Why?

The particle is like a little air-plane. The fuselage goes along the raypath. The airplane is steered by the wings, which extend at right angles to each side. The left wing enters the high-velocity layer first, whereas the right wing is still in the low-velocity layer. This swings the plane to the right, and the raypath bends at the interface. How much does it bend?

Ray vector AB in layer 1 has length and makes angle with the vertical. Ray vector BC in layer 2 has length and makes angle with the vertical. By Snell’s law, we have . This equation says that the horizontal projections of the two ray vectors are the same. We expect this result because the slowness does not change in the horizontal direction. The slowness does E change at the interface in the vertical direction. The shallow medium has slowness , and the deep medium has a smaller slowness . Because the interface is level, it follows that slowness is constant along any horizontal line, so . However, slowness does change at the interface. The difference of the slowness at the interface is , which is negative because . The gradient of slowness at the interface is proportional to . This vector points straight into the low-velocity layer. The difference vector BD = BCAB must be proportional to the gradient of the slowness, and hence vector BD must point straight down. Thus, , which says that the raypath bends to the right at the interface. In this section, we have given a preview of the ray equation.

Figure 10.  Ray EFG, which is at right angles to the interface, does not bend at the interface. Ray ABC, which is skewed to the interface, does bend.


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