Velocity is the most important variable in seismic analysis. For example, let us assume that the earth is isotropic and that the seismic velocity is continuously varying. Slowness is defined as the reciprocal of the velocity. Thus, the slowness function is . Because we are so used to velocity, it is a little difficult to get used to slowness. Think of velocity as swiftness. For example, a speeding car goes with large swiftness and small slowness. A seismic wave goes with small swiftness and large slowness in shallow layers. Conversely, a seismic wave goes with large swiftness and small slowness in deep layers.
In the treatment of wave motion, a region of approximation exists in which the wavelength is considered to be small compared with the dimensions of the components of the system involved. That region of approximation is treated by the methods of geometric optics, geometric acoustics, or geometric seismology, as the case may be. When the wave character cannot be ignored, then the methods of physical optics, physical acoustics, or physical seismology apply.
In physical optics, physical acoustics, or physical seismology, the waves carry energy along all sorts of paths. A ray is the path along which most of the energy is transmitted from one point to another. The other paths are called diffraction paths. Within the approximation represented by geometric optics, geometric acoustics, or geometric seismology, the diffraction paths are, in effect, discarded. Hence, all the energy travels along rays. The ray is a mathematical device rather than a physical entity. In practice, one can produce very narrow beams of light (as, for example, laser beams) that can be considered physical manifestations of rays.
Because the wavelength of light is very small compared with the size of ordinary objects, geometric optics can describe the behavior of a light in commonplace situations. When we turn to a seismic wave, the wavelength is not particularly small compared with the dimensions of geologic layers within the earth. However, the concept of a seismic ray fills an important need. Geometric seismology is not nearly as accurate as geometric optics, but ray theory can be used to solve many important practical geophysical problems. In particular, popular forms of prestack migration are based on tracing the raypaths of the primary reflections.
For example, a person in the shadow of a house is protected from the direct beams of the sun. However, that person is not completely protected from the noise of traffic on the other side of the house because sound waves with their larger wavelengths can diffract around the house. In other words, the house does not cast a distinct shadow for sound. Seismic energy does not travel exclusively along raypaths because some energy would reach points by diffraction even if the raypath were blocked. In other words, a buried obstacle does not cast a distinct shadow for seismic waves.
A raypath can be represented by a parametric curve in space as given by the vector . The parameter t is the traveltime, and are three real-valued functions of the time t.
What is ray tracing? When a propagating wave encounters an inhomogeneity, the wavefront will change direction. Rays are curves that trace the motion of the wavefront over time. Recall the story of Hansel and Gretel. As they went into the depths of the trees, Hansel dropped a little white pebble here and there on the mossy green ground. Night fell. However, the tiny white pebbles gleamed in the moonlight, and the children found their way home. In other words, Hansel and Gretel used the pebbles to backtrack their way to their home. Ray tracing can be used to backtrack the paths of the received reflected events back into the earth and place their energy at the point of reflection. We now will describe the behavior of the rays and wavefronts in media with a continuously varying velocity.
To simplify the mathematics in the rest of this chapter, we will consider only two spatial coordinates: the horizontal coordinate x and the depth coordinate y. We can use this simplification because it is just a mathematical formality to add the third coordinate, and there is no reason to carry all that baggage along the way. When we do the mathematics, we plot y in the upward direction so that we stay within the first quadrant. We learned mathematics in the first quadrant, so we should do mathematics in the first quadrant whenever possible. However, when the final picture is displayed on the computer, then depth y would be plotted in the downward direction.
A vector represents the geometric idea of a directed line segment. A vector is a quantity that has direction as well as magnitude. Boldface letters usually are used to denote vectors. The vector represents a raypath as a function of traveltime t. The foot of this vector is at the origin (0,0), and the arrowhead is at the point on the ray-path. For ease of notation, often the functional dependence on traveltime is not written explicitly but is understood implicitly. A 2D straight ray starting from the source point and going at constant speed v in the direction of the constant angle (as measured from the horizontal) has the parametric representation
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Also in this chapter
- Reflection seismology
- Digital processing
- Signal enhancement
- The unit tangent vector
- The gradient
- The directional derivative
- The principle of least time
- The eikonal equation
- Snell’s law
- Ray equation
- Ray equation for velocity linear with depth
- Raypath for velocity linear with depth
- Traveltime for velocity linear with depth
- Point of maximum depth
- Wavefront for velocity linear with depth
- Two orthogonal sets of circles
- Migration in the case of constant velocity
- Implementation of migration
- Appendix B: Exercises