The principle of least time
The gradient is a concept that is familiar to everyone who has climbed a mountain (Figure 8). As a climber goes up the mountain, he experiences the rate of change of elevation. the gradient is the vector (because it has both magnitude and direction) that points in the steepest direction (i.e., the direction in which the rate of change is greatest). The magnitude of the gradient vector is the rate of change of the elevation along this path. Thus, if the climber continually follows the gradient, he will take the steepest path to the top of the mountain. This steepest path is called a flow line.
Using another analogy, a skier wants to descend on the steepest path available to him at his present position on a hill. The negative gradient is the vector that points in the steepest direction down the hill. This steepest path down is called a fall line. The fall line is always in the direction of the negative gradient of the function whose graph represents the surface of the hill. The fall line and the corresponding flow line coincide. The fall line will change direction as the skier goes downhill from one position to another. A skier feels the fall line and gets visual hints about its location. The skier’s eyes can detect local changes in the steepness of the hill. The skier’s legs indicate whether they are on the fall line or off it. When a skier is skiing the fall line, he feels equal pressure on both legs. All forces are channeled in a direction parallel to the skis. Skiing in any other direction results in pressure on one leg that differs from the pressure on the other. In other words, skiing the fall line means that the skier follows the direction of the negative gradient all the way down the hill.
The Mississippi River provides a geologic example. By depositing sand and silt, the Mississippi has created most of Louisiana. The river’s purpose is to get to the Gulf of Mexico in the least time possible. The river wants to follow the direction of the negative gradient, which is the path of steepest descent. In other words, like all rivers, the Mississippi is destined to follow a flow line. If the river had kept to one channel, southern Louisiana would be a long narrow peninsula reaching into the Gulf of Mexico. However, the river has deposited sediment over a wide area. Over time, as it continues to carry and deposit more sediment, the river lengthens and its mouth advances southward. The steepness of its path declines, the current slows, and sediment builds up the riverbed. Eventually, the bed builds up so much that the river surges over the left or right bank and goes off in a new direction, to follow what has newly become the steepest way down.
Southern Louisiana exists in its present form because periodically the Mississippi River has radically changed course by jumping here and there within an arc about 200 miles wide. Major shifts of that nature have tended to occur about once a millennium. About 1000 years ago, the Mississippi’s channel shifted to the river’s present course. Today, the Mississippi River has advanced far past New Orleans and out into the Gulf of Mexico. The Mississippi River wants to change its course again to follow a shorter and steeper route, but engineers have built a levee system to keep the river from jumping its banks and changing course.
Pierre de Fermat (1601–1665) formulated the rule known as Fermat’s principle of least time. In his original statement, Fermat asserted that the raypath taken by light traveling between any two points is such that the time taken is a minimum. In other words, the ray-paths are the flow lines. Eventually, Fermat’s original statement underwent some expansion. Fermat’s principle is expressed now as: The raypath taken by light traveling between any two points is such that the time taken is stationary with respect to variations of that path. Stationary means that the traveltime can be a minimum or can be a maximum or can be a point of inflection having a horizontal tangent. More specifically, the traveltime of the true trajectory (i.e., the raypath) will equal, to a first approximation, the traveltime of paths in the immediate vicinity. Energy traveling along these neighboring paths will arrive at the destination at about the same time by routes that differ only slightly. Thus, these neighboring paths will tend to reinforce one another. Energy taking other paths arrives out of phase and therefore tends to cancel out. The net result is that energy effectively propagates along the raypaths (i.e., the paths that satisfy Fermat’s principle). In this way, Fermat’s principle helps explain why light is so clever in its meanderings.
A lifeguard on the beach at A sees a drowning person in the water at E (Figure 9a). Which path should the lifeguard take to rescue the drowning person in the least time? The lifeguard might be tempted to take the straight-line path ACE because it represents the minimum distance he must travel. However, he knows that he can run faster on the beach than he can swim in the water. As a result, it pays to go a longer distance AD on the dry land to go a reduced distance DE in the water. Thus, by running to D and then swimming to E, the lifeguard does not minimize the distance he must travel, but he does minimize the time required to reach the drowning person. On the other hand, suppose the lifeguard were a seal. The seal waddles slowly on land but swims rapidly. The seal would take the path ABE. In fact, by instinct, a person in water will make a beeline for land, whereas a seal on land will make a beeline for water.
Let us take an isotropic medium in which the velocity varies continuously so that there is no interface at which reflection and/or refraction could occur. We now show that in such a case, Fermat’s principle holds in its original formulation (i.e., traveltime is a minimum). Let an arbitrary path between two wavefronts be given (Figure 9b). This arbitrary path is called the test path. Let us first figure out what happens between two closely spaced wavefronts. Because the wavefronts are so close together, we might consider the parts of them within a small region to be two parallel straight lines. The test path AC would be a straight line between the two wavefronts, and the flow line AB would be a straight line orthogonal to both wavefronts. If is the angle between the test path and the flow line, if ds is equal to the length of the flow line, and if is the length of the test path, then .
We now use the fact that the medium is isotropic, i.e., that the velocity at any point is the same in all directions. Thus, the time that energy takes to traverse the flow line is , where the slowness n is defined as the reciprocal of velocity. Likewise, the time it takes to traverse the test path is
The traveltime along the flow line between the two wavefronts is . The traveltime along the test path is . Because cos at any point is always less than or equal to one, it follows that . Thus, the traveltime along the flow line is less than the traveltime along any test path except along the flow line itself. Finally, the flow line - that is, the line whose direction at any point coincides with the direction of the gradient - is the least-time path.
|Previous section||Next section|
|The directional derivative||The eikonal equation|
|Previous chapter||Next chapter|
Also in this chapter
- Reflection seismology
- Digital processing
- Signal enhancement
- The unit tangent vector
- The gradient
- The directional derivative
- 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