The unit tangent vector

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

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

The problem now is to find an expression for the unit tangent vector of the seismic ray. Let the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathbf{r}=\left(x,y\right) represent a point on a given ray (Figure 7). Let s denote arc length along the ray. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathbf{r+}d\mathbf{r}=\left(x{\rm +}dx{\rm ,\ }y{\rm +}dy\right) give an adjacent point on the same ray. The vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): d\mathbf{r}=\left(dx{\rm ,\ }dy\right) is (approximately) a tangent vector to the ray. The length of this vector is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \sqrt{{\rm (}dx^{{\rm 2}}{\rm +}dy^{{\rm 2}}}{\rm )} which is approximately equal to the increment ds of the arc length on the ray. As a result, the unit vector tangent to the ray is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathbf{u=}&\frac{d\mathbf{r}} {ds}{\rm =}\frac{dx}{ds}\mathbf{i}+\frac{dy}{ds}\mathbf{j}=\left(\frac{dx}{ds},\frac{dy}{ds}\right) . \end{align}

(2)

Here, i and j are the unit vectors in the horizontal and vertical directions, respectively. In other words, these unit vectors are defined as the vectors that have magnitude equal to one and have directions lying along the x- and y- axes, respectively.

Figure 7.  The unit tangent vector to a ray. The ray is specified by the radius vector (i.e., position vector) from the origin to a point on the ray. The wavefronts are orthogonal to the ray.

At this juncture, we want to clear up one confusing aspect of the mathematics of rays. The usual convention in the case of a raypath is to measure the angle of the tangent line from the vertical. This usage goes right back to the original seventeenth-century statement of Snell’s law. However, in mathematics, the convention is to measure the angle of the tangent line from the horizontal. It is better to stick with conventional mathematics and then convert to raypath usage when the results are displayed. A raypath is a curve along which the seismic energy travels. The unit tangent vector can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \mathbf{u=}&\left({\rm \ cos\ }\theta {\rm ,\ \ sin\ }\theta \right) , \end{align}

(3)

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta is the angle that the ray makes with the horizontal.

Let us review the results. Let the vector r be a point on the raypath. A closely spaced point on the raypath can be represented by the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathbf{r}+d\mathbf{r} . Their difference is the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): d\mathbf{r} , which is the vector that connects the two points in question. Let the path length between the two points be ds. The unit tangent to the raypath is the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): d\mathbf{r}/ds as the points approach each other. The length of the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): d\mathbf{r} is approximately equal to the path-length difference ds. As a result, the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): d\mathbf{r}/ds is the unit vector. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta be the angle that the tangent makes with the horizontal axis. The unit tangent vector is then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \mathbf{u=}\left({\rm \ cos\ }\theta {\rm ,\ \ sin\ }\theta \right) . The vector u is directed along the tangent to the curve in the direction of increasing values of the arc length s.

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Table of Contents (book)

Also in this chapter

• Reflection seismology
• Digital processing
• Signal enhancement
• Migration
• Interpretation
• Rays
• Traveltime
• 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

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