The gradient

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 traveltime function is like a hill whose height at the point $\mathbf {r} =\left(x,y\right)$ is $t\left(x,y\right)$. The gradient of $t\left(x,y\right)$ at a given point is a vector that points in the direction of the steepest slope at that point. The magnitude of the gradient vector gives the steepness of the slope. The gradient depends only on the partial derivatives of $t\left(x,y\right)$ evaluated at the point in question. The gradient is the vector defined by the equation

${\begin{aligned}\mathrm {grad} {\textit {t}}&{\rm {=}}\left({\frac {\partial t}{\partial x}}{\rm {,\ }}{\frac {\partial t}{\partial y}}\right){\rm {=}}{\frac {\partial t}{\partial x}}\mathbf {i} +{\frac {\partial t}{\partial y}}\mathbf {j} .\end{aligned}}$

(4)

Here i, j are the unit vectors in the x-, y-directions, respectively. The gradient operator

${\begin{aligned}\mathrm {grad} \equiv {\frac {\partial }{\partial \mathbf {r} }}&{\rm {=}}\left({\frac {\partial }{\partial x}}{\rm {,\ }}{\frac {\partial }{\partial y}}\right){\rm {=}}{\frac {\partial }{\partial x}}\mathbf {i} +{\frac {\partial }{\partial y}}\mathbf {j} \end{aligned}}$

(5)

is a generalization of the familiar differentiation operator. When the gradient operator acts on a function $t\left(x,y\right)$, it produces a vector, namely the gradient.

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Wave Motion	Visualization
Table of Contents (book)

Also in this chapter

• Reflection seismology
• Digital processing
• Signal enhancement
• Migration
• Interpretation
• Rays
• The unit tangent vector
• Traveltime
• 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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