# The gradient

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 **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/":): {\displaystyle \mathbf{r}=\left(x,y\right)}**
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/":): {\displaystyle t\left(x,y\right)}**
. The gradient of **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/":): {\displaystyle 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 **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/":): {\displaystyle t\left(x,y\right)}**
evaluated at the point in question. The gradient is the vector defined by the equation

**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/":): {\displaystyle \begin{align} \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{align} }****(**)

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

**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/":): {\displaystyle \begin{align} \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{align} }****(**)

is a generalization of the familiar differentiation operator. When the gradient operator acts on a function

## Continue reading

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Traveltime | The directional derivative |

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Wave Motion | Visualization |

## 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