The gradient

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
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} } (4)

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} } (5)

is a generalization of the familiar differentiation operator. When the gradient operator acts on a function 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)} , it produces a vector, namely the gradient.


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