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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 2 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

The traveltime function is like a hill whose height at the point ${\displaystyle \mathbf {r} =\left(x,y\right)}$ is ${\displaystyle t\left(x,y\right)}$. The gradient of ${\displaystyle t\left(x,y\right)}$ 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 ${\displaystyle t\left(x,y\right)}$ evaluated at the point in question. The gradient is the vector defined by the equation

 {\displaystyle {\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

 {\displaystyle {\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 ${\displaystyle t\left(x,y\right)}$, it produces a vector, namely the gradient.