GMS:Gradient Plane Nodal Functions: Difference between revisions
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A limitation of [[GMS: | A limitation of [[GMS:Shepard's Method|Shepard's method]] is that the interpolating surface is a simple weighted average of the data values of the points; the surface is constrained to lie between the minimum and maximum data values. In other words, the surface does not infer local maxima or minima implicit in the data values. This problem can be overcome by generalizing the basic form of the equation for Shepard's method in the following manner: | ||
<!-- | <!--<math> F(x,y) = \sum_{i=1}^n w_{i}Q_{i}(x,y)</math>--> | ||
<math> F(x,y) = \sum_{i=1}^n w_{i}Q_{i}(x,y)</math> | :[[Image:gradplanenode1.jpg]] | ||
where Q<sub>i</sub> are nodal functions or individual functions defined at each | where ''Q<sub>i</sub>'' are nodal functions or individual functions defined at each point (Franke 1982; Watson & Philip 1985). The value of an interpolation location is calculated as the weighted average of the values of the nodal functions at that location. The standard form of Shepard's method can be thought of as a special case where horizontal planes (constants) are used for the nodal functions. The nodal functions can be sloping planes that pass through the data point. The equation for the plane is as follows: | ||
<math> {\displaystyle Q_{i} (x,y) = f_{x} (x-x_{i})+f_{y} (y-y_{i})+f_{i}}</math> | <!--<math> {\displaystyle Q_{i} (x,y) = f_{x} (x-x_{i})+f_{y} (y-y_{i})+f_{i}}</math>--> | ||
:[[Image:gradplanenode2.jpg]] | |||
where f<sub>x</sub> and f<sub>y</sub> are partial derivatives at the | where ''f<sub>x</sub>'' and ''f<sub>y</sub>'' are partial derivatives at the data point that have been previously estimated based on the geometry of the surrounding points. Gradients are estimated in GMS by computing the gradient at each point as the average of the gradients of nearest ''n'' points. | ||
The planes represented by the above equation are sometimes called "gradient planes". By averaging planes rather than constant values at each | The planes represented by the above equation are sometimes called "gradient planes". By averaging planes rather than constant values at each point, the resulting surface infers extremities and is asymptotic to the gradient plane at the point rather than forming a flat plateau at the point. | ||
3D Interpolation | ==3D Interpolation== | ||
The 3D equivalent of a gradient plane is a | The 3D equivalent of a gradient plane is a gradient hyperplane. The equation of a gradient hyperplane is as follows: | ||
<math> {\displaystyle Q_{i}(x,y,z) = f_x (x-x_i)+f_y (y-y_i)+f_z (z-z_i)+f_i} </math> | <!--<math> {\displaystyle Q_{i}(x,y,z) = f_x (x-x_i)+f_y (y-y_i)+f_z (z-z_i)+f_i} </math>--> | ||
:[[Image:gradplanenode3.jpg]] | |||
where f<sub>x</sub>, f<sub>y</sub>, and f<sub>z</sub> are partial derivatives at the | where ''f<sub>x</sub>'', ''f<sub>y</sub>'', and ''f<sub>z</sub>'' are partial derivatives at the point that are estimated based on the geometry of the surrounding points. The gradients are found using a regression analysis which constrains the hyperplane to the point and approximates the nearby points in a least squares sense. At least five non-coplanar points must be used. | ||
==Related Topics== | |||
* [[GMS:Inverse Distance Weighted|Inverse Distance Weighted]] | |||
{{Navbox GMS}} | {{Navbox GMS}} | ||
[[Category:Interpolation]] | [[Category:Interpolation]] | ||
[[Category:Equations|Gradient]] | |||
Latest revision as of 14:00, 12 September 2017
A limitation of Shepard's method is that the interpolating surface is a simple weighted average of the data values of the points; the surface is constrained to lie between the minimum and maximum data values. In other words, the surface does not infer local maxima or minima implicit in the data values. This problem can be overcome by generalizing the basic form of the equation for Shepard's method in the following manner:
where Qi are nodal functions or individual functions defined at each point (Franke 1982; Watson & Philip 1985). The value of an interpolation location is calculated as the weighted average of the values of the nodal functions at that location. The standard form of Shepard's method can be thought of as a special case where horizontal planes (constants) are used for the nodal functions. The nodal functions can be sloping planes that pass through the data point. The equation for the plane is as follows:
where fx and fy are partial derivatives at the data point that have been previously estimated based on the geometry of the surrounding points. Gradients are estimated in GMS by computing the gradient at each point as the average of the gradients of nearest n points.
The planes represented by the above equation are sometimes called "gradient planes". By averaging planes rather than constant values at each point, the resulting surface infers extremities and is asymptotic to the gradient plane at the point rather than forming a flat plateau at the point.
3D Interpolation
The 3D equivalent of a gradient plane is a gradient hyperplane. The equation of a gradient hyperplane is as follows:
where fx, fy, and fz are partial derivatives at the point that are estimated based on the geometry of the surrounding points. The gradients are found using a regression analysis which constrains the hyperplane to the point and approximates the nearby points in a least squares sense. At least five non-coplanar points must be used.
Related Topics
GMS – Groundwater Modeling System | ||
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