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3D INTERPOLATION OPTIONS
[[File:3DinterpolationOpts.jpg|thumb|500 px|The ''3D Interpolation Options'' dialog.]]

[[GMS:3D Scatter Point Module|3D scatter point]] sets are used for [[GMS:Interpolation|interpolation]] to other data types such as grids and meshes. Interpolation is useful for such tasks as isosurface rendering or setting up input data for a model. Since no interpolation scheme is superior in all cases, several interpolation techniques are provided in GMS.

The basic approach to performing an interpolation is to select an appropriate interpolation scheme and interpolation parameters, and then interpolate to the desired object using one of the interpolation commands (to [[GMS:3D Grid Module|3D grid]], to [[GMS:3D Mesh Module|3D mesh]], etc.) described below.

The interpolation options are selected using the ''3D Interpolation Options'' dialog which is accessed through the '''Interpolation Options''' command in the ''Interpolation'' menu. The ''Interpolation'' menu appears when a scatter set item is active in the Project Explorer. Once a set of options is selected, those options are used for all subsequent interpolation commands. The options in the ''3D Interpolation Options'' dialog are as follows:

==Active Dataset==
Interpolation is always performed using the active [[GMS:Datasets|dataset]] of the active scatter point set. The active dataset is normally selected in the [[GMS:The GMS Window|Project Explorer]]. The name of the current active dataset is listed at the top of the ''3D Interpolation Options'' dialog in the ''Interpolating from'' section.

''Object'' displays the name of the active 3D scatter item, and ''Dataset'' displays the name of the 3D scatter dataset.	

If the active dataset is transient then more interpolation options are available.

{{Steady State vs. Transient Interpolation}}

==Interpolation Methods==
{{GMS 3D Interpolation Methods}}

==Anisotropy==
Sometimes the data associated with a scatter point set will have directional tendencies. The ''Azimuth'' and ''Horizontal Anisotropy'' allow taking into account these tendencies. 

===Vertical Anisotropy===
In 3D, vertical anisotropy is also available. In previous versions of GMS it was possible to enter a Z scale. Vertical anisotropy is 1 over the Z scale. This notation was changed to be consistent with Kriging.

Occasionally, scatter point sets are sampled along vertical traces. In such cases, the distances between scatter points along the vertical traces are an order of magnitude smaller than the distances between scatter points along the horizontal plane. For example, if the scatter point set was obtained from borehole data, the distance between scatter points may be a few centimeters, whereas the distance between boreholes may be several meters. This disparity in scaling causes clustering and can be a source of poor results in some interpolation methods.

The effects of clustering along vertical traces can be minimized using the vertical anisotropy option in the ''3D Interpolation Options'' dialog. The Z coordinate of each of the scatter points is multiplied by 1 divided by the vertical anisotropy parameter prior to interpolation. Thus, if the vertical anisotropy parameter is less than 1.0, scatter points along the same vertical axis appear farther apart than they really are and scatter points in the same horizontal plane appear closer than they really are. As a result, points in the same horizontal plane are given a higher relative weight than points along the Z axis. This can result in improved accuracy, especially in cases where the horizontal correlation between scatter points is expected to be greater than the vertical correlation (which is typically the case in soils since soils are deposited in horizontal layers).

== Extrapolation ==
Although they are referred to as interpolation schemes, most of the schemes supported by GMS perform both interpolation and extrapolation. That is, they can estimate a value at points both inside and outside the scatter point set. Obviously, the interpolated values are more accurate than the extrapolated values. Nevertheless, it is often necessary to perform extrapolation. Some of the schemes, however, perform interpolation but cannot be used for extrapolation. Interpolation points outside the scatter set are assigned the ''Default extrapolation value''.

===Assign default extrapolation value to hidden objects===
This option will assign the default extrapolation value to all cells that are hidden using the '''Hide''' command in the ''Display'' | '''Visibility''' menu option (see [[GMS:Display Menu|Display Menu]]).

<!--===Assign default extrapolation value to unloaded subdomains===-->

==Truncation==
When interpolating a set of values, it is sometimes useful to limit the interpolated values to lie between a minimum and maximum value. For example, when interpolating contaminant concentrations, a negative value of concentration is meaningless. However, many interpolation schemes will produce negative values even if all of the scatter points have positive values. This occurs in areas where the trend in the data is toward a zero value. The interpolation may extend the trend beyond a zero value into the negative range. In such cases it is useful to limit the minimum interpolated value to zero. Interpolated values can be limited to a given range by selecting the ''Truncate values'' option in the ''3D Interpolation Options'' dialog. The minimum and maximum values can either set according to the minimum and maximum of the dataset or according to specified range.

== Related Topics ==
*[[GMS:2D Interpolation Options|2D Interpolation Options]]


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[[Category:Interpolation]]
[[Category:Interpolation Dialogs|3]]
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UGRID INVERSE DISTANCE WEIGHTED INTERPOLATION
[[File:Interpolate-IDW.png|thumb|300 px|The ''Interpolate &ndash; Inverse Distance Weighted'' dialog.]]
One of the most commonly used techniques for interpolation of point data is inverse distance weighted (IDW) interpolation. Inverse distance weighted methods are based on the assumption that the interpolating surface should be influenced most by the nearby points and less by the more distant points. The interpolated surface is a weighted average of the point data; the weight assigned to each point diminishes as the distance to the interpolation location increases. Several options are available for inverse distance weighted interpolation. The options are selected using the ''IDW Interpolation Options'' dialog. This dialog is accessed through the '''Options''' button next to the ''Inverse distance weighted'' item in the ''Interpolate UGrid to UGrid'' dialog. 

The options in the dialog are as follows:
*''Nodal function''
**[[GMS:Shepard's Method|Constant (Shepard's Method)]] &ndash; The simplest form of inverse distance weighted interpolation. Includes the option to ''Use classic weight function'' to enter a ''Weighting exponent''.
**[[GMS:Gradient Plane Nodal Functions|Gradient Plane Nodal Functions]] &ndash; Variation of Shepard's method with nodal functions or individual functions defined at each point
**[[GMS:Quadratic Nodal Functions|Quadratic Nodal Functions]] &ndash; Makes use of quadratic polynomials to constrain nodal functions. 
*''Computation of nodal function coefficients'' &ndash; Uses a subset of the data points. Using a subset of the data points drops distant points from consideration since they are unlikely to have a large influence on the nodal function. In addition, using a subset can speed up the computations since less points are involved. Two options are available for defining which points are included in the subset. In the first approach, only the nearest ''N'' points are used. In the second approach, only the nearest ''N'' points in each quadrant are used as illustrated in the figure below. The second approach may give better results if the data points tend to be clustered.
*''Computation of interpolation weights'' &ndash; Uses a subset of the data points as with the ''Computation of nodal function coefficients'' section.
*''Truncate values'' &ndash; This section allows for limiting the interpolated values to lie between the minimum and maximum value.
**''Truncate to min/max of dataset'' &ndash; Limits the interpolated values to the minimum and maximum values in the original dataset.
**''Truncate to specified range'' &ndash; Allows setting a user specified minimum and maximum value range.
**''Min'' &ndash; Manually sets a minimum value.
**''Max'' &ndash; Manually sets a maximum value.
*'''Advanced''' &ndash; This button will open the ''Interpolate &ndash; Advanced'' dialog where options for anisotropy and extrapolation can be adjusted.

==Related Topics==
* [[GMS:UGrid Interpolation|UGrid Interpolation]]


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UGRID interpolation
{{UGrid links}}
__TOC__
UGrid [[GMS:Datasets|datasets]] can be interpolated to other UGrids similar to how scatter point datasets can be interpolated to other objects. The '''Interpolate To''' command is found in the right-click menu of the UGrid dataset to be interpolated to another UGrid. This command opens the ''Interpolate UGrid to UGrid'' dialog:

[[File:Interpolate to UGrid.png|thumb|500px|none|''Interpolate UGrid to UGrid'' dialog]]

This dialog allows selecting the interpolation options to use, and the UGrid to interpolate to. See [[GMS:Interpolation|Interpolation]] for more information on interpolation. Options include:
* ''Source UGrid'' &ndash; The drop-down menu here will display a list of all available UGrids. Select the UGrid containing the source dataset.
* ''Source dataset'' &ndash; The drop-down menu will display a list of all available datasets under the source UGrid. Select the dataset to use for interpolation.
* ''Times'' &ndash; If the source dataset is transient, time steps should be specified. Options include:
** "Specified Time Step" &ndash; This option will use the select time step in the next field.
** "All Times" &ndash; This option will use all time steps available in the source dataset.
*''Interpolation method'' &ndash; This section specifies which [[GMS:Interpolation|interpolation process]] will be used.
**[[GMS:Linear|''Linear'']] &ndash; Uses data points that are first triangulated to form a network of triangles. The '''Options''' button for this method will bring up the ''Interpolate &ndash; Linear'' dialog where the interpolation values can be truncated or the Clough-Tocher method can be applied.
**[[GMS:Inverse Distance Weighted|''Inverse distance weighted'']] &ndash; Creates an interpolated surface that is a weighted average of the point data; the weight assigned to each point diminishes as the distance to the interpolation location increases. The '''Options''' button next to this option will bring up either the ''2D IDW Interpolation Options'' dialog or the ''3D IDW Interpolation Options'' dialog.
**[[GMS:Natural Neighbor|''Natural neighbor'']] &ndash; Based on the Thiessen polygon network of the point data. The '''Options''' button next to this option will bring up the ''Natural Neighbor Options'' dialog.
**[[GMS:Kriging|''Kriging'']] &ndash; Based on the assumption that the parameter being interpolated can be treated as a regionalized variable. The '''Options''' button next to this option will bring up the [[GMS:Kriging Options|''Kriging Options'']] dialog.
**Dimension &ndash; Options in this drop-down determine whether the interpolation will be two-dimensional or three-dimensional.
***''2D'' &ndash; Designates the interpolation as two-dimensional data. This is the default interpolation method.
***''3D'' &ndash; Specified the interpolation as three-dimensional data. Only the inverse distance weighted and Kriging options are available with this method.
* ''Target UGrid'' &ndash; The drop-down menu will contain a list of available UGrids where the source dataset can be interpolated.
* ''Target dataset name'' &ndash; Enter a name for the new dataset that will appear under the target UGrid.
* ''Target dataset location''  &ndash; Specifies where the new dataset will be located: at the points or at the cells.
**''Points'' &ndash; Specifies the new dataset will be located at the points.
**''Cells'' &ndash; Specifies the new dataset will be located at the cells.

==UGrid Interpolation Options==
Clicking the '''Options''' button will open a dialog specific to the interpolation method being used.
===Linear===
The Linear interpolation scheme uses data points that are first triangulated to form a network of triangles. See [[GMS:UGrid Linear Interpolation|UGrid Linear Interpolation]] for more details.

===Inverse Distance Weighted===
Inverse Distance Weighted (IDW) is one of the most commonly used techniques for interpolation of point data. Its methods are based on the assumption that the interpolating surface should be influenced most by the nearby points and less by the more distant points. See [[GMS:UGrid Inverse Distance Weighted Interpolation|UGrid Inverse Distance Weighted Interpolation]] for more details.

===Natural Neighbor===
Natural neighbor interpolation is based on the Thiessen polygon network of the point data. The Thiessen polygon network can be constructed from the Delaunay triangulation of a set of points. A Delaunay triangulation is a network of triangles that has been constructed so that the Delaunay criterion has been satisfied. As with IDW interpolation, the nodal functions can be either constants, gradient planes, or quadratics. See [[GMS:UGrid Natural Neighbor Interpolation|UGrid Natural Neighbor Interpolation]] for more details.

===Kriging===
Kriging is based on the assumption that the parameter being interpolated can be treated as a regionalized variable. A regionalized variable is intermediate between a truly random variable and a completely deterministic variable because it varies in a continuous manner from one location to the next. Therefore points that are near each other have a certain degree of spatial correlation, but points that are widely separated are statistically independent. Kriging is a set of linear regression routines which minimize estimation variance from a predefined covariance model.

See [[GMS:UGrid Kriging Interpolation|UGrid Kriging Interpolation]] for more details.

===Advanced===
[[File:Interplate-Advanced.png|thumb|350 px|The ''Interpolate &ndash; Advanced'' dialog]]
The ''Interpolate &ndash; Advanced'' dialog is accessed through the '''Advanced''' on the interpolation method dialogs. Options in this dialog include:
*''Anisotropy'' &ndash; Options in this section allow taking into account for directional tendencies in the original dataset.
**''Horizontal anisotropy''
**''Azimuth'' &ndash; Sets angle of degrees between the projected vector and a reference vector on the reference plane.
**''Vertical Anisotropy (1/z mag)'' &ndash; Available with the 3D option. Vertical anisotropy is 1 over the Z scale.
*''Extrapolation''
**''Value''
**''Assign extrapolation value to hidden objects'' &mdash; Assigns the default extrapolation value to all cells that are hidden using the '''Hide''' command in the ''Display | Visibility'' menu.
<!--10.3 and older
UGrid [[GMS:Datasets|datasets]] can be interpolated to other UGrids similar to how scatter point datasets can be interpolated to other objects. The '''Interpolate To''' command is found in the right-click menu of the UGrid dataset to be interpolated to another UGrid. This command opens the ''Interpolation Options'' dialog:

[[File:UGridInterpolation.png|thumb|600px|none]]

This dialog allows selecting the interpolation options to use, and the UGrid to interpolate to. See [[GMS:Interpolation|Interpolation]] for more information on interpolation. Options include:
*''Interpolating from'' &ndash; This section gives information on the dataset being interpolated to the new dataset.
**''Object'' &ndash; Shows the dataset being used for the interpolation.
**''Dataset'' &ndash; Shows the type of dataset being used for the interpolation.
**''Time step'' &ndash; If the dataset being used has time steps, shows which time step is being used.
**''Use all time steps'' &ndash; If there are time steps, this option will interpolate all time steps in the source dataset.
*''Interpolating to'' &ndash; This data tree shows all objects that can receive the interpolated data. Select the object that will receive the new interpolated dataset.
*''Dimension'' &ndash; Options in this section determine whether the interpolation will be two-dimensional or three-dimensional.
**''2D'' &ndash; Designates the interpolation as two-dimensional data. All interpolation methods are available with this option.
**''3D'' &ndash; Specified the interpolation as three-dimensional data. Only the inverse distance weighted and Kriging options are available with this method.
*''Anisotropy'' &ndash; Options in this section allow taking into account for directional tendencies in the original dataset.
**''Horizontal anisotropy''
**''Azimuth'' &ndash; Sets angle of degrees between the projected vector and a reference vector on the reference plane.
**''Vertical Anisotropy (1/z mag)'' &ndash; Available with the 3D option. Vertical anisotropy is 1 over the Z scale. 
*''Interpolation method'' &ndash; This section specifies which [[GMS:Interpolation|interpolation process]] will be used.
**[[GMS:Linear|''Linear'']] &ndash; Uses data points that are first triangulated to form a network of triangles. 
**[[GMS:Inverse Distance Weighted|''Inverse distance weighted'']] &ndash; Creates an interpolated surface that is a weighted average of the point data; the weight assigned to each point diminishes as the distance to the interpolation location increases. The '''Options''' button next to this option will bring up either the ''2D IDW Interpolation Options'' dialog or the ''3D IDW Interpolation Options'' dialog.
**[[GMS:Clough-Tocher|''Clough-Tocher'']] &ndash; A finite element method that has origins in the finite element method of numerical analysis.
**[[GMS:Natural Neighbor|''Natural neighbor'']] &ndash; Based on the Thiessen polygon network of the point data. The '''Options''' button next to this option will bring up the ''Natural Neighbor Options'' dialog.
**[[GMS:Kriging|''Kriging'']] &ndash; Based on the assumption that the parameter being interpolated can be treated as a regionalized variable. The '''Options''' button next to this option will bring up the [[GMS:Kriging Options|''Kriging Options'']] dialog.
**''Log interpolation'' &ndash; Takes the log of each data value in the active dataset prior to performing interpolation.
**''Set data value <=0 to'' &ndash; Assigns a value to points where the current data value is less than or equal to zero. Typically, a small positive number should be used.
*''Default extrapolation value''
*''Assign default extrapolation value to hidden objects'' &mdash; Assigns the default extrapolation value to all cells that are hidden using the '''Hide''' command in the ''Display | Visibility'' menu.
*''Truncate values'' &ndash; This section allows for limiting the interpolated values to lie between the minimum and maximum value.
**''Truncate to min/max of dataset'' &ndash; Limits the interpolated values to the minimum and maximum values in the original dataset.
**''Truncate to specified range'' &ndash; Allows setting a user specified minimum and maximum value range.
**''Min'' &ndash; Manually sets a minimum value.
**''Max'' &ndash; Manually sets a maximum value.
*''Create dataset at'' &ndash; Specifies where the new dataset will be located: at the points or at the cells.
**''Points'' &ndash; Specifies the new dataset will be located at the points.
**''Cells'' &ndash; Specifies the new dataset will be located at the cells.
*''New dataset name'' &ndash; Assigns a name to the new dataset created from the interpolation process.

It is also possible to drag a UGrid dataset and drop it on to another UGrid to open this dialog and to indicate what the interpolation target should be.-->

==Points vs. Cells==

UGrids can have datasets associated with both cells and points. Thus there is an option to specify where the new dataset will be located: at the points or at the cells.


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UGRID LINEAR INTERPOLATION
[[File:Interpolate-Linear.png|thumb|350 px|The ''Interpolate &ndash; Linear'' dialog]]
The Linear interpolation scheme uses data points that are first triangulated to form a network of triangles. The network of triangles only covers the convex hull of the point data, making extrapolation beyond the convex hull not possible.

If the linear interpolation scheme is selected, the data points are first triangulated to form a network of triangles. The equation of the plane defined by the three vertices of a triangle is as follows:

<!--<math>\ Ax+By+Cz+D = 0</math>-->
:[[Image:linear_eq1.jpg]]

:where ''A'', ''B'', ''C'', and ''D'' are computed from the coordinates of the three vertices ''(x1,y1,z1)'', ''(x2,y2,z2)'', and ''(x3,y3,z3)'':

:<math>\ A = y_1(z_2-z_3) + y_2(z_3-z_1) + y_3(z_1-z_2)</math>

:<math>\ B = z_1(x_2-x_3) + z_2(x_3-x_1) + z_3(x_1-x_2)</math>

:<math>\ C = x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)</math>

:<math>\ D = -Ax_1 - By_1 - Dz_1</math>

The plane equation can also be written as:

:<math>z = f(x,y) = -\frac{A}{C}x-\frac{B}{C}y-\frac{D}{C}</math>

:which is the form of the plane equation used to compute the elevation at any point on the triangle.

Since the network of triangles only covers the convex hull of the point data, extrapolation beyond the convex hull is not possible with the linear interpolation scheme. Any points outside the convex hull of the point data are assigned the default extrapolation value entered in the ''Interpolation Options'' dialog. The figure below shows a network of triangles created from point data.

:{{hide in print|[[File:convex_hull.jpg|thumb|none|left|300 px|Network of triangles]]}}
:{{only in print|[[File:convex_hull.jpg|275px|frame|center|Network of triangles]]}}

If the Linear interpolation method is selected in the ''Interpolate UGrid to UGrid'' dialog, options can be set in the ''Interpolate &ndash; Linear'' dialog. This dialog has the following options:
*''Truncate values'' &ndash; This section allows for limiting the interpolated values to lie between the minimum and maximum value.
**''Truncate to min/max of dataset'' &ndash; Limits the interpolated values to the minimum and maximum values in the original dataset.
**''Truncate to specified range'' &ndash; Allows setting a user-specified minimum and maximum value range.
**''Min'' &ndash; Manually sets a minimum value.
**''Max'' &ndash; Manually sets a maximum value.
*''Clough-Tocher'' &ndash; When on, the [[GMS:Clough-Tocher|Clough-Tocher]] interpolation technique will be used.
*'''Advanced''' &ndash; This button will open the ''Interpolate &ndash; Advanced'' dialog where options for anisotropy and extrapolation can be adjusted.

==Related Topics==
* [[GMS:UGrid Interpolation|UGrid Interpolation]]
* [[GMS:Clough-Tocher|Clough-Tocher]]

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GMS: SHEPARD'S METHOD
The simplest form of inverse distance weighted interpolation is the constant nodal function sometimes called the "Shepard's method" (Shepard 1968). The equation used is as follows:

<!--<math>F(x,y) = \sum_{i=1}^n w_if_i</math>-->
:[[Image:shep_eq1.jpg]]

where ''n'' is the number of points used to interpolate, ''f<sub>i</sub>'' are the prescribed function values at the points (e.g., the dataset values), and ''w<sub>i</sub>'' are the weight functions assigned to each point. The classical form of the weight function is:

<!--<math>w_i = \frac{h_i^{-p}}{\displaystyle \sum_{j=1}^n h_j^{-p}}</math>-->
:[[Image:shep_eq2.jpg]]

where ''p'' is an arbitrary positive real number called the weighting exponent and is defaulted to 2. The ''Use classic weight function'' can be turned on, and the weighting exponent modified, by turning on the ''Use classic weight function'' option in the ''IDW Interpolation Options'' dialog. ''h<sub>i</sub>'' is the distance from the point to the interpolation location or

<!--<math>h_i = \sqrt{(x-x_i)^2+(y-y_i)^2}</math>-->
:[[Image:shep_eq3.jpg]]

where ''(x,y)'' are the coordinates of the interpolation location and ''(x<sub>i</sub>,y<sub>i</sub>)'' are the coordinates of each point. The weight function varies from a value of unity at the point to a value approaching zero as the distance from the point increases. The weight functions are normalized so that the weights sum to unity.

Although the weight function shown above is the classical form of the weight function in inverse distance weighted interpolation, the following equation is used in GMS:

<!--<math>w_i = \frac{ \left [ \frac{R-h_i}{Rh_i} \right ] ^2}{\displaystyle \sum_{j=1}^n \left [ \frac{R-h_i}{Rh_i} \right ] ^2}</math>-->
:[[Image:shep_eq4.jpg]]

where ''h<sub>i</sub>'' is the distance from the interpolation location to the point ''i'', ''R'' is the distance from the interpolation location to the most distant  point, and n is the total number of points. This equation has been found to give superior results to the classical equation (Franke & Nielson, 1980).

The weight function is a function of Euclidean distance and is radially symmetric about each point. As a result, the interpolating surface is somewhat symmetric about each point and tends toward the mean value of the point data between the points. Shepard's method has been used extensively because of its simplicity.

== 3D Interpolation ==

The 3D equations for Shepard's method are identical to the 2D equations except that the distances are computed using:

<!--<math>h_i = \sqrt{(x-x_i)^2+(y-y_i)^2+(z-z_i)^2}</math>-->
:[[Image:shep_eq5.jpg]]

where ''(x,y,z)'' are the coordinates of the interpolation location and ''(x<sub>i</sub>,y<sub>i</sub>,z<sub>i</sub>)'' are the coordinates of each point.

==Related Topics==
* [[GMS:Inverse Distance Weighted|Inverse Distance Weighted]]


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