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GS+ Interpolation

Interpolation in GS+
Interpolation means estimating values for points not actually sampled, thereby producing a map or some other spatial model for an area that was not exhaustively sampled. There are many different interpolation techniques, ranging from simple linear techniques that average the values of nearby sampled points, to more complex techniques like kriging that use nearby points weighted by distance from the interpolate location plus the degree of autocorrelation for those distances.

GS+ provides three broad types of interpolation. All are nearest-neighbor techniques in which values at locations close to the interpolation point are used to estimate the interpolation point value. They differ in the way that nearby locations are weighted. The techniques are:

  • Kriging, in which interpolation estimates are made based on values at neighboring locations plus knowledge about the underlying spatial relationships in a data set. Variograms provide knowledge about the underlying relationships. Kriging is usually superior to other means of interpolation because it provides an optimal interpolation estimate for a given coordinate location, as well as a variance estimate for the interpolation value.
  • Inverse Distance Weighting (IDW) and Normal Distance Weighting (NDW), in which interpolation estimates are made based on values at nearby locations weighted only by distance from the interpolation location.
  • Conditional Simulation, in which interpolation estimates are based on a form of stochastic simulation for which measured data values are honored at their locations. This allows one to map sharp spatial discontinuities such as contamination hotspots or fault lines. Punctual and block kriging as well as IDW will smooth out local details of spatial variation, especially as interpolated locations become more distant from measured locations.

GS+ allows you to define the interpolation grid (including any polygon or blanking areas) and other aspects of the analysis, and then GS+ runs a 32-bit interpolation engine that cuts hours off of former methods. The ASCII output file can be written in GS+, ArcView®, or Surfer® formats.

A Cross Validation command performs a jackknife analysis in which every measured point in the data set is temporarily deleted from the data set then estimated to provide an indication of the appropriateness of a given variogram model.

Kriging in GS+
Kriging provides a means of interpolating values for points not physically sampled using knowledge about the underlying spatial relationships in a data set to do so. Variograms provide this knowledge. Kriging is based on regionalized variable theory and is superior to other means of interpolation because it provides an optimal interpolation estimate for a given coordinate location, as well as a variance estimate for the interpolation value. Other interpolation techniques in GS+ include conditional simulation and inverse distance weighting.

The Krig tab is part of the larger Interpolation Window.

Cokriging in GS+
Cokriging is an interpolation technique that allows one to better estimate map values if the distribution of a secondary variate sampled more intensely than the primary variate is known. If the primary variate is difficult or expensive to measure, then cokriging can greatly improve interpolation estimates without having to more intensely sample the primary variate.

Soil carbon, easier to measure than uranium, was sampled at the same locations as uranium and additionally at another 60 locations as noted in the quartile map below left. Regression of carbon against uranium showed that the variates were highly correlated (right), suggesting that cokriging might improve the map of uranium.

Using carbon as a covariate to produce a cokriged map of uranium results in the below-right map, plotted next to the original map above. Note the substantial improvement in the definition of contour (isoline) differences, especially in the upper right quadrant of the map where the uranium was sampled most sparsely:

How do you perform cokriging? Prior to cokriging you must have

  1. defined a covariate in the Field Assignment dialog, of the Data Worksheet;
  2. performed semivariance analysis (including variogram modeling) for both the primary variate and the covariate; and
  3. performed cross-semivariance analysis (including variogram modeling).

Confirm that the covariate is in fact correlated with the primary variate by viewing the Regression Window in the Data Summary Window. Note that there is no advantage to cokriging (over ordinary kriging) if the sample density of your primary variate is the same as for the secondary variate, or if the variates are uncorrelated.

Once the three variograms are modeled, you can choose the Cokrig tab in the Interpolation window:

Conditional Simulation in GS+
Conditional simulation is an interpolation technique for which Z estimates are based on a form of stochastic simulation in which measured data values are honored at their locations. Other interpolation methods, including kriging and inverse distance weighting, will smooth out local details of spatial variation, especially as interpolated locations become more distant from measured locations. This can be a problem when you are trying to map sharp spatial discontinuities such as contamination hotspots or fault lines. GS+ uses a sequential gaussian simulation method.

The Simulate tab is part of the larger Interpolation Window.

Inverse Distance Weighting (IDW) in GS+
Inverse Distance Weighting (IDW) and Normal Distance Weighting (NDW) are interpolation techniques in which interpolated estimates are made based on values at nearby locations weighted only by distance from the interpolation location. Neither IDW nor NDW make assumptions about spatial relationships except the basic assumption that nearby points ought to be more closely related than distant points to the value at the interpolate location. IDW applies stronger weights to nearby points than does NDW. Other interpolation techniques in GS+ include kriging and conditional simulation.

The IDW tab is part of the larger Interpolation Window.

The GS+ Interpolation Grid
Use the interpolation grid to define the region to be interpolated as well as the interpolation or grid intensity. A Distance Interval value provides the distance between grid intersections (where interpolation occurs). Polygon areas are specified in a separate polygon definition window. The interpolation grid window is accessed from the Interpolation Window.

GS+ Define Polygon Outlines
Irregular shapes can be interpolated or excluded from being interpolated by defining polygons prior to kriging. Define the polygons to be used by entering coordinates of polygon vertices, i.e. the coordinate pairs that define the polygon outline. An unlimited number of vertices can be specified per polygon (up to several billion), and polygons may be nested within one another as in the example below. The Map command allows you to check the shape of the polygon while it is being defined.

The example below defines two polygons: the first is a 6-sided area that is excluded from interpolation, the second defines an inclusive 4-sided area (rectangle) inside the 6-sided area. The Polygon Map Window shows the shapes of the

GS+ Polygon Map
When defining polygons it is helpful to confirm the shape of the masks being defined by drawing the polygons prior to interpolation. The overall area is the interpolation area specified in the Kriging window. Polygons are defined in the Polygon Definition Window. Exclusive polygons appear in red, inclusive in blue. In the map below, the area between the red outer (exclusive) polygon and the inner blue (inclusive) rectangle would remain unmapped. A more complex polygon (e.g. the shape of a continent or a lake) could be defined by a more complex set of vertices in the Polygon Definition window.

GS+ Cross-Validation Analysis
Cross-validation analysis allows you to evaluate alternative models for kriging. In cross-validation analysis, each measured point in a data set is individually removed from the set and its value is then estimated via kriging as though it were never there. This provides a graph of estimated vs. actual values for each sample location; values can also be listed in a separate window.