# Gridding (spatial interpolation)

“Gridding” is the spatial interpolation of scattered 2D data points onto a regular grid. Three columns with position (x,y) and corresponding data values are required.

Gridding makes a map showing a continuous spatial estimate of some variate such as fossil abundance or thickness of a rock unit, based on scattered data points. The user can specify the size of the grid (number of rows and columns). The spatial coverage of the map is generated automatically as a square covering the data points. When plotting, this can be reduced to the convex hull of the points.

*Cross validation*: This option will remove each data point in turn and re-compute the surface based on the remaining points (“jackknife”). The differences between the original data values and the cross-validated values indicate the prediction accuracy of the surface model. These differences are reported for each point, together with the mean squared error (MSE) over all points.

Four interpolation algorithms are available:

#### Inverse distance weighting

The value at a grid node is simply the average of the N closest data points, as specified by the user (the default is to use all data points). The points are weighted in inverse proportion to distance. This algorithm is fast but will not always give good (smooth) results. A typical artefact is “bull’s eyes” around data points. One advantage is that the interpolated values will never exceed the range of the data points. By setting N=1, this algorithm reduces to the nearest-neighbour method, which sets the value at a grid node to the value of the nearest data point.

#### Thin-plate spline

Maximally smooth interpolator. Can overshoot in the presence of sharp bends in the surface. This is a radial basis method with radial basis function φ = r ln r.

#### Multiquadric

Radial basis function φ = r. Popular for terrain modelling.

#### Kriging

The user is required to specify a model for the semivariogram, by choosing one of four common models and corresponding parameters to fit the empirical semivariances (the residual sum of squares should be as small as possible). The semivariogram is computed within each of a number of bins. Using the histogram option, choose a number of bins so that each bin (except possibly the rightmost ones) contains at least 30 distances.

The nugget parameter is a constant added to the model. It implies a non-zero variance at zero distance, and will therefore allow the surface to not pass exactly through the given data points. The range controls the extent of the curve along the distance axis.

The “Optimize all” button will select the model and parameters giving the smallest residual sum of squares in the semivariogram. This may not be what you want: For example you may wish to use a specific model or to have zero nugget in order to ensure exact interpolation. This will require setting the values manually.

The kriging procedure also provides an estimate of standard errors across the map (this depends on the semivariogram model being accurate). Kriging in PAST does not provide for anisotropic semivariance.

Warning: Kriging is slow, do not attempt it for more than ca. 1000 data points on a 100x100 grid.

#### References

Davis, J.C. 1986. Statistics and Data Analysis in Geology. John Wiley & Sons.

de Smith, M.J., M.F. Goodchild & P.A. Longley. 2009. Geospatial Analysis, 3rd ed. Matador.