The G3GRID Procedure

# Concepts

• two horizontal variables, (x, y)

• one or more vertical variables, z through z-n, that will be interpolated or smoothed as if it were a function of the two horizontal variables.

The procedure can process multiple vertical variables for each pair of horizontal variables that you specify. If you specify more than one vertical variable, the G3GRID procedure performs a separate analysis and produces interpolated or smoothed values for each vertical variable. If more than one observation in the input data set has the same values for both horizontal variables, x and y, a warning message is printed, and only the first such point is used in the interpolation.

By default, the interpolation is performed after both variables are similarly scaled because the interpolation methods assume that the scales of x and y are comparable.

If the horizontal variable points are collinear, the procedure interpolates the function as constant along lines perpendicular to the line in the plane that is generated by the input data points.

You can control both the number of x and y values in the output data set and the values themselves. In addition, you can specify an interpolation method.

### Default Bivariate Interpolation

This default method of interpolation works best for fairly smooth functions with values given at uniformly distributed points in the plane. If the data points in the input data set are erratic, the default interpolated surface can be erratic.

This default method is a modification of that described by Akima (1978). This method consists of

1. dividing the plane into nonoverlapping triangles that use the positions of the available points

2. fitting a bivariate fifth degree polynomial within each triangle

3. calculating the interpolated values by evaluating the polynomial at each grid point that falls in the triangle.

The coefficients for the polynomial are computed based on

• the values of the function at the vertices of the triangle

• the estimated values for the first and second derivatives of the function at the vertices.

The estimates of the first and second derivatives are computed using the n nearest neighbors of the point, where n is the number specified in the GRID statement's NEAR= option. A Delauney triangulation (Ripley 1981, p. 38) is used for the default method. The coordinates of the triangles are available in an output data set if requested by the OUTTRI= option in the PROC G3GRID statement.

### Spline Interpolation

The function u, formed when you specify the SPLINE option, is determined by letting

and

where

The coefficients c1, c2,..., cn and d1, d2, d3 of this polynomial are determined by these equations:

and

where

E
is the n × n matrix E(ti , tj )

I
is the n × n identity matrix

is the smoothing parameter that is specified in the SMOOTH= option

c
is (c1 ,..., cn )

z
is (z1 ,..., zn )

d
is (d1, d2, d3)

T
is the n × 3 matrix whose ith row is (1, xi, yi).

See Wahba (1979) for more detail.

### Spline Smoothing

To produce a smoothed spline, you can use the GRID statement's SMOOTH= option with the SPLINE option. The value or values specified in the SMOOTH= option are substituted for in the equation that is described in Spline Interpolation. A smoothed spline trades closeness to the original data points for smoothness. To find a value that produces the best balance between smoothness and fit to the original data, you can try several values for the SMOOTH= option.