Surfer® Profiling Methods 40

Golden (2007) pointed out that Surfer® is a contouring and 2-D surface mapping program that quickly and easily transforms random surveying data into continuous curved face contours using interpolation. In particular, the new version, Surfer® 8.0, provides more than twelve interpolation methods, each having specific functions and related parameters. Below is a brief description of the most popular methods.

4.2.1 Inverse Distance to a Power

The Inverse Distance to a Power gridding method is a weighted average interpolator and can be either exact or smoothing. With this method, a weighting power is assigned to data that defines how the factors will decline as distance from a grid node increases. The higher the weighting power, the less effect there is on the estimation point further away from the initial grid node point. Davis (1986) found that the equation used for Inverse Distance to a Power is:

1 1 2 2 1 n i i _ij j n i _ij ij ij Z h Z h h d β β β = = = = + δ

∑

∑

Where:

hij The effective separation distance between grid

node “j” and the neighboring point “i” Zj The interpolated value for grid node “j”

Zj The neighboring points

dij The distance between the grid node “j” and the neighboring point “i”

β The weighting power (the Power parameter) δ The Smoothing parameter

Normally, Inverse Distance to a Power behaves as an exact interpolator. To calculate the grid node, data points are assigned a fractional weight and the sums of all weights are equal to 1.0. When a known point aligns with a grid node, the distance between that known point and the grid node is 0.0, and that known point is given a weight of 1.0, while all other points are given weights of 0.0. Thus, the grid node is assigned the value of the known point (Franke, 1982). One disadvantage is that the known points are not uniformly spaced among the interpolation points. Because of this, some clusters of points tend to carry an unnaturally large weight. To minimize this effect, no point is given an overpowering weight. No point is given a weighting factor equal to 1.0.

4.2.2 Kriging

Kriging is a geostatistical gridding method that has proven useful and popular in many fields. This method produces visually appealing maps from irregularly spaced data (Cressie, 1990). Kriging is a very flexible gridding method. Kriging defaults can be used to produce an accurate grid of data, or Kriging can be custom-fit to a data set by specifying the appropriate

variogram model. Within Surfer®, Kriging can be either an exact or a smoothing interpolator depending on the user-specified parameters. It incorporates anisotropy and underlying trends in an efficient and natural manner (Journel, 1989).

4.2.3 Minimum Curvature

Minimum Curvature is widely used in the earth sciences. The interpolated surface

generated by Minimum Curvature is analogous to a thin, linear elastic plate passing through each of the data values with a minimum amount of bending. Minimum Curvature generates the

smoothest possible surface while attempting to honor the data as closely as possible. Minimum Curvature is not an exact interpolator, however. This means that data are not always honored exactly.

Minimum Curvature produces a grid by repeatedly applying an equation over the grid in an attempt to smooth it. Each pass over the grid is counted as one iteration (Franke, 1982). The grid node values are recalculated until successive changes in the values are less than the

Maximum Residuals value, or the maximum number of iterations is reached.

4.2.4 Modified Shepard’s

According to Shepard (1968), the Modified Shepard’s Method uses an inverse distance weighted least squares method. As such, Modified Shepard’s is similar to the Inverse Distance to a Power interpolator, but the use of local least squares eliminates or reduces the appearance of the generated contours. Modified Shepard’s can be either an exact or a smoothing interpolator.

Franke and Nielson (1980) state that the Modified Shepard’s starts by computing a local least square fit of a quadratic surface around each observation.

4.2.5 Natural Neighbor

Sibson (1980) and (1981) reported that the Natural Neighbor gridding method is quite popular in some fields. Natural Neighbor is as simple to use as Nearest Neighbor and provides more precise results; however, it is only available for 2-D interpolations. Natural Neighbor requires that a grid be defined.

Natural Neighbor interpolation is a weighted moving average technique that uses geometric relationships in order to choose and weight nearby points. The equation for the Natural Neighbor interpolation is (Isaaks & Sirvastava, 1989):

G(x, y) = ) ( 1 ,

∑ ∫

• G(x, y) is the natural neighbor estimation at (x, y);

• n is the number of nearest neighbors used for interpolation; • f(xi , yi) is the observed value at (xi ,yi); and

• Wi is the weight associated with f(xi, yi)

According to the Surfer® Users Guide, sometimes with nearly complete grids of data, there are areas of missing data that a user might want to exclude from the grid file. In this case, the search ellipse can be set to a value so the areas of no data are assigned the blanking value in the grid file. By setting the search ellipse radii to values less than the distance between data values in the file, the blanking value is assigned at all grid nodes where data values do not exist.

4.2.6 Polynomial Regression

Polynomial Regression is used to define large-scale trends and patterns in data. Polynomial Regression is not really an interpolator because it does not attempt to predict unknown Z values. Several options can be used to define the type of trend surface (Draper & Smith, 1981).

4.2.7 Radial Basis Function

The Surfer® Users Guide states that the Radial Basis Function interpolation is a diverse group of data interpolation methods. In terms of the ability to fit the data and to produce a smooth surface, the multiquadric method is considered by many to be the best. Powell (1990) noted that all of the Radial Basis Function methods are exact interpolators, so they attempt to honor the data. A smoothing factor can be introduced to all the methods in an attempt to produce a smoother surface.

4.2.8 Local Polynomial

Lee and Schachter (1980) stated that the Local Polynomial gridding method assigns values to grid nodes by using a weighted least squares fit with data within the grid node’s search ellipse.