Stereonet Gridding Options

In the topics below we discuss different schemes for computing point densities, the "basic" method and several "customized" methods.

Basic Gridding Settings

If you have chosen to include contours in your diagram (either Line Contours or Color-Filled Contours or both) and wish to have the density grid computed using "standard" or "basic" settings, enter these settings under Gridding Options in the Stereonet Options window.

• Grid Type: Step Function
  With the Step Function gridding method, numerical densities are calculated upon the distribution of points after they have been projected onto the stereonet. For this option, an imaginary grid of 67 by 67 cells is placed over the stereonet. Each grid node acts as a counting station that is incremented by a whole integer amount if a plotted point lies within a chosen circular area around it. This size of this circle is typically 1% of the area of the stereonet, but can be set to any size. If a point lies within this circle, the grid node value is incremented by one; otherwise it is not affected. The number of points that affect each grid node are counted and converted to a percentage of the total number of points per the chosen fractional area.
• Density Units: Percent
  Using this setting, the density values will be reported as percentages of the total number of points per the selected area. This can be used with either the Step Function or Spherical Gaussian method.

Spherical Gaussian Gridding

With the Spherical Gaussian gridding method, numerical densities are calculated using a Gaussian weighting function. As with the Step Function method, a 67 x 67 grid is used for determining density values. Each node or counting station is incremented by an amount proportional to the angular distance between each point and the grid node. With this method, we are no longer determining our density values on a flat surface, but on the sphere itself. Because of this, our grid node locations are re-calculated to find their 3D spatial coordinates on the sphere.

The value assigned to a node for each point affecting it is given by:

w = exp ( k ( cos q - 1 ) )

where q is the angle between the data point and the counting station. The closer a point is to a counting station (small q) the larger the value of w. If the angle is 0 then w equals 1. Larger angles result in smaller values until finally a point has no effect on a counting station. Compare this to the Step Function method where the only results are 0 or 1: with the Spherical Gaussian method, the counting station will be incremented by a fractional amount for any point within the "lens" of influence. This results in much smoother, and more realistic density values.

The Gaussian method does take roughly FIVE TIMES as long to execute as the Step Function method.

• Weighting Factor:
  • The weighting factor is the variable k in the above equation. You can think of k as being somewhat equivalent to the search area in the Step Function method, although the numbers themselves are quite different. The "equivalent" value of k for a search value of 0.01 is "100." Decreasing k has the same result as increasing the search value.
• Gaussian Minimum Value:
  • The minimum value is the smallest acceptable number to be used when incrementing each counting station. As was discussed above, the larger the angle between the counting station and the point, the smaller the value of w. When the program determines the density values for the counting stations it only considers those stations within the "lens" of influence of the point. The minimum value declaration allows you to determine how large this lens is. The larger the lens, the longer the program will take to run since it has to check many more stations.
  • Depending on the data set, a minimum value of "0.1" is probably okay. For example, if the minimum value were set to 0.1 and, for a particular counting station, 10 points that could have affected the station were left out because they were too far away to add a value of 0.1, then that station would have a total summed value of 1 (or 0.1 * 10) less than what it could have had with a smaller minimum value. After the final percentages are calculated, this is unlikely to have a large effect.
  • Depending on what number you type in for the minimum value, the program will determine what the search lens should be for a particular value of k. The smaller the value of k, the larger the lens will be. (In other words, smaller values of k will increase the execution time.)

Density Units in Standard Deviations

Using this setting, the density values will be reported as standard deviations from an expected number of points. This can be used with either Step Function or Spherical Gaussian gridding methods.

Estimated Number of Points in Data Set: In this Kamb Calculator prompt, enter the approximate number of points in the data sheet that you are processing.

Calculate: Click on this button to tell the program to compute the default calculations for the search parameter and expected number of points to be found within it. The Kamb method will produce different results, depending on whether you are using the Step Function or Spherical Gaussian gridding method.

Step Function

For Step Function gridding, utilizing the Kamb method involves using a much larger search value. Kamb theorized that when a larger counting circle is used, many spurious, insignificant concentrations would be eliminated. According to Kamb's method, the fractional area of the counting circle is found by:

search area = 9 / ( N + 9 )

where N is the total number of points in the sample that you have entered in the dialog box. Assuming that the data points are sampled from a uniform population, the number of points that is expected to be found within the fractional area is:

expected number of points = N * search area

The standard deviation, s, from the expected value is:

s = ( ( N * search area ) * ( 1 - search area ) ) 1/2

The equation for the expected value shown above actually yields a value of 3s or three standard deviations.

If you feel that for your particular data set the calculated values for the expected number of points or the search value should be different from that displayed after the Kamb Calculate button is clicked, feel free to change them. The value for sigma (i cannot be changed and will always be the value calculated from the above equation.

Example: With a data set of 75 points, the program would calculate the variables as follows:

• search area: 0.1071
• expected number of points: 8.04
• sigma: 2.678

The density grid values will be reported in terms of the number of positive standard deviations from 8.04. If a cell had a value of 11.0, this would represent 1.105 standard deviations. ( ( 11.0 - 8.04 ) / 2.678 = 1.105 )

Because the Kamb method involves larger search areas, it will take longer to calculate the grid using this option.

Spherical Gaussian

You can also utilize the Spherical Gaussian equation with the Kamb method. The expected number of points and k are calculated a little differently, however.

As with the Step Function gridding method, the search area for the Kamb method is much larger. The equation used to find k is as follows:

k = 2 ( 1 + N / 9 )

The expected value for 3s is given by:

expected number of points = 0.5 * ( 9N / ( N + 9 ) )

s is calculated as follows:

s = ( N ( 0.5 - ( 1/k ) ) / k ) 1/2

Note that the expected value listed here is much smaller than that calculated for the same data using the Step Function method. This is because the Gaussian function is more sensitive to departures from the expected value and they are therefore more significant. You can, of course, change these default values (computed when the Calculate button is selected) to whatever you feel is most appropriate.

Example: With a data set of 75 points, the program would calculate the variables as follows:

• k: 18.667
• expected number of points: 4.02
• sigma: 1.79
  
  

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