# Solved – the difference between Kamb and 1% area contours

I have been doing a bunch of spherical statistics using Stereonet and there is a function for producing contour plots of data density.

One of the options in this function is a choice between Kamb contours and 1% area contours. What is the difference between these two and what are the benefits/disadvantages of each?

The main difference seems to be that 1% area contours fit much more tightly to the data.

Kamb contours: 1% area contours: Contents

1% area contours refers to the Schmidt – and sometimes Mellis – Method of Contouring where by densities of point data measured as a percentage of total number of points per 1% area of the stereogram. This is measured by a cardboard overlay with two circular holes that occupy 1% of the equal-area projection and, in the case of the Schmidt method, square grid with grid points 0.75 cm apart (Marshak and Mitra 1988). It should be noted that the Schmidt is most commonly used if statistical significance isn't considered (e.g., Stereonet; Allmendinger et al. 2013, Cardozo and Allmendinger 2013).

The Schmidt or Grid Method of Contouring is most commonly used and works well for (>400) and/or high concentrations of points. The method employs a square grid, so it is sometimes also called the "grid" method. Contours are determined by counting number of points in the 1% area circular overlay at each grid node and converting number of points (n) at each node to a percentage using the equation [n(100)/N=%]. Then draw contours at intervals corresponding to he appropriate point density (Marshak and Mitra 1988).

The Mellis Method of Contouring is convenient for a small number of points (<100) and for populations of points that do no show local high concentrations. It's particularly useful for determining the contour of minimum density (i.e., usually the one-point contour). The contours are determined by tracing the 1% area circle around each datum point. Areas of overlap are given multipliers (e.g., 2x, 3x, and etc.) given the number of overlapping circles. The percentage of the total number of points that is represented by one point can be calculated. Multipliers are added to that percentage at areas of respective overlap (Marshak and Mitra 1988).

The Kamb Method, proposed by Kamb (1959), is a contouring method that permits the graphic analysis of the statistical significance of point concentrations on an equal-area plot whereby you can choose the area of the counting circle overlay to be any fraction of the total area of the equal-area plot, from 0 to 1. And the spacing of grid points on the counting net is also calculated by the radius of the counting circle. Without going into the statistics, if the number of points (n) that fall within the counting circle is significantly greater than the expected number, then you have a significant cluster. The distribution of n values is a binomial distribution, and the mean and standard deviation of such a distribution can be calculated for each data set that is plotted (Marshak and Mitra 1988).

The Kamb method is the same as the Schmidt or Mellis methods. the observed densities are contoured at intervals of standard deviation. Generally, the area of a counting circle used in constructing a Kamb contoured plot is larger than that used for conventional contouring; therefore, the Kamb method smooths out the contour irregularities that are of no statistical significance (Marshak and Mitra 1988). This is the default method used in modern stereographic contouring practices (e.g., Stereonet; Allmendinger et al. 2013, Cardozo and Allmendinger 2013).

*References:

Allmendinger, R. W., Cardozo, N. C., and Fisher, D., 2013, Structural Geology Algorithms: Vectors & Tensors: Cambridge, England, Cambridge University Press, 289 pp.

Cardozo, N., and Allmendinger, R. W., 2013, Spherical projections with OSXStereonet: Computers & Geosciences, v. 51, no. 0, p. 193 – 205, doi: 10.1016/j.cageo.2012.07.021

Marshak, S., and Mitra, G., 1988, Basic Methods of Structural Geology, Prentice-Hall, p.446

Kamb, W.B., 1959, Petrofabric observations from Blue Glacer, Washington, in relation to theory and experiment: J. Geophys. Res., v.64, p.1908-1909.

*A significant portion of this material is directly pulled from Chapter 8 of Marshak and Mitra 1998 and stitched together with personal insight from my ongoing geo-environmental engineering PhD research.

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