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==Computer contouring versus hand contouring==

==Computer contouring versus hand contouring==

 

[[file:introduction-to-contouring-geological-data-with-a-computer_fig1.png|thumb|{{figure number|1}}(a) Random points. (b) Clustered points.]]

file:introduction-to-contouring-geological-data-with-a-computer_fig1.png|{{figure number|1}}(a) Random points. (b) Clustered points.

Contour maps that represent three-dimensional geological surfaces are prepared by time-honored procedures involving estimation methods. Prior to the advent of fast computers and computational algorithms, maps showing geological variation were prepared by hand. Hand-contoured maps represent a geologist's best approximation of the shape of a surface under investigation. Ideas based on the regional geological framework and the geologist's bias arising from prior experience are an inherent part of the hand-contoured map. Hand-contoured maps cannot be reproduced exactly, and values implied by the contours cannot be recovered.

Contour maps that represent three-dimensional geological surfaces are prepared by time-honored procedures involving estimation methods. Prior to the advent of fast computers and computational algorithms, maps showing geological variation were prepared by hand. Hand-contoured maps represent a geologist's best approximation of the shape of a surface under investigation. Ideas based on the regional geological framework and the geologist's bias arising from prior experience are an inherent part of the hand-contoured map. Hand-contoured maps cannot be reproduced exactly, and values implied by the contours cannot be recovered.

In contrast, high speed computing facilities have engendered methods whereby an “objective” surface can be created by applying mathematical interpolation procedures to a control point set. These methods are free of any geological bias or interpretation introduced during map preparation because they produce a representation of a surface that is constructed by an “unbiased” and decidedly ungeological mathematical formulation from data measured at selected control points. Computer contoured maps can be reproduced easily by presenting the program with the same data and options as those used to create the original map. Values underlying the contoured representation can be obtained by the same interpolation procedure used to generate it.

In contrast, high speed computing facilities have engendered methods whereby an “objective” surface can be created by applying mathematical interpolation procedures to a control point set. These methods are free of any geological bias or interpretation introduced during map preparation because they produce a representation of a surface that is constructed by an “unbiased” and decidedly ungeological mathematical formulation from data measured at selected control points. Computer contoured maps can be reproduced easily by presenting the program with the same data and options as those used to create the original map. Values underlying the contoured representation can be obtained by the same interpolation procedure used to generate it.

Procedures usually used in hand contouring require that the geologist choose a contour interval that best displays ideas to be conveyed by the map. Computer-contouring methods, in contrast, require that the geologist select parameters that will ultimately determine the mathematical basis that computes and draws the finished map. Many sets of parameters can be used to produce a contoured representation of a surface sampled by a sparse set of control points. Maps will be similar in overall appearance, but will differ in specific areas because each set of parameters causes different mathematical procedures to be invoked. Each procedure produces a different map. (For example, compare figures 1 and 2 of <ref name=Philip_and_Watson_1982>Philip, G. M., and D. F. Watson, 1982, A precise method for determining contoured surfaces: Australian Petroleum Exploration Society Journal, v. 22, p. 205-212.</ref> and figure 11.07 of <ref name=pt08r4>Clarke, K. C., 1990, Analytical computer cartography: Englewood Cliffs, NJ, Prentice-Hall, 290 p.</ref> with Figures 3, 4, 5, and 7 of this article). Suitable parameters for a particular mapping project are selected by carefully inspecting both density and distribution of control points from which the map will be made.

Procedures usually used in hand contouring require that the geologist choose a contour interval that best displays ideas to be conveyed by the map. Computer-contouring methods, in contrast, require that the geologist select parameters that will ultimately determine the mathematical basis that computes and draws the finished map. Many sets of parameters can be used to produce a contoured representation of a surface sampled by a sparse set of control points. Maps will be similar in overall appearance, but will differ in specific areas because each set of parameters causes different mathematical procedures to be invoked. Each procedure produces a different map. (For example, compare figures 1 and 2 of <ref name=Philip_and_Watson_1982>Philip, G. M., and D. F. Watson, 1982, A precise method for determining contoured surfaces: Australian Petroleum Exploration Society Journal, v. 22, p. 205-212.</ref> and figure 11.07 of <ref name=pt08r4>Clarke, K. C., 1990, Analytical computer cartography: Englewood Cliffs, NJ, Prentice-Hall, 290 p.</ref> with [[:file:introduction-to-contouring-geological-data-with-a-computer_fig3.png|Figures 3]], [[file:introduction-to-contouring-geological-data-with-a-computer_fig4.png|4]], [[file:introduction-to-contouring-geological-data-with-a-computer_fig5.png|5]], and [[:file:introduction-to-contouring-geological-data-with-a-computer_fig7.png|7]] of this article). Suitable parameters for a particular mapping project are selected by carefully inspecting both density and distribution of control points from which the map will be made.

Data values between control points are obtained by some form of interpolation for both manual and computer contouring. For hand-contoured maps, interpolation required to estimate position and shape of individual contours is accomplished by eye or by simple averaging techniques. A triangular mesh or a rectangular grid provide a basis on which to interpolate data from control points. These frameworks rely on a complex mathematical interpolating function (bicubic splines, high order polynomials) to estimate contour positions between data points. This function is a polynomial that is “flexible” and can represent a wide variety of curve shapes. However, these functions have no direct geological significance. They have a continuous derivative everywhere within a triangle or a rectangle and therefore are at least once differentiable. This ensures that slope information implied by the control point set will be more faithfully rendered by the computational procedure.

Data values between control points are obtained by some form of interpolation for both manual and computer contouring. For hand-contoured maps, interpolation required to estimate position and shape of individual contours is accomplished by eye or by simple averaging techniques. A triangular mesh or a rectangular grid provide a basis on which to interpolate data from control points. These frameworks rely on a complex mathematical interpolating function (bicubic splines, high order polynomials) to estimate contour positions between data points. This function is a polynomial that is “flexible” and can represent a wide variety of curve shapes. However, these functions have no direct geological significance. They have a continuous derivative everywhere within a triangle or a rectangle and therefore are at least once differentiable. This ensures that slope information implied by the control point set will be more faithfully rendered by the computational procedure.

==Triangulation==

==Triangulation==

Triangulation connects control points into a mesh of locally equiangular (Delaunay) triangles ([[:file:introduction-to-contouring-geological-data-with-a-computer_fig2.png|Figure 2]]). Contour positions within the bounds of each triangle are estimated by interpolating from control point values that are triangle vertices. Each member of the triangular mesh is handled separately, and the surface is created by assembling triangles. Interpolation and contouring on a triangulated mesh requires few decisions from the geologist. Control point data are presented to the method along with a contour interval, and a contoured representation of the required surface is produced. [[:file:introduction-to-contouring-geological-data-with-a-computer_fig3.png|Figure 3]] is a contour representation of control point data presented by Davis<ref name=pt08r6 /> produced by interpolating on a triangular mesh ([[:file:introduction-to-contouring-geological-data-with-a-computer_fig2.png|Figure 2]]).

Triangulation connects control points into a mesh of locally equiangular (Delaunay) triangles ([[:file:introduction-to-contouring-geological-data-with-a-computer_fig2.png|Figure 2]]). Contour positions within the bounds of each triangle are estimated by interpolating from control point values that are triangle vertices. Each member of the triangular mesh is handled separately, and the surface is created by assembling triangles. Interpolation and contouring on a triangulated mesh requires few decisions from the geologist. Control point data are presented to the method along with a contour interval, and a contoured representation of the required surface is produced. [[:file:introduction-to-contouring-geological-data-with-a-computer_fig3.png|Figure 3]] is a contour representation of control point data presented by Davis<ref name=pt08r6 /> produced by interpolating on a triangular mesh ([[:file:introduction-to-contouring-geological-data-with-a-computer_fig2.png|Figure 2]]).