Changes

Shapes of geological surfaces are complex and not readily approximated by simple mathematical functions because they result from a multitude of interacting processes that vary at different spatial scales. Ideally, spatial data should be examined with a spatial sample of regular geometric design. These designs can capture the range of variation exhibited by most spatial phenomena. However, such designs are, for all practical purposes, impossible for most geological work, although in some instances recent developments in satellite imagery allow their economic implementation. In most cases, subsurface geological features are sparsely sampled relative to their complexity and the samples are highly biased to geophysical and/or geological anomalies. Therefore, values of a variable across an area of interest must be estimated by interpolating from a sparse, irregular control point set.

Shapes of geological surfaces are complex and not readily approximated by simple mathematical functions because they result from a multitude of interacting processes that vary at different spatial scales. Ideally, spatial data should be examined with a spatial sample of regular geometric design. These designs can capture the range of variation exhibited by most spatial phenomena. However, such designs are, for all practical purposes, impossible for most geological work, although in some instances recent developments in satellite imagery allow their economic implementation. In most cases, subsurface geological features are sparsely sampled relative to their complexity and the samples are highly biased to geophysical and/or geological anomalies. Therefore, values of a variable across an area of interest must be estimated by interpolating from a sparse, irregular control point set.

Several control point patterns are commonly encountered in geological practice. These include random patterns or clusters (Figure 1). Geophysical data that contribute to a geological study are gathered in lines. Lines are a special case of clustered points. Each pattern has its own spatial characteristics and must be understood before a meaningful contoured representation can be constructed. Most geological data usually exhibit properties of both end-member patterns. Gridded patterns are rare in geological practice. Most commercial contouring packages compute statistics that when used with visual inspection of the pattern on a base map can greatly aid selection of an appropriate contouring method.

Several control point patterns are commonly encountered in geological practice. These include random patterns or clusters ([[file:introduction-to-contouring-geological-data-with-a-computer_fig1.png|Figure 1]]). Geophysical data that contribute to a geological study are gathered in lines. Lines are a special case of clustered points. Each pattern has its own spatial characteristics and must be understood before a meaningful contoured representation can be constructed. Most geological data usually exhibit properties of both end-member patterns. Gridded patterns are rare in geological practice. Most commercial contouring packages compute statistics that when used with visual inspection of the pattern on a base map can greatly aid selection of an appropriate contouring method.

 

[[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|thumb|{{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.

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 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.

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 (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. Figure 3 is a contour representation of control point data presented by Davis<ref name=pt08r6 /> produced by interpolating on a triangular mesh (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]]).

Contours prepared on a triangular mesh will always strictly honor all data points used for the interpolation. Triangulated meshes are easily updated, therefore adding new control points and updating maps is simplified. However, contours prepared on this mesh often look “rough” and are less desirable in appearance. Some more sophisticated mapping packages provide smoothing procedures to render maps of more acceptable appearance. Figure 4 is a portion of a surface contoured using a triangulation scheme. Notice the irregular shape of the contours. The original surface that was sampled to produce this map is a fourth order polynomial. The shape of this surface is characterized by smooth contours. Irregularities seen in Figure 4 are artifacts of the interpolation procedure used to estimate the contours within individual triangles. However, the relative positions of the contours are a good approximation to those for the original surface.

Contours prepared on a triangular mesh will always strictly honor all data points used for the interpolation. Triangulated meshes are easily updated, therefore adding new control points and updating maps is simplified. However, contours prepared on this mesh often look “rough” and are less desirable in appearance. Some more sophisticated mapping packages provide smoothing procedures to render maps of more acceptable appearance. [[:file:introduction-to-contouring-geological-data-with-a-computer_fig4.png|Figure 4]] is a portion of a surface contoured using a triangulation scheme. Notice the irregular shape of the contours. The original surface that was sampled to produce this map is a fourth order polynomial. The shape of this surface is characterized by smooth contours. Irregularities seen in [[:file:introduction-to-contouring-geological-data-with-a-computer_fig4.png|Figure 4]] are artifacts of the interpolation procedure used to estimate the contours within individual triangles. However, the relative positions of the contours are a good approximation to those for the original surface.

Triangulation is not a method of choice when surfaces derived from several horizons for the same area of interest are desired. Operations between surfaces (such as subtraction of the lower from the higher) require that data exist at each control point for both horizons. Often, this is not the case with well data and requires an estimated point to be submitted where data are missing.<ref name=pt08r11>Jones, T. A., Hamilton, D. E., Johnson, C. R., 1986, Contouring geologic surfaces with the computer: New York, Van Nostrand Reinhold Company, 314 p.</ref>

Triangulation is not a method of choice when surfaces derived from several horizons for the same area of interest are desired. Operations between surfaces (such as subtraction of the lower from the higher) require that data exist at each control point for both horizons. Often, this is not the case with well data and requires an estimated point to be submitted where data are missing.<ref name=pt08r11>Jones, T. A., Hamilton, D. E., Johnson, C. R., 1986, Contouring geologic surfaces with the computer: New York, Van Nostrand Reinhold Company, 314 p.</ref>

==Rectangular gridding==

==Rectangular gridding==

Rectangular gridding, in contrast to triangulation, first uses data at measured control points to interpolate values to a set of grid nodes at a predefined spacing. These values are then used to estimate positions of contours crossing each grid rectangle. The complete surface is assembled from contiguous grid rectangles. For most geological applications, grid squares are used rather than the more general rectangle. Interpolation and contouring of an irregularly spaced control point set on a rectangular grid requires many decisions from the geologist.

Rectangular gridding, in contrast to triangulation, first uses data at measured control points to interpolate values to a set of grid nodes at a predefined spacing. These values are then used to estimate positions of contours crossing each grid rectangle. The complete surface is assembled from contiguous grid rectangles. For most geological applications, grid squares are used rather than the more general rectangle. Interpolation and contouring of an irregularly spaced control point set on a rectangular grid requires many decisions from the geologist.

Because geological data are rarely presented on a uniform grid and are most often irregularly distributed across the map area, the number of control points used to estimate values at grid nodes is an important consideration. Several search procedures have been devised and are included in most mapping packages. These include nearest neighbor, circular, quadrant, and octant searching.

Because geological data are rarely presented on a uniform grid and are most often irregularly distributed across the map area, the number of control points used to estimate values at grid nodes is an important consideration. Several search procedures have been devised and are included in most mapping packages. These include nearest neighbor, circular, quadrant, and octant searching.

Nearest neighbor searching uses the nearest neighbors of a grid node to estimate nodal values. The number of neighbors to use can be decided arbitrarily or can be taken as nearest neighbors defined by a Delaunay triangulation of the control point set. The number of nearest neighbors determined from irregularly spaced control points can vary so that each grid node can be estimated from different numbers of control points depending upon their distribution across the map area. Figure 5 is a contour representation of the same data used in Figure 3 using nearest neighbor search and a 13 × 13 grid (Figure 6).

Nearest neighbor searching uses the nearest neighbors of a grid node to estimate nodal values. The number of neighbors to use can be decided arbitrarily or can be taken as nearest neighbors defined by a Delaunay triangulation of the control point set. The number of nearest neighbors determined from irregularly spaced control points can vary so that each grid node can be estimated from different numbers of control points depending upon their distribution across the map area. Figure 5 is a contour representation of the same data used in [[:file:introduction-to-contouring-geological-data-with-a-computer_fig3.png|Figure 3]] using nearest neighbor search and a 13 × 13 grid ([[:file:introduction-to-contouring-geological-data-with-a-computer_fig6.png|Figure 6]]).

 

[[file:introduction-to-contouring-geological-data-with-a-computer_fig6.png|thumb|{{figure number|6}}A13 × 13 grid showing the relationship between grid nodes and control points for the Davis<ref name=pt08r6 /> data set.]]

Circular, quadrant, and octant neighborhood searching procedures attempt to balance the number and distribution of control points used to estimate each grid node. Most mapping packages include procedures to estimate density and control point spacing, and these statistics should be examined carefully before deciding on search criteria for a particular project.

Circular, quadrant, and octant neighborhood searching procedures attempt to balance the number and distribution of control points used to estimate each grid node. Most mapping packages include procedures to estimate density and control point spacing, and these statistics should be examined carefully before deciding on search criteria for a particular project.

Trend projection methods are an adaptation of a linear regression technique called ''trend surface analysis''. This method has been devised because geological subsurface sampling rarely provides observations at the highest or lowest points on a surface, and it is sometimes desirable to allow the interpolation procedure to exceed the measured maximum and minimum. Trend projection methods use one of the search criteria previously described to select points that are taken in groups of three and fitted exactly to a plane using a least squares or bicubic spline methods. The grid node estimate is obtained by averaging projections of these planes. This method can be quite effective for smooth surfaces where regional dip orientation remains relatively constant over a large area of the map. This method can produce a surface that is more highly textured than the actual surface in highly deformed areas where the dip direction changes rapidly over small distances. Sampson<ref name=pt08r19>Sampson, R. J., 1978, Surface II graphics system (revision 1): Lawrence, KS, Kansas Geological Survey, Series on Spatial Analysis, n. 1, 240 p.</ref> reviews this method in detail.

Trend projection methods are an adaptation of a linear regression technique called ''trend surface analysis''. This method has been devised because geological subsurface sampling rarely provides observations at the highest or lowest points on a surface, and it is sometimes desirable to allow the interpolation procedure to exceed the measured maximum and minimum. Trend projection methods use one of the search criteria previously described to select points that are taken in groups of three and fitted exactly to a plane using a least squares or bicubic spline methods. The grid node estimate is obtained by averaging projections of these planes. This method can be quite effective for smooth surfaces where regional dip orientation remains relatively constant over a large area of the map. This method can produce a surface that is more highly textured than the actual surface in highly deformed areas where the dip direction changes rapidly over small distances. Sampson<ref name=pt08r19>Sampson, R. J., 1978, Surface II graphics system (revision 1): Lawrence, KS, Kansas Geological Survey, Series on Spatial Analysis, n. 1, 240 p.</ref> reviews this method in detail.

Figure 7 is the same portion of the surface shown in Figure 4. The map in Figure 7 was produced by a gridding method with nearest neighbor search. Contours for this map are smooth, and their shape closely approximates those of the original fourth-order polynomial surface from which control points were obtained. However, contours are not in the same geographic positions as in the original surface, and some control points are not strictly honored.

[[file:introduction-to-contouring-geological-data-with-a-computer_fig7.png|Figure 7]] is the same portion of the surface shown in [[:file:introduction-to-contouring-geological-data-with-a-computer_fig4.png|Figure 4]]. The map in [[:file:introduction-to-contouring-geological-data-with-a-computer_fig7.png|Figure 7]] was produced by a gridding method with nearest neighbor search. Contours for this map are smooth, and their shape closely approximates those of the original fourth-order polynomial surface from which control points were obtained. However, contours are not in the same geographic positions as in the original surface, and some control points are not strictly honored.

==Statistical methods==

==Statistical methods==