Changes

Following is a brief description of the range of volume calculation techniques used. This discussion uses a grid-based system, although similar procedures could apply to triangulated or other systems. The discussion assumes that volumes are calculated within a bounding area (polygon).

Following is a brief description of the range of volume calculation techniques used. This discussion uses a grid-based system, although similar procedures could apply to triangulated or other systems. The discussion assumes that volumes are calculated within a bounding area (polygon).

====Volume by point count====

====Volume by point count====

<disp-quote>Number of nodes × Average value × ''x''-grid increment × ''y''-grid increment = Volume inside the polygon

<disp-quote>Number of nodes × Average value × ''x''-grid increment × ''y''-grid increment = Volume inside the polygon

</disp-quote>This approach assumes that each grid cell extends from a base up to a flat top, which is the grid node's value, and that the node is in the center of the cell. If the center of the cell is inside the polygon, it is counted; if it is outside, it is not (Figure 18).

</disp-quote>This approach assumes that each grid cell extends from a base up to a flat top, which is the grid node's value, and that the node is in the center of the cell. If the center of the cell is inside the polygon, it is counted; if it is outside, it is not ([[:file:using-and-improving-surface-models-built-by-computer_fig18.png|Figure 18]]).

[[file:using-and-improving-surface-models-built-by-computer_fig18.png|thumb|{{figure number|18}}The cell is centered on the grid node and lies either inside or outside the polygon. The cell's area is multiplied by its z value (thickness) and that volume is added to volumes for all other cells inside the polygon.]]

[[file:using-and-improving-surface-models-built-by-computer_fig19.png|left|thumb|{{figure number|19}}The cell's corners are defined by grid nodes. The top is defined by two or more planes passing through the node z values and lie inside the polygon. The prism of volume under each plane is calculated and added to volumes for all other prisms inside the polygon.]]

====Volume by simple plane fits====

====Volume by simple plane fits====

Grid nodes occupy corners of grid cells, and a line is drawn diagonally across a cell dividing it into two triangles. The value at each corner of the triangle is known; therefore, a flat plane can be fit through these points. The base above which volumes are to be calculated is also a plane, and its value is known. Since the dimensions of the sides of the triangular prism are known from the grid increments, all of the information needed to calculate the prism's volume is available. The volume of all triangle prisms totally inside the polygon are calculated and summed. Any prisms partially within the polygon are subdivided into smaller prisms with the values at the corners of the smaller prisms linearly interpolated from the three original triangle corners. The volumes of these partial values are calculated, summed, and added to the volume of prisms totally within the polygon (Figure 19).

Grid nodes occupy corners of grid cells, and a line is drawn diagonally across a cell dividing it into two triangles. The value at each corner of the triangle is known; therefore, a flat plane can be fit through these points. The base above which volumes are to be calculated is also a plane, and its value is known. Since the dimensions of the sides of the triangular prism are known from the grid increments, all of the information needed to calculate the prism's volume is available. The volume of all triangle prisms totally inside the polygon are calculated and summed. Any prisms partially within the polygon are subdivided into smaller prisms with the values at the corners of the smaller prisms linearly interpolated from the three original triangle corners. The volumes of these partial values are calculated, summed, and added to the volume of prisms totally within the polygon ([[file:using-and-improving-surface-models-built-by-computer_fig19.png|Figure 19]]).

[[file:using-and-improving-surface-models-built-by-computer_fig19.png|thumb|{{figure number|19}}The cell's corners are defined by grid nodes. The top is defined by two or more planes passing through the node z values and lie inside the polygon. The prism of volume under each plane is calculated and added to volumes for all other prisms inside the polygon.]]

[[file:using-and-improving-surface-models-built-by-computer_fig20.png|thumb|{{figure number|20}}A mathematical surface is fit to the grid cell. Calculus is used to integrate the volume under the curve, inside the grid cell, and inside the polygon. All cell volumes inside the polygon are added together.]]

====Volume by mathematical fit and integration====

====Volume by mathematical fit and integration====

This approach integrates under a smooth mathematical surface passed through the grid nodes in and around the cell for which volumes are being calculated. As with the simple plane fit described previously, node values are at cell corners and volumes are calculated and summed for each cell and partial cell within the polygon. A common approach is to use 16 grid node values, 4 from the cell of interest and 12 from the adjacent cells. A third-order polynomial is fit to these 16 node values. By applying calculus to this polynomial over the area of the grid cell, the exact volume under the surface is determined (Figure 20). If only a portion of the cell is contained within the polygon, then the integration is performed only within that part of the cell.

This approach integrates under a smooth mathematical surface passed through the grid nodes in and around the cell for which volumes are being calculated. As with the simple plane fit described previously, node values are at cell corners and volumes are calculated and summed for each cell and partial cell within the polygon. A common approach is to use 16 grid node values, 4 from the cell of interest and 12 from the adjacent cells. A third-order polynomial is fit to these 16 node values. By applying calculus to this polynomial over the area of the grid cell, the exact volume under the surface is determined ([[:file:using-and-improving-surface-models-built-by-computer_fig20.png|Figure 20]]). If only a portion of the cell is contained within the polygon, then the integration is performed only within that part of the cell.

 

[[file:using-and-improving-surface-models-built-by-computer_fig20.png|thumb|{{figure number|20}}A mathematical surface is fit to the grid cell. Calculus is used to integrate the volume under the curve, inside the grid cell, and inside the polygon. All cell volumes inside the polygon are added together.]]

====Discontinuities and other constraints====

====Discontinuities and other constraints====