Construction of isarithms and isarithmic maps by computers

Bengtsson, Bengt-Erik; Nordbeck, Stig

doi:10.1007/bf01939851

Cited by 32 publications

(5 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Bengtsson and Nordbeck [6], Heap [7], and Lawson [8l suggested methods based on partitioning the x-y plane into a number of triangles (each triangle having projections of three data points in the x-y plane as its vertexes) and on fitting a plane to the surface in each triangle. These methods are useful for some applications.…”

Section: Introductionmentioning

confidence: 99%

A Method of Bivariate Interpolation and Smooth Surface Fitting for Irregularly Distributed Data Points

Akima

1978

ACM Trans. Math. Softw.

688

379

View full text Add to dashboard Cite

A method of blvariate interpolation and smooth surface fitting is developed for z values given at points irregularly distributed in the x-y plane. The interpolating function is a fifth-degree polynomial in x and y defined in each triangular cell whmh has projections of three data points in the x-y plane as its vertexes. Each polynomial is determined by the given values of z and estimated values of partial derivatives at the vertexes of the triangle. Procedures for dividing the x-y plane into a number of triangles, for estimating partial derivatives at each data point, and for determining the polynomial in each triangle are described A simple example of the application of the proposed method is shown.

show abstract

Section: Introductionmentioning

confidence: 99%

A Method of Bivariate Interpolation and Smooth Surface Fitting for Irregularly Distributed Data Points

Akima

1978

ACM Trans. Math. Softw.

688

379

View full text Add to dashboard Cite

show abstract

“…This method was used extensively in early computer algorithms for automatic grid interpolation of data (IBM 1965;Bergtsson and Nordbeck 1964;Shepard 1968;Green and Sibson 1978). Successive linear interpolation is then applied to each facet to determine the value at a missing data point.…”

Section: Cartographic and Related Methods Of Interpolationmentioning

confidence: 99%

“…This procedure assumes that the linear slope between two data points is a good estimate for interpolation. This method was used extensively in early computer algorithms for automatic grid interpolation of data (IBM 1965;Bergtsson and Nordbeck 1964;Shepard 1968;Green and Sibson 1978). It has the disadvantage that there is no unique set of results, as there is no unique way of partitioning the space (Duppe and Gottschalk 1970).…”

Section: Cartographic and Related Methods Of Interpolationmentioning

confidence: 99%

The Problem of Missing Data on Spatial Surfaces

Bennett

Haining

Griffith

1984

Annals of the Association of American Geographers

View full text Add to dashboard Cite

Although the problem of missing data arises in most branches of the discipline, it has received little systematic treatment in the geographical literature. In an effort to overcome this deficiency, this paper reviews a number of methods for approaching the problem. Of the three classes of solutions-ad hoc, cartographic interpolation, and statistical-the statistical approaches appear to be preferable. In this review a modified version of the Orchard and Woodbury missing-information principle receives the greatest emphasis because it combines classical statistical theory with trend surface and spatial autoregressive models. Although the best solution of all is to return to the experimental situation in order to collect supplementary data, this is often impractical or impossible. The analyst should then consider the estimation techniques presented here. The methods used to address the missing data problem thus become an important stage in the overall process of experimental design, sampling, and hypothesis testing.

show abstract

“…Perhaps the first attempt at computer-assisted contour maps was by Bengtsson and Nordbeck (1964), where a coarse grid was superimposed on the map, weighted average values were estimated at the grid nodes and contour lines then threaded through the cells.…”

Section: Further Readingmentioning

confidence: 99%

Points

2016

International Society for Photogrammetry and Remote Sensing (ISPRS) Book Series

View full text Add to dashboard Cite

We have seen how dominance, the assignment of spatial locations to the closest generator, leads to a fundamental spatial partition -the Voronoi diagram. In some cases only the generator location is known, and in others some attributes, such as colour, may be assigned.In forest mapping, the Voronoi diagram clearly gives a good estimate of the context of an individual tree in a forest: its constraining neighbours are found directly if tree crowns may be detected from aerial imagery. The locations of the tree crowns may then be used as generators, and the area of each associated Voronoi cell gives a reasonable first estimate of the tree's spread, in competition with its neighbours, and tables may be constructed to compare this with the tree height and wood volume. (This applies where the trees are in close competition, and does not apply at the edge of the forest stand.) Summing these volume estimates for a particular area may be used to estimate harvestable timber. In some cases the tree type may also be estimated, and the resulting volume totals used for harvest planning.The same approach may be used for other types of natural resource assessments. Coal thickness for example, or permeable aquifer thickness, is often assessed by shallow drilling -and the total thickness observed at each location. (This may often involve several thin strata, and often only the total thickness is needed -this is difficult by modelling the geological 'tops' of each stratum. Experience suggests that stable regional estimates may be made by assuming the local thickness at any location is best estimated by the closest real observation -the Voronoi cell generator. Summing the thickness times the cell area often gives satisfactory results for minimum effort. The same holds true for total rainfall estimation based on rain gauges -as originally suggested by Mr. Thiessen.A related, and common, problem is that of point density mapping. This is usually attempted by counting circles -counting how many points fall within a given circle, and then moving it and trying again: a very small circle, or a very large one, would give implausible results. Noting that point density (points per unit area) is the reciprocal of the Voronoi cell area (area per unit point) greatly simplifies this process. In this case the local density around each generator is 1/(cell area). Mapping this directly will often give a highly variable density surface, and some traditional smoothing function is often appropriate. Nevertheless, the whole problem of using counting circles has been eliminated.A significant application of this approach may be found in the work of van de Weijgaert described earlier. A major catalogue of a significant portion of the cosmos was performed by the 2dF Galaxy Redshift Survey, completed in 2002, extending over a huge number of galaxies, and light-years, in 3D ( Figure 78). In order to speculate on the cosmic structure they needed to produce a density map rather than just galaxies/points, and so they developed the 'Delaunay Tessellation Field Esti...

show abstract

Construction of isarithms and isarithmic maps by computers

Cited by 32 publications

References 1 publication

A Method of Bivariate Interpolation and Smooth Surface Fitting for Irregularly Distributed Data Points

A Method of Bivariate Interpolation and Smooth Surface Fitting for Irregularly Distributed Data Points

The Problem of Missing Data on Spatial Surfaces

Points

Contact Info

Product

Resources

About