TOWARDS 3/4-D GIS
Oleg R. Musin
Department of Cartography & Geoinformatics
Moscow State University Moscow, 119899, Russia
e-mail:
INTRODUCTION
Many forms of geoscientific analysis seek to collect data setswhich are truly tri-dimensional. The dynamic behavior of geo-systemsimplies that time has also to be considered as a further dimension.Recent development in the design and the implementation of 3D-GIShave been reported in the field of oil exploration, oceanology,meteorology, hydrogeology, geological modeling and environmentalmonitoring (Bak & Mill, 1989; Jones, 1989; Turner, 1989; Musin &Saveliev, 1994; Yi-Hsing Tseng & Chang-Jan Lin, 1996; Berlyant etal., 1997). To date, only a few GIS development deal with thefourth dimension. These systems all have limitations forgeoscientific applications. Many of these systems generate highquality visualizations of the studied features but which cannotoften be interactively interrogated. Moreover the 3/4-D dataanalysis abilities provided by these systems are not well developed.This limits the usefulness of GIS techniques for the development,the calibration and the validation of a mathematical models forgeo-systems.
There is a need to develop and improve the specialized GISfunctions that would support the analysis of patterns and trends andto add the means of linking GIS analysis with theoretical andexplanatory models (Clark et al., 1991). The coupling of GIS anddynamic explanatory models into integrated GIS requires thedevelopment of four-dimensional interpolation techniques to deriveinformation layers/volumes from experimental data sets. Suchinterpolated information layers/volumes could serve as inputinformation for dynamic explanatory models. Moreover it would easethe analysis of complex space-time processes which is often impairedby the low availability of geographic data for environmentalstudies. Despite international research efforts in the study ofglobal change, most collection strategies are generally sparse,point-located in nature, and irregularly distributed in both spaceand time. The ability to interpolate in both space and time istherefore needed to develop tools to analyze these complex spatialprocesses and to interface GIS products with mathematical models.
There is a need to develop and improve the specialized GISfunctions that would support the analysis of patterns and trends andto add the means of linking GIS analysis with theoretical andexplanatory models (Clark et al., 1991). The coupling of GIS anddynamic explanatory models into integrated GIS requires thedevelopment of four-dimensional interpolation techniques to deriveinformation layers/volumes from experimental data sets. Suchinterpolated information layers/volumes could serve as inputinformation for dynamic explanatory models. Moreover it would easethe analysis of complex space-time processes which is often impairedby the low availability of geographic data for environmentalstudies. Despite international research efforts in the study ofglobal change, most collection strategies are generally sparse,point-located in nature, and irregularly distributed in both spaceand time. The ability to interpolate in both space and time istherefore needed to develop tools to analyze these complex spatialprocesses and to interface GIS products with mathematical models.
3D GEOMETRIC MODELLING
The 2D geometric data are represented either by vectors or byelementary areas. In the efficient systems the vector data aremanaged in topological structure, while the systems based onelementary areas as a rule operate with elementary squares (pixels)called rasters. The most up to date systems can handle bothgeometrical structures.
The 3D analogues of raster model is the voxel model. The basicelement of the model is the cube that can be divided or aggregatedlimitless into similar smaller or larger cubes. For storage andmanipulations the voxels are usually organizes in some modificationof octree (Samet, 1990).
To represent 3D objects, geometric modeling methods have beendeveloped for Computer Graphics and CAD systems (Foley et al.,1990). Among them, the most popular are the boundary representationmodel and CSG model.
An object in 3D space can be represented by its boundaries, whichare defined as separation of parts inside and outside the object.This representation method called boundary representation (B/R).A boundaries (surfaces) in this model are polyhedrons and itscombinatorial structure represented by topological relations. Basedon this geometric modelling can be set up a data model for 3Dspatial data (Molenaar 1992; Yi-Hsing Tseng & Chang-Jan Lin, 1996).The B/R model can used to represent a very wide class of objects andtheoretically is compatible with contemporary models of 2D vectormap data.
The Constructive Solid Geometry (CSG) is a special tessellationmodel and is commonly used in CAD systems. The CSG constructs theobjects from elementary regular solids (cubes, cylinders, prismsetc.) by means set theory operations (union, intersection andsubtraction) and introducing some new operations (e.g. gluing).
Unfortunately, I do not know about practical implementation thesemodels and data structure in GIS.
Developed from traditional mapping techniques, current GISs aredesigned based on a 2D representation of spatial objects. The thirddimensional data are modelled by using z attributes for an x,yposition. This approach is called 2.5D representation, which cannothandle multiple z-values that happen frequently in 3D space.
The Seam Thickness (ST) model has its origin in oil and gasgeology, where seam top and bottom represented on the maps bycontours of a depth.
ST model is not comprehensive 3D geometric model and can be usefor solid that have a structure of geological seam (Musin &Saveliev, 1994). Some of the solids in geology, meteorology andoceanology can have more complicated geometry. For example, marineisotherms can have a torus form and be knotted. However, from ourpoint of view most of the solids in GIS can be represented by STmodel.
A "seam" is solids between two surfaces: seam top and seam bottom. (All together seams look like puff-pastry.) For seam top and bottom we used two 2.5D models. For construction of a 2.5D model by contours can be used one of the algorithms of interpolation (Dowd, 1985; Musin, 1990).
3.5D AND DYNAMIC MODELS.
Let us call as 3.5D geometric model a three dimensional surfacew = f(x,y,z) that is modelled a spatial distribution of afourth attribute w for an x,y and z position; where z is height(or depth) and w is a some value like temperature, salinity ofwater, ozone, pollutants etc. This is the direct analogues of 2.5Dmodel for 2-dimensional surface.
All of the 2.5D methods of interpolation from scattered three -dimensional data points can be generalized to 3.5D case. Two of themare most popular: kriging and Delaunay triangulation. The krigingestimates the unknown values w=f(x,y,z) with minimum variances ifthe measured data fulfil some condition of stationarity (Dowd,1985). This method is not depend on dimension and can be use in 3D.
The Delaunay triangulation is tessellation of convex hull of datapoint set dual to the Voronoi diagram. In 2D this structure is usednot only for interpolation and for representation of a terrainmodel (TIN model). It is widely used for 3D case too. My recentresearch shows that Delaunay triangulation may be non appropriate in3D (Musin, 1997). Thus the problem of funding "good" triangulationin 3D is open and more detailed consideration is necessary.
There are a several methods for representation of these kindmodels. One of the most popular is the 3D grid representationmodel, when for each node of 3D XYZ grid is attributed fourthdimension w. For this model of representation is necessary a largevolume of computer memory.
The traditional method of representation and mapping of fourthdimension value w in oceanology is to creating of maps for a finiteset of depths. Usually, these depths is not uniformly distributed.For example, in Black Sea GIS sequence of depths is 0, 30, 50, 100,500, 1000 meters (Berlyant et al., 1997). It is in consequence ofthe fact that if deeply that less reliable data can be get.Representation by horizontal sections is available for 3.5D model.
For each level z=h of finite sequence of values (levels) of coordinate Z corresponds 2.5D distribution model W(h) where w=f(x,y,h). W(h) is horizontal section z=h of 3.5D model w=f(x,y,z). For modelling w between two level z can be used linear interpolation.
The dynamics of a geographic objects is one of the most priorof geographic researches. If the time factor is taken intoconsideration with two-dimensional model then this model becomesthree-dimensional and 3D model becomes four-dimensional.
From mathematical point of view the consideration of dynamicmodel of some object make sense if and only if this object changedin time continuously i.e. if w=f(x,y,t) (or w=f(x,y,z,t) then f iscontinuous function. For these kind models the same representationlike for 3.5D models is available i.e. the linear interpolation between two dates can be used.
COMPUTATIONS BASED ON 3/4D MODELS
For computations based on 3/4D images (models) A.M. Berlyantsuggested to use three branches of metric disciplines (Berlyant,1996):
- geoplanimetry, measurements of 2D geoimages;
- geostereometry, measurements of 3D and 2.5D images;
- geochronometry (or dynamic geoiconometry), measurements of 3D and 4D dynamic models.
The analysis of numerous problems of geoplanimetry, geostereomeryand geochronometry shows that its represent problems ofComputational Geometry and Computer Graphics. Among them aredetermination of mutual location and intersection of objects (pointand polyhedron, point and set of polyhedrons, two set ofpolyhedrons etc); the interpolation by spatial and temporal data ofdigital 3/4D models. In order to present graphic images (e.g.projection, animation, isosurfer 3D maps, horizontal and verticalsections etc) in both vector and raster formats a number ofalgorithms of Computer Graphics can be used.
The application of Computation Geometry algorithms to theaforementioned as well as to a number of other tasks allowsto reduce of required computer memory and time complexity and toimprove quality of graphic images. The broad applications ofComputational Geometry and Computer Graphics algorithms providesways to develop GIS for complex and nontraditional requirements ofusers.
CONCLUSIONS
This paper discussed different ways of efficient construction of3/4D models for GIS. Most of the models have been partiallyimplemented in different GIS projects (Musin & Saveliev, 1994;Berlyant et al, 1997).
A data model is built up for 3/4D GIS applications. These modelsare constructed through the analysis of the geometricalrepresentations and the topological relations of 3/4D objects. It isprovides the fundamental principles of 3/4D GIS.
References