DTM Based on an Ellipsoidal Squares

Transcription

1 DM Based on an Ellipsoidal Squares KRZYSZOF NAUS Institute of Navigation and Hydrography Polish Naval Academy Śmidoicza 69, 8-3 Gdynia POLAND Abstract: - he paper presents the description of the quality analysis folloing the most popular orld standards used for the description of terrain and sea bottom shape: o DED - Digital errain Elevation Data, used by NAO, o SRM model - Shuttle Radar opography Mission, orked out ithin the confines of the international mission of the space shuttle Endeavour, o DBDB-V - Digital Bathymetric Data Base - Variable Resolution, used for the description of sea and ocean bottoms, o NMB - Netork Model Bathymetry, as one of the AML - Additional Military Layers for WECDIS - Warship Electronic Chart Display and Information Systems. In the main part of the article, a ne method of the description of terrain and sea bottom shape have been shon. he method is based on the matrix model of the GRID type, here DM array consists of geographical squares. Key-Words: - Digital errain Model, ellipsoidal squares, 3D navigation Introduction Currently there are many standards orldide for describing the shape of terrain and sea bottoms, hich use the matrix model. From the most important of these, e can recognize: DED, SRM, DBDB-V and NMB. hey are defined by a matrix model based on a grid, hose approximated fragments (geographical grid mesh are ellipsoidal trapezoids ith fixed angle dimensions. Because the size of the trapezoids changes along ith the change of their parallel position, these models do not allo the maintenance of uniform precision in the projection of the actual surface of the terrain/sea bottom in the hole area on the reference ellipsoid. In consideration of the above, attempts made to solve this problem through the use of scientific research, resulted in the studied matrix model based on a grid, hose approximated fragments are approximately square ellipsoids determined as ellipsoid. he model designed takes into consideration its future use in sea navigation and also the necessity to maintain a consistently high level of precision of the projected area of terrain/sea bottom in the hole area on the reference ellipsoid. Worldide evaluations according to DM standards Every DM - Digital errain Model is an approximation of reality, hich is hy the choice of a most suitable model and the definition of its parameters depends on the demanded requirements for interpreting the phenomenon. In vie of the simplicity of building a matrix model, the most often described phenomenon is the character of the area (also the shape of the terrain/sea bottom. he faithfulness of the presented actual area of terrain/sea bottom by means of this model depends above all on the quality of the source material and size of aggregate data of height/depth. he DED standard classifies the matrix model according to hat is knon as information. Each of the numerical products is rated at a conventional level: Level (DED suitable ith regards to at a scale of :, ISBN:

2 Level (DED suitable ith regards to at a scale of :, Level (DED suitable ith regards to at a scale of :, Level 3 (DED 3 suitable ith regards to at a scale of :. In addition, ithin the confines of each level, the Earth s area is divided into zones, in hich the matrix has various s []. ab. Division into zones and precision levels in the DED standard Zone Zone borders [geographic latitude] Parallel Meridian I II III IV V he numerical model of terrain SRM (Shuttle Radar opography Mission is developed ith the angle = = =.7777º. Currently publicly available SRM -3 data has been divided into segments, of hich each one incorporates a sector of the Earth s surface of the dimensions º by º, and the exact sides of the sector measure.83333º and as a result these neighboring segments overlap each other negligibly. he data in each segment makes up a matrix of numbers corresponding to a grid of a size of ros and columns composed of cells of dimensions = = 3 =.83333º []. he technical specifications of DBDB-V (Digital Bathymetric Data Base - Variable Resolution developed by the National Imagery and Mapping Agency, describe a numerical model of the sea bottom by means of geographical grids of the s = =,, = =, = =, = = [3]. he technical specifications of NMB developed by the Geospatial Maritime Working Group appointed by the NAO Geographic Conference, describe a numerical model of the sea bottom by means of geographical grids ith 9 levels of [4]. ab. Division of levels of precision in the NBM standard Resolution level Parallel Meridian 3 Distortion of projected areas of the terrain/sea bottom in DED, SRM, DBDB-V and NMB All the represented standards define a matrix model based on a grid, hose approximated fragments (grid mesh, are ellipsoidal trapezoids. P Fig. Ellipsoidal trapezoid he size of the trapezoid changes along ith change in its parallel position in the ellipsoid. A change in the position of the trapezoid causes a change in the length of the arcs meridians and parallels, hich make up the borders of the trapezoid. he area of the ellipsoidal trapezoid P can be determined using the relationship: P = b φ cos ( e sin , 3,,,, 3,,, d d ( here b - length of the semi-minor axis of the ellipsoid, e - square of the first ellipsoidal eccentricity. ISBN:

3 he use of this relationship ( can determine the area of the ellipsoidal trapezoid P in relation to geographic latitude for the chosen level of established by the DED, SRM, DBDB-V, NMB standards (this calculation as also done using the Simpson numerical method hich gave approximately the same results. P [m ] P [m ] [ ] Fig. 4 Area of ellipsoidal trapezoid in relation to geographic latitude of the of the DBDB-V, NMB grid ith the = = (on a WGS-84 ellipsoid [ ] Fig.. Area of ellipsoidal trapezoid in relation to geographic latitude of the DED grid (on a WGS-84 ellipsoid P [m ] [ ] Fig. 3 Area of ellipsoidal trapezoid in relation to geographic latitude of the SRM -3 grid (on a WGS-84 ellipsoid After analysis of the obtained results, it can be stated that ellipsoidal trapezoids of fixed angle dimensions throughout the hole area of the ellipsoid, change their linear dimensions. he greatest changes undergone are in the length of the trapezoid base. he effects of this are that the area of the approximated trapezoid fragment of terrain/sea bottom area differs at different geographic latitudes. DED can be recognized as the best model. In this model, the changing base length of the trapezoid is compensated for by the creation of zones of differing angle s. Hoever, this model only minimizes and does not eliminate the changing phenomenon of the ellipsoidal trapezoid area produces change in its parallel position on the ellipsoid. 4 DM based on an ellipsoidal squares I propose introducing the DM model, hich ould enable the maintenance of uniform precision of projections of the actual area of terrain/sea bottom in the hole area on the ellipsoid. his model ill be a matrix model of the GRID type in hich grid mesh ill be ellipsoidal squares, positioned one next to the other covering the hole area of the respective ellipsoid. he linear length of the sides of each ellipsoidal square x ill correspond to the length of the equator arcs defined for each fixed ISBN:

4 angle and ill be calculated by means of the folloing formula: = a arc [ m] ( here π arc =, 6 a - length of the semi-major axis of the ellipsoid in meters, - the fixed angle value expressed in minutes, arc - the angle value expressed in radians. j P( i, j i he starting coordinate system for grid ill be found at points specified by the geographical starting coordinate system. he position of the remaining grid nodes ill be defined in accordance ith these points. Fig. Relation beteen geographical and DM grid o determine the geographical coordinates of each grid node P(, the folloing formula ill be applied: -i -j = - j = = ( e sin ( / a( e ( / 8 [( e ( e sin( / ] 3 cos ( tg ( / ( sin ( / tg / 4e sin / e ( ( e sin ( / a( e ( e sin ( / a 3 3 ( e 3 (3 here e' - second ellipsoidal eccentricity, = for =, geographic latitude is accepted as north for j > and south for j <, ISBN:

5 i [ ( e sin ] = (4 here = geographic longitude is accepted as east for i > and est for i <. Summary Considering their simplicity in building, matrix models should be applied to DM descriptions. Until no, applied orld standards of DM have not guaranteed uniform precision of actual projections of the area of terrain/sea bottom in the hole area of the reference ellipsoid. I propose the application of an DM model based on ellipsoidal squares. his model allos the maintenance of a uniform level of precision ith projections of the terrain/sea bottom in the hole area of the respective ellipsoid. o use this in navigation, it is recommended to apply a model based on a UM or UPS grid projection. hese models are characterized by true angles and high precision of projected distances. A particular quality of these is also the possibility of easily and precisely accounting for projection distortions. References: [] Department of the Interior, U.S. Geological Survey (USGS, Performance specification digital terrain elevation data (DED, National Imagery and Mapping Agency (NIMA, Reston,. [] [3] Department of the Interior, U.S. Geological Survey (USGS, Performance specification digital bathymetric data base - variable (DBDB-V, National Imagery and Mapping Agency (NIMA, Reston, 998. [4] Geospatial Maritime Working Group of the North Atlantic reaty Organization (NAO, Additionally Military Layers, Netork Model Bathymetry (Draft, United Kingdom Hydrographic Office, 7. ISBN: