GIS professionals and technicians both work extensively with geospatial data in providing services to society. Spatial data are the raw material while people, businesses, agencies, and organizations are the clients. The Global Spatial Data Model (GSDM) provides an efficient geometrical bridge between the raw data (observations) and the end uses (applications). Coordination of policies (professionals) and practices (technicians) will enhance efficiency and provide additional benefits for all parties working with 3D digital spatial data. While reasons for adopting the GSDM as a matter of policy have been discussed elsewhere [1], the focus of this article is applications of spatial data in 2D practice.

Acknowledging that policy and practice are interrelated and that professional/technical considerations are not mutually exclusive, it is anticipated that additional discussions will be needed in developing standards and specifications associated with a transition to using a “3D model for 3D data”. Questions of coordinate uniqueness, distance distortion, and meridian convergence are inherent in using a 2D map projection. Those issues are addressed by the GSDM which, by performing computations in 3D space, avoids the need for a conformal map projection and directly connects the spatial data primitive to local horizontal distances, directions, and elevations.

The digital revolution is characterized by transition from analog to digital in how information is collected, processed, stored, and used. Of many types of information affected, spatial data are the focus of this article. “Disruptive innovation” is encountered as methods conforming to properties of digital information supplant more traditional procedures.

Understanding that the ability to do something does not necessarily justify doing it, the transition to “using a 3D model for 3D data” deserves careful evaluation and justification. In addition to the practice procedures highlighted in this article, prior work provides a solid foundation on which to build a rigorous model for 3D digital spatial data.

Appendix C in Bomford [2] provides an authoritative summary titled “Cartesian Coordinates in Three Dimension”. Rigorous solid geometry equations for spatial data computations are long-standing.In 1978, world-renowned geodesist Helmet Moritz (1933-2022) [3] discusses “The Basic Rectangular Coordinate System” and in “The Reference Ellipsoid” section, he comments on the simplicity of using the basic rectangular X/Y/Z system. Now, nearly 50 years later, the digital revolution and advances in technology (including AI) make using a 3D datum more feasible than before. The Earth-Centered Earth-Fixed (ECEF) rectangular X/Y/Z coordinate system is the foundation of the World Geodetic System 84 (WGS84) defined and maintained by National Geospatial-Intelligence Agency (NGA) [4]. The International Earth Rotation and Reference Service [5] also observes satellite orbits and publishes ECEF coordinates based on the International Terrestrial Reference Frame (ITRF). The WGS84 and the ITRF are computed independently and although the two are not identical, subsequent positioning differences are minimal. The defining document for the Global Spatial Data Model (GSDM) [6] was filed with the U.S. Copyright Office in 1997. The GSDM has two components—a functional model that covers the geometry of location and a stochastic model that employs error propagation methods to compute the standard deviation of any derived geometrical quantity (given appropriate covariance matrices are available).

The approach employed herein is to provide a practical example showing how the GSDM is used to improve the workflow when bridging the gap between field observations and a ground-level 2D survey plat showing directions and distances. Four scenarios will show that horizontal distance distortion is eliminated and that the need for a conformal projection is obviated.

Scenario 1—The local Point of Beginning (P.O.B.) origin is taken to be the SW Corner of Section 31, T23S-R1E, NM Principal Meridian as shown in Figure 1.Scenario 2—The P.O.B. origin is taken to be the SE Corner of Section 31, T23S-R1E, NM Principal Meridian.Scenario 3—The P.O.B. origin is taken to be NGS HARN station “REILLY” on the NMSU campus, 13 kilometers distant from Section 31. Scenario 4—There is no P.O.B. origin. Each vector forming the loop around Section 31 is computed as a geodetic inverse (both forward and back) yielding local directions and distances for each vector.

Since the Earth-Centered Earth-Fixed (ECEF) system, X/Y/Z, is “right handed”, the east/north/up convention for local right handed coordinates is adopted.Mathematically, coordinate differences define a vector (ΔX/ΔY/ΔZ) while a baseline is defined by GPS/GNSS measurements. Although technically different, the terms “baseline” and “vector” are sometimes used interchangeably.Coordinate differences, called “delta” (Δ), are computed as Point 2 minus Point 1. Point 1 is the standpoint and called “here”. Point 2 is the forepoint and called “there”. When reversing direction on a line, Point 2 becomes the new “here” and Point 1 becomes the new “there”. Preserving positive/negative signs is important.On the globe, longitude is counted from the Greenwich Meridian as 0˚ to 180˚ East (Eastern Hemisphere) and 0˚ to 180˚ West (Western Hemisphere). Although mathematically correct to use west longitude as a negative value, many prefer to use east longitude 0˚ to 360˚. The standpoint latitude and longitude are used with a rotation matrix to convert ΔX/ΔY/ΔZ geocentric coordinate differences to local Δe/Δn/Δu differences (and vice versa). Those local differences are the same as plane surveying coordinates [7].

Figure 1. Plat of survey—Section 31, T23S-R1E, New Mexico Principal Meridian.

Plane surveyors and other spatial data users have a long history deserving of appropriate recognition. Many professional practices flourish under assumptions of local flat-Earth geometry, but the digital revolution drives a larger (global) view. Although acquiring a 3D mindset may require effort, an expanded mindset will enhance a user’s ability to connect GSDM coordinates with physical reality—the geometrical equivalent of a digital twin.

Plumb lines are not parallel. The right triangle component of a slope distance is commonly used as the horizontal distance and is called HD(1) [8]. The tangent plane distance between plumblines is defined as HD(2) and differs slightly from HD(1). Other definitions of HD may depend on the elevation at which the HD is computed.Meridians converge at the Poles and are not parallel. Computations in most GISs are simplified because the meridians defined by the underlying map projection (grid north) are parallel. Convergence, the difference between true north and grid north at a point, is also the difference (±180˚) between forward and back azimuths between “here” and “there”. Meridians and plumblines are both assumed to be parallel in flat-Earth (plane surveying) computations. Map projections used in GISs make it possible to extend the validity of flat-Earth assumptions through the use of grid distances and grid azimuths. Although those procedures have served the user community superbly well, the digital revolution pushes the acceptable limit of flat-Earth assumptions. The GSDM accommodates both non-parallel plumb lines and meridians because computations are performed in 3D space—providing a direct connection between actual physical features (ground level) and their GIS representation. The rules of solid geometry embedded in the GSDM offer greater flexibility and give the user more computational choices. Those options (e.g., choosing an appropriate definition of horizontal distance) enable a user to solve a greater range of problems. An esoteric example is computing the shadow height of a blocking mountain at a NEXRAD installation [9][10].

When asked to identify the essential elements of a GIS, Microsoft Copilot listed:

HardwareSoftware Data PeopleMethods

Acknowledging the benefit of that AI overview, the AI response displays a disturbing absence of human insight. What about the central role of a clearly defined spatial data model which serves as the foundation of the spatial data infrastructure and is fundamental to modern implementation of a GIS?

Change can be difficult and change for the sake of change should be avoided. In some cases, it might be better to stick with an imperfect system that “works” than to incorporate change fostered by disruptive innovation. Given that end users will participate in trade-off discussions, the avowed purpose of this article is to articulate the advantages of improving the spatial data workflow from observation to end use. Two important points: First, a 3D database, utilized by the GSDM, provides a common data exchange place for both generators and users of 3D digital spatial data [11]. Second, using the GSDM as a GIS foundation is compatible with responsible adoption of AI (algorithmic integrity).

Geometrical integrity is the foundation of professional practice in engineering, surveying, mapping, and other uses of digital spatial data. Without criticizing map projections and use of separate horizontal and vertical datums, the following issues are associated with the design and implementation of current GISs.

Mathematical uniqueness for coordinate positions: Different plane coordinate origins help prevent duplication of plane coordinate values zone to zone, including units of meters and feet. Two additional considerations are 1) large coordinate values on a map projection are encountered when covering a large area and 2) Low-Distortion Projections (LDPs) require the use of multiple zones and origins. Careful of location of each LDP origin and assigning appropriate false eastings/northings to each origin makes this a non-problem.Distance distortion: Map projections are flat and strictly 2-dimesional. It is impossible to represent a curved distance on the Earth on a flat map without distortion. The solution is to use a Low-Distortion Projection (LDP) in which zone coverage is limited and nominal levels of distortion are of no consequence. Conformality: Map projections can be mathematically complex, and cartographers have devised numerous ways of portraying a curved Earth on a flat map. It is critical for spatial data users that an angle on the curved Earth is accurately portrayed on a flat map. A conformal projection based on the Cauchy-Reimann differential equations preserves the angles ground to map. The mathematical complexity of conformal projections can be challenging, but mathematicians and programmers have developed software for converting curvilinear latitude and longitude to rectangular map coordinates and vice versa.The GSDM, using a 3D database of ECEF geocentric X/Y/Z coordinates, addresses all three concerns by performing computations in 3D space. A triplet of X/Y/Z geocentric coordinates is unique worldwide. A vector of geocentric coordinate differences (ΔX/ΔYΔZ) rotated to the local perspective at a standpoint chosen by the user yields local plane coordinate differences (Δe/Δn/Δu) “here” to “there”. The horizontal distance, HD(1) = sqrt(Δe² + Δn²), computed using those local differences, lies in the tangent plane through the designated P.O.B. The azimuth “here” to “there”, computed as α = arctan(Δe/Δn), with due regard to quadrant, is the true geodetic azimuth of the line with reference to the meridian through the standpoint [12]. There is no distortion of the horizontal distance and GSDM azimuth differences to multiple forepoints yield the same angle as observed on the ground. Preservation of the angles is demonstrated in the examples cited in this paper and the complexity of a conformal projection for directions is avoided. The rotation matrix used to convert between geocentric and local differences is derived in Appendix A of [13] and [14] and shown in [7]. The following links document the rigor of GSDM distances and azimuths. Horizontal: http://www.globalcogo.com/HD-Options.pdf [8].Azimuths: http://www.globalcogo.com/3DGPSAZ.pdf [12].

The scenarios described in this article are part of a project completed nearly 20 years ago [15]. Primary computations are based on ECEF control points and GPS baselines. Latitude, longitude, and ellipsoid height are computed by a standard least squares adjustment. Distances are in meters except that (in deference to USPLSS convention), distances on the 2007 plat are shown in U.S. Survey Feet. Metric distances are used in this paper.

The NMSU class project, GPS observations, network adjustment, and plat, is documented in [15]. Note that the stations and coordinates, listed in Figure 2, represent a “hand-over” by the generators of 3D digital spatial data (NMSU SUR 292 surveying students) for the benefit of subsequent spatial data users (current readers) [11].

Figure 2. Project control points and section corner coordinates.

The 2007 paper [15] includes a final plat (Figure 1) based on the 3D GPS survey. The 2D plat was computed assuming a local P.O.B. at the SW Corner of Section 31 and the local eastings and northings for each point are computed with respect to that P.O.B. The bearings and distances around the section lie in the tangent plane through the P.O.B. The distances are local ground level, and the bearings are with respect to the true meridian through the P.O.B. The sum of the interior angles adds up to a multiple of 180˚ and there are no issues about it being a conformal map. Conceptually, the details are identical to plane surveying practices traditionally employed by most surveyors. All four scenarios discussed herein are based on the same ECEF coordinate values listed in Figure 2.

The rotation matrix, described previously, is used frequently with the GSDM. The following printout shows an inverse between the SW Corner of Section 31 and the SE Corner of the same section. Several logical points are:

The project data shown in Figure 2 include both coordinates and standard deviations. Including standard deviations is beyond the scope of this paper.Figure 3 is a portion of a BURKORD^TM printout showing inverses, forward and back, between the SW Corner and the SE Corner of Section 31.Except for a change in sign, the ΔX/ΔY/ΔZ values are identical. However, because the standpoint meridians are not parallel and because there are two slightly different tangent planes, the Δe/Δn/Δu values are somewhat different between forward and back computations. But, regardless of forward or reverse, the 3D spatial distances are identical, and the horizontal distances are nearly the same.

3D = √(ΔX² + ΔY² + ΔZ²) = √(Δe² + Δn² + Δu²) fwd = √(Δe² + Δn² + Δu²) rev (1.0)

The difference between forward and back azimuths (180˚ + 32.8 seconds) is the convergence of the meridians between the SW Corner and the SE Corner of Section 31. That same convergence shows up in the difference of bearings (azimuths) in Scenarios 1 and 2.

Figure 3. BURKORDTM printout of directions and distances—SW and SE Corners, Sec 31.

A word of caution—the GSDM can be easily misused. As with careless use of matches or manipulating weights in a least squares adjustment to get an acceptable answer, the GSDM can be misused (unwittingly or deviously) with little chance of being detected. In particular, horizontal distance varies with elevation. A user can easily traverse to any height at the P.O.B. by computing X/Y/Z coordinates for an “imaginary” height. Accordingly, subsequent computations referred to that P.O.B. will be affected [8]. The GSDM provides many tools for the user, but it also requires accountability for proper use.

Because meridians are not parallel, the bearing of each line is different in each scenario. However, the computed interior angles are identical in each case, and the computed local horizontal distances are the same (within a small tolerance).

Input from the Editor and anonymous reviewers is greatly appreciated. Mr. Scott Farnham, PE, PS, former City Surveyor for Las Cruces, NM, also shared insights that were used to improve this article.