Earlier, spatial data were defined as distances. Spatial data types were also listed as distances represented by coordinates or coordinate differences in one of several coordinate systems. And, unless attempting to convert from one datum to another,it should be understood that equations for converting spatial data from one coordinate system to another should have little or no mathematical uncertainty associated with them. Any uncertainty should be the result of an imperfect measurement, not a defective model, equation, or algorithm. With that said, primary spatial data are defined as geocentric X/Y/Z coordinates, their associated covariance matrices, and point-pair correlation matrices. Primary spatial data are created by a specific measurement process or determined on the basis of some prescribed geometry. Measurements have standard deviations and covariances associated with them, while errorless quantities have zero standard deviations. The GSDM accommodates both measurements and errorless quantities by using standard deviations of all three components at each point, covariances between components, and correlations between points.
Observations and Measurements
Mikhail (1976) describes how measurement and observation are very similar and, in fact, are used interchangeably. A mathematical distinction made here is that observations are independent, while measurements may be correlated. Stated differently, an observation (whether it is the process or the numerical outcome) is taken to be the actual comparison of some quantity with a standard, while a measurement is taken to be the same as either an observation or a subsequently computed quantity after corrections are applied as dictated by observation conditions. For example, a horizontal distance is said to be measured by an EDM. Actually, an EDM uses (1) the observed phase difference of two electromagnetic signals on several frequencies (the transmitted and received signal, and an internal reference signal), (2) the estimation (observation) of air temperature and barometric pressure for the atmospheric correction, and (3) the measurement of the vertical (or zenith) angle. These observations are used to compute the horizontal distance, which is called a measurement, when, in fact, physical quantities other than length were observed. Also note that the same observations are used to compute the vertical component of the slope distance. If one of the observations is changed, it may affect both computed values. Hence, the horizontal and vertical measurements are correlated and not independent. Slope distance and zenith directions are the independent observations.
Having made a distinction between observations and measurements, several other points also need to be made:
• In the strictest sense, primary spatial data should include only errorless quantities and independent observations. However, given the multitude of sensors used to make observations and the number of steps often needed to convert observations into spatial data components (measurements), it would be onerous indeed for each spatial data user to assume responsibility for the integrity of his or her data all the way back to the observation. It is hereby suggested the GSDM will conveniently serve two distinct groups: those responsible for generating quality spatial data and those who use spatial data. The work of scientists, physicists, electrical engineers, and programmers is completed upon delivering a measurement system that can be used to generate quality spatial data. If cartographers, geographers, planners, and other spatial data users know they can rely on the quality of data provided, they need not be so concerned with the science of measurement, computations, and adjustments but are free to focus their energies on spatial analysis and other chosen applications. The geomatician (geodesist, surveying engineer, photogrammetrist, etc.) provides a valuable service to society by interacting with and serving both groups.
• All primary spatial data have covariance matrices associated with them. In the case of errorless quantities, the covariance matrix is filled with zeros. Otherwise, the covariance matrices are obtained by formal error propagation from basic observations through a competent network adjustment.
• Computation of measurements often results in correlation between computed spatial data components. That correlation is defined and determined by the error propagation computation procedure applied to independent observations and the mathematical equations used to obtain the spatial data components. For that reason, it is necessary to store the full (3 x 3 symmetric, six unique values) covariance matrix along with the XIYIZ coordinates of each point.
• As used here, errorless quantities and unadjusted measurements are the basis of primary spatial data. But, in reality, primary spatial data are the XIYIZ coordinates and associated covariance values stored following rigorous network adjustments and successful application of appropriate quality control measures.
A statement of the obvious is that primary spatial data having small standard deviations are more valuable than primary spatial data with large standard deviations. Whether a standard deviation is large or small is dependent upon the measurements made and the correct propagation of the measurement errors to the spatial data components. The GSDM handles 3-D spatial data the same way, component by component, regardless of the magnitude of the standard deviations, and each user has the option of deciding what level of uncertainty is acceptable for a given application. Additionally, the GSDM is strictly 3-D and makes no mathematical distinction between horizontal and vertical data. But, the GSDM readily provides local Δ e/Δ ηΙΔ u components that can be used locally as flat-Earth (local tangent plane) distances.
