Taping
A calibrated tape is laid flat on a horizontal surface at some specified tension and temperature. The measurement involves a visual comparison of the unknown length with uniform markings on the tape (a fundamental physical quantity). The observation is recorded as a measurement. If the temperature (another physical quantity) is different than the specified calibration temperature or if the tension (a derived physical quantity) is not what it should be, these other measurement conditions must also be noted. Using this additional information and appropriate equations, corrections to the taped distance are computed and applied to this otherwise direct measurement. Whether the computed distance is a direct or an indirect measurement is left to the reader.
Leveling
A level rod with graduations marked on it is held erect in the field of view of an observer looking through the telescope of an automatic (or tilting) level, and the distance from the bottom of the rod to the cross-hair intercept is read and recorded. Separate readings are made with the rod resting on other objects. In this case, the difference of two direct readings provides an indirect determination of the relative heights of the two objects. Among others, the accuracy of such an indirect measurement is affected by (1) whether the line of sight is perpendicular to the plumb line, (2) the presence of parallax, (3) whether or not a vernier was used to refine the reading, (4) the plumbness of the level rod when the readings were made, and (5) the distance from the instrument to the rod (curvature and refraction correction). Modern barscale reading instruments are becoming commonplace.
Electronic Distance Measurement
An electronic distance-measuring (EDM) instrument emits electromagnetic radiation, which is modulated with a known frequency (giving a known wavelength). The signal is returned by a retro-reflector from the forepoint end of a line, and the phase of the returned waveform is electronically compared to that of the transmitted signal. The measurement of phase differences on several modulated frequencies provides information used to compute the distance between the EDM and a reflector. Other quantities such as temperature and barometric pressure are also measured to determine corrections that account for the signal traveling through the atmosphere between the standpoint and forepoint at some speed slower than it would have traveled through a vacuum. The point is that several physical quantities are measured and the physical environment is modeled with equations before a collection of observations can be converted into spatial data.
With later-generation pulse laser instruments and scanners, the physical distance between EDM and object is determined using the time interval required for a pulse to travel from EDM to object and back. Of course, atmospheric delay must be modeled and direction to the target must be known before a slope distance can be resolved into rectangular components.
Angles
Although not a fundamental physical quantity, angles are commonly measured and used in computing spatial data components. Two examples are (1) using a vertical angle to resolve a slope distance into horizontal and vertical rectangular components, and (2) using the bearing of a line to find the latitudes and departures of a traverse course. Looking beyond the obvious where an angle is measured directly with a protractor on paper or on the ground using a total station surveying instrument, angles are also measured indirectly as the difference of two directions such as might be observed with a compass, a gyroscope, or GPS. Whether an angle was measured in the horizontal, vertical, or some other plane is also an important consideration, especially when using angles to resolve the hypotenuse of a triangle into its rectangular components. Two examples are resolving slope distances into horizontal and vertical and resolving traverse courses into latitudes and departures.
GPS
GPS is very versatile in that several kinds of fundamental observations can be used to determine spatial data quantities. An oversimplified view of GPS measurements includes three concepts: (1) distance is the product of rate and a time interval obtained from code phase observations, (2) the Doppler shift of a frequency recorded on the ground as compared to the frequency transmitted by a satellite, and (3) interferomet-ric interaction of the carrier phase signal as recorded simultaneously at two different antennas (carrier phase observations). Without going into detail, the point is that portable handheld code phase GPS equipment routinely determines absolute geocentric ECEF coordinates and converts them to absolute geodetic coordinates before displaying them. The accuracy of code phase instruments is typically less than that obtained using carrier phase instruments. GPS carrier phase data must be collected at two points simultaneously and can be processed to give a very precise 3-D space vector between two antennas in terms of relative geocentric coordinate differences. The relative accuracy of such GPS observed vectors (with operator care and diligence) routinely exceeds one part in a million. If such a vector is attached to a known control point, a precise 3-D position of the second point can be easily computed.
This brief description implies that GPS positions determined using portable handheld equipment are not as accurate as those collected using a receiver mounted on a tripod. That may be, but is not necessarily, true because whether or not a GPS receiver collects code phase or carrier phase data is not determined by whether or not it is portable. For example, differential corrections (from a base station) may be used to improve upon the accuracy of code phase solutions, and radio connections between portable carrier phase receivers mean that relative differences may be obtained in real time (such as real-time kinematic, or RTK). Even though the two GPS technologies are quite different, in either case, once a GPS position is determined, answers can be viewed in a coordinate system of the user’s choice.
Photogrammetric Mapping
Relative spatial data, both local and geocentric, can be determined efficiently and precisely using geometrical relationships reconstructed from stereoscopic photographs of a common image. A photogrammetric measurement is the relative location of an identifiable feature on a photographic plate with respect to fiducial marks on the same plate as determined with a comparator. A more complex measurement of 3D spatial relationships based upon principles of photogrammetry requires mechanical reconstruction of the stereoscopic image by achieving the proper relative and absolute orientation of the stereo photographs in a mechanical stereo plotter. A 3-D contour map of the ground surface is the end result of the plotting operation. That traditional photogrammetric mapping process has been computerized and automated and now comes under the banner of softcopy photogrammetry. The end result of the modern computerized process is a 3-D digital model of the terrain. Hardcopy maps, computer displays, and other products, both digital and analog, are made from a common digital spatial data file.
Remote Sensing
The American Society of Photogrammetry and Remote Sensing (1984) describes remote sensing as the process of gathering information about an object without touching or disturbing it. Photogrammetry is an example of remote sensing, and ray tracing based upon stereo photographs of a common image is very geometrical. Bethel (1995) also discusses remote sensing and describes interpretative (less metrical) applications of remote sensing, which include use of nonvisible portions of the electromagnetic spectrum to record the response of an object or organism to stimuli from a distant source. Sensors include infrared film, digital cameras, radar, satellite imagery, and so on, and information is stored pixel by pixel in a raster format. Determining the unique spatial location represented by each pixel can be a daunting challenge and typically requires enormous storage capacity.
Other measurement methods are also used to create spatial data. But, regardless of the technology used to measure fundamental physical quantities, the GSDM provides a common universal foundation for expressing fundamental spatial relationships. Various equations (models) are used to convert observations into spatial data components, which are then used as measurements in subsequent operations, for example traverse computations, network adjustments, and plotting maps. The GSDM also accommodates fundamental error propagation in all cases, and that information, if available, is stored in the covariance matrix for each point. From there, each user can make informed decisions about whether or not the spatial data accuracy is sufficient to support a given application.
Errorless Spatial Data Must Also Be Accommodated
Several cases exist in which spatial data are considered errorless. Examples include (1) spatial data created during the design process, (2) physical dimensions (such as the width of a street right-of-way) defined by ordinance or statute, and (3) spatial data whose standard deviations are small enough to be judged insignificant for a given application. In the case of a proposed development such as a highway, bridge, skyscraper, or other civil works project, the planned location of a feature and the numbers representing the size and shape of each feature qualify as spatial data. But, they are the result of a design decision instead of a measurement process. Such design dimensions are without error until they are laid out. After being laid out and constructed, the location of the feature or object is determined by measurement and typically recorded on as-built drawings or in project files. Considered that way, the perfection of a design dimension is transitory and ceases to exist when laid out during construction.
An exception to the transitory nature of an errorless design dimension exists when a dimension is established by ordinance or statute. Such a dimension may be fixed by law, but the physical realization of that dimension is still subject to the procedures and quality of measurements used to create it. Under ideal conditions there will be no conflict between a statutory dimension and its subsequent remeasurement if the layout process was more accurate or reliable than the measurements made to document its location. For example, a 100-foot right-of-way may have been monumented very carefully and current measurements between the monuments are all 100.00 feet, plus or minus 0.005 feet. In that case, the right-of-way width can be shown as 100.00 feet, measured and recorded. Under less-than-ideal conditions, several possible dilemmas are as follows:
1. The right-of-way monuments really are separated by 100.00 feet, but the survey is based upon a state plane coordinate grid and the measured grid distance is 99.97 feet, plus or minus 0.005 feet (possible at elevations over 4,200 feet). Understandably, the monuments are not to be moved, so they are separated by a grid distance of 100.00 feet, but some users may not be willing to accept the implication that a foot is not really a foot. The apparent discrepancy arises from the use of two different definitions for horizontal distance, local tangent plane distance or grid distance.
2. The right-of-way monuments appear stable and undisturbed, but the measurement between them is consistently 99.96 feet, plus or minus 0.005 feet (it could happen if the monument locations were staked during cold weather and no temperature corrections were applied to the steel tape measurements at the time). The conflicting principles are that the statutory dimension (of 100.00 feet) must be honored and that the original undisturbed monument controls, even if originally located with faulty measurements.
The intent here is not to solve those problems, but to acknowledge the potential for conflicting principles when working with so-called errorless spatial data.The point here is that the GSDM offers a consistent standard environment in which to make comparisons between conflicting data. The GSDM does not distort horizontal distances as does the use of map projections and/or state plane coordinates.
When the coordinates of any point are held “fixed,” the result is the same as assuming the standard deviations associated with the coordinates are very small or are zero. In many cases, such an assumption is reasonable and defensible because the standard deviations of a point are small enough to be insignificant and the implied statement “with respect to existing control” is acceptable. However, each spatial data user making decisions about which control points are held “fixed” should document such decisions specifically so that subsequent users may always know “with respect to what.” With the accuracy of spatial data becoming ever more important, criteria for judging the quality of spatial data should be unambiguous and easy to understand. The stochastic model portion of the GSDM uses 3-D standard deviations to describe the accuracy of spatial data and accommodates errorless spatial data as those data having zero standard deviations.An answer to the question “Accuracy with respect to what?” is determined by (1) the control points used as primary data, (2) the 3-D standard deviations of those control points, (3) the quality of measurements used to establish new positions, (4) competent determination of the covariance matrix of each new point, and (5) the manner in which equation 1.36 is used in subsequent computations.
