Paper presented at the MSPS 57th Annual Meeting, St. Cloud, MN, 28–30 January, 2009.

Instrument Calibration for the 21st Century N.W.J. Hazelton

Department of Geography, SH 352A, St. Cloud State University, 720 Fourth Ave., South, St. Cloud, MN, 56301-4498. Ph: 320-308-6686 E-mail: nwhazelton@stcloudstate.edu

Abstract

This paper presents a brief overview of surveying instrument calibration in general, before moving on to the theory of EDM calibration in particular. The need for calibration is discussed, together with a quick look at available facilities. An overview of what is required to calibrate different types of instrument is provided.

Errors in EDM are discussed, together with modern designs for Calibrated Base Lines (CBLs). This is an area of on-going research, and Saint Cloud State University hopes to be in the forefront of developments in this area.

Introduction

The purpose of the paper is to provide a brief overview of calibration procedures and concepts, as they currently exist. The perspective is designed to be close to the needs of practicing surveyors, who need to consider how to calibrate measurement equipment. A broad overview of calibrating different types of instruments is provided, while the more focused discussion is on calibration of EDM.

Of obvious importance is a rationale for calibration. At present there appear to be no strict legal requirements to calibrate surveying equipment in the USA, something that is not the case in many other developed countries. A rationale for calibration must be provided, if it is to be done at all, and this rationale needs to be something that works for practicing surveyors.

Why Calibrate Instruments?

There are many reasons why measurement instruments should be maintained in a state of proper calibration. To some extent, these may be self-evident, but the main ones relate to the need to ensure different measurements fit together, and that measurements have a definite quality and are fit for the purposes for which they were made.

Many different instruments may be in use by the one organization, and these may need to be closely linked together. One simple example would be using different total stations on the one project, where all the measurements need to be the same, but mixing different instrument types produces far more complex issues. How do we ensure that the measurements from a total station, laser scanner, GPS and handheld camera can all fit together, especially when all of them are used as part of the one project?

Calibration should ultimately provide a definite chain from a given instrument back to the national measurement standards. This means that in any dispute or similar situation, a measurement can be shown to be derived directly from the national standards. This also provides a guarantee of measurement quality.

National standards provide an end point to the problem of what to compare a measurement to, if you want to know that it’s correct. I can compare this distance to what’s on the plan, but how was it determined? And how was that determination checked, and so on? This is the problem of ‘infinite regress,’ where there is no end to the chain of comparisons. With national standards, there is a basis for consistency of measurements across space and time, and a way to keep the quality of measurements independent from other considerations, such as who manufactured the instrument. Within one nation, there should be one standard of measurement, to which everyone has access. This leads to another issue, dealt with below, of how to disseminate the national standards across the nation.

Calibration provides a means of determining the errors and uncertainties associated with an instrument. While this information may be supplied by the manufacturer, as a generic value for that model of instrument, it is important to ensure that the instrument meets those specifications. The quality data is critical to the use of the measurements in least squares adjustments, as a first application of the measurements to derive information beyond the measurements themselves.

One of the claims of the surveying profession is that ‘we are the measurement experts.’ Today, anyone can make a measurement: GPS and total stations are quite easy to operate and data flows at prodigious rates. But measurement experts produce not only the best measurements, but measurements with the correct meaning. The best measurements ultimately require a link to national standards, hence the need for calibration.

Measurements are usually made so as to derive additional information, ranging from co-ordinates and areas, through to complex decisions relating to land use and management. The measurements form a foundation for this later work, and as a structure is only as a good as its foundation, the quality of later decisions is directly related to the quality of the foundations. This author has seen a number of cases where the later derived information was rendered unfit for its purposes because of insufficient quality of original measurements, including one project where the current $250 million cost overruns and three year delay can be tied fairly closely to insufficient quality of the control survey. (It should be noted that in this case, this problem was NOT the fault of the surveyors who undertook the original control survey.)

Naturally, no-one wants to discover that their measurements are inadequate in the middle of a large court case. Having a demonstrable connection to the national measurement

In summary, calibration of measurement instruments is simply good professional practice. It builds professional credibility, simply by ensuring that the measurements that are produced are high quality and connected to the national standards. Calibration allows errors and uncertainties to be quantified, which allows their proper use in deriving later information, and ultimately forming a high-quality foundation for the decision-making that results from survey work. Finally, calibration of various instruments allows measurements from different instruments and different instrument types to be integrated, a critical issue as new technologies become more widely used.

How to Calibrate Different Instruments

Calibration varies with different instruments, as they tend to work in different ways. Some instruments can be calibrated by anyone at almost any time. For instance, it is quick and simple to check that levels are working properly, and it can be done in almost any situation. Similarly, it is fairly easy to check that angle-measuring equipment is operating properly and providing reliable results.

Other equipment needs some kind of support equipment. EDM needs lines of known length, usually set out with specific distances, and connected to the national standards, to provide a full calibration. While anyone can undertake EDM calibration, the calibration baseline needs to be constructed and maintained.

Other equipment may need laboratory calibration, in particular equipment that relies upon high-precision oscillators, such as Mekometer EDMs. Other equipment can only be adjusted in laboratory conditions, such as most laser plummets in total stations.

The following sections will briefly examine a range of instruments and how to calibrate them, as well as highlighting some areas of concern for specific areas often overlooked.

Leveling

Levels can be checked independently. All that’s needed is a bit of space and a couple of stable points. Traditional levels can be checked and adjusted using the traditional two-peg test. Digital levels use much the same procedure, but it is individualized to each instrument’s software (check the manual). Similarly, a few repetitions of a run of leveling can test that the instrument’s closure capabilities match the manufacturer’s specifications, and that operators are capable of achieving those specifications.

Difficult areas with calibrating leveling equipment are primarily calibrating leveling rods. This comes back to a comparison tape, and how it was checked, for conventional rods. Barcode rods present a major problem, if the barcode can’t be interpreted in terms of distances along the rod. For precise work, the ability to determine the temperature of the rod may also be important, although not easy, and the quality of that temperature determination, along with the associated change in length of the rod, must be known.

One of the more complex areas in leveling is stability of marks. Various marks tend to move in a variety of ways, but the quality of leveling (in terms of its usefulness for later work) is directly tied to the stability of the benchmarks used. Calibrating different types of marks for their stability is an area of practice that is not widely examined, but is a matter for concern.

Angle-Measuring Instruments

Traditional angle-measuring instruments, such as transits, theodolites and total stations, can usually be checked for correct angle reading without any special equipment or skills. The methods of angle measurement described in the international standards, and used by manufacturers to obtain their instrument specifications, will indicate if there are problems with angle measurements.

Most of the small errors in instruments can be cancelled out or minimized by using the correct reading procedures, such as the international standards describe, and this deals with most problems. Most total stations and electronic theodolites have a means of reducing collimation error in angles, and with non-electronic instruments this can be done by adjusting the graticule (although there is rather more to this when dealing with vertical collimation error).

Errors that go beyond these traditional errors usually require a trip to a repair facility. This author has seen one electronic instrument that occasionally dropped 2' in the horizontal circle readings, and one that was so far out of adjustment that it could not deal with vertical collimation error correctly, but these cases are fairly rare. But without calibration and checking, you may never know that the error was occurring.

Distance Measurement Calibration

Tapes are much less commonly used for critical measurements these days, and so the need to calibrate tapes is less critical. However, if there is access to a standard measurement line, such as an EDM baseline, it is possible to calibrate a tape. Many EDM baselines include a 100-foot mark for tape calibration. Tapes can also be compared to EDM distances, if the EDM has been calibrated. With all tapes, care must be taken to ensure that the zero point is correctly located with respect to the zero value on the tape.

EDM calibration is a major issue, as much of today’s distance measurements are done using EDM. While EDM calibration will be dealt with later in this paper, a critical point is that EDM cannot be calibrated without reference to national standards. This requires access to lines measured that have been calibrated with instruments that have themselves been calibrated with respect to the national standards.

GPS

Antenna calibration is one of the main calibration concerns with GPS. This can be done independently, but it is not always easy to do. ‘Zero offset’ calibration does provide one means to do this, assuming that one antenna has been fully calibrated.

Of more concern with GPS is overall system performance. Unlike more traditional measurement equipment, GPS errors can occur in ways that are difficult to see at the time, such as multipath and poor satellite geometry. It is a wise move to measure a selection of lines in a GPS network more than once, so that comparisons between measurements can be made. This will give an idea of the errors present in overall measurements, which include centering and antenna height errors, as well as errors caused by variations in the signal. This latter can be done without additional equipment by planning the GPS network to include redundancies and to connect to known control for comparison.

Laser Scanners

Scanners measure both angles and distances. Distances are measured using EDM, often using time-of-flight methods, while horizontal angles are measured using either a traditional electronic circle like a total station, or determined from a stepping motor that drives the scanner in a slow circle or arc. Vertical angles are measured from the position of the rotating prism or mirror, and are subject to greater potential for error. In fact, the errors in scanners tend to be acute for steep sights, an area that seems well-suited to their use.

Calibration of a scanner therefore requires a field of known points scattered throughout the hemisphere of view of the scanner. This requires the precise location of points in 3-D over a substantial distance, both horizontally and vertically. Such a field of control points is a non-trivial exercise to place and maintain, and is hardly the sort of thing that a single survey firm would undertake. Such a facility would need to be available for a wide range of people to be justified.

The entire scanning system is not just the measurement, but also the determination of surfaces from the scanning, using software. As there are too many points generated to be dealt with individually, they need to be condensed by fitting surfaces to the points, but these derived surfaces, which constitute part of the initial results, are part of the ‘measurement’ process, and also need to be calibrated. This means that known surfaces need to be set up in the measurement calibration space and compared to the finished product, both for point positioning and surface generation.

At present, error handling techniques for surfaces are not as well developed as those for points, so this is an area needing additional work.

Terrestrial Cameras

Terrestrial digital cameras are becoming extremely common, as almost every mobile electronic device is getting one these days. These cameras, thought small, have the advantage of fixed optics, which allows for long-term stability of calibration. Even the most interesting distortions can be modeled and corrected, allowing the camera to be used for measurement.

Setting up a suitable test range for a digital camera is quite simple, as all that is needed is being able to set out a rectangular grid on a flat surface, such as a wall. This can be done indoors or outdoors, and getting an image can be done very quickly. Calibration can be done very quickly, with the help of suitable software.

The ability to calibrate the camera (if only partly) using nothing more than straight lines in the image allows the calibration to be monitored with almost every frame. Self-calibration software should be able to be included in various packages to enable this process to proceed almost automatically for terrestrial cameras.

Terrestrial digital cameras represent an area where low-cost instruments can produce quality results very quickly, if the support systems are in place. The support systems include suitable software, calibration facilities and the appropriate know-how among users.

Aerial Cameras

Aerial cameras can be calibrated very successfully by several methods. The easiest to manage independently are in-flight calibration and self-calibration.

In-flight calibration involves taking some images of known points. This can be a test range that has been constructed on the ground. A simple array of visible points with known locations can be placed in a suitable area, perhaps fairly close to the airfield where the survey aircraft are based. The ground points can be paint marks on roads, where the center of the mark has its location determined to better than centimeter level, by GPS or traditional methods. The points are then measured on the images and the differences between the measured and known values are used to determine corrections to the camera’s calibration constants. The test array can be photographed at the start and end of missions for maximum precision in the calibration values, hence the advantage of having the test array near the home airfield.

Self-calibration involves including calibration parameters in the unknowns that are solved for in the aerotriangulation of a block of images. If there is sufficient ground control, this can be very successful, and it can be combined with in-flight GPS/INS control to provide good results for the calibration.

Of course, the camera can also be calibrated in a laboratory, in the traditional manner, but the other methods allows more frequent calibration in field conditions.

Aerial LiDAR

LiDAR can be calibrated using known ground points. It has been found that it is possible to get good return signals from known points by using ordinary 3" or 4" diameter reflectors (the 50c ones from hardware stores) that can be placed directly on a point, facing upwards. The approach is similar to in-flight calibration for aerial cameras, but allow the angular and distance measurements of the LiDAR unit to be compared to known values.

The critical issues for LiDAR concern GPS and INS calibration, as this is critical to the overall measurement system. The GPS/INS unit in the aircraft determines the location and orientation of the LiDAR unit, which then determined the orientation and lengths of vectors from the unit to the ground. The LiDAR unit and its GPS/INS unit must be calibrated as a single measurement unit.

A current research topic in LiDAR is knitting together strips of scanned areas. While this has some similarities to strip aerotriangulation, it also has significant differences, in particular the variations that can occur along the strip, which can occur instant by instant, rather than between instantaneous frames in aerial photography. The need for better ground control is critical here, as well as means of calibrating LiDAR/GPS/INS as a single measurement system.

Aerial Scanners

Aerial scanners were once concerned with something more like satellite imaging, i.e., attribute information rather than positional information. But the latest generation of airborne scanners are being marketed as alternatives to aerial cameras, and provide color imagery at a resolution close to what can be reasonably used from aerial camera imagery. Their big advantage is a much more direct digital workflow than a conventional aerial camera, although a conventional camera can produce greater precision, if pushed in the photogrammetry.

Aerial scanners have some issues that are akin to satellite images, in that they have a ‘principal line,’ rather than the principal point of an aerial photo. This changes the geometry of the image, as we have to deal with it, and changes the nature of the mode of calibration. In effect, very line of the image has its own principal point, which requires location and orientation in space. GPS/INS can do this, but the inclusion of ground points can also help.

In the same way as LiDAR, the scanner/GPS/INS unit must be calibrated as a whole, as they form a single measurement system. Ground control arrays are of less use than with aerial cameras, as the scanner is only part of the overall system. As the scanner usually has no moving parts in the optical path, being a pushbroom scanner, the calibration of the scanner’s optics would be expected to be reasonably stable over time, but the GPS/INS part is a different case.

It is still very early in the development and use of these systems, so we still have to learn about the individual foibles of the technology. But it still has to be calibrated.

Synthetic Aperture Radar (SAR)

Synthetic Aperture Radar (SAR) is very difficult to calibrate, largely because the results are very highly correlated because of the signal processing used to develop the ‘image’ data. Timing and frequency shift measurements are involved, as well as extensive processing, and this makes it very difficult to isolate specific parts of the measurement system for calibration. In addition, the ‘image’ that is produced is based on the assumption of flat terrain during processing, which makes it somewhat problematic about what is actually being detected and ‘imaged.’

SAR data is usually internally self-consistent, especially that from satellite platforms that have their location and orientation determined to a high precision. This allows Interferometric SAR (InSAR) measurements to produce remarkably precise results of surface differences over extensive areas. That said, it is very difficult to connect that information to points. Airborne SAR has the same issues of GPS/INS positioning and orientation, but it is more frequently use for attribute measurement, rather than positional masurement.

Summary of Instrument Calibration Requirements

The following instruments can be calibrated fairly easily with minimal equipment and expertise:

• Levels • Angle-measuring equipment • Cameras (with one proviso, see below)

They require no major investment in calibration facilities and can be done by almost anyone.

The following instruments can be calibrated with a suitable test range:

• EDM • Laser Scanners • Cameras • GPS

The test range can by fairly simple (for GPS) to quite complex (for laser scanners). All these test ranges require calibration themselves, and this can mean a connection back to a national standard.

The following instruments require major calibration efforts, in particular because of the integrated GPS/INS that provides a significant part of the overall measurement system:

• LiDAR • Aerial scanners

These are complex systems with highly connected components and correlated outputs.

Calibration of SAR, beyond calibration of the various electronic components, is still a major research area. If using SAR, it may be best to do such ground control as can be done, and try for the best solution.

Calibration of any distance measurement, whether EDM, laser scanners or level rod measurement, cannot be done without connection to the national standards if it is to be meaningful in any way. After all, it is hardly a strong point in defending one’s survey in court to state that the EDM is calibrated against a line of unknown length, no matter how well it agrees. However, the connection through EDM calibration to the national standards enables these standards to be transferred to other measurement systems, such as cameras, GPS and laser scanners, with relatively little trouble.

As a note, the greater the complexity of a measurement system, the more difficult it is to calibrate it. In the past, the most complex systems were usually aerial cameras and EDM, and these require complex systems to ensure proper calibration. Many of today’s measurement systems involve the tight integration of several measurement systems, and as such produce highly correlated results, which can hardly be considered to be raw measurements. Unraveling the error contributions of each system and correcting it becomes increasingly complicated as the system gains more components. It may mean that laboratory calibration of components is required in many cases, until we work out ways to make the calibration process simpler and more straightforward.

Finally, measurement systems must be calibrated as single entities. EDM has to be calibrated with a specific set of reflectors. Cameras need to be calibrated at specific lens positions, if there is the potential for moving the lens. Systems involving GPS/INS cannot be calibrated without including the GPS/INS unit, as it is an integral part of the entire unit.

EDM Calibration

EDM calibration has a pivotal role in the calibration of a wider range of measurement equipment, for several reasons. The primary reasons are the ability to obtain a direct connection to the national measurement standards, and the versatility of EDM in providing calibration facilities for other measurement systems.

EDM, largely through total stations, can provide calibrated measurements to support camera and laser scanner calibration, as well as ground control for a range of airborne systems. EDM can usually provide better positioning over shorter distances than GPS; in fact, for distances under about 500m, EDM is the more precise method of determining relative positions.

As a consequence, calibrating EDM is critical for calibrating a wide range of other measurement equipment, as EDM’s role is very broad. Therefore it is important that EDM calibration facilities are widely available to support instrument calibration generally.

If EDM calibration is widely available, the only instruments that really need remote calibration facilities are level rods (which usually need special facilities for dealing with barcode rods) and equipment that needs an electronics laboratory for calibration. Special facilities for laser scanner calibration will be needed, but total stations (with their EDM calibrated) are sufficient to calibrate these test ranges.

EDM Calibration Facilities in the USA

NGS and Existing CBLs

In days gone by, the National Geodetic Survey (NGS) established and monitored a series of Calibrated Base Lines (CBLs) across the nation. These lines were established in conjunction with various state bodies, and NGS at one time had two survey crews engaged full-time on placement, calibration and re-calibration of CBLs. All state had at least one CBL, and most states had several. The calibration data was readily available, in later years via the Internet, and it was fairly straightforward for any surveyors familiar with EDM calibration to use a CBL to calibrate their EDM.

In the early 1990s, NGS decided to cut back their involvement with CBLs. There was a belief among some NGS people that the days of EDM were over, and that GPS would become the universal survey instrument. While this decision was criticized, it also had some support. Within their own domain, NGS were quite right: EDM did largely disappear from geodetic work, and as the National Geodetic Survey, they foresaw the future of geodetic measurement quite clearly.

However, for the remainder of the surveying profession, EDM use simply burgeoned. EDM appears in total stations, which became the usual instrument for almost all 2-D and 3-D survey work. The baseline calibration process fell into decline, and NGS provided support only upon request, and then largely in terms of equipment. States with an active NGS State Advisor tended to have their CBLs recalibrated from time to time, but otherwise, not much happens.

Minnesota had several of its CBLs recalibrated in 2002, but North Dakota, South Dakota, Nebraska, Iowa, Wisconsin, Illinois and Indiana between them have had one new CBL placed since 1993, and it is also the only one calibrated since 2000. In 2004, Indiana’s sole CBL was found to have been destroyed.

The standard CBL used in the USA uses ground marks, and is based on the Aarau design (see below). The CBLs do not conform to the Aarau design, in that the distances are not always integral multiples of 10 m, and there is no 10 m rail or equivalent for cyclic error determination. The 10 m unit distance used

was based on the usual EDM instrument from the 1960s and 1970s, and is no longer the sole unit distance in use today. The value of the CBL process also depends on the precision with which the instrument can be placed over the ground mark, and unless the tribrachs and related gear are also carefully calibrated, significant errors may be introduced into the EDM calibration process. So the current CBLs have significant design flaws, in addition to the question of their stability over time, given the long periods between re-calibration of the line. That said, any decent CBL is much better than no CBL.

The Base Lines and their Use

When NGS calibrates a base line, they undertake a very thorough job or work. The tribrachs are very carefully plumbed over the ground marks, using a special high-precision optical plummet. The tribrach height is very carefully measured through the center of the tribrach. A single prism is used for all measurements, ensuring that there is no variation in the EDM/prism combination. A thermistor array is used to determine the temperature at each end of the line, and barometers are used at each end of the line, as well.

The distances are measured using a pair of Wild DI-2002 EDM units, which have an internal precision of 1 mm + 1 ppm, according to the manufacturer’s specifications. This makes them among the most precise EDM that were generally available, although this model was manufactured only during the period 1989 to 2001. The EDM units are mounted on a Wild T2000 electronic theodolite.

Given the base line design and the available equipment, NGS calibrates baselines to the absolute best of its ability. The only way to improve upon this, given the current base line design, would be to use an instrument such as a Kern Mekometer, or a Geomensor (precision of both around 0.1 mm), but these instruments have also been out of production for many years, and are very hard to come by.

When users go to a CBL to calibrate their EDM, they commonly make a number of mistakes in how they approach the process. They often don’t calibrate their tribachs or other centering equipment, thereby introducing significant centering errors into the measurements. If thermometers and barometers are used at all, they are rarely calibrated, introducing errors into the distances from meteorological errors. Individual prisms and their mountings are not compared and so calibrated, so there may be differences between the measurements made to individual prism within the calibration process.

Calibrating an EDM on a CBL can take several hours, so there sometimes an element of impatience creeps in. The process is hurried and less care is taken. Finally, not all users know how to compare their measurements to those from the CBL data. Slope distances as measured are compared to slope distances between the marks, without allowance for instrument and reflector heights, among other efforts noted, and in many cases, the process is treated as a simple comparison, rather than as an error determining process. This lessens the value of the CBL to the user.

The current CBLs tend to have uncertainty in their distances that increase over time from the last calibration. This makes it more difficult to determine scale error in the EDM. The current design is often incompatible with current EDM design, in particular EDM unit distances, which means it is more difficult to use the CBL to determine the zero or offset error in the EDM. When users fail to calibrate different reflectors used in the calibration process, it makes this situation even worse. With the absence of the Aarau design’s rail, or other additions to the CBL, the cyclic error of EDM cannot be determined. In summary, current CBLs cannot guarantee any part of the EDM calibration process, and therefore have limited value for modern EDM.

But there isn’t anything else out there.

NGS Policy

NGS is aware of these problems, but budget constraints limit their action. There is also the question of whether this should be part of NGS’s mandate, or if some other body should handle it. NGS do plan to build a modern CBL at Corbin, VA, within a year or two, but budge issues may prevent that.

NGS is uncertain about a national CBL policy, because of budge issues, because of mandate questions, and also because of user indifference. There is no point in spending federal money on infrastructure if no one uses it. Regardless of the importance of instrument calibration, until there is a major perceived risk in not calibrating instruments (i.e., someone wins a major lawsuit because of poor instrument calibration), or it is required by law, it appears unlikely that the profession will consider it worthwhile to calibrate their instruments.

What Happens Overseas?

In Canada, CBLs are calibrated using Mekometers at regular intervals, In Finland, CBLs are calibrated to better than 0.1 mm, and have been for over 50 years. In Australia and New Zealand, CBLs are re-calibrated annually using Mekometers, and the data and software for individual user calibration are available free over the Internet, updated early each year, after the re-calibration. Australia also has a requirement for annual EDM calibration. CBLs in Australia are of a more modern design, allowing faster EDM calibration to a higher standard, with far less chance of error.

EDM Calibration Theory

Errors that can be Determined by Calibration

Apart from the meteorological systematic errors, EDM has three main systematic errors. All of these can be determined during a thorough calibration of the instrument–reflector system. It is recommended that EDM be calibrated annually, perhaps every 6 months for certain applications, and that all corrections be determined and applied in high precision work. Some corrections may be able to be overlooked for lower-precision work, but for high-precision work, all corrections must be made. Figure 1 shows where these errors occur in a distance measurement.

The first systematic error is the constant, offset, zero, reflector or similarly named error. This is a combination of the effect of the zero point of the instrument being elsewhere than centered over the mark from which the measurement is being taken, together with the reflector’s zero point being other than over the ground mark at its end of the line. This error should be less than 1 cm if the correct reflector offset is used. The offset or zero error tends to be constant for a particular type of reflector, so to correct this constant error, it can be included in the reflector constant to be set for the instrument, such as a total station that allows a variable reflector offset to be set.

The mathematical model for the zero or offset error is:

cz = d – dt

where cz is the zero or offset error, d is the measured distance, and dt is the ‘true’ distance.

If a single type of reflector is in use, i.e., the surveyor or organization has standardized on just one type of reflector, e.g, all Leica or Geodimeter reflectors, this error can be controlled very easily. A single offset covers all equipment, and this only needs to be updated after each calibration. If using a mixture of

reflectors, the possibility exists for mixing up the reflector settings. The sudden appearance of about 30 mm differences in line lengths may suggest this as the problem.

Figure 1 Systematic Errors in EDM: Where They Occur

The transmitter sends the carrier wave with the superimposed waveform (shown) towards the reflector. The reflector returns the signal along a parallel path. The receiver collects the incoming wave. The phase difference is what is actually measured by the EDM instrument.

The zero or offset error is the difference between the electrical zero point of the EDM and the physical zero point, which is placed over the ground mark. The difference between the reflector’s optical zero point and its physical zero point, which is placed over the ground mark, is the reflector offset. The combination of these two offsets is the systematic offset error, designated ➀ in the Figure. If the superimposed wavelength differs from the required fundamental unit that the instrument is designed to use, the measurement will have a scale error, designated ➁ in the Figure. Systematic variation in how the instrument measures the phase difference creates cyclic error, designated ➂ in the Figure.

It is a wise move to check that all reflectors of a particular type have the same reflector offset. The table below shows the results of measurements carried out on a series of Leica-type reflectors, which were each placed at the same distance from the EDM. Each prism was measured 10 times and the mean distance used. The different prisms are shown in Figure 2. Reduced horizontal distances were used to remove the effect of the higher prism mounting for units four and five.

Prism Number 1 2 3 4 5 6

Distance (meters) 10.3760 10.3773 10.3770 10.3786 10.3791 10.3766

The range of distances is 3.1 mm for all the prisms. Some of this may be caused by small amounts of tilt in units 4 and 5. If these units are not considered, the variation is 1.3 mm. It should be noted that the prisms do have a little forward and back movement possible in their holders. It is a sensible move to check all an organization’s reflectors in this way and see if those with significant variation can be fixed. Individual prisms with a significant offset difference from similar prisms should not be used for high-precision work. In this example, prisms four and five may be left out of the equipment sets used for high-precision work.

Figure 2 The Six Leica-type Prisms Measured for Variation Between Prisms. All prisms use the same basic prism unit, which is held the same nominal distance from the center of the mounting spindle. The prisms are numbered 1 to 6 from left to right, matching the above table.

The second error is scale error. This is usually caused by the EDM’s oscillators producing the wrong frequency, leading to errors that are directly proportional to distance. This error can be corrected by replacing the oscillators, or by monitoring and correcting the error by applying a scale factor to all distances. It is important to check if the scale error changes significantly over time, i.e., between calibrations. If this is the case, the instrument should have the oscillators replaced. If the EDM is working properly, the scale error should be significantly less than the offset and cyclic errors.

The mathematical model for the scale error is:

�

cs = ddt

Where cs is the scale error, d is the measured distance and dt is the ‘true’ distance.

The meteorological correction to the measured distance is also a scale correction, so errors in the thermometer and barometers used to determine this correction also produce a scale error. This is why it is important to calibrate thermometers and barometers. Any scale error discovered in the instrument can be corrected by applying a change to the meteorological correction, i.e., by changing the ppm setting to correct for the known scale error. For example, if the meteorological correction is +5 ppm for a given set-up, but we know that this particular instrument has a scale error of +2 ppm (i.e., it reads 2 ppm too long), the actual ppm correction set in the instrument can be changed to +3 ppm, to allow for both the meteorological correction and the scale error. Most total stations that allow temperature and pressure to be entered directly will show the ppm correction computed, allowing the operator to correct this value manually. Be aware that it is very easy to get this ‘correction’ of the ppm setting wrong, so there are advantages to setting the meteorological correction only in the instrument, and correcting for any scale factor afterwards.

The third error is cyclic error, and is caused by variation in the way that the phase difference is measured by the instrument. In most cases, this can be approximated by a sine wave with a wavelength the same as the length of the basic measuring wavelength (commonly 10m, but see the discussion, below, of different instruments’ fundamental units). The amplitude is noted (generally less than 5mm) and the offset from zero of the sine wave is noted. It is then possible to determine the correction based on the measurement itself.

The mathematical model for cyclic error is:

�

cc = a sin f0 + 2π d mod uu

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Figure 3 Cyclic Error Trace

where cc is the cyclic error, a is the amplitude of the cyclic error, ƒ0 is the location of the zero point of the sine wave that represents the cyclic error, d is the measured distance, and u is the fundamental distance unit of the instrument

The modulo (mod) operator (in d mod u) is a ‘remainder’ operator. If the distance measured, d, can be expressed as d = n u + r, where n is an integer, u is the EDM fundamental distance unit, and r is between 0 and u, then r = d mod u. It is this remainder, r, that the EDM instrument determines directly from the measured phase difference. For example, if u = 10 m and d = 1,234.567 m, then r = d mod u = 4.567 m.

Cyclic error is a direct consequence of the measurement of phase difference and so repeats any pattern of variation with a cycle length equal to the fundamental distance unit of the individual EDM instrument; hence the cyclic nature of the error and its name.

Cyclic error will vary between instruments of the same make and model. Modern EDM is expected to have very small cyclic errors, but these must be checked to ensure that the cyclic error actually is very small. Older EDM instruments often had cyclic errors up to 10 mm in amplitude. Cyclic errors this large are often larger than the random errors in the measurement, and so must be determined and removed.

Additional checks that should be carried out include calibrating the thermometer and barometer that are used to determine the meteorological corrections, and checking the tribrachs and optical plummets used for centering equipment over marks. Thermometers and barometers should be compared to standard instruments, not weather report data, as pressures are reduced to sea level and temperatures may have been estimates for a region. It may take some effort to locate suitably calibrated instruments for comparison. Local meteorological stations are usually able to help.

Calibrating EDM

Because there are three different errors to quantify, there are three components of the calibration process. These can be undertaken separately, but it is much more efficient if they are combined into a single measurement program.

To determine the combined zero and reflector offset, the following property of a measurement is used. Assuming that all other errors are zero, the value that the EDM gives for a measured distance will be:

D = d + c

where D is the distance the EDM reports that it measures, d is the ‘true’ distance and c is the combined zero and reflector offset correction. Note that this combined offset is a basic factor of EDM measurement.

If it is properly determined and taken into account, the error involved is small enough to be ignored among the random errors. If the combined offset correction is ignored, it becomes an error.

If a distance is measured in two pieces, d1 and d2, and then in one piece, d, the following situation occurs. Each of the smaller pieces is measured by the EDM as D1 and D2, respectively, while the overall distance is measured as D. However, the combined zero and reflector offset correction is the same for all measurements. So, since

D1 = d1 + c D2 = d2 + c D = d + c d = d1 + d2

and

D1 + D2 = d1 + d2 + 2c

therefore

(D1 + D2) – D = c

If a line is measured in more than two pieces, the process can be extended. Measuring several lines, or better, one long line in many pieces, will give a better estimate of the combined zero and reflector offset.

If the various distances measured are multiples of the EDM’s fundamental distance unit, the cyclic error will be the same for all measurements, and so will be included in the combined zero and reflector offset.

Scale error cannot be determined independently, in the way that the zero error can be. If all lines are scaled by the same amount, there is no way to determine the scale factor using just one instrument. If two instruments are compared, there is no way to know which one, if either, is correct. Scale error must be determined by measuring lines of known length and comparing the differences between the measured lengths and the known lengths. Scale error can also be detected by laboratory calibration of the instrument’s oscillators, but this procedure tends to be more expensive and time-consuming than a well-organized system of instrument calibration.

Measuring several different known lines, and preferably longer lines, will give a better result for the scale factor. The longer the line, the larger the error becomes, so the more it will stand out from random errors in the measurements.

Finally, cyclic error can be determined in two ways. The first is to move a reflector along a line and measure at many different distances, such that the distances are spread evenly across the fundamental distance unit of the instrument. For example, if the instrument’s fundamental distance unit is 10 m, then the measurements are made over a 10 m length. A rail with a tape or set stops may be used, as may a tape along the top of a wall. However, the distances need to be very carefully taped to ensure that the taping errors are very small; generally about 0.001 m or better is required.

The second way to measure cyclic error is to build the measurement process into the test lines being measured to determine the combined offset and scale error. If the various lines and components are set up so that they have components that cover the instrument’s fundamental distance unit with a good range of measurements (i.e., d mod u for all lines gives a good spread of values across u), the cyclic error can be deduced from the collection of all the measurements.

The ideal EDM calibration baseline will be a series of points located along a line of at least a kilometer. To make setting up easier and more repeatable, it is a good idea to have concrete pillars at each point with a standard screw thread in the top that will allow a tribrach to be screwed directly to the pillar. This eliminates errors caused by poorly adjusted plummets and setting tripods over ground marks. To minimize the labor and time involved in running an instrument over the baseline, it is advisable to have the pillars offset slightly (either horizontally or vertically) in a shallow curve, so that all the pillars can be seen from any other pillar without getting a return signal from more than one reflector. The small differences in distance caused by the offset can be included in the computations that reduce the measurements.

If a permanent baseline is established using pillars, it becomes quick to do the actual calibration process. Typically it can be done easily within a couple of hours, even more quickly for a Hobart-type baseline (see below). Dedicated software to do the reduction of the calibration measurements for a specific baseline can also be developed.

Calibrating the Baseline

Ideally, all the distances in a baseline should be known in terms of the national standard of length. This allows the scale error in tested EDM instruments to be determined. The way that this is done is to use a high-precision EDM to measure a line that is considered ‘known,’ such as a line that has been compared directly with a national standard of length. This EDM is then used to transfer this length to the working baseline, by virtue of its errors being known. After doing this, the EDM is compared to the known line again to ensure that its calibration didn’t change.

The instruments used for this type of work have been the Kern Mekometer, which had the advantage of being able to measure to 0.1 mm and be less affected by meteorological effects than most EDM, and the Geomensor. Unfortunately, the Mekometer has been out of production for some years.

Baselines should be re-calibrated annually. This allows for monitoring of the stability of the pillars. Sometimes a pillar is found to be unstable and should be removed or replaced.

Baseline Designs

Over the years since EDM became commonly used, there have been a number of baselines developed and used, but the various designs come down to three basic types. These basic types are theoretically sound and designed for easy use. Other designs should be avoided, as they may not allow determination of all the systematic errors in the EDM, but involve similar costs in establishment and maintenance.

The first design is the ‘Aarau’ baseline, named for the town in Switzerland where Kern instruments were made. This design places measurement stations at a range of distances along a line, with the characteristic that all distances to be measured are integral multiples of the fundamental measurement unit of the EDM. In the past, this was most commonly 10 meters. All components of the test range are measured. Today, using multiples of 60 meters for the component distances will allow a wide range of different EDM fundamental distance units to be used on the baseline.

When the measurements are made, the cyclic error should be the same for all measurements and is included in the combined zero and reflector offset. If the lines have known lengths, the scale factor can be determined. The actual cyclic error is determined using a short line, wall or rail, where the prism is moved along a tape.

The second design is the ‘Schwendener’ or ‘Heerbrugg’ design, named for the author who described it (Schwendener, 1972), or the city in Switzerland where Wild (now Leica) is based. In this design, the test line has the measured distances set out so that the remainder component (d mod u) for each line falls into an even spread of values along the fundamental measurement unit. For example, if the points are at distances of 0, 5, 58, 126, 254, 362, 510 and 1021 meters along the line, the remainder component of the distances from the 0 point for a 10-meter EDM are 5, 8, 6, 4, 2, 0 and 1. Combined with the other distances between points, these component distances are spread out across the 10-meter fundamental measurement unit. With a suitable choice of distances for the pillars, it is possible to provide a good coverage of fundamental measurement units for a wide range of instruments. For example, placing pillars at distances of 0, 5.0, 58.75, 126.85, 254.1, 362.55, 510.15 and 1021.45 meters provides rather better coverage across all EDM fundamental distance units, especially 1.5, 2 and 3 meter EDM instruments.

This baseline design allows the combined zero and reflector offset and cyclic error to be determined, whether the baseline has been calibrated or not, and the scale error if the baseline distances are know. The procedure is to measure all components of the test range.

The third design is the ‘Sprent/Zwart’ or ‘Hobart’ design, named after the authors who developed it (Sprent and Zwart, 1978), or the city in Australia where they were faculty members of the University of Tasmania. This design has a line of points, usually 8 to 11 (including the zero point), whose distances from the zero point have components that cover half of the cyclic error range, based on 10-meter EDM. An additional point is placed at 5 m along the line. Measurements made to the various distant points from the 5-meter point will now each have a cyclic error component that is equal and opposite to the equivalent distances measured from the zero point. By combining the matching measurements, the cyclic error can be canceled out, allowing determination of the combined zero and reflector offset. Once this is known, the variation between the original measurements and the measurements corrected for the combined offset will show the cyclic error. A plot of the cyclic error can be made to determine its parameters, or they can be computed. If the baseline distances are known, the scale factor can also be determined.

If the pillars for a Hobart baseline are located at distances of 0, 5, 10, 72, 134, 236, 378, 640, 980 and 1160 meters, the remainders for distances measured from the zero pillar are 5, 0, 2, 4, 6, 8, 0, 0 and 0 meters, while those measured from the 5-meter pillar are 5, 7, 9, 1, 3, 5, 5 and 5 meters.

Referring to Figure 4, if the measurements with remainders 1 and 6 meters are combined, their cyclic errors will cancel, being equal and opposite. Similarly, combinations of 2 and 7, 3 and 8, and 4 and 9 meters will also cancel the cyclic errors. It will be appreciated that even if the cycle does not start exactly at zero (as with the black curve), but is offset (as with the green line), the cyclic error still cancels out.

By using these combinations, the combined zero and reflector offset can be determined without any cyclic error being included. This is a significant improvement on the Aarau design, which always has some cyclic error included, even though it is constant for all measurements.

Figure 4 Cyclic Error Curves for a 10-meter EDM instrument.

The ‘Hobart’ design has the advantage of requiring fewer measurements, and those at two points just meters apart. The overall precision of the determined calibration parameters is a little poorer than those determined from the ‘Heerbrugg’ design, simply because fewer measurements have been made, but still quite sufficient to meet all requirements for EDM calibration. The disadvantage of the ‘Hobart’ design is that it is restricted to instruments with a small selection of possible fundamental measurement units. In the past, when most instruments used a 10-meter fundamental measurement unit, this was less of a problem.

If additional pillars are placed at distances half the fundamental measurement distance of various EDM instruments from the zero pillar, then this arrangement of points can support other EDM than 10-meter units, but those with fundamental distance units of 2, 7.5, 20, 30.769 and 33.333 meters are not suitable. This is because there is a poor distribution of points through the cyclic error range and hence a poor determination of the cyclic error. Altering the location of other pillars on the baseline may allow the ‘Hobart’ design to support a wider range of EDM.

The EDM Calibration Process for each Baseline Design

Aarau Design

Aarau baselines have between four and nine points. The baselines are usually straight, so that it is necessary to move the reflectors to avoid measuring to the wrong one. Suitable spacings along the line would allow points at, say, 0, 60, 180, 360 and 1080 meters, as shown in Figure 5. All these distances are integral multiples of 60 meters, This allows EDM with fundamental distance units of 1.5, 2, 3, 3.333, 5, 7.5, 10 and 20 meters to be calibrated over a single range. As only a handful of instruments do not fall into this group (those with fundamental distance units of 10.101, 30.769 and 33.333 meter), and most of them are older units, such a test range will work with almost all EDM that will be encountered.

Figure 5 Aarau Baseline Design. Using distances that are multiples of 60 meters, most EDM

fundamental distance units can be accommodated. The rail is used for the cyclic error determination.

Prisms are placed over all the points on the baseline. The instrument is set up at the zero point, and measurements of temperature and pressure are made. Instrument and reflector heights are measured. The instrument is warmed up and measurements are made to all the other points along the baseline. Each distance is measured several times.

When the measurements are complete at the zero point, the instrument is moved to the next point along the line, and measurements are made to all the points ahead of it along the line. This process is repeated until the last line segment is measured. Temperature and pressure are monitored throughout the measurement process.

When the baseline has been measured, the instrument is moved to the cyclic error test line. This is a shorter distance with the facility to move the reflector along a rail, the level top of a wall, or a line of points. The distances are measured with the EDM as the reflector is moved along the line, while the location of the reflector is carefully determined using a tape. A minimum of 12 points, spread evenly along the fundamental distance unit of the EDM, are required.

Heerbrugg Design

In the particular instance of a ‘Heerbrugg’ design baseline discussed here, there are eight concrete pillars placed along level ground in a gentle curve. The site of this particular line is within the grounds of the airfield of a rural city. It is assumed that the prism units have been checked to ensure that they are all the same, as discussed above. If there is any variation between prisms, a single prism unit may have to be moved along the pillars as the calibration measurements proceed.

The pillars for this particular baseline are placed at distances of 0, 5, 58.5, 126.5, 254, 362.4, 510 and 1021.5 meters (approximately), as shown in Figure 6. The line is along the boundary fence of the airfield and largely shaded by adjacent trees, although this is not required for baselines in general. Because this particular baseline is a public resource, a surveyor wishing to calibrate equipment reserves the baseline for a specific time by phoning the local authority responsible for management of the baseline.

Figure 6 Heerbrugg Baseline Design. This arrangement of pillar distances provides a good

coverage of remainder distances for EDM using fundamental measurement distances of 5, 7.5, 10, 10.101, 20 and 30.769 meters, and average coverage for EDM using 3.333 and 33.333 meters.

On arrival at the baseline, the crew drives along the line from the 1021 m pillar, placing a tribrach and reflector on each pillar. At the 0 m pillar, the crew places the instrument on the pillar (screwing its tribach onto the pillar and allowing it to warm up) and after recording air temperature and pressure readings, measures the distances to each of the reflectors. Instrument and reflector heights above the pillar tops are recorded. Each distance is measured several times (a minimum of four) to get a good mean value. The curved line allows all the reflectors to be measured without the closer reflectors obstructing the rest of the line.

When all seven measurements are complete, the instrument is moved to the 5 m pillar, and all the distances ahead of the instrument (now six) are measured. When this is completed the instrument is moved to the third pillar and five measurements are made there. This process continues until the last set-up on the 510 m pillar, when the distance to the 1021 m pillar is all that is needed.

The temperature and pressure are monitored throughout the measurement process and changes recorded. All the measurements are recorded on special data sheets designed for this particular baseline, which helps ensure that nothing is forgotten.

As the reflectors and tribrachs have been gathered as the crew proceeded along the baseline, and the pillars’ protective caps have been replaced, the reflector on the 1021 m pillar is the only one left, and it is picked up as the crew leaves the site.

The data sheets can be sent to the responsible authority for reduction, or can be reduced on a software package that can be downloaded for the purpose. The software is updated annually after the baseline is recalibrated, so that the latest calibrated measurements are used.

Once the calibration values for the instrument are computed, they are then available to be applied to all measurements made by that instrument.

While the measurements for a baseline can be reduced manually, it is a lot easier to do this with a computer program. It is relatively simple to create a dedicated least squares adjustment program to do this.

Hobart Design

The Hobart baseline design requires the instrument to be set up at just two points: the zero pillar and a pillar at half the fundamental distance unit of the EDM from the zero pillar. The Hobart design baseline shown in Figure 7 was designed for 10-meter EDM instruments.

Figure 7 Hobart Design Baseline. The 640, 980 and 1160 meter pillars may be placed at other

distances, as they are mainly used for checking longer distances for scale error. Sometimes an additional pillar is placed, e.g., 800 meters, or some of these pillars may be omitted.

Hobart baselines commonly have a gentle curve along their length to allow more efficient measurement. Most have horizontal curves, but a few are constructed on gentle slopes that get slightly steeper as they get further up the hill.

Reflectors are placed on each of the pillars and the instrument is placed on the zero pillar. Meteorological data is collected and the instrument warmed up. Distances are measured to all the points on the line. The instrument is then moved to the 5-meter pillar and measurements are made to all the pillars. Meteorological conditions are monitored throughout the measurement process.

Once the measurements are completed, the reflectors are collected and the pillar protective tops replaced. Given the large number of pillars in some baselines, it may not be possible to have sufficient reflectors for all pillars at once, so reflectors may have to be moved. The use of pillars means that placing the reflector at the same location every time is easy to achieve.

EDM Fundamental Distance Units

EDM have been developed with a wide range of fundamental distance units. The reason for this is that as the fundamental distance unit is made smaller, the more finely the measurement can be made with oscillators of equivalent quality and price.

The various fundamental distance units that have been used over the years include the following:

1.5, 2, 3, 3.333, 5, 7.5, 10, 10.101, 20, 30.769, 33.333 meters.

It should be noted that different instruments from the same manufacturer can have different fundamental distance units, and that changes may occur within variants of the same model of instrument. There appear to be no instruments that were set up to use the foot as a fundamental measurement unit, with the exception of the microwave Tellurometers.

Instruments that use a pulsed laser measurement system, such as many reflectorless systems, do not have a fundamental measurement unit, since there is no superimposed signal. As a consequence, these instruments do not appear to exhibit cyclic error.

Determining the Offset and Reflector Constants: A Quick and Easy Method

It is apparent that without lines of known length, a full calibration cannot be performed. But a partial calibration can be done to determine the combined zero and reflector offset, for any combination of EDM and reflectors. This is a useful step to take with all EDM and reflector combinations, and should be done regularly to check for stability of the various offset values. The procedure described below is simple enough to be done by any survey group and takes only a few minutes to perform. It can be done before a series of measurements at a specific site to ensure that the equipment is in good order and the reflector constants are correct, or it can be done regularly on a line near the survey organization’s offices.

Using this procedure, scale error cannot be determined, as this is a constant scale factor for all lines. Measurements should be made so that the same portion of the fundamental distance unit is used for all measurements, as in the ‘Aarau’ baseline design, thereby ensuring that the cyclic error is the same for all measurements. Because of the many different fundamental distance units, the best compromise for a single test range is one where all the line components are multiples of 60 m. This allows EDM with fundamental distance units of 1.5, 2, 3, 3.333, 5, 7.5, 10 and 20 meters to be checked over a single range. As only a handful of instruments do not fall into this group, and most of them are older units, such a test range will work with almost all EDM that will be encountered. Even EDM instruments with 10.101- and 30.769-meter fundamental distance units could be checked using this approach with relatively little impact from variation in the cyclic error.

The simplest way to check the offset and reflector constants is to set up three tripods in a line, so that the separation between them is in multiples of 60 m. It is simpler if the line is on level ground and reasonably stable ground marks are placed. Such a line may be used to check all combinations of EDM and reflectors that are to be used, whether on a specific project or on all jobs. As each set of measurements is independent of any other, being used for calibration, it is of no significance if the points move by small amounts. If the tripods are not moved and the instrument and reflectors are placed in the same locations during the measurement process, the effect of any error in the optical plummets is reduced.

In the example given here, the overall line length was 180 m, with an intermediate point 120 m from the north end of the line. The instrument is set up over the mark at one end of the line, while reflectors are set on the tripod at the 180 m point. The temperature and pressure are measured and the meteorological corrections set in the instrument. The distance to the 180 m point is measured several times, in this case eight times to allow a good estimate of the mean distance. However, any reasonable number of observations may be made.

Figure 8 Test Line for Quick and Easy Determination of Combined Zero and Reflector Offset. Distances are in meters. There is no need for the first distance to be 120 m; 60 m would have been satisfactory.

The instrument is then moved to the intermediate point and set up there. A reflector is placed at the 0 m point, using the same tribrach in the same position as for the EDM instrument. The two parts of the line are then measured eight times each. The data collected are as given in the worked example below. It was originally developed as a simple Excel spreadsheet to facilitate reduction of the calibration process.

When this ‘quick and easy’ calibration was done, several different reflectors were tested at the same time. In addition, the instrument was tested in reflectorless mode to a flat target, so that the zero offset in reflectorless mode could also be determined. These results are not shown here.

Other EDM Errors

There are a handful of other errors in EDM. These include short-range error (which appears in measuring distances less than a few meters in length), time-of-operation errors (EDM can produce erroneous results before it has had a chance to warm-up all the components) and range-dependent errors. These errors are usually very small and cannot be readily determined without a more extensive test line. Fortunately, they rarely cause any problems, provided that the EDM has a chance to warm up (which may take just a few measurements these days) and that very short-range measurements are not taken too seriously.

Bibliography

BUCKNER, R. B. 1983. Surveying Measurements and Their Analysis. Landmark Enterprises, Rancho Cordova, CA.

CIDDOR, P. 1996. Refractive Index of air: new equations for the visible and near infrared, Applied Optics (Lasers, Photonics and Environmental Optics), Volume 35, Issue 9, March, 1996, pp 1566–1573.

EDM and Reflector Calibration 15th May, 2005.

Instrument: Leica TCR307 total station, Serial Number 646536.

Reflectors: Leica GPH1A prism set Fundamental Distance Unit: 1.5 m

Temperature: 12°C Pressure: 1010 mb

Full Line Mid to South Mid to North

179.938 59.953 119.988 179.938 59.953 119.989 179.938 59.953 119.989 179.938 59.953 119.989 179.938 59.953 119.989 179.939 59.953 119.989 179.938 59.953 119.989 179.939 59.953 119.989

Mean 179.938 25 59.953 00 119.988 875

179.941 875 Sum of parts of line

+ 0.003 625 Combined zero error and reflector constant

Therefore all measurements made using this Leica TCR307 total station to this set of reflectors must have 0.004 m subtracted from the measured slope distance.

GOFF, J.A., 1957. Saturation pressure of water on the new Kelvin temperature scale. Transactions of the American Society of Heating and Ventilating Engineers, presented at the semi-annual meeting of the American Society of Heating and Ventilating Engineers, Murray Bay, Que. Canada. Pp. 347-354,

GOFF, J.A., and GRATCH, S., 1946. Low-pressure properties of water from –160° to 212° F. Transactions of the American Society of Heating and Ventilating Engineers, presented at the 52nd annual meeting of the American Society of Heating and Ventilating Engineers, New York.

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LAND VICTORIA, 2002. EDM Calibration Handbook. Edition 7. Land Victoria, Department of Natural Resources and Environment, State of Victoria, Australia. 37pp. Download the PDF at: http://www.land.vic.gov.au/land/lcnlc2.nsf/646e9b4bba1afb2bca256c420053b5ce/d1b4a135085ffcdfca256e6a0018615e/$FILE/EDM%20Handbook%20V7.pdf

MURRAY, F.W., 1967. On the computation of saturated vapor pressures. Journal of Applied Meteorology, 6, pp. 203-204.

RÜEGER, J.M., 1996. Electronic Distance Measurement: An Introduction. 4th edition. Berlin, New York : Springer–Verlag.

RÜEGER, J.M., 1999. Report of the Ad-Hoc Working Party on Refractive Indices of Light, Infrared and Radio Waves in the Atmosphere of the IAG Special Commission SC3—Fundamental Constants (SCFC). http:// www.gfy.ku.dk/~iag/Travaux_99/wp51.htm 15th June, 1999.

SCHWENDENER, H.R., 1972. Electronic Distancers for Short Ranges: Accuracy and Checking Procedures. Survey Review. Vol. 21, No. 164, pp. 273–281.

SPRENT, A., and ZWART, P.R., 1978. E.D.M. Calibration—A Scenario. The Australian Surveyor. Vol. 29, No. 3, September, 1978, pp. 157–169.