Figure 1. Circular vial with bubble 2 milli-meters out of center.

Q: How can I quantify the errors inherent in optical plummet tribrachs?

A: You don’t actually say what you are doing with your optical plummet tribrach. However, there are two kinds of principal errors to be aware of. The first relates to how well a tribrach can be leveled, and a corollary has to do with lack of adjustment of the circular vial of the tribrach. Another error arises, even when the leveling is perfect, due to the optical plummet’s line of sight not being along the vertical (the direction of gravity). To properly discuss these errors, we will subdivide this answer into sections.

Figure 2. Tubular vial with bubble 2 millimeters out of center.

Tribrach Circular Vial

Check the circular vial for adjustment. Typical circular vial sensitivity of optical plummet tribrachs is on the order of 10 arc minutes per 2 millimeters of bubble movement. This is only a general statement; you should determine the sensitivity yourself or find out what it is from the manufacturer. To understand the errors from mis-leveling, including lack of adjustment of the level vial, place an instrument with a more sensitive vial (for example, most theodolites and total stations have tubular vials with sensitivity in the range of 20 to 40 arc seconds per 2 millimeters of bubble movement), and level the instrument carefully (it must be on a stable platform such as a tripod). Now view the position of the optical plummet’s bubble. If it is perfectly centered, it is in adjustment and you can continue the experiment. If it is not centered, the distance in millimeters (a in Figure 1) between the centers of the bubble and the centering ring, divided by two and multiplied by the circular bubble’s sensitivity, is the leveling bias. Use the appropriate adjusting tools to bring the bubble to the center. Be careful to ensure that the setup of the instrument is not disturbed when doing this. It is always wise to repeat the initial test. If you do not make this adjustment, all your leveling will be biased.

Determine typical leveling error. Now remove the instrument and throw the tribrach out of level by turning the footscrews, and then re-level it. Carefully replace the instrument and determine how level it really is. You can calculate the state of mis-leveling by measuring the distance between the centers of the instrument’s tubular level vial and the bubble in millimeters (b in Figure 2), dividing it by two and then multiplying by the level vial sensitivity of the instrument. Repeat this several times, perhaps about 10. Be sure to use your “normal” care in leveling the tribrach each time. The average mis-leveling (c in Figure 3) will give you an idea of how well you typically level your tribrach. You can also calculate the standard deviation to further improve your analysis, although the application of that statistic will not be discussed here.

Apply circular vial leveling error. The error in your surveying due to error in leveling your tribrach depends on what you are using it for. The most obvious error is in your line of sight of the optical plummet (making the assumption that it is in perfect adjustment) due to the leveling platform not being truly horizontal. The line of sight of the optical plummet is adjusted so that it is perpendicular to the horizontal plane of the tribrach. For example, if the average mis-leveling you determined above (c), is 2.5 arc minutes, and the average set up distance of your tribrach over the ground point is 4.5 feet, the average mis-centering will be 0.003 feet or about 1 millimeter. This has implications for setting up GPS antennae, theodolites, EDMs, total stations, and targets for angle and distance measurement. In the case of GPS, the average error in the position of the antenna due to this error will be 1 millimeter, but the direction in which it occurs will tend to be random. For angle measurement, the direction in which the error is manifested, and the direction of the backsight and foresight in relation to the direction of the leveling error, all impact the error in the angle. The distance to the target must also be considered. For example, the worst-case error, due to a 1 millimeter error in centering when the sight distance is 100 meters, will be 2" (arc seconds). The angular error contributed by target mis-centering is also random in direction, so the maximum could be the sum of the maximum error at each end or 4". For EDMs, the maximum distance error will be 1 millimeter at each end, but the direction of the error will be random and the errors at the two ends could combine for a maximum of 2 millimeters or a minimum of zero. If your instrument does not have so-called “dual-axis” tilt compensation, you could have an additional error due to the vertical axis not being truly vertical. This error is a function of the vertical angle of the telescope as the target is sighted and the direction in which the error occurs. If, after the instrument is placed in the tribrach, no re-leveling with the instrument’s tubular vial is done, the error for the backsight and foresight are calculated using: 
                                                                 EHA = a tan v
Where EHA is the error introduced in the horizontal direction (circle reading), a is the transverse leveling error of the theodolite (i.e., measured perpendicular to the direction of the telescope) and nis the vertical angle of the sighting (i.e., measured with respect to the horizontal).

For example, if the sighting to a target has a vertical angle of +10 degrees and the transverse leveling error is 2.5 arc minutes, the error in the circle reading will be +26".

Figure 3. Deviation in optical plummet line of sight due to imperfect leveling.

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