START First steps Deutsch Guide : Sets of angles and distances

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Measurement values Set means and accuracy measures Instrumental corrections Further computations with set means Loading adjustment models in Adjustment with observation equations Pure processing of horizontal angles Processing of zenith angles with slope distances Joint processing of all measurements
On a station there may be measurements in sets to the same targets. Measurement values may be given in arbitrary succession and may be arbitrarily missing. Set means, instrument errors and accuracy estimates are computed. The results may be processed with further IN DUBIO PRO GEO computation tools.

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When measuring with a tacheometer or a theodolite, target points are usually not sighted and measured only once, but repeatedly and in both telescopic faces. Such measurements are called sets of angles and distances. IN DUBIO PRO GEO uses geodetic adjustment to compute the best solution for angles and distances as well as for instrument errors of the tacheometer or theodolite. The scale circle must not be moved during the sets. All measurements must be taken with one and the same instrument immediately one after the other, such that we can rely on unchanged instrument errors. If steep sightings are due, than the instrument must not contain an uncorrected horizontal axis error.

START First steps Deutsch Measurement values

Measurements are given as a . For the processing of the measurements the succession of the rows of the measurement list is arbitrary. However, for the practical measurement it is recommended to adhere to the classical rule: In every set all targets should be measured first clockwise in face I and then in reverse succession in face II. Measurement values may be almost arbitrarily missing . This may happen either in case of eliminated gross errors or in case of forgotten sightings.

For each horizontal angle r there must be given a zenith angle v such that it is possible to compute, how a collimation error acts on this measurement. The effect of the collimation error only slightly depends on the zenith angle, such that an approximate value is sufficient, e.g. 100 gon = 90° oder 300 gon = 270° for only slighlty inclined sightings. Additionally, at least one target point must have been sighted in both faces, otherwise no instrument errors are computable. Normally, for all target points there should be sightings in both faces.

Distances may be slope distances s or horizontal distances e .

All target heights th of one and the same target point must coincide, if given multiple times. If target heights are not specified for all sightings, it is assumed that the missing target heights coincide with the specified values. If for some target point no target height is given then the target height of this point remains undefined.

START First steps Deutsch Set means and accuracy measures

The set means of the horizontal angles r and of the zenith angles v are computed by two separate adjustments by observation equations . All horizontal angles are equally weighted, and all zenith angles as well. As a result of both adjustments the a posteriori standard deviations of all measured values σr, σv and of all set means σr, σv are obtained. If a set is complete, i.e. no angle values are missing, then the standard deviations of all set means coincide.

For distances no standard deviations are computed because electronic distance measurements mostly show only systematic measurement errors, such that these accuracy measures are much too optimistic. Instead of this, set means s or e and ranges Δs or Δe are computed.

START First steps Deutsch Instrumental corrections

The instrument typically shows so-called instrument errors. Corrections for two kinds of such instrument errors can be computed from the measurement values: collimation error c and vertical index error i . The more targets are sighted in both faces, the higher the accuracy of the computed instrumental corrections. The set means are already corrected for those errors. If you want to correct further uncorrected measured values r',v' then you have to compute

in telescopic face Ir = r´ + c / sin(v) v = v´ + i
in telescopic face IIr = r´ - c / sin(v) v = v´ - i

Note that the obtained values represent corrections in telescopic face I. (''Errors'' would actually get the opposite sign.) If the measurement values are already corrected for those errors, then one would only get the updates for those corrections.

Horizontal axis errors are not computed because for this purpose exclusively steep sightings are required, which are rarely found in normal sets of angles.

Sets of angles and distances may also be processed by the . However, no instrument errors are computed and no accuracies are estimated. The accuracy of the results is therefore usually worse.

START First steps Deutsch Further computations with set means

Set means can be transfered to the or in for further computations, e.g. for the determination of coordinates of station and/or some target points. If there is not yet any measurement list defined in the target computation tool then such a list can be created from the set means. Otherwise it can be overwritten by the current set means, or the set means are appended as a new measurement list. In the latter case only those data are appended which fit into the current format specification of station and target row of the target computation tool.

Note that some data may not be transfered. If the succession of the data disagrees then it is adapted. Ideally you choose the formats of the target rows consistently.

Set means can also be transfered to for further computations. But here an existing measurement list is always overwritten because this target computation tool only supports one station setup at a time.

START First steps Deutsch Loading adjustment models in Adjustment with observation equations

and can be re-adjusted with . This yields the following advantages:

Forthcoming: More tools will provide this option.

The parameters of the adjustment are the set means in the succession of the set means table with appended instrument error, either the collimation error c or the vertical index error i. The observations are the measured horizontal or zenith angles in the succession of the measurement input area.

START First steps Deutsch Pure processing of horizontal angles

If only horizontal angles need to be processed and all sightings are all gently inclined then

should be specified. The following horizontal angles (all gently inclined) on the station S0, from which one measured value is evidently grossly erroneous due to a target point mix-up, need to be processed:

targetset 1 [gon]set 2 [gon]
pointtelescopic face Itelescopic face II telescopic face Itelescopic face II
T116.1063216.110416.1083216.1139
T217.1165223.071223.0697223.0787
T391.0214291.027791.0312291.0303
and Compute

As a result one obtains the three set means: 16.10973 gon; 23.07251 gon; 91.02765 gon . The posterior standard deviation for the mean of two faces is obtained as 2.6 mgon . The posterior standard deviations of the set means amount to 1.8 mgon for T1 and T3 and to 2.1 mgon for T2. This value is worse due to the missing measurement. However, the three not grossly erroneous measurement values have been used. The standard deviations of the zenith angles are all zero, because these measurements are faked, and must be ignored.

Load this adjustment model of the horizontal angles into and test if the specification of the manufacturer of a priori standard deviation for the mean or two faces of 2 mgon (corresponds to a priori standard deviation for a uncorrected single measurement of 2.8 mgon) with a probability of type I decision error of 0.01 = 1% may be considered as met. At the same time, check that no further gross errors are detected.

START First steps Deutsch Processing of zenith angles with slope distances

If only zenith angles need to be processed then all other measurement values can be omitted.

In the following example on the station S0 zenith angles in two faces and slope distances are measured:

targetzenith angle [gon]slope distancetarget
pointtelescopic face Itelescopic face II telescopic face Itelescopic face IIheight
T190.1866309.815717.58917.5901.40
T298.5077301.497923.69723.6971.40
T394.9949305.006614.29114.2941.40
and Compute

As a result one obtains for the three set means: 90.18545 gon; 98.50490 gon; 94.99415 gon. The a posteriori standard deviations of the set means amount to 1.1 mgon . The a posteriori standard deviations of the set means has the same value because only one set has been measured and it is complete. The vertical index error amounts to -1.6 mgon and its a posteriori standard deviation is obtained as 0.6 mgon . Horizontal angles and zenith angles are not computed.

Transfer the set means by the button into and add some arbitrary coordinates of the station S0 there and an instrument height. Start the computation and note that heights are computed also for the target points. If the station height would have been set to 100.0000 and instrument and target heights set to zero, then the heights of the target points are obtained as 101.30 99.15 99.72. If coordinates of a target point would have been specified, then the heights of all other points would have been obtained in the same manner.

START First steps Deutsch Joint processing of all measurements

The measurement of both previous examples may be introduced in a joints processing. However, the second set of the horizontal angles must be dropped because no faked zenith angles can be used here. At best you could repeat the zenith angles from set 2, this would not change the set means, but the standard deviations would be computed too small. Since in one target point row a horizontal angle is missing now, you could shift the column of horizontal angles to the end of the measurement list . This can be avoided by letting the missing value empty. For this purpose you write '';;'' in the measurement list, i.e. two separating characters, which are not joined to one.

and Compute

Repeat the exercises from the previous examples with the entire data set.
Although only one set of horizontal angles is given and it is on top of that incomplete, an adjustment is possible. However, the total redundancy is only 1, which makes adjustment results unreliable. Note that the horizontal angle to point T2 is now fully uncontrollable.
In the universal computer the target coordinates are now fully computed in the local station system. For example, using the station coordinates S0 100.000 100.000 100.000 and with instrument height and orientation angle in the station row being both 0.000 one obtains

PNameXYZ
T1 116.827452 104.351096 101.30098431
T2 122.152065 108.397866 99.15647232
T3 102.002168 114.106965 99.72268593

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