Adjustment with observation equationsAlso interesting
All kinds of vertical networks are adjusted, both spirit levelling networks (height differences, lengths of lines) as well as trigonometric vertical networks (zenith angles, distances), both free as well as connected networks, even single levelling lines.
For a connected vertical network specify the list of points with known heights. The input of the two to four columns complies with the general rules of tabular input . The pointnames may be chosen as in the Coordinate lists .
If heights should be subject to adjustment, please specify an accuracy measure, either a standard deviation or a weight. If a default value is given for the accuracy measure, it pads all missing values in this column. Otherwise all heights without accuracy measure are treated as error-free in the adjustment. This is equivalent to zero standard deviation.
Points not present in the network are ignored. Points without height and network points missing in the list are treated as new points. A free vertical network consists exclusively of new points, e.g., if the list of known points is empty.
If you specify coordinates X and Y then the vertical network is displayed on the canvas. Points without X and Y are not displayed. These coordinates are not used for network adjustment.
Measurement values are given as Measurement lists . See also the rules of tabular input. Each measurement record starts with two pointnames, i.e. startpointname and endpointname of the network line. They are followed by up to five measurement values or accuracy measures in user-defined order.
Some or all network lines may be measured twice, i.e.
In this case you simply specify two columns for height difference dh or the zenith angle v . The other quantities in the measurement record must not appear twice here. For the third case the following is assumed: The values of the instrument height ih and the target height th are only specified for AE and in the backsight EA they must be exactly interchanged. If this condition is not met, please specify means or use the method described below.
It is not necessary to change the sign of the height difference in backward direction EA. For the height difference in forward direction AE the sign of the first value is assumed. Therefore, two forward directions could have been measured. Your case is recognised by the signs of the first pair of values. (Theoretically it may happen that the height differences of the first pair are so small, that the recognition fails. In the extreme case they may be zero. Then a different record of the measurement list should be shifted to the front.)
The network lines measured twice may also be specified in two separate measurement records. This practice allows greater flexibility. In the same way you may specify network lines measured more than twice: each measurement in a separate record.
Examples: The following inputs yield identical results.
geometric network | trigonometric network (Angle unit: Grad) |
---|---|
// from to dh1 dh2 l pdh A E 16.10 16.11 2306 1 |
// from to v1 v2 e ih th pdh A E 92.1402 92.1418 88.2306 1.54 1.66 1 |
// from to dh1 dh2 l pdh A E 16.10 -16.11 2306 1 |
// from to v1 v2 e ih th pdh A E 92.1402 107.8582 88.2306 1.54 1.66 1 |
// from to dh l pdh A E 16.10 2306 1 E A -16.11 2306 1 |
// from to v e ih th pdh A E 92.1402 88.2306 1.54 1.66 1 E A 107.8582 88.2306 1.66 1.54 1 |
Multiple measured network lines enter the adjustment as separate observations , which is generally recommended. If you want to let them enter as mean values, please get them yourself. In the left example above the record could read:
geometric network | trigonometric network |
---|---|
// from to dh l pdh A E 16.10/2+16.11/2 2306 1 |
// from to v e ih th pdh A E 92.1402/2+92.1418/2 88.2306 1.54 1.66 2 |
(Note that in arithmetic expressions brackets are not yet allowed.)
Take care not to specify the multiple measured network lines as a constraint, otherwise there wil be a contradiction of the probably different values.
In the adjustment we need accuracy measures for the observations.
These are the given height differences dh
or the height differences computed from zenith angles v
and distances e or s
as well as the heights of known, adjustable points.
⚠ Also for trigonometric vertical networks these
values refer to the adjustable height difference dh,
i.e. they include measurement errors in the instrument height and target height.
Correlated observations are not supported at the moment.
There are multiple opportunities, how to define the stochastic model via the selection of column format:
as given above or from standard deviations or all unity
For the definition of weights for height differences you can indirectly use
Then, for each network line please specify such a value. For this purpose you select the respective column format. You can specify how the weights are derived from distances or lengths of lines. Each height difference without length of line is treated as invariable, just as having infinite weight or standard deviation zero. Lengths of lines must not be specified in the same unit as the heights and height differences. Distances must have the same unit as heights and height differences. All values must be positive.
If you selected such an option and in addition you specify explicite standard deviations or weights then you obtain an error message.
For heights of known, movable points you specify a standard deviation in the same unit as the heights or a weight, as described above. Heights without accuracy measure or with standard deviation zero are treated as error-free and fixed in the adjustment. ⚠ The weights of heights and height differences must agree. A variance component estimation is not yet implemented.
If zero standard deviations, i.e. infinite weight is assigned to a height difference then they act as constraints. It may happen that these constraints contradict. This case arises if these height differences
These cases cause an error. Use height differences as constraints with care.
The computation is performed such that either the startpoint or the endpoint of each such line is eliminated from the list of adjustable points and is added after the adjustment using the constraint.
The adjustment is performed according to the method of least squares. Free vertical networks are adjusted with the datum constraint that the height of all adjusted heights is equal to zero.
The documented results contain the residuals, the adjusted values, the a posteriori standard deviations and the redundancy parts.
In the third semester of the curriculum Surveying/Geoinformatics of the parts of the campus vertical network are measured by spirit levelling. As a result the height differences dh in metres and lengths of the lines l in kilometres given below have been obtained. Each levelling line has been measured multiple times, here between two and six times. These observations should be adjusted as a free vertical network.
As a result for a single line of length 1 km (unit weight) we obtain a posteriori
standard deviation of 0.47 mm
. All partial redundancies are above 0.7
, such that the network enjoys sufficient reliability.
All residuals amount to less than 0.5 mm
. For the point heights this yields a posteriori standard deviations up to a maximum of
0.15 mm.
from | to | Δh [m] | l [km] | from | to | Δh [m] | l [km] | from | to | Δh [m] | l [km] |
---|---|---|---|---|---|---|---|---|---|---|---|
2580 | 2644 | -0.05638 | 0.37 | 1011 | 2575 | 9.70155 | 0.29 | 822 | 2644 | -2.91135 | 0.28 |
2580 | 2644 | -0.05592 | 0.28 | 1000 | 1011 | 8.80803 | 0.29 | 822 | 2644 | -2.91116 | 0.30 |
2580 | 2644 | -0.05659 | 0.37 | 1000 | 1011 | 8.80839 | 0.30 | 184 | 822 | 3.11445 | 0.25 |
2580 | 2644 | -0.05609 | 0.27 | 1000 | 1011 | 8.80838 | 0.33 | 184 | 822 | 3.11462 | 0.27 |
1490 | 2575 | 9.79288 | 0.25 | 1000 | 1011 | 8.80852 | 0.29 | 184 | 822 | 3.11540 | 0.25 |
1490 | 2575 | 9.79311 | 0.26 | 1000 | 1490 | 8.71692 | 0.28 | 184 | 822 | 3.11474 | 0.26 |
1012 | 2580 | 0.60757 | 0.32 | 1000 | 1490 | 8.71712 | 0.29 | 125 | 184 | 0.61625 | 0.11 |
1012 | 2580 | 0.60767 | 0.31 | 1000 | 1490 | 8.71700 | 0.28 | 125 | 184 | 0.61652 | 0.10 |
1012 | 2580 | 0.60785 | 0.32 | 1000 | 1490 | 8.71724 | 0.33 | 125 | 184 | 0.61656 | 0.10 |
1012 | 2580 | 0.60800 | 0.31 | 1000 | 2580 | -0.60227 | 0.32 | 125 | 184 | 0.61632 | 0.11 |
1012 | 2644 | 0.55162 | 0.16 | 1000 | 2580 | -0.60220 | 0.38 | 125 | 184 | 0.61656 | 0.10 |
1012 | 2644 | 0.55167 | 0.16 | 1000 | 2580 | -0.60257 | 0.29 | 125 | 184 | 0.61669 | 0.10 |
1012 | 2644 | 0.55163 | 0.16 | 1000 | 2580 | -0.60284 | 0.36 | 125 | 822 | 3.73164 | 0.46 |
1012 | 2644 | 0.55177 | 0.16 | 1000 | 2644 | -0.65811 | 0.47 | 125 | 822 | 3.73182 | 0.46 |
1011 | 1490 | -0.09163 | 0.10 | 1000 | 2644 | -0.65810 | 0.51 | 125 | 2644 | 0.82011 | 0.21 |
1011 | 1490 | -0.09118 | 0.12 | 1000 | 2644 | -0.65860 | 0.49 | 125 | 2644 | 0.82042 | 0.22 |
1011 | 1490 | -0.09169 | 0.10 | 822 | 2644 | -2.91118 | 0.28 | 125 | 2644 | 0.82039 | 0.28 |
1011 | 1490 | -0.09140 | 0.12 | 822 | 2644 | -2.91111 | 0.30 | 125 | 2644 | 0.82062 | 0.22 |
1011 | 2575 | 9.70153 | 0.31 |
from | to | height difference |
slope distance | target height |
---|---|---|---|---|
1 | A | 105.545 | 55.454 | 1.400 |
1 | 2 | 95.112 | 71.689 | 1.400 |
3 | 2 | 103.230 | 49.528 | 1.400 |
3 | 4 | 92.751 | 65.666 | 0.005 |
4 | 3 | 107.239 | 65.666 | 0.005 |
4 | 5 | 96.345 | 78.300 | 0.005 |
5 | 4 | 103.645 | 78.300 | 0.005 |
5 | E | 99.300 | 45.650 | 1.400 |
Let us consider the displayed levelling line with a leap station at point 2, i.e. no instrument has been erected at this point and consequently at point 1 no reflector was needed. At the points 3,4,5 there was in turn a station with instrument and a reflector position. Heights are given for the levelling bechmarks A by H=116.10 and E by H=123.06 . The reflector at A, 2 and E always has a target height of th=1.40 . The points 1,3,4,5 are not marked, such that we refer the heights to the tilt axis of the instrument: ih=0.00 . However, the reflection point is always 0.005 above the tilt axis, such that on the points 3,4,5 there is a target height of th=0.005 to consider. The obtained measurement values are shown at the right.
Residuals are computed for the height differences with magnitudes up to 0.0017 . The resulting height differences of the new points 1,2,3,4,5, obtain a posteriori standard deviations up to 0.0012.
The same example is also used for the Universal computer . However, the results are not fully identical because the Universal computer uses a robust adjustment algorithm. The deviations in the final heights amount up to 0.0004.
point | Vertical networks | Universal computer | ||
---|---|---|---|---|
name | adj. height | std.dev. | height (median) | range |
1 | 122.3238 | 9.1e-4 | 122.3238 | 0.0046 |
2 | 126.4233 | 0.0012 | 126.4230 | 0.0046 |
3 | 130.3350 | 0.0012 | 130.3347 | 0.0046 |
4 | 137.7910 | 0.0011 | 137.7906 | 0.0033 |
5 | 142.2778 | 7.7e-4 | 142.2779 | 0.0046 |
Dieses Beispiel wurde zuvor mit dem Universal computer bearbeitet. Wir gleichen die Messwerte jetzt als trigonometisches Höhennetz aus. Dazu benutzen wir die im Universalrechner erhalten Schrägstrecken. Die bekannten Punkthöhen von 515 und 632 werden unveränderlich gehalten. Außerdem benutzen wir die Lagekoordinaten, jedoch nur für die Darstellung des Netzes auf der Leinwand.
Bei der Berechnung erhält man die Warnung, dass Beobachtungen zwischen unveränderlichen Festpunkten gefunden und ignoriert wurden. Das sind die beiden Beoachtungen zwischen den Festpunkten 515 und 632. Alle Beobachtungen sind gut kontrollierbar ([Redundanzanteil]e≥0.45). Die Höhe des unzugänglichen Punktes 1121 erhalten wir mit 201.1118 und einer Standardabweichung von 0.0014.
Zum Vergleich: Mit dem Universal computer erhielten wir 201.1106 mit einer Spannweite von 0.0084. Dieser Unterschied erklärt sich aus den unterschiedlichen mathematischen Modellen, die diesen Rechenwerkzeugen zugrunde liegen.
Adjustment with observation equations
Vertical networks and Sets of angles and distances can be re-adjusted with Adjustment with observation equations . This yields the following advantages:
Forthcoming: More tools will provide this option.
The names of the observations are A°E and the names of the parameters are the names of the adjusted points. If you use ⇑ height differences as constraints then they do not show up as observations, but as constraints for parameters.
First steps Settings |
Vertical networks Adjustment with observation equations |
World Geodetic System 1984 Library search |