The ‘a priori’ mean error of levelling. Computation of heighting lines and joints Budapesti Műszaki és Gazdaságtudományi Egyetem Általános- és Felsőgeodézia Tanszék

The mean oscillation of the line-of-sight:  The mean error of a single staff reading:

The determination of the mean oscillation of the line-of-sight (automatic level)

precise level (staff reading is taken with optical micrometers) The determination of the line-of-sight

The mean error of a single elevation difference The mean error of the backsight and foresight readings are the same: m The elevation difference in a single station:  m = back – fore The mean error of a single elevation difference: Why?

The mean error of all the single elevation differences are constant for a levelling line: The mean error of the total elevation difference (according to the law of error propagation) Substitutin the mean error of the staff reading:. The elevation difference of the endpoints:,

When the instrument-staff distance is: d Then the total length of the levelling line: n should be substituted to the eq. of the mean error:

Thus the total mean error of the total elevation difference is: Substituting the formula of the mean error of a single staff reading: the following formula is obtained:

The one-way ‘a priori’ kilometric mean error of levelling (L = 1 km) The two-way ‘a priori’ kilometric mean error of levelling:

The preliminary elevation difference of the endpoints for the i-th section: The observed elevation difference between the endpoints (K-V): The given elevation difference of the endpoints: (  m) =  (BS – FS) =  BS –  FS Computation of heighting lines (levelling)

The closure error: The mean error is proportional to the squareroot of the length. Thus the weights are inverse proportional with the length. Thus the corrections are proportional with the length of the sections.

The corrected elevation difference: The final elevation of the i-th point: The correction of the elevation difference (i-th section):

PointDFWBWMeanCorr Corr. Elev. Diff. Elevation , , ,233454, , ,258452, , ,250451, , ,137450, , ,031449,481 Σ5, ,443

The preliminary elevation difference of the endpoints for the i-th section The observed elevation difference of the endpoints (K-V) The true elevation difference of the endpoints (K-V): Computation of heighting lines (trig. Height.)

The closure error: Normally mainly on the mean error of the zenith angle: How shall we distribute this error? What depends the mean error of the trig. heighting on?

Note that the mean error is proportional with the length in this case (note the squareroot of the length as in case of levelling) The weights of the sections are inverse proportional with the square of the length! Thus the corrections should be proportional to the square of the length of the section.

The corrected elevation difference: The final elevation of the i-th endpoint: The correction of the i-th elev. diff.:

PointD D2D2 FWBWMean Cor r. Corr Elev Diff Elevation , ,52,2+1,25-1,28+1,26-2+1,24154, ,0 -2,56+2,58-2,57-2,58151, ,62,6+1,97-2,01+1,99-3+1,96153, ,0 +5,65-5,63+5,64+5,63159, ,50,2-0,03-0, ,16 Σ–7,0––+6,31-7+6,24 -(+6,31) -0,07

Computation of heighting joints (j=10, 20, 30) Based on the three independent observations, three preliminary values for the elevation of 99 can be computed. The adjusted elevation of the joint is the weighted mean of the three preliminary values:

How shall we determine the weights? Levelling:Trigonometric heighting:

Point Distance [km] Weight p Preliminary Elevation of point[m] L’ [mm] p×L’ 105,10,20(105,572)122,40 207,90,13(105,565)50,65 302,70,37(105,568)82,96 15,70, ,01 The final elevation of 99:105,569 m The computation of joints (in case of levelling)