The ‘a priori’ mean error of levelling. Computation of heighting lines and joints

Budapesti Műszaki és Gazdaságtudományi EgyetemÁltalános- és Felsőgeodézia Tanszék

The mean oscillation of the line-of-sight:

dm

The mean error of a single staff reading:

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

nmmm ,...,2

,1

imnm 1

immiv

1

2

niv

mm

2m

mm

d

m

precise level (staff reading is taken with optical micrometers)

n ,...,,21

in 1

iiv

1

2

nivm

d

m

The determination of the line-of-sight

The mean error of a single elevation differenceThe mean error of the backsight and foresight readings

are the same:

m

The elevation difference in a single station: m  = back – fore

2mmm

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)

nm

mm

mnnm

mm

mm

mmm 22...2

2

2

1

Substitutin the mean error of the staff reading:

mm

nmm

mm

mm ...

21The elevation difference of the endpoints:

nmmmm ...21

,

nmmm 2)2( mmm

When the instrument-staff distance is: d

Then the total length of the levelling line:

dnL 2 dLn 2/

nm

mmm

n should be substituted to the eq. of the mean error:

dLm

dLm

22

dLmmm

Thus the total mean errorof the total elevation difference is:

Substituting the formula of the mean error ofa single staff reading:

dm

the following formula is obtained:

dLmm

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

dm 1000km

The two-way ‘a priori’ kilometric mean error of levelling:

dmm 5002km

(km)

The preliminary elevation difference of the endpoints for the i-th section:

2)( ,, backwardiforwardii

mmm

The observed elevation difference between the endpoints (K-V):

n

iimm

1)()(

The given elevation difference of the endpoints:

kv MMm

(m) = (BS – FS) = BS – FS

Computation of heighting lines (levelling)

The closure error:

)( mm

The mean error is proportional to the squareroot of the length.

Thus the weights are inverse proportional with the length. 2

2

im

ip

Thus the corrections are proportional with the length of the sections.

The corrected elevation difference:

iii mm )(

The final elevation of the i-th point:

iii mMM 1

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

in

ii

i tt

1

Point D FW BW Mean CorrCorr. Elev. Diff.

Elevation

81526 452,924

81523 1,5 +1236 -1238 +1237 -4 +1,233 454,157

81522 1,0 -1256 +1254 -1255 -3 -1,258 452,899

81521 1,6 -1244 +1248 -1246 -4 -1,250 451,649

81520 1,0 -1135 +1133 -1134 -3 -1,137 450,512

81525 0,5 -1030 +1030 -1030 -1 -1,031 449,481

Σ 5,6 -3428 -15 -3,443 -3,443

The preliminary elevation difference of the endpoints for the i-th section

2)( ,, visszaiodaii

mmm

The observed elevation difference of the endpoints (K-V)

n

iimm

1)()(

The true elevation difference of the endpoints (K-V):

kv MMm

k1R2

dzcotdhm 2

Computation of heighting lines (trig. Height.)

The closure error:

)( mm

Normally mainly on the mean error of the zenith angle:

zzm mz

dm

z

dm

sinsin

22

2

How shall we distribute this error?

What depends the mean error of the trig. heighting on?

k1R2

dzcotdhm 2

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:

iii mm )(

The final elevation of the i-th endpoint:

iii mMM 1

The correction of the i-th elev. diff.:

2

1

2 in

ii

i tt

Point D D2 FW BW MeanCorr.

Corr Elev Diff

Elevation

3025 152,92

301 1,5 2,2 +1,25 -1,28 +1,26 -2 +1,24 154,16

302 1,0 1,0 -2,56 +2,58 -2,57 -1 -2,58 151,58

303 1,6 2,6 +1,97 -2,01 +1,99 -3 +1,96 153,54

304 1,0 1,0 +5,65 -5,63 +5,64 -1 +5,63 159,17

3026 0,5 0,2 -0,03 -0,01 -0,01 0 -0,01 159,16

Σ – 7,0 – – +6,31 -7 +6,24 +6,24

-(+6,31)

-0,07

Computation of heighting joints

ji

n

i

jj mMM

199

(j=10, 20, 30)

30

10

30

1099

99

jp

pMM

j

j

jj

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:

n

i

ji

j

tp

1

12

1

j

j

tp

How shall we determine the weights?

Levelling: Trigonometric heighting:

Lmm

2

2

im

ip

Lmm

2

2

im

ip

PointDistance

[km]Weight

p

PreliminaryElevation

of point[m]

L’[mm]

p×L’

10 5,1 0,20 (105,572) 12 2,40

20 7,9 0,13 (105,565) 5 0,65

30 2,7 0,37 (105,568) 8 2,96

15,7 0,70 ----- ----- 6,01

The final elevation of 99: 105,569 m

The computation of joints (in case of levelling)