Transcript The mean error of a single staff reading

Budapesti Műszaki és Gazdaságtudományi Egyetem
Általános- és Felsőgeodézia Tanszék
The ‘a priori’ mean error of levelling.
Computation of heighting lines and joints
The mean oscillation of the line-of-sight:

The mean error of a single staff reading:
m  d

The determination of the mean oscillation of the
line-of-sight (automatic level)
m ,m ,..., mn
1 2
v2
i
m 
m
n 1
m  1
n  mi
m
m  m

2
v  m  m
i
i
m
 
d
The determination of the line-of-sight
precise level (staff reading is taken with optical
micrometers)
1, 2,..., n
  1n  
i
v  
i
i
v2
i
m 

n 1
m
 
d
The mean error of a single elevation difference
The mean error of the backsight and foresight readings
are the same:
m
Why?
The elevation difference in a single station:
m = back – fore
The mean error of a single elevation difference:
m m 2
m 
The mean error of all the single elevation differences are
constant for a levelling line:
m
m
... m
m
m
m
mn m
1
2
The elevation difference of the endpoints:
m  m  m ... mn
1
2
The mean error of the total elevation difference
(according to the
law of error propagation)
,
mm  m2  m2 ... m2  nm2  m
n
m
m
mn
m m
1
2
Substitutin the mean error of the staff reading:
(m  m 2)
m 
mm  m 2n

When the instrument-staff distance is:
d
Then the total length of the levelling line:
L  n2d
n  L / 2d
n should be substituted to the eq. of the mean error:
mm  m
n  m 2 L  m L
m
 2d
 d
Thus the total mean error
of the total elevation difference is:
mm  m L
d
Substituting the formula of the mean error of
a single staff reading:
m  d
the following formula is obtained:
mm  Ld
The one-way ‘a priori’ kilometric mean error
of levelling
(L = 1 km)
mkm  1000d
The two-way ‘a priori’ kilometric mean error of
levelling:
mkm
m(km) 
 500d
2
Computation of heighting lines (levelling)
(m) = (BS – FS) = BS – FS
The preliminary elevation difference of the endpoints for the i-th section:
mi, forward mi,backward
(mi) 
2
The observed elevation difference between the endpoints (K-V):
n
(m)  (mi )
i1
The given elevation difference of the endpoints:
m M vM k
The closure error:
  m  (m)
The mean error is proportional to the squareroot of the
length.
2

p
Thus the weights are inverse proportional with the length. i m 2
i
Thus the corrections are proportional with the length of the
sections.
The correction of the elevation difference (i-th section):
i  n ti
ti
i1
The corrected elevation difference:
mi  (mi) i
The final elevation of the i-th point:
Mi  Mi1 mi
Point
D
FW
BW
Mean
Corr
Corr.
Elev.
Diff.
81526
Elevation
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
Computation of heighting lines (trig. Height.)




2 
d
m  h   dcot z  1 k 

2R 




The preliminary elevation difference of the endpoints for the i-th section
mi,oda  mi,vissza
(mi ) 
2
The observed elevation difference of the endpoints (K-V)
n
(m)  (mi )
i1
The true elevation difference of the endpoints (K-V):
m M vM k
The closure error:
  m  (m)
How shall we distribute this error?
What depends the mean error of the trig. heighting on?




m  h   dcot z  d2 1 k 


2R 
Normally mainly on the mean error of the zenith angle:
2
mm
 d  2  d 
  2  mz  
m z
 sin z 
 sin z 
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 correction of the i-th elev. diff.:
i  n ti2
ti2
i1
The corrected elevation difference:
mi  (mi) i
The final elevation of the i-th endpoint:
Mi  Mi1 mi
Point
D
D2
FW
BW
Cor
Mean
r.
Corr
Elev
Diff
3025
Elevation
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




n

j
j
M99   M  mij

i1
(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:
30
j  p j
M
 99 
M 99  j10




30
j
p

j10
How shall we determine the weights?
Levelling:
Trigonometric heighting:
mm  L
mm  L
2

p
i
p j  n1
ti j
i1
2

p
i
m2
i
pj  12
 j
t 




m2
i
The computation of joints (in case of levelling)
L’
[mm]
p×L’
Point
Distance
[km]
Weight
p
Preliminary
Elevation
of point[m]
10
20
5,1
7,9
0,20
0,13
(105,572)
(105,565)
12
5
2,40
0,65
30
2,7
15,7
0,37
0,70
(105,568)
-----
8
-----
2,96
6,01

The final elevation of 99:
105,569 m