reciprocal observations

Recipro observations are observations from A and B the arithmetic mean result being accepted if one assumes a symmetrical line of sight from each and the observations are taken simultaneously 

Then the effect of curvature and refraction is cancelled out for instance for elevated sights (c-r) is added to a positive value to increase the height difference , for depressed or downhill sights (c-r) is addes to a negative of curvature and refraction . this statement is not entirely true as the assumption of symmetrical line of sight from each is dependent on uniform ground and atmosphereic conditions at end , at the instant of simultaneus observation .

In practice over short distance , sighting into each other's object lens form an excellent target with some form of communication to ensure simultaneous observation.

The  following numerical example is taken from an actual survey in which the elevation of A and B had been obtained by precise geodetic levelling and was checked by simultaneous reciprocal trigonometrical levelling .

WORKED EXAMPLE 

As the observations are reciprocal the corrections for curvature and refraction are ignored :

This value compares favourably with 2.311 m obtained by precise levelling , However , the disparity between the two values 0.846 and - 3.722 shows the danger inherent in single observations uncorrected for s single for a single observation .

from A to B 

2.284  =  4279.446 cos   89ُ  59 َ   18,7 ً  +  1.290  - 1.300 + (c   -  r  )

where ( c - r )  =  S2  ( 1  - K  ) 2 R 

and the local value of R for the area of observation  =  6 364  700 m

2.284  =  0.856  -  0.010  +  S2  (1 - K ) / 2R 

1.438  =  4279.4462 (1  -  K  ) / 2  ×  6 364 700 m

K  =  0.0006

From B to A :

2.284  =  - 3.732  +  1.300   -  1.290  +  S2  (1  - K ) / 2R

K =  0.0006

Now this value for K could be used for single ended observations taken within the same area , at the same time , to give improved results .

A variety of formulae are available for finding K directly . For example , using zenith angles :

                     Za  +  Zb  -  180ُ             R

 K  =  1  -  ______________     ×   _____

                             180ُ / ∩                    S

and using vertical angles :

K =  ( ⊖  +  ∝ o +  B o ) / ⊖

where ⊝ ًً  =  Sp/R   where P = 206265

In the above formulae the values used for the angles must be those which would have been observation had 

hi  =  ht  and , in Case of  vertical angles , entered  with their appropriate  sign  .

By sine rule :  ∝ - e    and  for an angle of depression it becomes bo  = B + e .

                      ht  - i sin (90ُ  -  ∝  )

sin   e  =   ___________________

                                  S

                                        ht  -  i  cos ∝

        e   =    sin  - 1   (  ______________  )

                                                 S

                         ht  - 1                          ht3  - i

              =   _______  cos    ∝   +    ________  cos  3  ∝  + ............

                          S                                 6S3

∴   e   =  (  ht  -  i cos  ∝ ) / S

For  zenith  angles :

e  =  ( ht  -  i  sin Z ) / S