curvature and refraction

show two point A and B at exactly the same level . an instrument set up at X would give a horizontal line of sight through X . if a graduated levelling staff is held vertically on A the horizontal line would give the reading A . theoretically , as B is at the same level as A , the staff reading should be identical 

(B) this would require a level line of sight ; the instrument , however , gives horizontal line and a reading at b ; ( ignoring refraction ) . subtracting vertical heigh AA ' from BB '' indicates that point B is lower than point A by the amount BB '' this error (c) is caused by the curvature of the Earth and its value may be calculated as follows ; 

in which the instrument heights are ignored and earth is assumed to be sperical with a radius of R ; 

(X' B) 2 = ( OB'')2 - ( OX' )2 = ((R + C)2 - R2 = R2 + 2RC + c 2 - R2 = (2Rc + C2)

As both c and the instrument heights have relatively small values distance X ' B may be assumed to the are distance XB = D . therefore

D = ( 2 R c + C2 ) 1/2

Now as C is very small compared with R, C2 may be ignored , giving 

C = D2/2R

taking the distance D in Kilometers and an average value for R equal to 6370 Km , we have 

C = ( D × 100 )2   / 2 × 6370 × 1000

C = 0.0785D2

with the value of C in metres , when D is in kilometres .

would not be at B '' at Y due to refraction of the line of sight  through the atmosphere , in general it is considered that the effect is to bend the line of sight down .

horizontal and level lines

refraction effect

reducing the effect of curvature by 1/7 th thus the combined effect of curvature and refraction (C - R ) is (6/7) ( 0.0785D2) , I.e . 

( C - r ) = 0.0673D2

thus if D is 122 m the value of ( c - r ) is only 1 mm So in tertiary levelling , where the length of sights are generally 25 - 30 m , the effect may be ignored .

it should be noted that although the effect of refraction has been shown to bend the line of sight down by an amount equal to 1/ 7 th that of the effect of curvature , this is most unreliable assumption for precise levelling .

Refraction is largely a function of atmospheric pressure and temperature gradients , which may cause the bending to be up or down by extremely variable amounts .

There are basically three types of temperature gradient (dT/dh)

(1) Absorption ; occurs mainly at night when the colder ground absorbs heat from the atmosphere . this causes the atmosheric temperature to increase with distance from the ground and dT/dh >0 

(2) Emission ; occurs mainly during the day when the warmer ground emits heat into the atmosphere resulting in negative temperature gradient , i . e , dT / dh  <0 

(3)  Equilibrium ; no heat transfer takes place (dT / d h = 0 ) and occurs only briefly in the evening and morning .

the result of dT/dh < 0 is to cause the light ray to be convex to the ground rather than concave as generally shown , this effect increases increase the closer to the ground the light ray gets and errors in the region of 5 mm / km have occurred .

Thus , wherever possible , staff reading should be kept at least 0.5 m above the ground , using short observation distances (25 m ) equalized for backsight and foresight . 

equipment