Page 45 - 4798

Zenith angles are measured down ward from the vertical, and Vertical angles are
measured up or down from the horizontal.

Fig. 5.12 Trigonometric leveling for plane surveying

Thus, in figure 5.12 If slop distance S and zenith angle Z or vertical angle α
between a&b are measured, then h, the elevation difference between A and B, is
given by
h  h  S cosZ  or h  h  S sin  h .
h
1 2 1 2
Where h 1 is the height of the instrument above point A and h 2 is the reading on
the rod held at B when zenith angle z or vertical angle α is read. If h 2 is made equal
to h 1, then these two values cancel in equation above, and simplifies the
computations.
For shorter lines (up to about 1000ft or m in length) elevation differences
obtained in trigonometric leveling are appropriately computed and depicted by the
above equations. For longer lines, however, earth curvature and refraction become
factors that must be considered.

5.8 Error of Closure and Precise Leveling
The relative accuracy required for a vertical control or leveling survey
depends on its purpose. In countries where there are sets of standards and
specifications as a guide line for surveyors, standards are expressed in terms of
an allowable error of closure instead of a relative accuracy ratio.
The allowable error of closure is a function of the length or total horizontal
distance of the leveling line or circuit. The function is expressed in the
following form:
m  C  D ,
h
where m h - error; C – constant; D – distance
The higher the order of the accuracy, the smaller the constant.
For example, for a level circuit with a total length of 2000 m, at the third
order accuracy, the maximum error of closure would be 12√2 = 17 mm.

43