Traverse positional error from bearing uncertainty If θ is the probable error (in radians) of the observed bearing of each line of length l, what is the probable position error at the end of a traverse of n equal lines (each of length l)? Assume independent random bearing errors.

Introduction / Context:
In traverse surveying, small random errors in observed bearings cause a displacement of the closing point. Understanding how these angular errors accumulate along multiple lines is crucial for estimating positional accuracy and for designing observation redundancy. This question focuses on the probable position error caused solely by bearing (direction) uncertainty when line lengths are equal.

Given Data / Assumptions:

• Each traverse side has length l.
• The probable error in the observed bearing of each side is θ (small angle in radians).
• There are n lines, and bearing errors are independent, random, and zero-mean.
• Linear measurements are assumed exact for this analysis.

Concept / Approach:

For a small angular error δ on a line of length l, the lateral displacement contributed by that side is approximately l * δ. If errors on successive lines are independent, the resultant displacement behaves like a random walk: the root-mean-square (or probable) resultant grows as the square root of the number of steps. Therefore, the combined probable position error after n equal lines is proportional to l * θ * sqrt(n).

Step-by-Step Solution:

Verification / Alternative check:

If all direction errors were identical and of the same sign (a worst-case systematic scenario), the accumulation would be linear, n * l * θ. But for independent random errors, the statistical combination is by root-sum-square, leading to sqrt(n) scaling, which matches empirical traverse-closure behavior.

Why Other Options Are Wrong:

Option A assumes perfect correlation (systematic), not probable error. Options C, D, and E ignore one or more governing parameters and thus are dimensionally or conceptually incorrect.

Common Pitfalls:

Confusing systematic and random accumulation; mixing degrees with radians (use radians for displacement estimates).

Final Answer:

l * θ * sqrt(n)