Error theory in chaining and traversing: Accidental (compensating) errors associated with a measured length L are generally proportional to which function of L?

Introduction / Context:
In surveying measurements, accidental (random or compensating) errors arise from small, unpredictable causes such as slight pointing errors, minor tape misalignment, or reading scatter. Understanding how these random errors grow with the total length L is essential for estimating probable accuracy and for planning checks and redundancies.

Given Data / Assumptions:

• Error sources are random and unbiased (zero-mean).
• Length L is made up of many similar segments or observations.
• Systematic errors have been corrected or are negligible in this context.

Concept / Approach:
By the theory of random errors, the standard deviation of a sum of many independent, identically distributed errors grows with the square root of the number of observations. If the total measured length L is built up from segments of similar length, the number of segments n is proportional to L, and hence the standard deviation (accidental error) grows as sqrt(n) ∝ sqrt(L). Thus accidental (compensating) errors scale as sqrt(L), not linearly with L.

Step-by-Step Solution:

Verification / Alternative check:
Traverse error of closure often follows a root-mean-square behavior, and practical specifications commonly quote tolerances proportional to sqrt(perimeter).

Why Other Options Are Wrong:

• L: Linear growth suggests systematic bias, not random error.
• 1/L: Implies larger lengths reduce random error, which is incorrect.
• L^2: Unrealistically rapid growth; not supported by error propagation theory.
• Constant: Ignores accumulation over multiple segments.

Common Pitfalls:
Confusing systematic scale/bias errors (often ∝ L) with accidental errors (∝ sqrt(L)).

Final Answer:
sqrt(L)