Quantification of accuracy degradation in differential leveling under unfavorable terrain conditions

Abstract

Differential leveling is a fundamental surveying technique valued for its simplicity and high intrinsic precision. The method’s accuracy relies on the midpoint principle, wherein the level instrument is placed equidistant between backsight and foresight staffs to eliminate distance-dependent systematic errors. However, in mountainous or complex terrains, this ideal configuration is often impractical. The instrument must then be set at a perpendicular offset from the baseline, which increases sight distances and error propagation. This study investigates the mechanisms of accuracy degradation under such unfavorable conditions, focusing on the geometric elongation of the optical sight path. We analytically examine the principal distance-dependent error sources, including collimation error, vertical index error, Earth curvature, atmospheric refraction, and staff-reading uncertainty, and show how they scale with sight length. Using the geometric relationship between the ideal baseline and actual sight lengths, we derive an accuracy degradation coefficient K to quantify the loss of efficiency caused by offset instrument placement. The coefficient is evaluated over a range of offset ratios (0.05 ≤ h/S ≤ 2.00), revealing a pronounced non-linear decrease in accuracy with increasing offset. The results indicate that degradation remains minor for h/S < 0.25, becomes significant for moderate offsets (0.25 < h/S < 0.9), and is severe when h/S > 1.5. Based on these findings, we propose practical guidelines: applying double-station leveling in moderate-offset conditions and resorting to alternative height-determination methods in extreme terrain. The proposed degradation coefficient provides a simple tool for survey planning and quality control, enabling engineers to assess expected accuracy loss and design appropriate observation strategies for differential leveling in challenging topographic environments.