GELE Surveying (Geomatics) — LevelingRevision Notes

Formulas

Exam Tips

• Memorize the two master equations: HI = Elev + BS and Elev = HI − FS. Every leveling problem reduces to these two lines.
• In a tabular level-run problem, always fill in the HI column first before computing any new elevations.
• If the problem gives you only ΣBS and ΣFS, use Δelev = ΣBS − ΣFS to find the net elevation change without computing intermediate values.
• Double-check sign: if ΣBS > ΣFS, the final point is higher than the start; if ΣFS > ΣBS, it is lower.

Key Points

• A datum is a reference surface of zero elevation; all elevations are measured above (or below) this surface.
• A benchmark (BM) is a permanent point of known elevation used as a starting reference in a level run.
• The line of sight of a properly adjusted level is horizontal (perpendicular to the direction of gravity).
• A backsight (BS) is a rod reading taken on a point of known elevation to establish the Height of Instrument (HI).
• A foresight (FS) is a rod reading taken on a point whose elevation is to be determined.
• An intermediate foresight (IFS) is a foresight taken at an intermediate station along a profile — it does NOT establish a new HI.
• A turning point (TP) is a temporary point that carries the elevation forward; it receives both a foresight (to close the old HI) and a backsight (to open the new HI).
• The Height of Instrument (HI) is the elevation of the line of sight above the datum — NOT the height of the instrument above the ground.

Definitions

Section Title

1. Fundamental Concepts and Terminology

Common Mistakes

• Confusing the height of instrument (elevation of line of sight) with the physical height of the level above ground.
• Adding FS to HI instead of subtracting it — remember: BS adds, FS subtracts.
• Treating intermediate foresights (IFS) as turning points — IFS values are NOT included in ΣBS or ΣFS for the arithmetic check.
• Forgetting to check that the starting benchmark and closing benchmark are different numbered points when performing a level loop.

Formulas

Exam Tips

• The arithmetic check is almost always asked in board-exam level-run problems. Set up a 5-column table: Station | BS | HI | FS | Elevation — and fill it systematically.
• After completing the table, always perform ΣBS − ΣFS and compare with Elev_last − Elev_first before submitting your answer.
• If the problem asks you to 'adjust' elevations, apply corrections proportional to distance; if distance is not given, apply them proportional to number of setups.
• For a loop problem asking for misclosure: compute ΣBS − ΣFS; since the loop must return to the starting elevation, the answer should be zero — any non-zero value is the misclosure.

Key Points

• Differential leveling determines the difference in elevation between two points using a series of BS and FS readings through one or more instrument setups.
• The arithmetic check verifies the correctness of all HI and elevation computations in a level run.
• For a level line from Point A to Point B: ΣBS − ΣFS = Elev_B − Elev_A.
• For a closed level loop returning to the starting benchmark: ΣBS − ΣFS = 0 (theoretically); any discrepancy is the misclosure.
• Only BS and FS values at turning points and benchmarks enter the arithmetic check — NOT intermediate foresights.
• The misclosure of a level loop is distributed among the TPs proportionally to the distance (or number of setups) for adjusted surveys.
• Acceptable misclosure for ordinary leveling: e = ±12√K mm, where K is total distance in km (DPWH standard); for precise leveling: e = ±4√K mm.

Definitions

Section Title

2. Differential Leveling and the Arithmetic Check

Common Mistakes

• Including intermediate foresight values in ΣBS or ΣFS — this will corrupt the arithmetic check.
• Applying the misclosure correction with the wrong sign — if the computed elevation is too high (positive misclosure), the correction applied to each TP is negative.
• Using number of instrument setups instead of distance for proportional correction when distance is given in the problem.
• Forgetting that the arithmetic check catches addition/subtraction errors only — it cannot detect errors caused by wrong rod readings in the field.

Formulas

Exam Tips

• In a profile leveling problem, identify which readings are BS, FS (on TPs), and IFS (at stations). Only BS and FS go into the arithmetic check.
• If the problem asks for the elevation of a specific station and gives HI and IFS, simply compute Elev = HI − IFS.
• Board exam problems on profile leveling often include one TP in the middle of the run — set up the table with all BM, TP, and station rows to avoid confusion.
• Cut/Fill = Grade Elevation − Ground Elevation; positive = cut (excavation needed), negative = fill (embankment needed).

Key Points

• Profile leveling determines ground elevations along the centerline of a proposed route (road, pipeline, canal) at regular stations (typically every 20 m full stations) and at significant terrain changes.
• Stations are designated as 0+000, 0+020, 0+040 (metric system) or equivalently in full and plus stations.
• The ground profile obtained from profile leveling is plotted on profile paper to design the vertical alignment (grade line) of the route.
• Cross-section leveling takes rod readings at points perpendicular (left and right) to the centerline at each station to determine the ground cross-section shape, which is essential for computing earthwork volumes.
• In profile leveling, the level is set up and a BS is taken on the last TP to get the HI; then IFS readings are taken at each station along the centerline without moving the instrument.
• Only when the rod cannot be clearly read (terrain change or distance too great) is the instrument moved to a new TP, at which point a new FS and BS are recorded.
• End-area method and Prismatoid formula use cross-section data to compute cut and fill volumes — tested separately under Earthworks.

Definitions

Section Title

3. Profile and Cross-Section Leveling

Common Mistakes

• Using IFS values in the arithmetic check (ΣBS − ΣFS) — IFS readings are excluded from this check.
• Mixing up profile stations — remember that 1+500 means 1,500 m from the start, not 1.5 m.
• Forgetting to record a FS on the turning point before moving the instrument — omitting the FS makes it impossible to continue the run.
• In cross-section notes, not labeling left (L) and right (R) sides correctly relative to the direction of stationing.

Formulas

Exam Tips

• The board exam almost always gives K in km directly — verify units before substituting.
• If asked for curvature correction only, use C = 0.0785K²; if asked for combined curvature and refraction, use h_cr = 0.0675K².
• Quick memory aid: h_cr ≈ 0.067K² — round to 0.0675 for exact computation.
• If a problem states 'balanced backsights and foresights,' the curvature-refraction correction cancels — do NOT apply it separately.
• For reciprocal leveling problems, check whether they ask for the corrected elevation difference (use average of both observations).

Key Points

• For short sights (< 300 m), the Earth's surface can be treated as flat and the line of sight as truly horizontal.
• For long sights, two systematic effects must be corrected: (1) Earth's curvature — makes the rod appear to read higher than it truly is; (2) atmospheric refraction — bends the line of sight downward, causing the rod to read lower than it truly would without refraction.
• Curvature correction (upward effect): C = 0.0785K² (m, K in km). This is always additive to what the rod reads (it makes the reading appear too high).
• Refraction correction (downward effect, partially cancels curvature): R = 0.011K² (m, K in km). This partially compensates for curvature.
• Combined (net) curvature-and-refraction correction: h_cr = C − R = 0.0785K² − 0.011K² = 0.0675K² (m, K in km).
• The net correction h_cr is always SUBTRACTED from the observed rod reading to obtain the true rod reading (because the observed reading is too large due to the dominant curvature effect).
• Alternatively, h_cr is added to the computed elevation if the correction is applied to elevations rather than rod readings.
• Practical significance: at K = 1 km, h_cr ≈ 0.068 m; at K = 2 km, h_cr ≈ 0.27 m; at K = 5 km, h_cr ≈ 1.69 m — significant for precise geodetic leveling.
• In standard differential leveling with balanced backsights and foresights (equal sight distances), curvature and refraction effects cancel out automatically — no correction needed.
• The correction becomes significant and must be applied when sights are unequal in length or very long (reciprocal leveling, trigonometric leveling).

Definitions

Section Title

4. Curvature and Refraction Correction

Common Mistakes

• Using K in meters instead of km in the formula h_cr = 0.0675K² — this gives an answer 10⁶ times too large.
• Adding h_cr to the rod reading instead of subtracting it — remember, the observed reading is already too large.
• Forgetting that in standard differential leveling with balanced sights, no curvature-refraction correction is needed.
• Mixing up the curvature correction (C = 0.0785K²) with the combined correction (h_cr = 0.0675K²).

Formulas

Exam Tips

• The two-peg test concept is tested conceptually: understand that (1) at the midpoint setup, collimation error cancels for both readings; (2) at the near-point setup, the far-rod reading includes a larger collimation error.
• Remember: balancing BS and FS automatically eliminates collimation error and curvature-refraction error — this is why field crews are trained to balance sight distances.
• For rod-tilt problems: true vertical reading R_true = R_obs × cos(θ), where θ is the tilt angle from vertical.

Key Points

• Instrumental errors: collimation error (line of sight not truly horizontal), rod graduation errors, bubble not centered.
• Natural errors: Earth's curvature, atmospheric refraction, wind vibration, heat shimmer (refraction near ground).
• Personal errors: incorrect rod readings, rod not held plumb (rod tilt), poor target setting, parallax in the eyepiece.
• The two-peg test (also called the 'peg test') detects and quantifies the collimation error of a level; it is the standard field check.
• Collimation error: the line of sight is inclined rather than horizontal — corrected by adjusting the instrument or by always balancing BS and FS distances.
• Rod not held plumb: always causes the reading to be too large (the rod appears longer than it is); use a rod level or wave the rod and take the minimum reading.
• Parallax: occurs when the image of the rod and the crosshair are not in the same focal plane — eliminated by careful focusing of the eyepiece and objective.

Definitions

Section Title

5. Sources of Error in Leveling

Common Mistakes

• Not balancing BS and FS distances — unbalanced sights allow collimation error and curvature-refraction errors to accumulate.
• Reading the rod incorrectly due to parallax — always focus the eyepiece on the crosshairs before focusing on the rod.
• Failing to hold the rod plumb — even a small tilt significantly increases the reading for tall rods.