9.3 Equipment Errors and Corrections

9.3 Equipment Errors and Corrections

Quick Answer: A recorded length is not yet the true horizontal distance. For a steel tape (US °F and feet in this section): C_t = α L ΔT with α = 6.45×10⁻⁶ /°F, C_p = (P − P0) L / (AE), C_s = −w² L³ / (24 P²), and C_h ≈ −h² / (2L). True length ≈ recorded + ΣC. EDM adds an atmospheric ppm scale. Angles: eccentricity, collimation, pointing. Levels: collimation plus combined C_r ≈ 0.574 M² ft (M in miles), equivalently 0.0206 K² ft (K in thousands of feet).

CES Domain III item A is measuring-equipment errors — distance, angular, and leveling. Chapter 6 told you not to skip the slope reduction. This section is every systematic correction that remains, each with a number, plus the random errors you do not “correct” so much as average out.

Units in this section stay US customary: feet, pounds, °F. Do not mix 6.45×10⁻⁶ /°F with a Celsius ΔT. For curvature and refraction, do not feed miles into a thousands-of-feet coefficient (or the reverse).

Systematic versus Random

Systematic errors have a sign and a model. Temperature expansion, extra tension, sag, slope, EDM ppm, collimation, curvature-and-refraction — you compute them, or you design the procedure so they cancel (balanced level sights, direct-and-reverse angles).

Random errors scatter both ways: pointing, reading the circle, a plumb bob in wind, a slight mis-estimate of target height. You do not plug them into C_t. You repeat, mean, and watch the spread.

A blunder (wrong hub, 100-ft tape dropped a foot, zenith booked as a horizon VA) is neither. No correction formula repairs it.

Sign convention for tapes: true distance = recorded (nominal) distance + correction. A too-long tape (hot, extra tension) records too little; the correction is positive. Sag and slope both make the tape reading longer than the horizontal chord; those corrections are negative.

Temperature — C_t = α L ΔT

Steel expands when it warms. Use α = 6.45×10⁻⁶ per °F. Standard temperature in these examples is 68°F (state whatever standard the tape or the problem prints).

C_t = α L (T − T_std)

Worked: a line recorded as L = 300.00 ft (three tape lengths) at T = 88°F, T_std = 68°F.
ΔT = +20°F.
C_t = 6.45×10⁻⁶ × 300.00 × 20 = 0.0387 ft.

Add it: the tape was long, so the true length is about 300.04 ft. Skipping this on a 300-ft control line is four hundredths — visible in a tight traverse.

Cold tape, opposite sign: L = 140.00 ft, T = 42°F, ΔT = −26°F.
C_t = 6.45×10⁻⁶ × 140.00 × (−26) = −0.0235 ft.
The tape is short; you recorded too much; subtract.

On a single 100.00 ft length at the same +20°F, C_t = 0.0129 ft. Temperature scales with L; it is not a constant “0.01 ft per tape.”

Tension — C_p = (P − P0) L / (AE)

Extra pull stretches the tape (Young’s modulus). P is the field tension, P0 the standardization tension, A the cross-section, E the modulus, L the length. With P in pounds, A in in², E in psi, L in feet, C_p is in feet.

Worked: P = 25 lb, P0 = 12 lb, L = 100.00 ft, A = 0.0030 in², E = 29,000,000 psi.

AE = 0.0030 × 29,000,000 = 87,000 lb
C_p = (25 − 12) × 100.00 / 87,000 = 1,300 / 87,000 = 0.0149 ft.

If you pull less than P0, C_p is negative (tape shorter than standard). A spring balance on a tape is not decoration; 13 lb extra on this section is a hundredth and a half.

Sag — C_s = −w² L³ / (24 P²)

Between supports the tape hangs in a catenary. The chord (what you wanted) is shorter than the tape length you paid out, so the recorded length is too large. The correction is negative. w is tape weight per foot, L the unsupported span, P the actual pull.

Worked: w = 0.018 lb/ft (the 100-ft tape weighs 1.80 lb), L = 100.00 ft fully unsupported, P = 20 lb.

C_s = −(0.018)² × (100)³ / (24 × 20²) = −0.000324 × 1,000,000 / 9,600 = −324 / 9,600 = −0.0338 ft.

Support the tape at mid-span and L in the formula becomes 50 ft:
C_s = −(0.018)² × (50)³ / (24 × 15²) at a 15 lb pull = −0.0075 ft.
Sag grows with L³ and shrinks with P². Breaking the unsupported length is more effective than muscling a little more pull.

Slope — C_h ≈ −h² / (2L)

This is the first-order reduction from slope length L to HD when you know the elevation difference h. It is the same triangle as Section 9.2, written as a correction.

Worked with the 4% tape: L = 215.00 ft, h = 8.59 ft.

C_h = −(8.59)² / (2 × 215.00) = −73.79 / 430 = −0.172 ft
HD ≈ 215.00 − 0.172 = 214.83 ft — the same HD as the exact √(1+g²) reduction.

Shorter check: L = 80.00 ft, h = 3.20 ft.
C_h = −(3.20)² / (160) = −10.24 / 160 = −0.064 ft
HD ≈ 79.936 ft. Exact: √(80.00² − 3.20²) = 79.936 ft.

Use the exact trig when Z or VA is given. Use C_h when you have h from a level and a taped slope length.

Combined 100-ft Tape (Each Term, Then the Sum)

Recorded 100.00 ft, T = 88°F (ΔT = +20°F), P = 25 lb, P0 = 12 lb, A = 0.0030 in², E = 29×10⁶ psi, unsupported w = 0.018 lb/ft at the 25 lb pull, already held horizontal so C_h = 0.

• C_t = 6.45×10⁻⁶ × 100 × 20 = +0.0129 ft
• C_p = 13 × 100 / 87,000 = +0.0149 ft
• C_s = −(0.018)² × 100³ / (24 × 25²) = −324 / 15,000 = −0.0216 ft
• Sum = +0.0062 ft → true ≈ 100.006 ft

Temperature and tension almost offset sag here. That is luck, not a reason to skip the arithmetic. Change P to 20 lb and sag becomes −0.0338 ft; the sum flips sign.

EDM Atmospheric ppm

An EDM (or the EDM in a total station) measures on a light path whose speed depends on air temperature, pressure, and humidity. The instrument applies an atmospheric correction in parts per million (ppm) relative to its standard atmosphere. You either key the field weather into the box or apply ppm in the office.

Correction (ft) = (ppm × 10⁻⁶) × SD

Worked: +12 ppm on SD = 2,000.00 ft.
12 × 10⁻⁶ × 2,000.00 = 0.024 ft.

Another: +8 ppm on 4,125.00 ft = 0.033 ft.

A neglected 10 ppm on a 1,000-ft shot is 0.01 ft; on a 5,000-ft shot it is 0.05 ft. Prism constant (typically 0, −30 mm, or −40 mm depending on the reflector) is a fixed offset, not ppm — apply it once per measurement, not scaled by distance. CES needs the ppm scale and the idea that weather belongs in the reduction; it does not need the full Barrel–Sears equation.

Angular: Eccentricity, Collimation, Pointing

Eccentricity — the instrument (or the target) is not plumbed over the point. Linear offset e at sight distance D subtends e/D radians ≈ (e/D) × 206,265 seconds.

Worked: e = 0.02 ft, D = 250 ft.
Angle = 0.02 / 250 = 0.000080 rad = 16.5″.
At D = 80 ft the same 0.02 ft is 52″. Short sights magnify plumbing error. Optical plummet checks are distance-error control as much as they are “setup neatness.”

Collimation — the line of sight is not perpendicular to the instrument’s horizontal axis (and a related plate-bubble / vertical-axis tilt). The error reverses sign when you plunge. Mean of direct and reverse cancels the collimation contribution to the horizontal angle. Direct-only topography sideshots leave it in; control angles should not.

Pointing — how steadily you bisect the target. Random. Repeat the pointing; do not invent a C_point formula.

Leveling: Collimation, Curvature, and Refraction

Collimation of a level is a small tilt of the collimation line relative to true horizontal. On a sight of length D, the elevation error is about C (radians) × D. Balanced backsight and foresight distances cancel it: the error in BS minus the error in FS is C × (D_BS − D_FS).

Worked: collimation 20″ = 20 / 206,265 = 9.70×10⁻⁵ rad. Unbalanced D_BS = 200 ft, D_FS = 80 ft.
Error ≈ 9.70×10⁻⁵ × (200 − 80) = 0.012 ft.
Balance the sights and this term drops out. That is why level notes (Chapter 7) care about BS and FS distances, not only about the rod readings.

Curvature and refraction, US survey feet. Name the distance unit before you square it.

Combined curvature and refraction (what leveling and trig-height reductions use):

C_r ≈ 0.574 M² ft with M in miles
C_r ≈ 0.0206 K² ft with K in thousands of feet (K = D_ft / 1000)

Those two agree. Mean terrestrial refraction (about 14% of curvature, opposite in sign) is already folded into 0.574 and 0.0206.

Curvature only (no refraction) is a different coefficient on the thousands-of-feet unit:

C_curv ≈ 0.0239 K² ft with K in thousands of feet — not miles.

Using 0.0239 with miles understates combined C+R by about 24× (0.574 / 0.0239 ≈ 24). That mix-up is the usual exam trap.

Earth curvature makes the rod read high; mean refraction takes a fraction of that back. Subtract combined C_r from the rod reading if you apply it explicitly. Many digital levels apply it internally — do not apply it twice. Balanced BS and FS distances cancel most of it the same way they cancel collimation.

Worked — combined C+R in miles: M = 1.25 miles (6,600 ft).
C_r = 0.574 × (1.25)² = 0.574 × 1.5625 = 0.897 ft ≈ 0.90 ft.

Same sight with K: K = 6,600 / 1,000 = 6.6.
C_r = 0.0206 × (6.6)² = 0.0206 × 43.56 = 0.897 ft. Same number.

Worked — combined C+R on a construction sight (Chapter 7.1 check): D = 800 ft, so K = 0.80.
C_r = 0.0206 × (0.80)² = 0.0206 × 0.64 = 0.013 ft.
Miles check: M = 800 / 5,280 = 0.1515; 0.574 × (0.1515)² = 0.013 ft.

At M = 2.00 miles, combined C_r = 0.574 × 4 = 2.30 ft.
At M = 0.50 mile, combined C_r = 0.574 × 0.25 = 0.14 ft.

Curvature-only on the 800 ft sight, for contrast: 0.0239 × (0.80)² = 0.015 ft — a little larger than combined, as expected before refraction takes about 14% off. 0.0239 does not belong next to a distance in miles.

Do not import a “0.574 D² metres, D in kilometres” cousin. The 0.574 coefficient belongs to feet and miles.

Exam Traps

1. Unsigned sag — C_s is negative; adding 0.0338 ft instead of subtracting lengthens a 100-ft chord the wrong way.
2. α in /°F with a Celsius ΔT — pick one system; this section uses °F.
3. ppm treated as millimetres — 12 ppm on 2,000 ft is 0.024 ft, not 12 mm.
4. C_r coefficient on the wrong distance unit — 0.0239 is curvature-only with K in thousands of feet, not combined C+R with M in miles. 0.0239 × (1.25 mi)² = 0.0373 ft understates combined C+R (0.90 ft) by about 24×.
5. Collimation “corrected” by a formula on a single face — plunge and mean, or balance level sights.