7.1 Differential Leveling and Level Notes

7.1 Differential Leveling and Level Notes

Quick Answer: HI = elev_BM + BS and elev = HI − FS. A turning point (TP) is a temporary mark that receives an FS, then a new BS from the next instrument setup. Plus readings are backsights; minus readings are foresights. Create the notes that way, then check them: ΣBS − ΣFS = end − start for the main line. If that identity fails, you have a book blunder, not a field misclosure.

This section is the field half of CES Domain II items B and D: locating or establishing an elevation by differential leveling, and creating and checking level notes. Trigonometric leveling is §7.2. Equipment selection is §7.3. Office closure, allowable error formulas, and the full curvature/refraction correction live with Domain III in later chapters — here you learn what must be true in the book before you leave the job.

Two different quantities share similar names. In this section HI means height of instrument: the elevation of the line of sight, in feet on the project vertical datum. It is not the 5-ft tape measurement from a hub to a total-station trunnion (h.i. in §7.2).

The HI Method (Plus and Minus)

A level (automatic, dumpy, or digital) sets a horizontal line of sight. A level rod held on a point of known or unknown elevation intercepts that line. The intercept is a rod reading.

• Backsight (BS, plus): rod reading on a point whose elevation is already held. Adding it raises you from the mark up to the line of sight.
• Foresight (FS, minus): rod reading on a point whose elevation you want. Subtracting it drops you from the line of sight down to that mark.

HI = elev_known + BS
elev_new = HI − FS

If BS > FS on that setup, the new point is higher than the old one (a rise). If FS > BS, it is a fall. The rise-and-fall (plus-minus) column is the same arithmetic without writing HI: rise/fall = BS − FS for each turning setup, and the algebraic sum of those rises and falls must equal end − start.

A turning point is any solid, well-defined temporary mark (a TP pin, a chiseled square, a hub with a tack, a fire hydrant flange) that can hold a rod while you move the instrument. It gets an FS from the current setup and a BS from the next setup. A point that gets only an FS — a catch basin invert, a finished-floor hub, a grate — is an intermediate foresight (IFS). Compute IFS elevations from the current HI; do not put IFS values in ΣFS of the main-line page check, because they never receive a matching BS.

Creating a Full Note Page (Four Setups)

Run a spur from BM A (published elevation 247.32 ft) to BM B (published elevation 250.18 ft) to check a pad. Four instrument setups, turning points TP1–TP3, plus one intermediate shot to a catch-basin invert from the second HI.

Setup 1 — rod on BM A, then on TP1:
HI₁ = 247.32 + 5.26 = 252.58
Elev TP1 = 252.58 − 3.67 = 248.91

Setup 2 — rod on TP1, then on TP2:
HI₂ = 248.91 + 6.41 = 255.32
Elev TP2 = 255.32 − 3.47 = 251.85
IFS to CB-12 = 255.32 − 6.88 = 248.44 (not a TP)

Setup 3 — rod on TP2, then on TP3:
HI₃ = 251.85 + 4.88 = 256.73
Elev TP3 = 256.73 − 6.02 = 250.71

Setup 4 — rod on TP3, then on BM B:
HI₄ = 250.71 + 5.15 = 255.86
Elev BM B (computed) = 255.86 − 5.62 = 250.24

Checking the Notes — Page Arithmetic First

Main-line backsights: 5.26 + 6.41 + 4.88 + 5.15 = 21.70
Main-line foresights (TPs and BM B only): 3.67 + 3.47 + 6.02 + 5.62 = 18.78
ΣBS − ΣFS = 21.70 − 18.78 = 2.92
End − start = 250.24 − 247.32 = 2.92

The identity holds, so the HI arithmetic is internally consistent. The IFS 6.88 was correctly left out of ΣFS; including it would have broken a page that is actually right.

Rise-and-fall check (same four turning setups):

Net = 1.59 + 2.94 − 1.14 − 0.47 = +2.92 ft, matching both the page check and end − start.

Field Misclosure versus a Book Blunder

Published BM B is 250.18. Computed is 250.24.
Misclosure = 250.24 − 250.18 = +0.06 ft (the line came out high).

That 0.06 ft is not an arithmetic error: the page already checks. It is a field discrepancy — rod reading, turning-point settlement, unbalanced sights, or a published elevation that does not match the monument you occupied. On a closed loop you would finish on BM A and compare the computed re-elevation of A with 247.32; a perfect loop still uses the same page identity (end − start should be 0 after you close, and ΣBS should equal ΣFS).

Do not treat 0.06 ft as a Board-published CES tolerance. Compare it with the project specification you adopted in planning (Domain I.E). As a numeric illustration only, a common construction form is 0.05√N feet with N = number of setups: 0.05 × √4 = 0.10 ft. The 0.06 ft field misclosure would pass that example spec; a 0.21 ft loop would not, and you would rerun before staking grade.

Finding an Arithmetic Blunder

Suppose the same rod readings were booked, but TP2’s elevation was written 251.58 instead of 251.85 (a swapped 8 and 5 after a sloppy subtraction of 3.47 from 255.32). If the notekeeper then carried HI from that wrong elevation:

HI₃ = 251.58 + 4.88 = 256.46
Elev TP3 = 256.46 − 6.02 = 250.44
HI₄ = 250.44 + 5.15 = 255.59
Elev BM B = 255.59 − 5.62 = 249.97

Now end − start = 249.97 − 247.32 = 2.65 ft, while ΣBS − ΣFS is still 2.92 ft (the BS and FS columns were not changed). Discrepancy = 0.27 ft. That is the signature of a book blunder. Field misclosure against 250.18 is meaningless until the page identity holds.

Hunt line by line: recompute HI = previous elev + BS and elev = HI − FS. Setups 1 and the first half of setup 2 still produce TP2 = 251.85. The book’s 251.58 is the first disagreement, and 251.85 − 251.58 = 0.27 ft, the same amount as the page-check failure. Correct TP2, recompute downstream, and only then look at published BM B.

Three-Wire Leveling (Precision Method)

On control-quality runs, read the upper, middle, and lower stadia wires on a Philadelphia rod. The elevation reading is the mean of the three (or the middle after a check). The mean of the upper and lower must match the middle within the rod’s least count; a mismatch is a reading blunder caught before it enters the HI column.

Example: upper 5.426, middle 5.218, lower 5.010.
(5.426 + 5.010) / 2 = 5.218 — the middle agrees exactly.
Stadia interval = 5.426 − 5.010 = 0.416. With stadia constant 100, sight length ≈ 41.6 ft. Short, nearly equal BS and FS lengths are what you want for hundredths. Three-wire does not replace the page check; it improves the readings that go into the page.

Curvature and Refraction — Field Awareness

A compensator (or vial) sets a horizontal line. A level surface follows Earth curvature, so on a long foresight the rod is slightly “too tall”: the reading is large and the computed elevation of that point is slightly low. Atmospheric refraction bends the line of sight downward and cancels about one-seventh of the curvature. The combined effect (C+R) is still a small, too-large rod reading.

Tiny illustration (formula and sign conventions are developed in Chapter 9): at 800 ft, combined C+R is about 0.013 ft; at 1,000 ft, about 0.021 ft. On a 40-ft three-wire sight it is in the thousandths. Balancing each setup’s BS and FS lengths makes the two effects nearly equal, so they cancel in the elevation difference. Unbalanced long foresights are how C+R leaks into a loop that otherwise looks careful.

Exam Traps

1. Adding FS — elev = HI + FS is the inverted-rod case for an overhead point, not a ground TP.
2. Putting IFS in ΣFS — the page check then fails on a correct line.
3. Calling a failed page check a loop misclosure — fix the book first; only a checking page can be compared with a published BM.
4. HI versus h.i. — leveling HI is an elevation; the 5.28 ft tape to a total station is §7.2.