Two special cases of levelling deserve their own treatment. Differential levelling carries a height from one point to another regardless of what lies between. Reciprocal levelling solves the problem that arises when an obstacle — a river, a valley — makes it impossible to place the instrument midway.
Both rest on the same underlying idea: arrange the observations so that errors cancel.
Differential Levelling
A direct method in which the difference in elevation of two points is determined regardless of the horizontal positions of the points with reference to each other. It is also known as fly levelling.
What “Regardless of Horizontal Positions” Means
This phrase defines the method. In differential levelling you do not care where the points are — only how high they are.
No horizontal distances are measured, no positions are plotted, and the route taken between the two points is irrelevant. You may wander around obstacles, take as many instrument set-ups as needed, and follow any convenient path. All that is carried forward is the height.
This is why fly levelling is the natural method for establishing bench marks. The task is purely to transfer a known elevation from an existing bench mark to a new location, and the geometry of the route between them adds nothing.
It also explains why the height of instrument method of booking suits fly levelling — there are few or no intermediate sights to check, since intermediate points are not of interest.
Balancing of Back Sights and Fore Sights
If the instrument is placed approximately midway between two successive staff stations in fly levelling, the errors due to curvature and refraction may be eliminated. This is called balancing of sights.
Why Equal Sight Lengths Cancel the Error
This is an elegant result and worth following carefully.
From the previous topic, the combined curvature and refraction correction is 0.06735 d2 — it depends only on the length of the sight.
Now suppose the instrument sits exactly midway between the back staff and the fore staff. Both sights have the same length d, so both readings carry the same error.
But the difference in level is obtained by subtracting one reading from the other — and subtracting two equal errors leaves zero.
Equal sight lengths → equal errors → the errors cancel on subtraction.
Note that this also eliminates collimation error for the same reason. If the line of sight is tilted slightly, it deviates by an amount proportional to the sight length — so equal sights produce equal deviations, which again cancel.
This is why balancing sights is standard practice in careful levelling: one simple habit removes three separate errors at no cost.
Reciprocal Levelling
Sometimes balancing sights is impossible. If a river or a deep valley separates the two points, the instrument cannot be placed midway — there is nowhere to stand.
Reciprocal levelling solves this by taking observations from both banks.
The true difference in elevation is equal to the mean of the two apparent differences in elevations, obtained by reciprocal observations.
The Formula
H = ½ { (ha − hb) + (Ha − Hb) }
| Symbol | Meaning |
|---|---|
| ha, hb | Staff readings at A and B when the instrument is near A |
| Ha, Hb | Staff readings at A and B when the instrument is near B |
Why Averaging the Two Works
Follow the logic of the two set-ups.
With the instrument near A, the sight to A is short and the sight to B is long. All the error therefore falls on the reading at B, in one direction.
Move to near B and the situation is exactly reversed — the short sight is now to B and the long sight to A. The same error now falls on the reading at A, in the opposite direction.
Take the mean of the two apparent differences and the errors, being equal and opposite, cancel.
Balancing sights achieves cancellation by making the two sights equal within one set-up. Reciprocal levelling achieves it by making them unequal in opposite ways across two set-ups. Same principle, different arrangement.
This is the same idea as changing face on a theodolite, or comparing fore and back bearings — observe twice in opposite senses and take the mean.
Worked Example
| Instrument at | Reading on A | Reading on B |
|---|---|---|
| A | 2.165 | 3.810 |
| B | 0.910 | 2.355 |
Difference in level = ½ { (3.810 − 2.165) + (2.355 − 0.910) }
= ½ { 1.645 + 1.445 }
= ½ × 3.090
= 1.545 m
Which Point Is Higher
If the reading on A is less than the reading on B, then B is at the lower level.
In this example both set-ups give a larger reading on B than on A. Since a higher staff reading means a lower point, B lies 1.545 m below A.
Note the Two Apparent Differences
The two individual results are 1.645 m and 1.445 m — they disagree by 0.200 m. That disagreement is the error, and averaging splits the difference to give the true value of 1.545 m.
The Total Error
e = ½ { (ha − hb) − (Ha − Hb) }
Note the contrast with the previous formula — the true difference uses a plus sign between the brackets, while the error uses a minus sign.
This makes sense: adding the two apparent differences and halving gives their average, which is the truth. Subtracting them and halving gives half their disagreement, which is the error.
What the Total Error Contains
e = ecol + ecur − eref
The total error includes the collimation error, the curvature error and the refraction error.
Note the signs, which match the previous topic exactly — curvature adds, refraction subtracts, because the two act in opposite directions.
Rearranging to find the collimation error alone:
ecol = e − ecur + eref
This is useful in practice, because it lets the instrument’s collimation error be isolated and measured once the curvature and refraction components are computed from the sight distance.
The Three Errors Eliminated by Reciprocal Levelling
- Error in the line of collimation
- Combined effect of the earth’s curvature and refraction
- Variation in the average refraction
Why the Third Is Listed Separately
Items 2 and 3 look like duplicates, but they are distinct — and the distinction matters.
Item 2 concerns the standard refraction effect, the one seventh factor built into the combined correction formula. Item 3 concerns the fact that real refraction departs from that average, varying with temperature, humidity and time of day.
A computed correction using the 1/7 factor can handle item 2. It cannot handle item 3, because the actual departure from average is unknown.
Reciprocal levelling handles both, because it does not rely on a computed value at all — it simply arranges for whatever error exists to appear equally and oppositely in two observations. You do not need to know the size of an error to cancel it this way.
This is the real strength of the reciprocal method: it removes errors without measuring them.
The Two Techniques Compared
| Balancing of Sights | Reciprocal Levelling | |
|---|---|---|
| Instrument position | Midway between the two staff stations | Near each station in turn |
| Number of set-ups | One | Two |
| Sight lengths | Equal | Unequal, but reversed |
| Used when | The midpoint is accessible | An obstacle prevents a midway set-up |
| Eliminates | Curvature, refraction, collimation | Collimation, curvature and refraction, and variation in average refraction |
Balancing sights is the everyday method and costs nothing extra. Reciprocal levelling requires twice the work, so it is reserved for situations where balancing is impossible — chiefly levelling across a river or a wide valley.
Quick Revision Notes
- Differential levelling determines the difference in elevation regardless of the horizontal positions of the points; also called fly levelling.
- Balancing of sights: placing the instrument approximately midway between successive staff stations eliminates curvature and refraction errors.
- It works because the error depends on sight length, so equal sights give equal errors that cancel on subtraction.
- Reciprocal levelling: the true difference in elevation is the mean of the two apparent differences.
- H = ½{(ha − hb) + (Ha − Hb)}, where h are readings with the instrument near A and H with it near B.
- Total error e = ½{(ha − hb) − (Ha − Hb)} — note the minus sign.
- e = ecol + ecur − eref, hence ecol = e − ecur + eref.
- Reciprocal levelling eliminates: collimation error, the combined effect of curvature and refraction, and variation in the average refraction.
- A higher staff reading means a lower point.
Mistakes Students Commonly Make
- Confusing the two formulas. The true difference adds the two apparent differences; the error subtracts them.
- Forgetting the ½ factor in either formula.
- Mixing up which readings are h and which are H. Lower case h is with the instrument near A; capital H is with it near B.
- Getting the error signs wrong. Curvature adds, refraction subtracts.
- Listing only two errors eliminated. There are three, with variation in average refraction being distinct from the combined curvature and refraction effect.
- Saying differential levelling requires horizontal distances. It is carried out regardless of horizontal positions.
- Using reciprocal levelling where balancing sights would do. It requires two set-ups and is reserved for obstacles.
Conclusion
Both techniques in this topic exploit the same principle — errors that depend on sight length can be made to cancel by arranging the observations symmetrically. Balancing sights does it within a single set-up by making both sights equal, which removes curvature, refraction and collimation error at no extra cost. Where an obstacle makes that impossible, reciprocal levelling observes from each bank in turn, so the same errors appear equally and oppositely and vanish on averaging. Its particular strength is that it cancels even the unpredictable variation in refraction, which no computed correction could handle.
Frequently Asked Questions
What is differential levelling?
A direct method in which the difference in elevation of two points is determined regardless of the horizontal positions of those points with reference to each other. It is also known as fly levelling.
What is balancing of sights?
Placing the instrument approximately midway between two successive staff stations, so that the errors due to curvature and refraction are eliminated.
Why does balancing sights eliminate these errors?
Because the curvature and refraction error depends on the length of the sight. Equal sight lengths produce equal errors in both readings, and since the difference in level is found by subtracting the readings, the equal errors cancel.
What is reciprocal levelling?
A method in which observations are taken from near each of the two points in turn, the true difference in elevation being the mean of the two apparent differences.
What is the reciprocal levelling formula?
H = ½{(ha − hb) + (Ha − Hb)}, where ha and hb are the staff readings at A and B with the instrument near A, and Ha and Hb the corresponding readings with the instrument near B.
Why does averaging the two observations work?
Because with the instrument near A the long sight is to B, while with it near B the long sight is to A. The error therefore appears equally but in opposite directions in the two observations, and cancels in the mean.
When is reciprocal levelling used?
When an obstacle such as a river or deep valley makes it impossible to place the instrument midway between the two points, so that sights cannot be balanced.
What is the formula for the total error?
e = ½{(ha − hb) − (Ha − Hb)}, which is half the disagreement between the two apparent differences.
What does the total error consist of?
e = ecol + ecur − eref, comprising the collimation error, the curvature error and the refraction error.
Which errors does reciprocal levelling eliminate?
Error in the line of collimation, the combined effect of the earth’s curvature and refraction, and variation in the average refraction.
Why is variation in average refraction listed separately?
Because a computed correction can allow for the standard one seventh refraction effect but cannot allow for departures from that average caused by changing atmospheric conditions. Reciprocal levelling cancels such errors without needing to know their size.