A precise height difference can coexist with uncertain absolute heights
Two nearby points have carefully reported heights, each with an uncertainty figure. You want their height difference. Adding the two printed uncertainties is not automatically the right calculation, and assuming every shared error cancels is no better. The answer depends on the quantity being compared and the relationships among its uncertainty components.
Geodesy distinguishes ellipsoidal and gravity-related heights. This guide develops the uncertainty question beneath their comparison: how a changing geoid term affects a height difference, when a common model component cancels, and why spatial correlation must come from evidence rather than a convenient assumption about nearby points.
All heights, gradients, correlations and uncertainty values below are fictional teaching inputs. They are not values for a Singapore datum, a real geoid product or a construction site. The examples explain measurement reasoning and do not establish drainage direction, engineering acceptance, a legal boundary or an operational survey tolerance.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1: Define the height difference · Chapters 1–3
Route 2: Propagate the relationships · Chapters 4–6
Derive the variance of a difference with covariance included
Route 3: Check assumptions and scope · Chapters 7–8
Route 4: Interpret evidence carefully · Chapters 9–10
Route 5: Transfer and deliver · Chapters 11–12
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Define the height difference · Chapters 1–3
Name the height quantity before subtracting
Derive the difference instead of cancelling the geoid by habit
Separate a shared offset from a spatially changing error
Propagate the relationships · Chapters 4–6
Derive the variance of a difference with covariance included
Work a correlated geoid-difference example
Keep the full height model when more terms are correlated
Check assumptions and scope · Chapters 7–8
Do not infer covariance from proximity alone
Keep model compatibility and epoch separate from uncertainty arithmetic
Interpret evidence carefully · Chapters 9–10
State what the uncertainty figure actually covers
Choose independent checks that test the intended claim
Transfer and deliver · Chapters 11–12
Ask which additional evidence would improve the decision
Deliver the difference and its dependence assumptions together
CHAPTER 1 OF 12
1. Name the height quantity before subtracting
An ellipsoidal height is geometrically referenced to an ellipsoid. A gravity-related physical height uses another reference arrangement. Under a compatible simplified relation, write H=h−N, where h is ellipsoidal height and N is the relevant geoid-height term. The equation is useful only when the point, reference conventions, model and intended height type belong together.
Do not infer compatibility from the common word elevation. A receiver output, a benchmark record and a model grid can use different references or physical endpoints. Subtracting them produces a number whether or not the subtraction has a justified meaning. Start with a ledger identifying the point, height type, frame, epoch, vertical reference and model used.
The NGS GEOID18 technical documentation is useful because it states a specific reference arrangement and the scope of its uncertainty estimates. Its United States model is not adopted here as a universal height reference. The general lesson is to preserve the actual product’s stated meaning rather than borrowing a familiar geoid number for an unrelated location or datum.
Our target is a difference between two compatible physical heights, not the height of one point in isolation. That target determines which terms enter and which dependencies matter. A comparison can be more precise or less precise than a reader expects from the separate absolute heights, but only after the relationship among their errors has been established.
CHAPTER 2 OF 12
2. Derive the difference instead of cancelling the geoid by habit
Let A and B be two fictional points. Subtract H_A=h_A−N_A from H_B=h_B−N_B. The result is ΔH=(h_B−h_A)−(N_B−N_A), or ΔH=Δh−ΔN. A geoid term cancels completely only when the relevant difference of those terms is actually zero under the intended model. The shared name of a model does not make its values identical at different locations.
Take h_A=50.000 and N_A=12.000 metres, giving H_A=38.000. At B, let h_B=50.300 and N_B=12.180, giving H_B=38.120. The ellipsoidal difference is0.300 metres, the modelled geoid difference is0.180 and the physical-height difference is0.120. All values are invented, but their relationship is directly checkable.
If a reader subtracts only the ellipsoidal heights, the answer differs by0.180 metres in this fixture. That is a missing reference term, not a random rounding discrepancy. Applying another decimal place to h cannot recover it. The appropriate repair is to establish and apply the compatible height relationship, not to average the two different answers.
Reverse the relation to check signs: Δh=ΔH+ΔN=0.120+0.180=0.300. A negative geoid difference would change the arithmetic accordingly. Keep the signed quantity rather than replacing it with its magnitude. The independent reverse check is especially useful when a height relation is being copied across tables or explained in another language.
CHAPTER 3 OF 12
3. Separate a shared offset from a spatially changing error
Suppose a fictional geoid model has the same added offset of+0.500 metres at both A and B. Both derived H values decrease by0.500, but their difference remains0.120. The common component cancels from the difference. That cancellation does not prove that either absolute height is correct, nor does it establish that every model error is a common offset.
Now add an error only at B. If N_B is0.200 metres too high while N_A is unchanged, the inferred ΔH becomes−0.080 rather than+0.120. The apparent sign changes. The spatially varying error did not cancel, even though both values came from the same named model. A common product identity is not a substitute for an error-correlation model.
This is why absolute and differential accuracy are different questions. A model can have a substantial shared offset and still describe some local differences well. It can also fit absolute values at selected controls while having an inadequately resolved local variation between them. Establish which property the available validation actually supports.
For a changed-context exercise, replace two heights with two readings from one instrument that shares a zero offset. The offset can cancel in their difference if it is genuinely common and stable. If the instrument is reset between readings, that assumption changes. The transferable reasoning concerns shared components; it does not make the instrument’s particular calibration identical to a geoid model.
CHAPTER 4 OF 12
4. Derive the variance of a difference with covariance included
For two uncertain quantities X and Y, the variance of Y−X is Var(Y)+Var(X)−2Cov(X,Y). The minus sign follows from expanding the square of the difference of their centred fluctuations. It is not an optional adjustment added only when the result seems too large. Covariance belongs to the relationship between the quantities being differenced.
If the quantities have standard uncertainties u_X,u_Y and correlationρ, their covariance isρu_Xu_Y. With equal uncertainties u, the difference variance is2u²(1−ρ). Positive correlation can reduce the uncertainty of the difference. Negative correlation can increase it. Zero covariance gives the familiar sum of variances, not a universal rule that all observations are independent.
A standard uncertainty is the square root of the corresponding variance. It should not be confused with a maximum error, a guaranteed bound or a coverage interval without additional assumptions. The variance identity itself does not require a normal distribution; turning a standard uncertainty into a particular probability statement requires more information about the distribution and estimation.
This formula describes the stated uncertainty model. It cannot compensate for a wrong datum, an omitted bias, an incorrect physical point or unsupported covariance values. A mathematically valid calculation can remain incomplete because its inputs do not represent all relevant sources of uncertainty. Name those limits alongside the result rather than allowing a small number to conceal them.
CHAPTER 5 OF 12
5. Work a correlated geoid-difference example
Stipulate standard uncertainty0.040 metres for each of N_A and N_B, with correlationρ=0.95. The covariance is0.95×0.040×0.040=0.001520 square metres. The variance of ΔN is0.001600+0.001600−2×0.001520=0.000160 square metres. Its standard uncertainty is about0.01265 metres.
Now give each ellipsoidal height a standard uncertainty0.010 metres and assume their errors are independent. The variance of Δh is0.000200 square metres. Also stipulate that the ellipsoidal-height difference and geoid-height difference are uncorrelated. Then the variance of ΔH is0.000200+0.000160=0.000360, yielding standard uncertainty about0.01897 metres.
Compare the alternative assumption that the two geoid errors are independent. The ΔN variance becomes0.003200, and the ΔH variance becomes0.003400, with standard uncertainty about0.05831 metres. The height difference itself remains0.120. A different supported dependence model can therefore change the uncertainty while leaving the central arithmetic unchanged.
Neitherρ=0.95 norρ=0 is a default established for any real geoid product by this example. They are deliberately stated alternatives. The exercise teaches which information changes the result. A reviewer should ask for the applicable covariance evidence rather than choosing whichever assumption produces a preferred precision claim.
CHAPTER 6 OF 12
6. Keep the full height model when more terms are correlated
The preceding example deliberately separated the ellipsoidal and geoid terms. A more general linear combination can be written with the input vector(h_A,h_B,N_A,N_B) and coefficient vector(−1,+1,+1,−1). Its variance is the coefficient vector times the full covariance matrix times its transpose, with the ordering stated consistently. All relevant covariance terms belong in that account.
You do not need matrix notation to understand the principle. Expand the height difference and ask which pairs of uncertain terms share information or errors. Reusing observations, common reference solutions or linked model inputs can create relationships that a list of four separate standard deviations does not describe. Do not assume independence merely because the quantities occupy different spreadsheet columns.
Location uncertainty can matter too when a geoid value is evaluated at a position. In a fictional one-dimensional model N(x)=12+0.0001x metres, a ten-metre x error changes the evaluated N by0.001 metre. That derivative describes sensitivity under the stated local model, not a measured gradient or recommended uncertainty allowance for any real region.
A demanding real analysis needs the appropriate spatial model, position uncertainty and relevant cross-relationships. Our simple derivative only makes the dependency visible. It also explains why a sharp-looking model grid does not automatically justify arbitrary precision at every interpolated point. Numerical evaluation and physical resolving power are different parts of the evidence chain.
CHAPTER 7 OF 12
7. Do not infer covariance from proximity alone
Nearby points may share some model errors, but distance alone does not supply a numerical correlation coefficient. The applicable covariance depends on the product, observations, modelling and region. An analyst cannot justifyρ=1 merely by calling two points close, orρ=0 merely because their coordinate records were exported as separate rows.
A correlation assumption should have a documented basis appropriate to the task. That can involve the provider’s uncertainty model, suitable validation or a carefully bounded analysis of the available evidence. If it is not established, state the uncertainty or investigate a range of justified possibilities without labelling a convenient scenario as a measured property.
Check mathematical consistency as well. A proposed covariance cannot be arbitrarily larger than the product of the two standard deviations. An impossible combination can produce a negative variance for a difference. Do not clamp that negative result to zero and claim extraordinary precision; investigate the incompatible inputs or model that produced it.
Also distinguish a known deterministic correction from a random uncertainty component. If a known offset is present, address it appropriately rather than automatically hiding it inside a standard deviation. An unresolved systematic effect may require a separate account. A covariance formula is not a licence to treat every kind of missing reference information as well-behaved random noise.
CHAPTER 8 OF 12
8. Keep model compatibility and epoch separate from uncertainty arithmetic
Two geoid values from incompatible models or reference arrangements cannot be made comparable merely by assigning large uncertainty bars. The first question remains whether they represent the terms required by the same height relationship. Establish frame, datum, epoch, physical point and model version before propagating their numerical uncertainties.
For a fictional negative test, let one height use a model with a shared reference offset and the other use another model without it. The intended cancellation no longer applies. A report that assumed the two errors were perfectly shared can now understate both the reference difference and its practical consequence. The model identity is part of the calculation, not an administrative footnote.
Time matters when the physical points or reference realisations change. A height difference formed from different epochs can include genuine differential movement. That contribution is not a geoid covariance correction. The connected epoch guide separates coordinate time from model parameter time and shows why an appropriate motion account may be needed before comparison.
Do not interpret an uncertainty analysis as permission to ignore these compatibility questions. It becomes meaningful after the target quantity and inputs are correctly specified. A result with several decimal places and a formally propagated standard uncertainty can still answer the wrong question if one input belongs to another reference or physical endpoint.
CHAPTER 9 OF 12
9. State what the uncertainty figure actually covers
A provider’s uncertainty product has a defined scope. The NGS GEOID18 technical details explicitly explain what its estimated uncertainties represent and which broader datum or underlying-model errors they do not include. That careful scope statement is as important as the numerical values. It prevents a fit statistic from silently becoming a universal claim about every height derived from the product.
For our teaching case, the reported0.01897-metre standard uncertainty covers the stipulated ellipsoidal and geoid terms under the stated correlations. It does not include an unidentified antenna-height error, a wrong monument, an unmodelled discontinuity or a datum mismatch. Adding those words makes the number more useful because the reader can see which dependencies remain open.
Do not automatically call two standard uncertainties a95% interval or treat a standard uncertainty as a hard limit. Coverage interpretation depends on distributions, degrees of freedom and other relevant assumptions. A classroom variance calculation is not an engineering acceptance rule, and this article does not supply a probability-of-failure claim for a real height comparison.
Likewise, a height difference alone does not establish water-flow direction for every practical situation. Hydraulic and engineering questions require their own physical variables, reference conditions and professional evidence. The examples concern compatible measurement quantities and uncertainty, not a shortcut from a small computed elevation difference to an operational safety decision.
CHAPTER 10 OF 12
10. Choose independent checks that test the intended claim
An absolute-height check and a differential-height check can reveal different weaknesses. If two points share a model offset, their difference may agree with an independent relative measurement while both absolute heights disagree with a suitable reference. Neither observation is contradictory. They test different combinations of the underlying terms.
A check is less informative when it reuses the same model assumption being tested. Comparing two height products derived from the same geoid grid can show implementation consistency while preserving the same underlying spatial error. Identify which source components are shared before describing the comparison as independent validation.
An independently observed height difference also needs its own reference, uncertainty and validity conditions. The existing levelling-loop guide explains why a small closure error does not establish an absolute datum or prove every observation error is absent. Use that evidence within its actual scope instead of replacing one over-broad accuracy claim with another.
For a practical fixture, test an equal common offset and an unequal spatial error separately. The first should cancel from the difference under the stipulated model; the second should not. Then change the reference arrangement and show why the earlier cancellation claim is no longer justified. These cases expose assumptions that a single successful comparison could conceal.
CHAPTER 11 OF 12
11. Ask which additional evidence would improve the decision
Once the uncertainty model is explicit, you can see which component matters. In ourρ=0.95 fixture, both the ellipsoidal difference variance0.000200 and geoid difference variance0.000160 contribute materially. Improving only one cannot eliminate the other. The model helps identify useful follow-up evidence rather than encouraging indiscriminate repetition of whichever observation is easiest to collect.
Ifρ were genuinely near1 for the relevant geoid errors, the differential geoid contribution could be small while absolute-height uncertainty remains substantial. If the correlation were unknown, better evidence about spatial dependence might matter more than another decimal place in the exported heights. The priority follows from the target quantity and the actual uncertainty budget.
Repeated measurements can reduce some random variation under appropriate assumptions. They do not automatically remove shared model bias, a reference mismatch or an incorrectly measured antenna offset. The connected antenna guide makes one physical reference-point dependency explicit. Preserve those non-interchangeable controls instead of assuming a larger sample size repairs every problem.
A useful next step can also be a narrower claim. A dataset may support an approximate regional comparison while not resolving a small local difference for a demanding task. State what remains useful and what additional evidence is needed for the stronger claim. That is a proportionate response, not an excuse to manufacture confidence or reject every imperfect observation.
CHAPTER 12 OF 12
12. Deliver the difference and its dependence assumptions together
A complete handoff states the two points, compatible height references, epochs, model versions, central difference, uncertainty components, covariance assumptions, omitted effects and independent checks. It distinguishes absolute-height evidence from differential-height evidence. A single number labelled accuracy cannot reliably carry all of that meaning.
Ask a learner to predict what happens when an identical offset is added to both geoid values, then when an offset is added to only one. Next ask what changes if the correlation assumption changes without changing the central height values. These are three different tests: common-mode cancellation, spatially varying error and uncertainty-model dependence.
The learner should also explain why a perfect numerical cancellation does not establish that the shared-error assumption is true. That fact must come from the measurement and model evidence. First-principles algebra reveals the consequence of an assumption; it does not independently certify the assumption for a real geodetic product.
The endpoint is a height comparison whose precision claim can be inspected. You can show which physical difference was calculated, why its reference terms are compatible, what cancellation is supported and which uncertainties remain. That is stronger than either adding uncertainty figures by habit or assuming every nearby point benefits from perfect shared-error cancellation.
Independent worked case and evidence checks
Try an independent fictional pair C and D. Let h_C=80.200 and N_C=25.100 metres, giving H_C=55.100. Let h_D=80.500 and N_D=25.250, giving H_D=55.250. The ellipsoidal difference is0.300, the geoid difference0.150 and the physical-height difference0.150 metres. Reverse the relation to check the sign.
Give each ellipsoidal height standard uncertainty0.020 metres, with independent errors. Their difference variance is2×0.020²=0.000800 square metres. Give each geoid value standard uncertainty0.050 metres and stipulate correlation0.80. The geoid-difference variance is2×0.050²×(1−0.80)=0.001000 square metres.
Assume the ellipsoidal and geoid differences are uncorrelated in this fixture. The combined variance is0.001800 and standard uncertainty about0.04243 metres. If the geoid errors were instead independent, the combined variance would be0.005800 and standard uncertainty about0.07616 metres. The central0.150-metre height difference does not change when only that uncertainty assumption changes.
Add the same fictional+0.300-metre offset to both N values. The derived heights become54.800 and54.950, preserving their0.150 difference. An independent differential check could still agree while an absolute check reveals the common offset. Explain why the two comparisons are testing different quantities rather than competing to provide one universal verdict.
Now add an extra error only to N_D. That term changes the difference and does not cancel. Ask which evidence would establish its size or exclude it. The fact that C and D are nearby, or that both values carry the same model name, is not enough to prove a particular correlation or absence of a local error.
For transfer, suppose the height records describe different physical endpoints: one monument and one antenna point that has not been reduced. The covariance arithmetic remains executable, but it no longer compares the intended pair of monument heights. Recover the physical reference-point relationship before interpreting the result as a resolved geodetic difference.
Finally write the conclusion without calling0.04243 a maximum error or an automatic confidence interval. State the standard uncertainty, its stipulated components and correlation assumptions, the reference compatibility and the unresolved effects. A recipient should be able to recompute the result and identify exactly which new evidence would justify a stronger claim.
Check the same uncertainty through the derived absolute heights. Each H has variance0.020²+0.050²=0.002900 square metres, giving standard uncertainty about0.05385 metres under the stated independence between height and geoid terms. The covariance between the two H values is0.002000 square metres, inherited here from the shared geoid errors. Differencing gives0.002900+0.002900−2×0.002000=0.001800, matching the earlier route.
The correlation of the derived H values is therefore0.002000/0.002900≈0.68966, not the original geoid-error correlation0.80. A coefficient belongs to the quantities it describes; it cannot be copied automatically from one stage of the measurement model to another. This cross-check is useful because the central difference can remain correct even when a correlation has been attached to the wrong variables.
If a reviewer substitutes0.80 as the H correlation, the resulting variance is0.001160 instead of0.001800. The disagreement is not evidence that one route is inherently better. It exposes inconsistent dependence assumptions. Restore the covariance relationships before describing the smaller result as improved precision.
Worked comparison
| Question | Required relationship | What does not follow automatically |
|---|---|---|
| What is the height difference? | ΔH=Δh−ΔN under compatible conventions | The geoid term always cancels |
| What is the difference variance? | Both variances and their covariance | Separate rows imply independence |
| Does a common offset cancel? | The offset must genuinely be shared | Absolute heights are therefore correct |
| Are nearby model errors correlated? | Applicable uncertainty evidence | A universal coefficient from distance alone |
| What does a standard uncertainty mean? | Defined quantity and covered components | A maximum error or default coverage interval |
| Is a comparison independent? | Different relevant evidence pathways | Two products using one shared model are independent |
Questions and practical checks
Can I subtract ellipsoidal heights to get every physical-height difference?
Not generally. Under the compatible simplified relation, include the difference in the relevant geoid term.
Do shared geoid errors always cancel?
Only the components actually shared in the difference. Spatially changing errors and incompatible references need separate treatment.
Does positive correlation increase uncertainty of a difference?
For fixed individual variances it reduces the variance of the difference. That mathematical consequence does not establish the actual correlation.
Can I assume perfect correlation for nearby points?
No. Use the applicable product or validation evidence and state unresolved dependence.
Does a small calculated uncertainty approve a real project?
No. Its scope, reference compatibility, omitted effects, independent checks and governing professional requirements remain essential.
Continue with the connected subjects
Keep physical height and instrument references compatible
Geodesy overview: ellipsoidal and gravity-related height references
Height and uncertainty terminology: preserve the source claim rather than expanding it
Use independent differences and observation histories
Levelling loops: closure, variance and the limits of an internal check
Campaign comparisons: shared and independent error components
Reference consistency: matching declarations do not independently prove the positions
Sources and scope
JCGM: published guidance on measurement models, uncertainty and interpretation
The numerical cases in this guide are original fictional teaching examples, not measurements from a real site. This is an educational explanation, not a field safety instruction, engineering approval or an official certification. For a real project, use qualified practitioners, current product documentation and site-specific evidence. Sources were checked on 31 October 2026.
Connect the geodetic checks
These guides examine different links in one position claim. A correct time model does not prove the antenna setup, and a correct instrument model does not establish the reference or uncertainty. Follow the unresolved link rather than treating one passing check as complete validation.
Coordinate epochs: separate the clocks and moving-frame model. Establish compatible frames and epochs before treating a coordinate difference as movement.
Antenna reference points: connect setup geometry with the reported point. Identify the actual physical endpoint and which reductions have already been applied.
Carrier-phase continuity: separate ambiguity changes from displacement. Preserve observation arcs and independent evidence before interpreting a phase jump as ground motion.