A larger residual does not automatically mean a noisier instrument.
Two groups of survey observations disagree. One group also has larger residuals after adjustment. Should you downweight it? First check whether the observations have comparable redundancy, whether a shared error moved them together, and whether the geometry can separate the noise of one group from the other.
This guide starts with a weighted mean, then shows what changes when positions and weights have to be estimated together. The arithmetic is fictional and uses one coordinate so you can see which conclusions come from the numbers and which need independent evidence.
Follow three questions: What did the weights assume? What can the remaining disagreements actually identify? Will the new weighting predict data that were not used to tune it?
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1 · Chapters 1–3: Understand the weights
Start with two measurements of the same quantity
Follow a simple adjustment, compare spread with bias and see why residuals are not raw errors.
Route 2 · Chapters 4–6: Diagnose the evidence
Read a residual as what remains after fitting
Test identifiability, correlation and isolated failures before fitting group variances.
Route 3 · Chapters 7–9: Review and transfer
Use a held-out check before calling the weights improved
Try a held-out check, preserve uncertainty and transfer the method to another survey.
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Understand the weights
Start with two measurements of the same quantity
Distinguish a variance component from one particular weight
Work the difference between group spread and group disagreement
CHAPTER 1 OF 9 · Understand the weights
1. Start with two measurements of the same quantity
Imagine that two independent methods estimate the same horizontal coordinate, measured as a correction from an agreed reference. One reports 10 millimetres and the other 18. Suppose each method is unbiased in the conditions of the example, and their standard uncertainties are 2 and 4 millimetres. The smaller uncertainty describes a more concentrated range of plausible random errors. It does not promise that the first individual measurement was closer to the truth.
In this simple independent case, a common least-squares choice is to use weights proportional to the inverse of the variances. Variance is the square of standard uncertainty, so the weights are proportional to one quarter and one sixteenth. The first measurement has four times the weight of the second. Using weights 4 and 1 gives the combined estimate (4 × 10 + 18) ÷ 5 = 11.6 millimetres. Equal weights instead give 14.
The two results answer different mathematical questions because they use different assumptions about the observations. If the first method has an unrecognised bias, the more heavily weighted estimate can be worse. If the errors are correlated, treating them as independent is also questionable. The calculation makes the consequences of the assumptions visible. It does not verify those assumptions.
A survey adjustment extends this logic to more observations and more unknowns. It uses equations relating measurements to the positions or other quantities being estimated. It also needs a model of uncertainty and relationships between observations. When a network disappoints, keep these two parts separate: the model of what the readings mean and the model of how their errors behave.
CHAPTER 2 OF 9 · Understand the weights
2. Distinguish a variance component from one particular weight
A variance component is a parameter used to describe one contribution to the covariance of the observations. For a first example, imagine one component for a group of independent distance observations and another for a group of independent direction observations. In practice, the known structures might include how precision changes with sight length, or patterns of correlation. One estimated component can scale an entire specified structure, not just a single observation.
Weights are what the adjustment uses to combine measurements. In the general correlated case, a weight matrix is related to the inverse of the assumed covariance matrix, where that inverse is appropriate. The entries do not have to reduce to a separate positive weight for each reading. If two observations share an error source, the covariance model can represent that dependence instead of counting them as fully independent pieces of evidence.
It is tempting to put all observations made by the same instrument into one group. Ask whether they actually share the same noise conditions. A short indoor distance and a long sight across a heated surface might have different sources of uncertainty. At the same time, creating a separate unknown variance for every small subgroup can leave too little information to estimate any of them reliably. Use a grouping justified by the observations and the physics.
A useful question for each component is: if this component were larger, where would the network show more variation after fitting its unknowns? If you cannot answer, the proposed group may be hard to test. Choose the components before adjusting the result to match what you hoped the network would show.
CHAPTER 3 OF 9 · Understand the weights
3. Work the difference between group spread and group disagreement
In a fictional repeat experiment, group A measures the same stable coordinate as 9, 10, 10 and 11 millimetres. Group B measures it as 14, 17, 19 and 22. The group means are 10 and 18. Around their own means, the first group has deviations −1, 0, 0 and +1; the second has −4, −1, +1 and +4. Their sums of squared deviations are 2 and 34 square millimetres.
If the repeats within each group were independent and had common variance within that group, the usual sample variance in this toy calculation divides by three, because each group mean was estimated from its four readings. The estimates are about 0.67 and 11.33 square millimetres. These are small-sample illustrations of within-group spread, not dependable instrument specifications and not a general network variance-component solution.
Now look at the difference between the means: eight millimetres. That difference is not the within-group spread. If the measurements were taken at the same point under a common definition, investigate whether an offset, reference error or changed condition accounts for the disagreement. Simply weighting group B less can make the combined result look tidy while leaving the cause untested.
Change one fact: group B used a different reference origin that is independently known to add eight millimetres. Correcting that origin in the example brings both means to 10. The within-group spreads do not change when the same constant is subtracted from every B reading. A bias correction and a variance estimate do different jobs. In real work, an independently uncertain correction may also contribute to the final uncertainty.
CHAPTER 4 OF 9 · Diagnose the evidence
4. Read a residual as what remains after fitting
A residual is the difference between an observed value and its value predicted from the fitted model, under a stated sign convention. It is not the same as the observation’s unknown error against the true quantity. The fitted positions have already adapted to the data. An observation that has strong influence can pull a solution toward itself and end up with a small residual even if it is wrong.
Use a two-reading thought experiment. Fit an unknown coordinate to two independent readings with equal weights. Both residuals are equal in magnitude and opposite in sign. A discrepancy can show that the pair disagrees, but the pattern alone does not tell you which reading was wrong. With strongly unequal weights, the residual is generally smaller on the more influential observation. That fact reflects the fitting rule, not an independent check of its accuracy.
In a larger network, different observation types may have different opportunities to be checked by other observations. A line that is the only route to a new point may fit with little visible discrepancy. A measurement in a highly redundant loop may expose a disagreement. Comparing raw residual sizes between these situations can confuse network geometry with instrument noise.
Before comparing observation classes, inspect redundancy and use residual scales appropriate to the fitted model. When quantities have different units, such as lengths and angles, do not compare their numerical residuals as though the units matched. Even a standardised residual is a diagnostic, not a physical diagnosis. Ask which setup, observation or assumption can actually explain the pattern.
CHAPTER 5 OF 9 · Diagnose the evidence
5. Ask whether the network can distinguish the components
Imagine there is one independent measurement from group A and one from group B of the same unknown coordinate. Their difference cancels the coordinate. Under independent zero-mean errors, the variance of that difference is the sum of the two variances. One such contrast does not tell you how to split the sum. Even many repeats of that same type of contrast can reveal the sum without separating the contributions.
For a simple extension, suppose independent contrasts between A and B have a variance estimated as 13 square millimetres. Separate same-condition comparisons between pairs of independent A observations give a difference variance of 8 square millimetres. Under the assumptions of this example, two A variances make 8, so A contributes 4. B then contributes 9 to the A–B difference. The extra comparison changed what was identifiable; it did not merely add more readings of the same kind.
A real survey network can have the same problem in less obvious forms. Its geometry and unknowns may leave residual combinations that respond similarly to two proposed components. A fitting routine may return numbers, but the data can still provide little independent support for the split. Examine sensitivity, the geometry of the checks and the stability of the estimates when observations are held out.
The lesson is not to search for a particular number of repeats. It is to ask whether the experiment contains observations that respond differently to the candidate sources of variation. If not, consider a simpler grouped model or report which combination is supported. That is more defensible than treating two poorly separated estimates as if they were independent measurements.
CHAPTER 6 OF 9 · Diagnose the evidence
6. Keep correlation, a bad reading and bias as separate possibilities
Readings taken on the same setup can share an error. For example, a mistaken centring condition may affect several directions at once. More readings from that setup can improve estimates of pointing repeatability without providing independent evidence that the instrument was centred. A model with only independent errors can overstate how much uncertainty was reduced by repetition.
A single gross error is a different problem. Suppose ten distance observations agree within the range expected for their geometry, while one is off by far more and has an ambiguous target identification. Inflating the group variance because of that one observation can downweight all eleven, including the ten useful ones. Investigate the suspect reading and keep its original record. A broad variance change should not be the default repair for a local mistake.
A systematic bias can produce the opposite visual pattern: a whole group repeats closely while being consistently displaced. Look for signed differences by setup, direction, time or instrument, and compare with independent checks. Do not expect an estimate of random scatter to discover every systematic error. If a bias model is proposed, test whether the network can distinguish it from a change in the unknown positions.
Ask which of three evidence changes would help: repeat under the same setup, reobserve independently, or add a contrast with different geometry. Same-setup repeats can illuminate repeatability. An independent setup can test a shared setup problem. A contrast that changes the geometry can help distinguish causes that previously looked alike. None is a universal substitute for the others.
CHAPTER 7 OF 9 · Review and transfer
7. Use a held-out check before calling the weights improved
Suppose a proposed adjustment increases the variance assigned to one group. Its residuals now look less surprising under the new model. That is not a sufficient acceptance test: widening an expected range can make almost any discrepancy look less unusual. Ask whether the adjusted solution predicts independent checks more appropriately and whether the new uncertainties agree with those checks over a meaningful range.
For a fictional illustration, two candidate adjustments predict a held-out coordinate check of 11.6 and 14 millimetres. The independent check is 12 millimetres, with its own stated uncertainty. The first result is closer on this occasion, but one point cannot prove its weights are right. Compare multiple withheld observations, check what errors the checks share with the adjustment and report sensitivity to any unusually influential check.
Do not choose the validation observations after seeing which ones favour the preferred result. Decide which check points or independent measurements answer the question in advance, or disclose when the selection was exploratory. If a check helped estimate the weights, it is not an independent validation of those same weights. It can still be useful as an internal diagnostic; label it accordingly.
A good review looks at both the residual pattern and the precision of the final coordinates. A model may reduce a systematic pattern at one group of observations while making an important coordinate weaker. The operational question is whether the result is fit for its stated purpose and conditions. A neater residual table is not necessarily a better survey.
CHAPTER 8 OF 9 · Review and transfer
8. Report an uncertain estimate without hiding the uncertainty
Record the observation model, proposed covariance groups, supporting repeat or control evidence, and the data used to estimate each component. If the method constrains variances to be nonnegative, record any boundary result and what it means for the comparison. A component estimated at or near zero need not mean a device makes no errors. The data or model may not distinguish that contribution well.
Keep the original and revised adjustment results together. Explain what changed: the observations, the treatment of a suspect reading, an independently established correction, or the weighting assumptions. A coordinate can change when you revise the weights even if no observation was added. A later reader should be able to tell whether the change was new evidence or a new interpretation of existing evidence.
For the worked examples, a careful note might say: “Independent repeats suggest different within-group spreads under the stated conditions. The eight-millimetre difference of group means was investigated separately. Two sets of contrasts were needed to distinguish the example variance components. Results are not instrument specifications.” Each sentence names the level of evidence it actually has.
If the adjustment is being used for real survey decisions, follow the relevant professional requirements and project tolerances. This guide does not provide acceptance limits. A defensible report gives enough information for a responsible reviewer to judge the fit between the measurements, assumptions and intended use. It does not replace that review with a generic cutoff.
CHAPTER 9 OF 9 · Review and transfer
9. Transfer: compare two mapping passes with a shared reference
Imagine two crews estimate the elevation of the same point from repeat observations. Crew A’s repeats cluster tightly and crew B’s are more dispersed. Both crews used a common benchmark whose elevation has a possible error. Can you treat all their observations as independent and claim that averaging both crews eliminates the benchmark uncertainty? No. A shared contribution may persist even when independent scatter shrinks.
Change one fact: crew B used an independent reference whose relationship to the first is documented and uncertain. This creates a potential check on reference agreement, but its uncertainty and scope need to be considered. It does not automatically reveal which crew is right. Compare the actual difference with what the reference and observation models predict. The independent reference changes the evidence available, not just the number of lines in the file.
Make a final change: one reading from crew A was taken against the wrong target. Should you increase the variance of every A reading? Investigate the wrong-target evidence and its effect on the adjustment first. A mistake with a specific explanation is different from broad random variation. This is the same distinction that kept the fictional distance group from being downweighted because of one bad reading.
To check your transfer, write a separate sentence for repeatability, shared uncertainty and evidence of a mistake. Then name one independent check for each. If all three sentences use the same residual as proof, you have not yet separated the mechanisms. The strongest next observation is often the one that makes two rival explanations predict different outcomes.
Independent worked case and evidence checks
In a fictional adjustment, two classes are compared through the difference between one A and one B observation of the same quantity. The observed spread of many comparable differences suggests a variance of 20 square millimetres under the model. One colleague proposes A = 5 and B = 15; another proposes A = 12 and B = 8. Both pairs give the same variance for the differences if the observations are independent. More of the same contrast can improve the estimate of the sum without choosing a split.
Now add independent B–B comparisons with difference variance 18 square millimetres under comparable conditions. Since an independent pair contributes two B variances, the illustrative estimate becomes B = 9. That leaves A = 11. If the two B readings shared a setup error, however, their difference could cancel some of that shared component. The same arithmetic would no longer estimate the full variance of a B observation.
Add an independent B observation from a different setup and compare it with the same-setup repeats. If between-setup differences are substantially larger, the original within-setup comparison may have estimated only one part of the uncertainty. It was not useless; it answered a narrower question. Decide whether to model the additional contribution, redesign the comparison or report the limit.
As a final check, ask another reader to state what would change if A and B errors were correlated. The variance of an A–B difference then also depends on their covariance. The simple sum is no longer sufficient. This makes an assumption in the arithmetic visible, and shows why the numerical exercise cannot be lifted unchanged into a correlated survey network.
Here is the bridge from the arithmetic to a real model. Suppose observations are collected in a vector y, and a linear observation model predicts their means from unknown coordinates. Form combinations of the observations that eliminate those unknown means. In the toy problem, A minus B and one A reading minus another A reading are such combinations. In a network, the useful combinations follow from the observation equations and can involve more than two readings. Their expected scatter still depends on the original observation covariance.
Write that covariance as a sum of known patterns multiplied by unknown component values. Each pattern says where one source of variance or covariance would appear. After taking the combinations, carry each pattern through the same operation. Two original patterns can become indistinguishable in the resulting comparisons. If that happens, an apparently reasonable list of components is not estimable from those data alone. Add a comparison that responds differently, or state the combined quantity that can be estimated.
Methods such as least-squares variance component estimation use this connection between squared or cross-product terms and their expected values. The full estimation needs more care than dividing each group’s squared residuals by its observation count. It accounts for what was fitted, how the covariance is structured and what information the network supplies. Iterating an estimate of the covariance and the positions does not add evidence that the geometry never contained.
Check that the fitted covariance is suitable for the intended adjustment. A set of numerical component estimates can be unstable, or produce an inadmissible covariance under a particular model. Investigate the model and its constraints instead of treating a returned number as an instrument property. If the weights are estimated from the same limited observations, a reported coordinate uncertainty that treats those weights as perfectly known can also be overconfident. Disclose the interpretation and use independent validation where available.
For a last desk check, suppose the estimated variance of the A–B contrasts was 5 square millimetres but the independent B–B result still implied a B variance of 9. The toy subtraction would give a negative A variance. That is not a physical variance. Check uncertainty in both estimates, whether the comparisons really were independent and whether their conditions matched. A constrained result might put A at a boundary, but the boundary alone does not repair a bad comparison. Report the inconsistency and its effect on the use you planned.
Worked comparison
| Pattern | Candidate interpretation | Independent check |
|---|---|---|
| One group has large residuals | Noise, redundancy or a model mismatch | Compare geometry and appropriately scaled residuals |
| Repeated readings are tightly grouped but shifted | A shared setup or reference bias | Use an independent setup or reference |
| One reading disagrees strongly | A local problem, not necessarily a noisy class | Verify the target and original observation |
| Only mixed contrasts are available | The sum may be identifiable but not the split | Add an informative same-class or different-geometry check |
Questions and practical checks
Is a variance component the same as an individual weight?
No. It describes a contribution to the observation covariance; the weight model reflects the resulting covariance assumptions.
Can a small residual hide a bad observation?
Yes. A highly influential reading may pull the fitted result toward itself. Check redundancy and independent evidence.
Do larger weights mean more accuracy?
They mean the adjustment assigns more influence under its uncertainty model. Bias or a wrong model can still make the result inaccurate.
Can I estimate two variances from their sum?
No. The sum alone does not identify both. Use additional contrasts with different sensitivity or report the supported combination.
Should I delete an outlier before estimating weights?
Do not remove observations solely to improve appearance. Investigate the evidence and document any exclusions under an appropriate procedure.
Can the worked figures be used as field limits?
No. They are illustrative values, not professional tolerances or a calibration result.
Continue with the connected subjects
Understand the adjustment and its checks
Residuals, redundancy and leverage: why some survey errors remain hidden
Free networks: distinguish relative precision from a datum tie
Error ellipses: see how covariance affects coordinate uncertainty
Check physical sources of disagreement
Centring and short sights: separate a setup fault from repeatability
Reference prisms and moving control: test a common source of error
Sources and scope
Teunissen and Amiri-Simkooei: Least-squares variance component estimation
Open research copy: models, estimability and precision of variance components
Amiri-Simkooei: theory and geodetic applications of least-squares variance component estimation
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 5 November 2026.