Combining two solutions requires their information, not just two coordinate averages
Two observing techniques produce estimates of a common coordinate. Can we average their answers and obtain a better one? Sometimes, but the answer depends on what each estimate means, which nuisance parameters were fitted, which observations or constraints they share, and whether their uncertainty descriptions refer to the same quantity.
This guide builds a small static adjustment from fictional observations. It then eliminates a nuisance parameter while preserving its effect, combines independent contributions and constructs a case where treating one shared nuisance parameter as two separate ones changes the information retained. You will also see how duplicating a normal equation can make uncertainty look better without adding any evidence.
The scalar coordinate and length-equivalent biases below are educational stand-ins, not actual station parameters or an operational terrestrial-frame combination. Real geodetic products require compatible frames, epochs, observation models, parameter definitions and stochastic information. No invented number here is a weight, acceptance threshold or calibration for an actual technique.
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Choose your reading route
Route 1 · Chapters 1–2: Define the information being combined
Route 2 · Chapters 3–5: Retain nuisance uncertainty
Route 3 · Chapters 6–8: Combine compatible systems
Combine independent reduced contributions to the same parameter
Route 4 · Chapters 9–10: Challenge constraints and dependence
Align reference values, units and parameter order before summing
Route 5 · Chapters 11–12: Transfer and report
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Define the information being combined
Identify common parameters before averaging estimates
Build the normal equations from a small observation model
Retain nuisance uncertainty
Verify the solution against the original observations
Eliminate the nuisance parameter without declaring it known
Keep marginal uncertainty distinct from a conditional shortcut
Combine compatible systems
Combine independent reduced contributions to the same parameter
Show why shared nuisance parameters must remain shared
Do not count the same observation information twice
Challenge constraints and dependence
Align reference values, units and parameter order before summing
Separate constraints, variance scales and model discrepancy
CHAPTER 1 OF 12 · Define the information being combined
1. Identify common parameters before averaging estimates
A common parameter is one physical or modelled quantity represented consistently in both adjustments. Two arrays labelled station position may instead refer to different instrument points, epochs or frames. The local-ties and coordinate-epoch companions explain why those distinctions must be settled before combination.
A nuisance parameter is an unknown needed to describe the observations but not the final quantity of interest. Examples can include appropriately parameterised timing, propagation or instrument contributions. Calling a parameter nuisance does not make it known, independent of the desired coordinate or safe to delete from the uncertainty calculation.
For our main example, x is a single coordinate correction in metres and a is a length-equivalent nuisance contribution, also in metres. The deliberately simple units avoid mixing seconds and metres. An actual clock parameter would need the corresponding conversion factor in its design column and covariance.
Begin by identifying the parameter order, reference values, units and model scope. A normal matrix is attached to these definitions. Adding two arrays with identical dimensions is not a valid combination if their rows and columns mean different things. Resolve the represented quantities before interpreting the apparent numerical agreement of their estimates.
CHAPTER 2 OF 12 · Define the information being combined
2. Build the normal equations from a small observation model
Use three fictional observations with independent errors, each of known unit variance in squared metres. Their predicted values are x+a, 2x and a, and their observed values are three, four and zero metres. The design rows are [1,1], [2,0] and [0,1]. All numerical scales are chosen for visible arithmetic, not realism.
Let A be that three-by-two design matrix and y = (3,4,0)ᵀ. Minimising |Aθ−y|² for θ = (x,a)ᵀ gives normal equations Nθ = b, where N = AᵀA and b = Aᵀy. Direct multiplication gives N = [[5,1],[1,2]] and b = (11,3)ᵀ.
The diagonal entries summarise sensitivity to each parameter, while the off-diagonal entry records their coupling in the adjustment. It is not an optional decoration. The determinant is nine, so this toy system is nonsingular. Solving its two equations gives x = 19/9 metres and a = 4/9 metres.
ESA Navipedia describes weighted parameter adjustment and the role of an observation covariance. Our unit-weight system is one special case. In general, N = AᵀR⁻¹A and b = AᵀR⁻¹y for a known positive-definite observation covariance R, with compatible residual and reference-value definitions. Those conditions belong with the algebra.
CHAPTER 3 OF 12 · Retain nuisance uncertainty
3. Verify the solution against the original observations
Substitute x = 19/9 and a = 4/9 into the three predictions. They are 23/9, 38/9 and 4/9 metres. With residual defined as predicted minus observed, the residuals are minus 4/9, plus 2/9 and plus 4/9 metres. Their squared sum is 4/9 square metres.
The normal equations require Aᵀv = 0 for this unweighted residual vector. The x column gives v_1+2v_2 = −4/9+4/9 = 0. The a column gives v_1+v_3 = 0. This is a valuable independent check on matrix construction, signs and parameter ordering.
Residuals need not vanish because three noisy observations generally cannot be satisfied exactly by two parameters. Their nonzero values do not by themselves invalidate least squares. Conversely, a small squared residual sum does not establish that the model omitted no systematic effect or that the formal observation variances were correct.
Preserve enough information to reconstruct this test. A compressed normal system can retain the parameter solution while losing access to individual observational diagnostics unless those are stored separately. Numerical compression and physical validation are different jobs. A method that solves the reduced system correctly can still be applied to an inadequate model.
The objective also contains a constant. Expanding the weighted residual square gives θᵀNθ−2θᵀb+c, where c = yᵀWy for the stated observation vector and weights. Our first unit-weight example has c = 25 square metres. N and b determine its minimiser, but retaining c is needed to recover the objective value from this compressed quadratic.
When a is eliminated, the constant becomes c_reduced = c−hᵀH⁻¹h. Here it is 25−9/2 = 20.5. The minimum reduced objective is 20.5−9.5²/4.5 = 4/9 square metres, matching the residual calculation. This provides another independent reduction check beyond comparing only the fitted coordinate.
That scalar constant still does not reconstruct every individual residual or its observational context. Observation counts, rank, weighting assumptions and diagnostic records have their own roles when estimating variance scales or testing a model. State what a compressed product preserves and what accompanying information is required, rather than assuming that a small normal matrix is a complete archive of the original experiment.
CHAPTER 4 OF 12 · Retain nuisance uncertainty
4. Eliminate the nuisance parameter without declaring it known
The normal equations are 5x+a = 11 and x+2a = 3. Solve the second for a = (3−x)/2 and substitute it into the first. The retained equation becomes (5−1/2)x = 11−3/2, or 4.5x = 9.5. It gives the same x = 19/9 as the full solution.
The reduced coefficient 4.5 is the Schur complement of the nuisance block. In a partitioned system with retained block G, nuisance block H and cross block B, the retained matrix is S = G−BH⁻¹Bᵀ, and its right-hand side is g−BH⁻¹h. This expression assumes the nuisance block can be solved consistently.
Elimination has not set a to zero or removed its effect. It has allowed a to take the value preferred by the joint observations for each trial x. The subtracted term in S records the information lost when uncertainty can be exchanged between x and a.
In numerical software the notation H⁻¹ should generally be implemented through a suitable linear solve rather than an unnecessarily explicit inverse. PETSc’s primary documentation describes the same block Schur-complement operation. Our two-number example makes its statistical meaning visible: removing a variable from the final system is not the same as pretending it was precisely known.
CHAPTER 5 OF 12 · Retain nuisance uncertainty
5. Keep marginal uncertainty distinct from a conditional shortcut
With the stated known unit observation variances, the full parameter covariance is N⁻¹ = (1/9)[[2,−1],[−1,5]]. The variance of x is 2/9 square metres. The reduced coefficient gives the same result: S⁻¹ = 1/4.5 = 2/9. Proper elimination preserved the uncertainty associated with estimating a.
If a were genuinely fixed by exact external information, the conditional variance for x in this model would instead be 1/5 square metres. That smaller value describes a different inferential situation. Reporting it after fitting a from the same uncertain observations would overstate the information available about x.
Fixing a to its fitted value happens to preserve the fitted x when the equations are evaluated consistently, but it does not make the fitted a exact. Recomputing only the retained diagonal block after that operation loses the uncertainty exchange recorded by the off-diagonal coupling.
The numerical difference here is modest, but the principle is general. When a nuisance parameter is strongly coupled to a coordinate, ignoring its uncertainty can matter substantially. Label whether a covariance is conditional on fixed quantities or includes jointly estimated ones. A smaller uncertainty is not automatically the more faithful description.
CHAPTER 6 OF 12 · Combine compatible systems
6. Combine independent reduced contributions to the same parameter
Create a second fictional technique with its own nuisance parameter b. Its predicted observations are x+b, 2x and b; their observed values are six, six and two metres. With the same independent unit-variance assumptions, its normal matrix is again [[5,1],[1,2]], while its right-hand side is (18,8)ᵀ.
Eliminating b gives S_2 = 4.5 and g_2 = 18−8/2 = 14. The separate coordinate estimate is 28/9 metres with variance 2/9. The first technique supplied S_1 = 4.5 and g_1 = 9.5. If their observation errors are independent and a and b are genuinely separate parameters, add the reduced information.
The combined equation is 9x = 23.5, giving x = 47/18 metres, approximately 2.611111. Its formal variance is 1/9 square metres. Back-substitution gives a = 7/36 and b = 97/36 metres. These can be checked against the full joint system with parameter order (x,a,b).
Equal separate variances make the combined x equal the simple mean in this particular example. That coincidence is not a universal combination rule. Unequal information, correlated observations, vector parameters or incompatible definitions require the actual weighted system. State the independence and parameter-sharing assumptions before interpreting a convenient average as a rigorous geodetic combination.
CHAPTER 7 OF 12 · Combine compatible systems
7. Show why shared nuisance parameters must remain shared
Now use a different four-observation example. Let the observations be x+a = 3, a = 1, x−a = 1 and a = 1, all with independent unit-variance errors. Here a is one physical model parameter shared between both observation groups, not two separate instrument biases.
The joint design rows are [1,1], [0,1], [1,−1] and [0,1]. They give N = [[2,0],[0,4]] and b = (4,4)ᵀ. Thus x = 2, a = 1 and Var(x) = 1/2 square metre. The opposite sensitivities to the same a let the first and third observations jointly constrain x without that nuisance contribution.
If the two groups are instead processed as though they have unrelated nuisance parameters, each group separately yields x = 2 with variance two square metres. Adding those reduced coordinate estimates as independent information gives variance one square metre, not one-half. The point estimates happen to agree while the retained information differs.
This counterexample shows why nuisance elimination must respect the actual sharing structure. Eliminating separately is valid for truly technique-local parameters under the stated independence conditions. It need not preserve the information of a shared unknown that should have connected the groups. Parameter names, identifiers and physical definitions matter as much as matrix dimensions.
CHAPTER 8 OF 12 · Combine compatible systems
8. Do not count the same observation information twice
Take the first reduced equation 4.5x = 9.5 and copy it into a second file without acquiring any new observation. Adding both as independent contributions gives 9x = 19, so the estimate remains 19/9 but its reported formal variance becomes 1/9 rather than 2/9. The uncertainty appears to improve solely because the evidence was duplicated.
Two analysis products can share observations, orbit information, calibration or prior constraints even when they come from different processing runs. Different filenames or software labels do not prove stochastic independence. The appropriate joint covariance may contain cross terms that an independent-information sum omits.
The same issue arises when a local tie or external prior has already entered one solution and is added again as though it were new evidence. Keep a provenance ledger identifying which observations and constraints each contribution contains. If dependence is unresolved, describe the limitation rather than reporting the independent-combination uncertainty as established accuracy.
A useful duplicate test deliberately inputs the same contribution twice and checks what the algorithm reports. The arithmetic may correctly halve a variance under its assumed independence model; that is exactly the warning. Software can solve the model it was given while the analyst’s evidence-accounting assumption remains wrong.
CHAPTER 9 OF 12 · Challenge constraints and dependence
9. Align reference values, units and parameter order before summing
Normal equations are often written for corrections to reference values rather than absolute parameters. Suppose one scalar system describes δ = x−x_0 through Nδ = b. If a new reference is x_0+d, its correction is δ′ = δ−d, so the same information becomes Nδ′ = b−Nd. The right-hand side must change with the reference.
For our reduced first system, N = 4.5 and b = 9.5. Shift the reference by one metre. The new right-hand side is five, giving δ′ = 10/9 metres. Adding the shifted reference recovers the original absolute result 19/9. Adding right-hand sides from different unaligned reference values would combine different correction variables.
Parameter reordering requires matching both rows and columns of the normal matrix and the entries of the right-hand side. A unit transformation likewise changes information through the corresponding parameter transformation; it is not enough to relabel metres as millimetres. The complete predicted observations must remain unchanged.
These are algebraic preparation steps, not physical improvements to the observations. A valid combination needs compatible epochs, frames and reference points as well. Use surveyed local ties and documented coordinate transformations where relevant, with their uncertainty. Do not make two inconsistent products appear compatible by editing labels while leaving the numerical model untouched.
CHAPTER 10 OF 12 · Challenge constraints and dependence
10. Separate constraints, variance scales and model discrepancy
An independent uncertain constraint contributes information through its own observation equation and covariance. An exact constraint changes the parameter space. A datum choice resolves a reference freedom. These operations are not interchangeable, and their role should be identified before interpreting the combined covariance.
The free-network companion explains datum defects. A Schur reduction does not magically repair every rank-deficient system. If the nuisance block or retained system lacks the required information, the problem needs an appropriate constrained or rank-aware treatment. Adding a tiny diagonal number makes a new assumption; it is not a measured reference.
The covariance values in our examples used known observation variances. When only relative weights are supplied, a variance scale may need estimation under a stated model. Scaling weights changes the formal information. It does not by itself identify whether a discrepancy came from random noise, an omitted physical term, a reference mismatch or duplicated evidence.
Keep residual diagnostics and independent validation beside the combined solution. Fitting an extra nuisance term may reduce residuals while changing coordinate information. Inflating uncertainty may acknowledge unexplained scatter without explaining its cause. Neither should be presented as a physical correction unless the additional model and evidence support that interpretation.
CHAPTER 11 OF 12 · Transfer and report
11. Transfer the reduction to a new observation set
Use a new fictional pair of observations x+a = 5 and a = 1, each with independent unit variance. Their normal matrix is [[1,1],[1,2]] and right-hand side (5,6)ᵀ. Eliminating a gives 0.5x = 2, so x = 4 with variance two square metres. The nuisance estimate is one metre.
Add one genuinely independent observation x = 3 with unit variance. It contributes one to the retained information and three to the retained right-hand side. The combined equation is 1.5x = 5, giving x = 10/3 metres and variance 2/3 square metre. Back-substitution gives a = 4/3 metres.
Check the full normal equations: [[2,1],[1,2]](10/3,4/3)ᵀ = (8,6)ᵀ. The predicted-minus-observed residuals are minus one-third, plus one-third and plus one-third metres. Their squared sum is one-third square metre, and the design-weighted residual sums vanish.
Now imagine a was fixed exactly at one metre before processing. The conditional coordinate problem would combine x = 4 and x = 3 as unit-variance observations, giving x = 3.5. That differs from 10/3 because the uncertain nuisance was no longer allowed to trade against x. Explain which model is supported before choosing the answer that looks more agreeable.
CHAPTER 12 OF 12 · Transfer and report
12. Report an information chain that another analyst can reconstruct
A reproducible combination identifies each observation contribution, the common parameter definitions, reference values, units, retained and eliminated blocks, covariance assumptions, constraints and provenance. It records which nuisance parameters are shared and which are genuinely local. A final coordinate table alone cannot reconstruct all of these choices.
Include independent numerical checks: solve a small full system and its reduced version, compare retained covariance, back-substitute eliminated parameters and test the original normal residual equations. Then deliberately change a reference value, duplicate a contribution and alter the parameter-sharing assumption. These reveal different errors that a single successful matrix inversion cannot expose.
Keep the physical claim proportional to the evidence. A combined estimate can have smaller formal uncertainty under a defensible model without proving that all participating techniques are unbiased or all reference points are stable. Disagreement between inputs remains evidence to investigate, not an inconvenience that combination automatically erases.
The gold standard is preserving the information that the observations genuinely supply while making every assumption visible. Elimination should remove computational burden, not uncertainty. Combination should add independent evidence, not duplicate it. This is how a concise normal system can remain connected to the physical measurements and limits from which it was built.
Independent worked case and evidence checks
Independent scalar check: compare three representations of the first example. The full matrix is N = [[5,1],[1,2]] with b = (11,3). Direct inversion gives (x,a) = (19/9,4/9). The retained Schur equation is S = 9/2 and g = 19/2, giving the same x. Back-substitution then reproduces a. Agreement establishes the reduction arithmetic for this defined model.
Recover its uncertainty independently from the inverse: N⁻¹ = (1/9)[[2,−1],[−1,5]]. The retained variance is 2/9, equal to S⁻¹. The cross-covariance between x and a is minus 1/9 square metre. The nonzero cross term explains why retaining only the original x diagonal would describe a different conditional problem.
For a changed reference, let δ = x−one metre. The reduced right-hand side becomes 19/2−9/2 = five. Solving gives δ = 10/9 metres, and restoring the reference gives 19/9. Repeat with a two-metre shift: the new right-hand side is one-half, giving δ = one-ninth. Again the absolute result is unchanged.
For a changed information scale, double all weights in the first observation model. Both N and b double, leaving the estimate unchanged while the formal covariance halves under that new assumed noise model. This is not evidence that the original data became more accurate. Explain what actual uncertainty justification would support the new weights.
For a genuinely independent new coordinate observation x = two metres with variance one square metre, add one to S and two to g. The result is 5.5x = 11.5, or x = 23/11 metres, with variance 2/11 square metre. Back-substitute a = (3−23/11)/2 = five-elevenths of a metre. Check these values in the augmented full normal system.
Now replace that new observation by another copy of an observation already used. The same independent-update formula is not justified merely because a fresh row was created in a spreadsheet. The data provenance and joint error model determine whether information can be added as independent. A numerical result without that distinction can look more certain than the evidence warrants.
Finally describe a real geodetic limitation not represented by the scalar algebra: two instrument products may refer to different physical endpoints or epochs. A mathematically correct weighted combination of those mismatched quantities would still answer the wrong question. Reconstruct the physical connection first, then decide which uncertain transformations and ties belong in the combined model.
For a final provenance check, write the objective constant and observation count beside each contribution. If a contribution is copied, those bookkeeping quantities may also be copied and make a residual summary look internally consistent. Internal consistency still cannot establish independence. The question remains which physical observations and prior information actually entered the two products.
Worked comparison
| Operation | Correct result in the fictional model | Common mistaken interpretation |
|---|---|---|
| Eliminate a from [[5,1],[1,2]] | S=4.5, g=9.5 | Set a to zero and retain only5x=11 |
| Retain x uncertainty | Var(x)=2/9 | Report conditional1/5 after estimating a |
| Combine independent reduced systems | Add compatible S and g | Average arbitrary coordinate values |
| Share one physical nuisance parameter | Retain its cross-group identity | Eliminate two unrelated copies |
| Duplicate the same contribution | No new independent evidence | Smaller formal covariance proves improvement |
| Shift reference by d | Transform b to b−Nd | Change the label but not the correction system |
Questions and practical checks
Does nuisance elimination mean the nuisance parameter is ignored?
No. A correct Schur reduction retains its effect on the kept parameters and their uncertainty. Declaring it fixed or deleting cross terms changes the inference.
When can reduced normal equations be added?
When they describe the same retained parameters with aligned references and units, compatible models and the required independence or joint-covariance treatment. Shared nuisance parameters and constraints must be handled consistently.
Why can duplicate information leave the estimate unchanged but shrink uncertainty?
The independent-information model treats the repeated contribution as fresh evidence. If it is actually the same observation information, that assumption is wrong even though the arithmetic follows the supplied matrices.
Can a small formal covariance certify physical accuracy?
No. It is conditional on the observation, reference, constraint and stochastic models. Bias, endpoint mismatch and other missing effects need independent evidence and validation.
What is a useful first implementation test?
Solve a small full system and its reduced version, compare both the retained estimate and covariance, then back-substitute and check the original observations. Add a reference-shift and duplicate-information test before trusting a large combination.
Continue with the connected subjects
Establish the geodetic quantities being combined
Connect technique reference points with uncertain surveyed local ties
Align coordinate and frame epochs before comparison
Distinguish datum freedom from observational information
Retain the uncertainty and provenance
Follow shared-observation covariance through geodetic differences
Interpret coordinate covariance without overstating coverage
Compare static nuisance elimination with the broader sensor-fusion and marginalisation setting
Sources and scope
ESA Navipedia: linear observation models, adjustment and residuals
ESA Navipedia: weighting by observation covariance and formal parameter uncertainty
PETSc primary documentation: block Schur-complement operation
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 1 November 2026.
Continue through observations, reference points and combined information
Use the measurement model to identify what a record represents, the local survey to connect physical endpoints, and the adjustment model to retain the information and uncertainty. These are complementary steps, not interchangeable corrections.
Compare the observation quantities
Radio-source structure: follow complex visibility into a group-delay slope
DORIS phase counts: preserve integration windows and frequency endpoints
Connect the reference and the estimate
Surveyed local ties: connect the instruments’ reference points