Find the question you need
Choose a route or read from the beginning. All teaching chapters stay open; the grouped index below is expandable.
Change the contrast
Chapters 6–8. Compare rates, shared errors and spatial averages.
Full chapter index · Practice and worked checks · Remote Sensing subject root
Two image changes can share the same starting observation. That common baseline makes their errors related, even when the original images have independent measurement errors.
This guide compares two baseline-relative reflectance changes, simplifies the contrast and calculates its standard uncertainty. We then change the time intervals to see when the baseline cancels and when it remains.
Begin with The Gold Standard of Remote Sensing. These are invented additive-error models for compatible observations of the same target. They do not establish the accuracy of a particular sensor or the cause of an environmental change.
Chapter index
Define the change estimates · Chapters 1–2
1. Define the observations before their differences
2. Keep the repeated baseline visible
Derive the covariance · Chapters 3–5
3. Calculate the individual change uncertainties
4. Derive the covariance between changes
5. Propagate the contrast without double counting
Change the contrast · Chapters 6–8
6. Compare rates over unequal intervals
7. Check the rate result through covariance
8. Add a shared error component cautiously
Check and practise · Chapters 9–10
9. Verify comparability before interpreting precision
10. Rebuild a case with a much noisier baseline
Read and continue · Chapters 11–12
11. Questions readers ask
12. Sources and useful reading routes
CHAPTER 1 OF 12 · Define the change estimates
1. Define the observations before their differences
Let r₀, r₁ and r₂ be reflectance estimates from a baseline image and two later images of the same defined target and spectral band.
For the first example, use r₀ = 0.30, r₁ = 0.34 and r₂ = 0.37. Define baseline-relative changes c₁ = r₁ − r₀ = 0.04 and c₂ = r₂ − r₀ = 0.07.
Each observed reflectance is modelled as the target quantity plus an additive estimation error. Assume the three raw errors have zero mean, independent of one another, and standard uncertainty u = 0.02 reflectance unit.
The uncertainty is an assumed input to this teaching model. It is not inferred from the three displayed values.
CHAPTER 2 OF 12 · Define the change estimates
2. Keep the repeated baseline visible
Subtracting the two changes gives D = c₂ − c₁ = (r₂ − r₀) − (r₁ − r₀) = r₂ − r₁.
The baseline is used twice with opposite coefficients and cancels exactly. The estimated contrast is 0.03.
The same cancellation applies to that baseline’s additive error. A perturbation of r₀ changes both c₁ and c₂ equally, leaving their difference unchanged.
The comparison asks how the later reflectance estimates differ. It does not say whether either baseline-relative change is well determined, and it does not establish a cause.
Write the full expression before assigning uncertainty terms. Algebra can reveal shared observations that a pair of summary columns hides.
CHAPTER 3 OF 12 · Derive the covariance
3. Calculate the individual change uncertainties
For independent raw errors, Var(c₁) = u₁² + u₀² and Var(c₂) = u₂² + u₀². With all three standard uncertainties equal to 0.02, each variance is 0.0008.
Thus each change has standard uncertainty √0.0008, approximately 0.02828.
| Estimated quantity | Value | Standard uncertainty under the model |
|---|---|---|
| Baseline reflectance r₀ | 0.30 | 0.02000 |
| First later reflectance r₁ | 0.34 | 0.02000 |
| Second later reflectance r₂ | 0.37 | 0.02000 |
| Change c₁ = r₁ − r₀ | 0.04 | about 0.02828 |
| Change c₂ = r₂ − r₀ | 0.07 | about 0.02828 |
These are standard uncertainties, not maximum errors. A coverage interval would require an appropriate distributional and coverage treatment; it should not be invented from the table.
Let e₀, e₁ and e₂ be the independent raw errors. The change errors are e₁ − e₀ and e₂ − e₀.
Their covariance is Cov(e₁ − e₀, e₂ − e₀) = Var(e₀) = u₀², because all cross-covariances of distinct raw errors vanish.
Here the covariance is 0.0004. Since each change has variance 0.0008, the change-error correlation is 0.0004/0.0008 = 0.5.
The derived changes are correlated even though the raw observations were assumed independent. Both contain the same negative baseline error.
Independence belongs to specified inputs, not automatically to every quantity calculated from them.
CHAPTER 5 OF 12 · Derive the covariance
5. Propagate the contrast without double counting
For D = c₂ − c₁, variance is Var(c₂) + Var(c₁) − 2Cov(c₁, c₂).
Substitution gives 0.0008 + 0.0008 − 2 × 0.0004 = 0.0008. Standard uncertainty is approximately 0.02828.
The shortcut treating the two changes as independent gives variance 0.0016 and standard uncertainty 0.04. It counts the shared baseline as if two separate baseline errors remained.
Direct calculation from D = r₂ − r₁ gives u₂² + u₁² = 0.0008, agreeing with the covariance method.
NIST’s law of propagation of uncertainty includes covariance terms for related inputs. In this linear additive model the variance algebra is exact, rather than a nonlinear first-order approximation.
Suppose r₁ is two years after baseline and r₂ is five years after baseline. Define average baseline-relative rates v₁ = (r₁ − r₀)/2 and v₂ = (r₂ − r₀)/5.
Their estimated values are 0.020 and 0.014 reflectance unit per year. The difference is v₂ − v₁ = −0.006.
Expand it: v₂ − v₁ = r₂/5 − r₁/2 + 3r₀/10. The baseline coefficient is now 0.3 per year. It does not cancel because the two changes were divided by different intervals.
| Contrast | Coefficient of r₀ | Variance with independent raw errors u = 0.02 |
|---|---|---|
| c₂ − c₁ | 0 | 0.0008 |
| c₂/5 − c₁/2 | 0.3 per year | 0.000152 per year² |
For the rate contrast, standard uncertainty is √0.000152, approximately 0.01233 reflectance unit per year.
Var(v₁) = 0.0008/4 = 0.0002 per year². Var(v₂) = 0.0008/25 = 0.000032 per year².
Their covariance is u₀²/(2 × 5) = 0.00004 per year². Subtract twice this covariance from the sum: 0.0002 + 0.000032 − 0.00008 = 0.000152.
The direct coefficient method agrees: 0.02²[(1/5)² + (1/2)² + (3/10)²] = 0.000152.
These are averages over overlapping baseline-relative intervals. A difference between them is not automatically an instantaneous acceleration, nor evidence of a particular physical mechanism.
Changing the comparison changes both the scientific question and the coefficient attached to each uncertainty source.
Real images need not have independent errors. A common processing reference, calibration component or model parameter can create further covariance.
If the same additive offset b enters all three reflectance estimates identically, it cancels in c₁, c₂ and both contrasts. If its coefficients differ among images, cancellation need not occur.
A shared multiplicative gain behaves differently: a common gain error multiplies a nonzero difference. Its uncertainty is not removed merely because an additive offset would cancel.
Spatial averages need similar care. Averaging many pixels reduces some independent components, while a shared image-wide component can remain. Counting pixels alone does not establish a reduction by the square root of the count.
Specify the error model and evidence for its relationships; do not label every uncertainty independent by default.
CHAPTER 9 OF 12 · Check and practise
9. Verify comparability before interpreting precision
The algebra assumes compatible measurements of the same target. Registration error, cloud masks, different spectral responses and changing sampled areas can defeat that comparison.
Read co-registration offsets and false boundary change before treating pixel differences as temporal changes.
For an index, use its actual measurement equation and relevant covariance. Averaging NDVI and reflectance shows why changing the order of operations creates another problem.
Keep a table of observation identity, time, target definition, product version, processing and shared inputs. Standard uncertainty around a quantity does not repair a quantity that answers the wrong question.
CHAPTER 10 OF 12 · Check and practise
10. Rebuild a case with a much noisier baseline
Keep independent raw errors but use u₀ = 0.05, u₁ = 0.01 and u₂ = 0.02.
The individual change variances are 0.0026 and 0.0029. Their covariance is 0.0025 because both use the same baseline.
For D = c₂ − c₁, the correct variance is 0.0026 + 0.0029 − 2 × 0.0025 = 0.0005. Standard uncertainty is about 0.02236.
Treating the changes as independent would instead give √0.0055, approximately 0.07416. Directly using r₂ − r₁ confirms the smaller result.
The noisy baseline matters greatly to each baseline-relative change while cancelling from their difference. It would reappear if unequal interval coefficients or different baselines were used.
Does a shared baseline always cancel?
It cancels when the final expression gives that same observation equal and opposite coefficients. Unequal interval rates provide a counterexample.
Does a small uncertainty prove a real environmental effect?
No. Comparability, the uncertainty model, relevant error sources and interpretation all need checking. Causal claims require additional evidence.
Is correlation a flaw in the data?
Not necessarily. Here it is an unavoidable consequence of reusing a valid observation. It should be represented correctly.
Can I use two different baseline products as if they were identical?
No. Their errors and relationship need explicit treatment. Matching labels or nominal dates do not establish a shared numerical observation.
Measurement mathematics
NIST Technical Note 1297, Appendix A explains propagation with covariance. The three-image populations, contrasts and numerical checks here are original teaching applications.
Follow the observation questions
Use the remote-sensing mechanism guide, co-registration and false change and NDVI averaging order.
Return to the learning map
Continue through Remote Sensing, The Gold Standard of Education and How X Works. Keep the identity of every reused observation visible through the calculation.
Sources checked 22 October 2026. Worked examples are educational models with the assumptions stated above.