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Choose a route or begin at Chapter 1. Teaching chapters stay open; the chapter index expands by section.
Define the objects
Chapters 1–2. Specify the target before reducing the images
Build the argument
Chapters 3–5. Work a two-period example
Compare and interpret
Chapters 6–8. Show why the missing surface cannot be identified
Check and practise
Chapters 9–10. Make the coverage evidence readable
Questions and next routes
Chapters 11–12. Questions readers ask
Full chapter index · Worked practice · Remote Sensing subject root
Removing cloudy pixels can improve the validity of the observations that remain while changing which days contribute to an average. A clean composite therefore needs two explanations: how invalid measurements were excluded, and which time population the surviving measurements represent.
This article follows a single pixel through a constructed two-season example. It separates the mean over valid acquisitions from a target mean over all scheduled dates. The numbers describe hypothetical surface states, not an actual satellite record or a retrieval product.
Chapter index
Define the objects · Chapters 1–2
1. Specify the target before reducing the images
2. Write a mask into the arithmetic
Build the argument · Chapters 3–5
3. Work a two-period example
4. Expose the weights behind the composite
5. Reweight periods and state the condition
Compare and interpret · Chapters 6–8
6. Show why the missing surface cannot be identified
7. Use a bound when imputation is unjustified
8. Distinguish the mean from the median
Check and practise · Chapters 9–10
9. Make the coverage evidence readable
10. Practise with unequal target periods
Questions and next routes · Chapters 11–12
11. Questions readers ask
12. Sources and connected learning routes
CHAPTER 1 OF 12 · Define the objects
1. Specify the target before reducing the images
Suppose the question is the average surface index over twenty equally weighted scheduled dates at one fixed ground location. Ten dates belong to period A and ten to period B. This finite target is the sum of the twenty underlying index values divided by twenty.
A cloud mask removes acquisitions whose surface signal is unavailable or unreliable. A temporal mean of the remaining valid values answers a narrower question: the average among dates that supplied a valid observation.
Neither quantity is automatically wrong. Trouble starts when the second is labelled as the first without evidence that the selection leaves the target unchanged. Name the pixel, variable, date window, sampling schedule and temporal weighting before discussing representativeness.
Let Xₜ be the underlying surface index at scheduled date t. Let Mₜ equal one when a valid surface observation is retained and zero otherwise. For binary masks and equal acquisition weights, the clear-sky sample mean is ΣMₜXₜ / ΣMₜ, provided at least one observation remains.
The all-date target is ΣXₜ/T. A missing observation is omitted from the first sum and its denominator. It is not assigned the index value zero.
The notation Xₜ for a masked date represents a latent surface state in our teaching model. It does not imply the sensor recovered that state through the cloud. Real retrieval products need their own quality and measurement definitions.
In period A, all ten underlying index values equal 0.8, but only two acquisitions are retained. In period B, all ten values equal 0.2 and eight acquisitions are retained. The construction makes within-period values constant so the selection effect can be seen without another source of variability.
The target over twenty dates is (10×0.8 + 10×0.2)/20 = 0.5. The clear-sky mean is (2×0.8 + 8×0.2)/10 = 0.32. The valid observations overrepresent the low-index period.
Every retained value is correct in this example. The difference of −0.18 comes entirely from temporal composition. Improving per-observation quality does not by itself guarantee that an average represents the intended season.
| Period | Target dates | Surface index | Valid observations | Valid sum |
|---|---|---|---|---|
| A | 10 | 0.8 | 2 | 1.6 |
| B | 10 | 0.2 | 8 | 1.6 |
| Total | 20 | Target mean 0.5 | 10 | Clear-sky mean 0.32 |
Let nₖ be the valid observation count in period k and μ̂ₖ its valid-observation mean. Pooling all valid acquisitions gives Σnₖμ̂ₖ/Σnₖ. Thus the periods receive weights proportional to their surviving counts.
If the target uses date counts Tₖ, its intended weights are Tₖ/ΣTₖ. In our example the intended weights are one half and one half, but the pooled clear-sky weights are one fifth and four fifths.
This explains why recording a total count is helpful but incomplete. Ten valid observations concentrated in one part of a season supply different coverage from ten spread across the window. Preserve the dates or at least suitable temporal strata.
An equal-period estimator in the example is (0.8 + 0.2)/2 = 0.5. It recovers the target because periods have equal target weights and each period is constant, so retained observations exactly represent its missing dates.
More generally use the target weights: Σ(Tₖ/T)μ̂ₖ. This addresses unequal sampling between periods only when the valid observations adequately represent the target values within each period. It does not automatically remove selection within a period.
If an entire period has no valid observations, its valid mean is undefined. Do not silently drop that period and renormalise the remaining weights while retaining the original seasonal label. Report missing coverage or introduce a justified model and its uncertainty.
CHAPTER 6 OF 12 · Compare and interpret
6. Show why the missing surface cannot be identified
Now observe two valid values of 0.8 in period A and eight valid values of 0.2 in period B, without knowing the other ten states. One possible completion has all missing A values at 0.8 and missing B values at 0.2, yielding the earlier target 0.5.
Another completion has the eight missing A values at 0.4 and the two missing B values at 0.6. The total is 2×0.8 + 8×0.4 + 8×0.2 + 2×0.6 = 7.6. Dividing by twenty gives 0.38.
Both worlds have exactly the same observed clear-sky data and masks. Their all-date means differ. The observations alone therefore do not identify the target. A correction needs additional information or assumptions, even when its numerical output looks precise.
| Completion | Missing A values | Missing B values | All-date mean |
|---|---|---|---|
| World 1 | Eight at 0.8 | Two at 0.2 | 0.50 |
| World 2 | Eight at 0.4 | Two at 0.6 | 0.38 |
CHAPTER 7 OF 12 · Compare and interpret
7. Use a bound when imputation is unjustified
If the surface index is known to lie between zero and one for this specific teaching case, the observed sum is 3.2 and ten values are missing. The target must lie between 3.2/20 = 0.16 and (3.2 + 10)/20 = 0.66.
These are worst-case bounds, not a confidence interval. They express the unresolved possibilities given the stated range, without a probability model for missing states. A narrower physical range or other evidence could tighten them.
Do not apply the zero-to-one range to every index. For example, a normalised difference index can have negative values. Use the documented valid range and physical interpretation of the variable actually being analysed.
A median composite changes the summary statistic; it does not automatically repair temporal selection. The valid sample in the first example contains two values of 0.8 and eight of 0.2. Its median is 0.2.
For the full twenty-date population, the conventional midpoint median of the two central order statistics is (0.2 + 0.8)/2 = 0.5. Again, the surviving acquisitions answer a different population question.
The median may be useful for particular robustness goals, but that goal must be stated. Similarly, a medoid, maximum-value composite or modelled gap-filled product has its own selection mechanism and interpretation. Do not transfer the meaning of an arithmetic seasonal mean to every composite.
Alongside a temporal estimate, retain a valid-count layer, temporal span, per-period coverage and the quality criteria used. Where useful, show the distribution of acquisition dates rather than a single first-to-last range.
Keep geometry and radiometry compatible before combining images. The correct date weights cannot repair a misplaced pixel, a scale-factor error or a changed sensor definition. The linked co-registration and index-aggregation guides address those separate problems.
For implementation, follow the platform’s documented treatment of masks and weights. This article’s equations use binary masks, one value per scheduled date and equal acquisition weights; fractional masks or overlapping observations require a more specific weighting account.
A target window has five dates in A and fifteen in B. The constant underlying values are 0.9 and 0.3. Two observations survive in A and three in B. Calculate the pooled clear-sky mean and the target mean.
The clear-sky mean is (2×0.9 + 3×0.3)/5 = 0.54. The target is (5×0.9 + 15×0.3)/20 = 0.45. An equal-period mean would be 0.6, which is also wrong for this target.
The appropriate stratum weights are one quarter and three quarters, yielding 0.25×0.9 + 0.75×0.3 = 0.45. This works because the exercise supplies constant within-period values. In an actual record, justify within-period representativeness before presenting reweighting as a correction.
Does cloud masking create bad data? It can exclude unsuitable observations. The issue is how the surviving sample is interpreted, not an instruction to retain cloud-contaminated measurements.
Does more imagery guarantee an unbiased seasonal estimate? No. Additional observations can repeat the same selection pattern. Their timing and relationship to surface conditions matter.
Can radar replace every missing optical value? Different sensors measure different physical responses. Combining them needs a model for the target quantity, not an assumption that one observation is a direct replacement for another.
Can I compare two clear-sky means? Yes, as clearly defined clear-sky summaries. A claim about full seasonal surface conditions requires checking whether coverage differences or selection could explain the comparison.
CHAPTER 12 OF 12 · Questions and next routes
12. Sources and connected learning routes
Google Earth Engine’s reducer documentation explains image-collection reduction and the exclusion of zero-masked pixels. The finite-population examples, alternative missing-state completion and bounds above are explicit original calculations illustrating the interpretation problem.
Follow the Remote Sensing root for the complete signal-to-inference chain. Use the observation-timing guide for short events between acquisitions and the shared-baseline guide for dependence in change comparisons. These are different sources of uncertainty and can occur together.
The transfer exercise is to name the target population before averaging attendance, survey responses or instrument readings. A valid observed value can still belong to an unrepresentative selection. Preserve that distinction whenever a filtered dataset becomes a broader claim.
References checked 22 October 2026
Earth Engine: reducer overview and mask weighting
Earth Engine: image-collection reductions
Continue through Remote Sensing
Revisit gaps, cloud cover and missed short events
A shared baseline and comparing changes
Why averaging NDVI differs from averaging reflectance
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