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Choose a route or read from the beginning. All teaching chapters stay open; the grouped index below is expandable.
Define the census
Chapters 1–2. Separate area, eligible craters and probability of detection.
Work the correction
Chapters 3–5. Derive a conditional estimate and compare two terrains.
Handle changing detectability
Chapters 6–7. Use diameter bins and distinguish missed craters from contamination.
Assess the evidence
Chapters 8–10. Check calibration, uncertainty and useful questions.
Practise and continue
Chapters 11–12. Solve changed cases and follow planetary evidence routes.
Full chapter index · Practice and worked checks · Planetary Geology subject root
A terrain with fewer counted craters can contain the larger retained crater population if its images hide more of the eligible features. Dividing counts by area fixes unequal survey size; it does not fix unequal detection.
This guide separates those two denominators. We build a small detection model, compare original fictional terrains and test why one correction factor can fail across diameter ranges. It extends Planetary Geology. None of the examples dates a real planetary surface.
Chapter index
Define the census · Chapters 1–2
1. Define what belongs in the count
2. Describe detection as a conditional model
Work the correction · Chapters 3–5
3. Derive a completeness-adjusted estimate
4. Compare two fictional terrains
5. Separate a density ranking from an age ranking
Handle changing detectability · Chapters 6–7
6. Correct diameter bins before pooling them
7. Do not use missed-feature correction to repair contamination
Assess the evidence · Chapters 8–10
8. Ask how completeness was established
9. Carry uncertainty and support into the comparison
10. Questions that keep the correction honest
Practise and continue · Chapters 11–12
11. Practise with changed survey conditions
12. Continue through planetary measurements
Begin with a mapped area A and an eligibility rule: a specified diameter range, boundary rule and geological population. Let N be the number of eligible craters actually retained in that area. Let n be the number identified in the supplied observations.
N is not necessarily the number of impacts that ever occurred there. Resurfacing can remove craters. Secondary craters can belong to a different impact-production history. Here we isolate the additional step between a retained eligible feature and its detection.
Write the observed density as n/A. Keep it as an observation before applying a completeness correction or drawing an age conclusion. Each later step requires evidence that the raw density alone does not contain.
Suppose each retained eligible crater has probability q of being detected under the chosen imaging and counting procedure. Assume the same q within this first teaching population, no false identifications and independent detections conditional on N.
The expected observed count is E[n | N,q] = qN. Expected observed density is therefore qN/A. A smaller q means that the same retained population tends to produce fewer identified craters.
Real q can depend on diameter, morphology, terrain, illumination and image resolution. Robbins, Kirchoff and Ostrach’s 2025 primary study of crater detection investigates observing-geometry and resolution effects. Its results do not supply a universal q for arbitrary terrain. Our q values below are stipulated teaching inputs.
If q is known independently and positive, define Nhat = n/q and Dhat = n/(qA). Under the stated model, E[Nhat | N,q] = N, so the estimate is unbiased for the retained eligible count.
Unbiased does not mean exact in one realised survey. Detecting each crater with a stated probability can produce different counts on repeated observations. Nhat can also be non-integer: it estimates a population size from incomplete observation rather than naming an additional fractional crater in an image.
The correction is undefined at q = 0. An instrument that cannot detect a class of feature supplies no count-based estimate of that class through division. A very small q can produce a large and fragile correction, especially when q itself is uncertain.
Use the same eligibility rules and area of 1,000 km² for each terrain. Independently supplied detection probabilities are qA = 0.50 and qB = 1.00. The latter is a deliberately ideal perfect-detection assumption for B, not a claim that a particular image resolution guarantees completeness.
| Quantity | Terrain A | Terrain B |
|---|---|---|
| Observed eligible count n | 40 | 60 |
| Mapped area | 1,000 km² | 1,000 km² |
| Observed density n/A | 0.040 per km² | 0.060 per km² |
| Supplied detection probability q | 0.50 | 1.00 |
| Adjusted retained-count estimate n/q | 80 | 60 |
| Adjusted density estimate n/(qA) | 0.080 per km² | 0.060 per km² |
The observed ranking is B above A. The adjusted estimate ranks A above B. Neither row is a numerical age. We have changed the interpretation of what the observations missed, not discovered a new impact-production chronology.
CHAPTER 5 OF 12 · Work the correction
5. Separate a density ranking from an age ranking
Even a defensible retained-crater density does not uniquely determine elapsed time. Appropriate crater-production and chronology models, diameter distributions, resurfacing, saturation and population interpretation matter. The simple correction addresses detection under stated assumptions, not every geological cause of a population difference.
If A’s q were actually 0.80, its adjusted density would be 40/(0.80 × 1,000) = 0.050 per km², below B’s 0.060. The revised ranking would arise from a changed completeness input. It would not be legitimate to select q merely to obtain a preferred terrain history.
The comparison therefore has three layers: observed density, model-adjusted retained density and geological age interpretation. Report each layer separately so another reader can see where the conclusion depends on an unmeasured quantity.
CHAPTER 6 OF 12 · Handle changing detectability
6. Correct diameter bins before pooling them
Now consider one 1,000 km² area divided into two disjoint eligible diameter bins. Detection is poorer in the smaller bin. Assume all counted objects are correctly classified and use independently supplied q values.
| Diameter bin | Observed count | Supplied q | Adjusted count estimate |
|---|---|---|---|
| Smaller eligible craters | 20 | 0.25 | 80 |
| Larger eligible craters | 20 | 1.00 | 20 |
| Total across the disjoint bins | 40 | No unweighted pooled q assumed | 100 |
Add the adjusted bin estimates to obtain total density 100/1,000 = 0.100 per km². Averaging the two q values gives 0.625, but dividing total observed count 40 by 0.625 gives only 64.
That average gives equal weight to the bins rather than the retained objects. In the example’s adjusted population, 80% of objects are in the smaller bin, so an object-weighted effective q would be 0.80 × 0.25 + 0.20 × 1 = 0.40. Correcting bins avoids pretending that those underlying weights were already known from an unweighted average.
CHAPTER 7 OF 12 · Handle changing detectability
7. Do not use missed-feature correction to repair contamination
False detections and ineligible craters are different from missed eligible craters. If an expected contaminating count b enters the observed catalogue, the elementary expectation becomes E[n] = qN + b. Dividing all n by q would amplify the contamination too.
A model with justified b might instead use (n − b)/q, with its uncertainty carried through. That algebra does not make b known. Visual artefacts, misclassified landforms, secondary-crater inclusion or inconsistent boundary rules need their own investigation.
Likewise, geological removal is not an image-detection failure. A crater erased by resurfacing is absent from the retained population. Applying an image-completeness correction cannot reconstruct every impact that the terrain no longer preserves.
Possible checks include comparison with suitably better observations, independently reviewed reference catalogues, controlled degraded-image tests and carefully designed detection experiments. Each comparison must define what counts as an eligible crater and whether the reference itself misses features.
Synthetic insertion tests can measure performance on the inserted examples. Their transfer to natural crater morphologies requires justification. A training set and its own evaluation set also need appropriate separation; recognising familiar labelled examples does not demonstrate reliable discovery on unfamiliar terrain.
Report the tested diameter range, observing conditions and detection method together. A completeness threshold established for one catalogue is not a correction factor to copy unchanged onto another sensor, world or terrain. The relevant evidence must describe the observations actually being counted.
CHAPTER 9 OF 12 · Assess the evidence
9. Carry uncertainty and support into the comparison
If q is treated as exact, dividing a noisy count by q also rescales its count uncertainty. If q is estimated, its uncertainty contributes as well. A narrow counting-error bar alone does not represent uncertainty in eligibility, detectability or geological interpretation.
Keep area uncertainty and survey support visible. A partially covered map, an irregular exclusion zone or a diameter-dependent usable area can change the relevant denominator. The largest displayed map rectangle is not automatically the area actually searched under comparable conditions.
Cumulative diameter counts share objects, so their errors are not independent merely because they appear on separate plot points. Our bin example used disjoint classes to make the addition transparent. A real size-frequency analysis should retain the statistical structure of the chosen counts.
CHAPTER 10 OF 12 · Assess the evidence
10. Questions that keep the correction honest
Does a corrected count identify the missed craters individually?
No. It estimates how many eligible retained craters may be represented under a detection model. It does not locate features absent from the catalogue or supply their precise diameters.
Does zero observed count establish a young surface?
No. Survey area, diameter range, detection probability and geological preservation all matter. Even the elementary random-detection model can produce zero detections from a nonzero retained population.
Can a bigger image display improve completeness?
Enlargement alone does not add information that was never resolved. Better source observations or an independently validated interpretation method may help; a larger screenshot by itself is not a completeness assessment.
CHAPTER 11 OF 12 · Practise and continue
11. Practise with changed survey conditions
Case A: count 30 eligible craters in 600 km² with independently supplied q = 0.50. The observed density is 0.050 per km² and adjusted density is 0.100 per km². Another terrain has 40 in 800 km² with q = 1.00, giving 0.050 per km² both observed and adjusted. Equal observed densities do not establish equal retained densities.
Case B: retain the original A count of 40 and area of 1,000 km² but allow q to range from 0.40 to 0.80 as a sensitivity exercise. Adjusted density ranges from 0.100 to 0.050 per km². This crosses B’s 0.060, so the ranking is not robust to those allowed q choices. This range is not a confidence interval unless supported by a probability model.
Case C: a report pools 10 small-bin detections with q = 0.20 and 30 large-bin detections with q = 1.00. Correcting separately gives 50 + 30 = 80. Explain why dividing 40 by the arithmetic mean 0.60 answers the wrong weighting problem.
CHAPTER 12 OF 12 · Practise and continue
12. Continue through planetary measurements
Terrain, instruments and chronology
Return to Planetary Geology for geological relationships and crater-history limits. Remote Sensing and its sensor-and-resolution guide develop the observation chain. Geochronology separates numerical ages from the events they constrain.
Compare worlds carefully
Planetary Science supplies the wider comparison among worlds and observing methods. A detection model should travel with its evidence, not simply with the name of a technique.
Transfer the evidence habit
The Gold Standard of Education and How X Works connect this example to wider learning. Before treating an observed count as a population, ask what could have entered the record and what the observation process could have missed.
Sources checked 21 October 2026. Worked examples are educational models with the assumptions stated above.