A coarse raster cell can report an average without revealing how values are arranged inside it. If a new zone cuts through that cell, knowing the overlap area does not automatically reveal the average inside the overlap. A finer-looking resampled map can conceal this missing information rather than recover it.
The short answer is that an exact subzone mean may be unidentified, but useful bounds can follow from the coarse mean, overlap area and justified lower and upper limits on values. We will derive those bounds and construct fields that attain them. The examples are original fictional spatial data, not environmental measurements or maps of actual neighbourhoods.
This is a focused inference problem within the Zonal Statistics subject guide. It differs from merely computing a weighted mean of supplied cell values: we ask what can be said about the unknown field within coarse cells when those supplied values are explicitly cell averages.
eduKateSG · Zonal Statistics
Find your next learning step
Separate a fine-looking map from information about values inside coarse cells.
ROUTE 2 · CHAPTERS 6–10
Build sharp cell bounds
Work a complete original example with stated assumptions.
ROUTE 3 · CHAPTERS 11–15
Combine cells and information
Change the conditions and identify what survives.
ROUTE 5 · CHAPTERS 21–25
Practise and continue
Check independent understanding and choose the next guide.
Chapter contents
Define spatial support · Chapters 1–5
1. Identify what the raster value represents
2. Define the subzone question precisely
3. See why area weighting can become an extra assumption
Build sharp cell bounds · Chapters 6–10
6. Derive the overlap's lower bound
7. Derive the overlap's upper bound
8. Prove that the endpoints can be attained
Combine cells and information · Chapters 11–15
11. Construct both extreme two-cell fields
12. Separate a sum of bounds from an unsupported independence claim
13. Distinguish a fine grid from new field information
Audit assumptions and decisions · Chapters 16–20
16. Include uncertain coarse means honestly
17. Do not treat missing cells as measured zero
18. Handle uncertain boundaries as another input
Practise and continue · Chapters 21–25
21. Work a changed overlap independently
22. Choose a practical learner pathway
23. Distinguish a point reading from an area-supported observation
CHAPTER 1 OF 25 · Define spatial support
1. Identify what the raster value represents
Before doing any spatial arithmetic, read the value’s meaning. It might be a measurement at a point, a model estimate at a grid location, a category code or an average over a stated area. These are different kinds of evidence.
Our model assumes each coarse cell reports the exact area-average of a bounded scalar field over that entire cell. The cell boundaries are known and cells do not overlap each other. This assumption is part of the exercise, not a default claim about every raster product.
If a source instead reports a point estimate, multiplying it by cell area does not automatically create an observed integral. If it reports a category code, averaging that code may not have meaningful physical units. Establish the spatial support and quantity before selecting an operation.
For an original learning question, hand a student a cell labelled “18.” Ask what that number means. A satisfactory answer needs the variable, units, time and support. The printed number alone cannot decide which later calculations are justified.
Let a target zone intersect part of a coarse cell. We want the field’s area-average over that intersection, not the original whole-cell mean and not an individual person’s exposure.
Write A for the whole-cell area, a for the overlap area and m for the known whole-cell mean. Let z denote the unknown average inside the overlap and w the unknown average in the remainder. For a strictly partial overlap, both areas are positive.
The cell’s total integrated amount is Am. Splitting it by the zone boundary gives the accounting identity Am=az+(A−a)w. This equation does not assume that z and w are equal.
Treat this as a spatial ledger. The coarse record supplies one total. The zone boundary divides that total between two parts, but the record does not specify the division. A drawing can show the areas precisely while leaving both subarea averages unknown. Geometry and field information answer different parts of the question.
CHAPTER 3 OF 25 · Define spatial support
3. See why area weighting can become an extra assumption
A common reconstruction assigns the coarse value m uniformly throughout its cell. Under that reconstruction, the overlap contributes am and the outside contributes (A−a)m. The total is correct because the pieces sum to Am.
But matching the total does not establish the uniform reconstruction. Other inside and outside values can produce the same total. If the actual question concerns the original unknown field, assigning m everywhere is a modelling choice.
This distinction is subtle because the resulting weighted calculation can be arithmetically flawless. It can exactly summarize the reconstructed piecewise-constant raster while failing to identify the original field’s subzone mean.
Report which question was answered. “The mean of the piecewise-constant reconstruction within this polygon is 11” is different from “the true field’s polygon mean is known to be 11.” Our worked examples will make the difference explicit by producing several compatible fields. They do not imply that uniform reconstructions are useless; they identify the assumption needed to interpret them.
Take a cell of area 2 square units with whole-cell mean 18 measurement units. Its integrated amount is 36 measurement-units times square units. The target zone occupies area 0.5; the remainder has area 1.5.
One compatible field has overlap mean 12 and outside mean 20. Its total is 0.5×12+1.5×20=6+30=36. Another has overlap mean 20 and outside mean 52/3, approximately 17.333. Its total is 10+26=36.
Both fields obey values between zero and twenty, have the same cell mean and use the same zone geometry. Yet the target overlap means differ: 12 versus 20. The coarse record cannot distinguish them.
These fields can be constructed by making the scalar constant on each of the two subareas. Their discontinuity at the boundary is permitted in our present model. If smoothness or another spatial constraint is later imposed, that changes the feasible fields and requires another argument. Do not silently add it while interpreting these bounds.
Suppose the field is known to lie between L and U throughout the cell. Then both z and w lie in that interval. The limits must be supported by the quantity and evidence, not chosen merely because they make the resulting interval narrow.
For our fictional examples L=0 and U=20. These limits are stipulated features of the teaching model. They should not be transferred to a real temperature, concentration or ecological raster without justification.
The known mean must also lie between L and U. If it does not, the model inputs are inconsistent. That inconsistency might arise from wrong units, a misunderstood raster value, a bad limit or measurement error. It is not a reason to force an apparently plausible answer.
Value limits make the inverse problem more informative because they constrain how much of the known total can sit outside the target. If the whole-cell mean is near the maximum, a large remainder cannot be uniformly low. Some of the target must then also be high. The next calculation makes that statement exact.
Define I as the unknown integrated amount within the overlap, so I=az. The outside area can hold at most (A−a)U. Because the full amount is Am, the overlap must hold at least Am−(A−a)U.
Independently, the overlap’s own lower value limit requires I≥aL. Both conditions must hold, so the strongest lower bound supplied by these two constraints is I_low=max[aL, Am−(A−a)U].
For A=2, a=0.5, m=18, L=0 and U=20, this is max[0,36−30]=6. Dividing by a gives z≥12. A zero lower field limit does not imply that the target mean can be zero, because the remaining area could not hold the required whole-cell amount.
Notice the units. I_low is an integrated amount, not a mean. Dividing it by the overlap area produces the target-average bound. Mixing those two stages can create a result with the wrong scale even when every individual multiplication was performed correctly.
The overlap cannot hold more than aU. Also, the outside area must hold at least (A−a)L. Subtracting that minimum outside contribution from Am gives another upper bound on I.
Together, I_high=min[aU, Am−(A−a)L]. Divide by positive a to obtain the upper bound on z. For our cell, min[10,36]=10, hence z≤20.
Combining both sides gives z in [12,20]. The coarse mean 18 lies in that interval, but it is only one feasible overlap mean. The interval does not say that all values are equally likely, or assign any probability to them.
This distinction matters for reporting. These are deterministic feasibility bounds under stated assumptions. They are not a confidence interval from random sampling and not a posterior distribution. A reader who wants probabilities over the possible fields needs an additional stochastic model and evidence for it. The accounting bounds answer a useful question without pretending to answer that different one.
CHAPTER 8 OF 25 · Build sharp cell bounds
8. Prove that the endpoints can be attained
A bound is sharp under a model when a compatible configuration can attain it. The earlier lower-end field sets z=12 and w=20, meeting the known total and both value limits. The upper-end field sets z=20 and w=52/3, again meeting every stated condition.
More generally, choose any I between the derived bounds. Set the field equal to I/a inside the overlap and to (Am−I)/(A−a) outside it. The bounds ensure both constants lie within [L,U]. Their integrated amounts sum to Am.
This constructs a feasible field for every intermediate I when 0<a<A. It establishes that the interval cannot be narrowed using only these particular constraints. A narrower answer would need additional information or assumptions.
Do not confuse sharpness under the model with a claim about realistic spatial texture. The construction deliberately permits a discontinuity along the target boundary. If a real physical field cannot behave that way, state and justify the extra constraint, then recalculate. The proof tells us exactly which freedom produced the interval.
CHAPTER 9 OF 25 · Build sharp cell bounds
9. Handle empty and complete overlaps separately
If a=0, the cell contributes nothing to the target zone. There is no overlap mean to calculate, and division by a is invalid. Record the zero contribution rather than inventing a mean of zero.
If a=A, the target contains the whole cell. Its integrated amount is exactly Am and its mean is exactly m under the model. There is no unknown outside allocation. The bound formulas also collapse to that amount when the inputs are consistent.
For a tiny but nonzero overlap, the mean interval can remain wide. The coarse total may say little about a very small piece because the remainder can absorb most allocations. A precise polygon boundary does not remove that information limitation.
These endpoint checks are useful for code and reasoning. Test zero overlap, full overlap and a partial overlap before processing a large map. They exercise different branches of the calculation and can reveal division errors, inappropriate zero filling or a mistaken assumption that every intersecting cell contributes its whole value.
CHAPTER 10 OF 25 · Build sharp cell bounds
10. Combine two cells into a target zone
Now add a second cell B with area 1, whole-cell mean 4 and overlap area 0.5. Keep the same bounds [0,20]. Its whole-cell amount is 4.
For B, I_low=max[0,4−0.5×20]=0. Its upper bound is min[0.5×20,4]=4. The target overlap’s mean can range from zero to eight.
The target consists of the 0.5-area overlap from cell A and the 0.5-area overlap from B. Its total area is 1. The amount from A lies in [6,10], while B contributes [0,4]. Therefore the target total lies in [6,14], and its mean lies in [6,14] measurement units.
The piecewise-constant reconstruction would produce 0.5×18+0.5×4=11 as the target mean. That is feasible but not identified. Both the interval and the reconstruction value can be reported, provided their meanings remain distinct. One describes all fields allowed by the present information; the other selects a particular field model.
| Fictional cell | Full area | Whole-cell mean | Target overlap area | Feasible overlap amount |
|---|---|---|---|---|
| A | 2 | 18 | 0.5 | [6,10] |
| B | 1 | 4 | 0.5 | [0,4] |
| Target combined | — | — | 1 | [6,14] |
CHAPTER 11 OF 25 · Combine cells and information
11. Construct both extreme two-cell fields
At the target’s lower endpoint, choose cell A’s overlap mean 12 and outside mean 20. In B, choose overlap mean zero and outside mean eight. The cell totals remain 36 and 4, while the target receives amounts 6 and zero, giving mean 6.
At the upper endpoint, choose A’s overlap mean 20 and outside mean 52/3. In B, choose overlap mean eight and outside mean zero. The target amounts are 10 and 4, giving mean 14.
Every subarea value remains between zero and twenty. Both complete fields have the same two reported coarse cell averages and the same zone geometry. Their different target means prove the lack of point identification directly.
This is stronger than merely saying that resampling “might introduce uncertainty.” It shows the two incompatible answers that the same supplied records permit, then establishes the full achievable interval under the declared model. A reader can inspect each multiplication and decide whether an additional spatial assumption is justified for the actual application.
CHAPTER 12 OF 25 · Combine cells and information
12. Separate a sum of bounds from an unsupported independence claim
For a target made of disjoint overlaps from different coarse cells, the integrated contributions add. Under our model, each cell’s internal allocation can be chosen separately. Therefore their lower endpoints can be attained together, as can their upper endpoints.
No probability independence assumption is used. This is a deterministic product of feasible sets: the constraints for one cell do not restrict how another cell is internally arranged. Calling the cells “independent samples” would introduce an unnecessary and different statistical claim.
If additional constraints couple cells, the separate extremes may no longer be jointly attainable. A continuity condition, shared gradient bound or known cross-cell total for another region could restrict combinations. In that case summing individual bounds remains a possible outer bound, but sharpness needs a new joint analysis.
Name the coupling rather than vaguely requesting more data. A calculation becomes easier to review when it states which fields are allowed and which relationships connect their unknown values. The same spatial ledger can then be extended without confusing deterministic restrictions with stochastic dependence.
CHAPTER 13 OF 25 · Combine cells and information
13. Distinguish a fine grid from new field information
Subdividing each coarse cell into many smaller displayed cells does not by itself determine the original subcell values. A resampling algorithm can assign values according to a rule, but the rule contributes assumptions or a reconstruction, not automatically new observations.
In the two-cell example, a fine grid filled uniformly with 18 inside A and 4 inside B preserves the coarse means. A different fine allocation approximating the lower-end field also preserves them. The image resolution alone cannot tell which field occurred.
The ArcGIS Pro zonal-statistics documentation describes software choices involving rasterization, cell alignment and resampling. Those implementation details need reporting. They should not be confused with the separate inference problem about unknown within-cell values developed here.
A useful caption might identify a result as a zonal statistic of a named reconstructed raster under a specified inclusion rule. Avoid describing the extra small pixels as newly measured detail unless there is evidence supporting that statement. Visual smoothness and evidential resolution are different properties.
CHAPTER 14 OF 25 · Combine cells and information
14. Keep categorical resampling as a different problem
Our bounds concern a bounded scalar field with meaningful averages and integrated amounts. They are not directly a method for averaging arbitrary land-cover labels such as 1=water and 2=forest.
The existing Remote Sensing guide on bilinear resampling and land-cover classes explains how interpolation can create invalid intermediate category codes. That is a different failure from the one here.
For a proportion field, such as a genuinely defined area fraction, the numbers may have a meaningful bounded interpretation. But first identify whether the source reports a proportion or a nominal label. A plausible numeric range does not decide the issue by itself.
For independent practice, present two rasters containing only zeros and ones. In one, they are measured fractions that happen to reach the endpoints; in the other they are category identifiers. Ask which arithmetic interpretations are justified and why. The lesson is to read metadata before applying formulas, rather than selecting a method solely from the appearance of the numbers.
CHAPTER 15 OF 25 · Combine cells and information
15. Add information about the outside area
Return to cell A. Suppose additional evidence establishes that the outside mean w lies between 17 and 19, while the original field bounds remain [0,20]. This is new information, not a deduction from the coarse mean.
The outside amount now lies between 1.5×17=25.5 and 1.5×19=28.5. Subtracting from the total 36 gives overlap amount between 7.5 and 10.5. Intersect with the overlap’s own range [0,10], producing [7.5,10]. The overlap mean narrows to [15,20].
With B unchanged, the whole target mean narrows from [6,14] to [7.5,14]. The lower endpoint rises because the added information prevents the outside of A from holding as much of its total as before.
Check feasibility after adding evidence. If a claimed outside upper bound were so low that even setting the overlap to U could not recover the known total, the constraints would conflict. A narrow or empty interval should prompt an input review, not an automatic declaration of precision.
CHAPTER 16 OF 25 · Audit assumptions and decisions
16. Include uncertain coarse means honestly
Suppose A’s whole-cell mean is known only to lie in [17,19] and B’s in [3,5]. Keep the areas and field limits fixed, with no further coupling between these uncertainties.
A’s possible total is [34,38]. The lowest feasible overlap amount uses the lower total: max[0,34−30]=4. The largest uses the upper total: min[10,38]=10. For B, the amount can range from zero to five.
The target’s feasible mean consequently lies in [4,15], since its area is one. These are bounds over the stated interval-valued inputs and allowed allocations. They are not automatically a confidence interval, even if some earlier measurement process produced its own uncertainty interval.
If the mean errors are jointly constrained, or the intervals have a specific probabilistic interpretation, preserve that information rather than treating every combination as independently possible. Our calculation intentionally uses the wider model in which every listed value combination is allowed. Clearly naming that choice prevents a deterministic feasibility result from acquiring an unsupported probability meaning.
CHAPTER 17 OF 25 · Audit assumptions and decisions
17. Do not treat missing cells as measured zero
A cell without a valid reported mean cannot be entered into the exact-mean ledger as though m=0. That would replace missing evidence with a specific field total and could narrow the target interval unjustifiably.
If the only information for an overlapping missing cell is that its field lies within [L,U], its overlap amount lies within [aL,aU]. Include that contribution and report the missing coverage. If even those limits are unsupported, a finite bound may be unavailable.
For a changed example, replace cell B’s mean with missing data while retaining the stipulated [0,20] field range. Its overlap amount now lies in [0,10], rather than [0,4]. Combined with A’s [6,10], the target mean lies in [6,20].
This widening has a clear reason: one constraint was removed. A map that silently fills B with zero would instead create an invented constraint. State which area lacks evidence and how that affects the conclusion. The reader can then decide whether another measurement is needed for the intended decision.
CHAPTER 18 OF 25 · Audit assumptions and decisions
18. Handle uncertain boundaries as another input
So far, overlap areas are exact. Real boundaries can be uncertain through survey error, coordinate transformations, changing definitions or an imperfect geographic model. Changing a changes both the target area and its possible integrated contribution.
Do not automatically insert a mean overlap fraction into the fixed-area formula and call the result a rigorous bound for all possible boundaries. The numerator and denominator can vary together, and geographic constraints can couple overlaps across cells.
A sound extension defines the allowed boundary family, calculates compatible overlap areas and optimizes the target mean over that family and the permitted fields. For a finite set of explicitly allowed boundaries, one can evaluate each boundary’s interval and take the union’s outer endpoints. That is a stated scenario calculation, not a claim to cover every unknown geometric error.
For learning, draw two possible zone boundaries and compute each interval separately. Ask whether their resulting difference reflects changed evidence about the field, changed definition of the target or both. This keeps map revision from being silently translated into physical environmental change.
CHAPTER 19 OF 25 · Audit assumptions and decisions
19. Examine the special case of whole-cell zones
If a target is exactly a union of complete coarse cells, the model identifies its integrated amount by summing Am over those cells. Divide by their total area for the target mean. Unknown within-cell arrangements no longer affect this particular average.
For all of A and B together, the amount is 36+4=40 and area is 2+1=3. The mean is 40/3, approximately 13.333. That conclusion is exact under our source and area assumptions, even though the detailed spatial field remains unknown.
This special case explains why aligning a reporting zone with measurement supports can change what is identified. It does not mean that changing a real policy or ecological zone to convenient grid cells always answers the original question. The target itself may have substantive meaning.
Also, equal overlap fractions in several cells do not automatically identify a partial-zone mean. Each selected part can differ from its remainder while preserving the coarse total. Full support matching is a stronger condition than merely giving every cell the same proportional coverage.
CHAPTER 20 OF 25 · Audit assumptions and decisions
20. Connect bounds to a stated decision threshold
Suppose an educational exercise asks whether the target mean exceeds 10. Under the original two-cell model, the interval is [6,14]. Some compatible fields are below 10 and others above it. The supplied evidence does not settle that threshold question.
If a different threshold is 5, every compatible target mean exceeds it. If the threshold is 15, none does. These are deterministic conclusions conditional on the assumptions, not risk estimates or recommendations for managing a real environmental hazard.
Use the interval rather than only the reconstruction value 11. That single estimate would place the target above 10, while the feasible-field analysis shows the threshold claim is unresolved. A modelling choice has practical consequences when the decision lies within the uncertainty range.
The next evidence should address the allocation that matters. A targeted observation within the overlap might help more than another attractive interpolation of the same coarse means. The best actual measurement strategy depends on the variable, costs, support and required accuracy; the toy bounds identify the missing relationship without prescribing a universal sampling plan.
Keep A’s area 2, mean 18 and field limits [0,20], but change its target overlap area to 1. The outside area is now also 1. Calculate the possible overlap amount before reading the check.
The lower bound is max[0,36−20]=16 and the upper is min[20,36]=20. Since the overlap area is one, its mean lies in [16,20]. A larger share of the high-mean cell cannot have a very low mean while leaving enough outside capacity to preserve the total.
If B’s overlap remains 0.5 with amount [0,4], the new target area is 1.5. Its total lies in [16,24], so its mean lies in [32/3,16]. Do not reuse the earlier denominator of one.
Construct the endpoints: for A’s lower overlap mean 16, outside mean is 20; for its upper overlap mean 20, outside mean is 16. Combine with B’s earlier lower and upper constructions. This verifies that the new interval is sharp under the same uncoupled-field assumptions.
A reporting sheet can use two explicit result columns. One gives the reconstruction-based point estimate, with its within-cell assignment rule. The other gives the feasible-field interval, with its value and support assumptions. In the original example these are 11 and [6,14]. They should not be averaged together or treated as two competing observations.
Ask which conclusion changes if the reconstruction assumption is removed. The feasible interval remains supported by the coarse totals and bounds; the unique value 11 no longer follows. Conversely, if new independent evidence supports uniformity throughout every cell, the selected reconstruction gains justification and its zonal mean may become identified under that stronger model.
This comparison teaches a practical way to communicate model dependence without discarding useful calculations. A number can be reproducible under a named reconstruction while the underlying field question remains unresolved. Naming the source of uniqueness is more informative than presenting an unexplained midpoint or a long decimal.
For repair, begin with the one-cell ledger Am=az+(A−a)w. Use a drawing and attach units to every term. Ask the learner to produce two different subarea means that preserve the same whole-cell average.
For consolidation, impose a lower and upper value limit and derive the minimum and maximum integrated contributions. Require explicit constructions at both endpoints. A memorized formula is less useful than explaining how outside capacity creates the lower bound.
For extension, combine cells, then add one new piece of information: an outside-mean restriction, an uncertain coarse mean, a missing cell or an alternative boundary. Predict whether the feasible set should narrow, widen or change target before calculating.
A strong final explanation reports the interval, the model it covers and what additional evidence would narrow it. It should distinguish deterministic feasibility from probability and distinguish the source field from a convenient raster reconstruction. Those distinctions make the method transferable to new mapped quantities without turning every coarse-grid number into false fine-scale certainty.
CHAPTER 23 OF 25 · Practise and continue
23. Distinguish a point reading from an area-supported observation
Suppose a sensor records value 14 at one point inside A’s overlap. Under our bounded-field model, changing a value at a single point does not change an area integral. The point has zero area. Without a further spatial relationship, that reading alone does not narrow the derived subzone-mean interval.
This does not make point measurements unimportant. It says that connecting a point to an area average requires a justified model, sampling design or spatial constraint. A continuity or gradient assumption may contribute information, but it must be stated and supported rather than silently inferred from the presence of a sensor symbol.
Now replace the point reading with a genuinely area-supported observation: a known subarea of 0.1 within A’s overlap has exact mean 14. Its integrated contribution is 1.4. The remaining overlap area is 0.4, whose contribution lies between zero and eight under [0,20]. Therefore the whole overlap amount lies in [1.4,9.4] before the coarse-cell constraint is reapplied.
Intersect that interval with A’s existing [6,10] to obtain [6,9.4]. Both constraints matter. The new target mean, including B’s unchanged [0,4] amount over total target area one, lies in [6,13.4]. The new observation narrows the upper endpoint while leaving the lower one unchanged.
Check attainability. At A’s lower endpoint I=6, the unmeasured 0.4 subarea needs amount 4.6, hence mean 11.5, while the outside mean is 20. At the upper endpoint I=9.4, the unmeasured subarea has mean 20 and the outside mean is (36−9.4)/1.5=266/15, approximately 17.733. All are allowed.
The contrast is instructive: the same printed value 14 carries different information when its support is a point versus a known area. Record the observation’s support, not merely its coordinates and numeric precision. That is the relationship needed to decide whether the measurement can legitimately narrow the spatial-average claim.
Consider a new teaching threshold of 13.5. Before the area-supported observation, the target interval [6,14] straddled that threshold. After the additional constraint, the upper bound is 13.4, so every permitted field lies below 13.5. The point reading alone did not establish this conclusion. The distinction comes from the measured support and its integration into the existing ledger, rather than from reporting 14 with more decimal places. If that area-supported mean is uncertain, propagate its actual allowed range instead of retaining the exact 1.4 contribution.
For a final consistency test, set m=U in a cell with positive overlap. The lower amount bound becomes AU−(A−a)U=aU, matching the upper bound. Under the model, a field bounded above by U whose whole-cell average is U must equal U almost everywhere, so every positive-area subzone has that same mean. The analogous statement holds at m=L. This special case explains why some coarse means can identify partial-zone averages even though the general case does not. Test these extremes separately from a middle-range mean and explain which constraint removes the internal allocation freedom.
Is an area-weighted mean wrong?
Not inherently. It can exactly summarize the values assigned by a specified raster model. The issue is whether those values identify the unknown original field within partially intersected cells.
Are these confidence intervals?
No. They are feasible ranges under declared exact or interval constraints. A probabilistic interpretation requires another model and supporting evidence.
Can interpolation recover the missing arrangement?
It can propose an arrangement using additional assumptions. The coarse means alone do not distinguish the compatible fields constructed here.
What makes the bounds sharp?
We explicitly construct allowed fields reaching both endpoints. If additional spatial constraints are imposed, sharpness must be reconsidered under that new feasible set.
Does a finer grid fix the problem?
More displayed cells do not by themselves add observations of the original within-cell field. State what new evidence or reconstruction assumption accompanies the finer grid.
CHAPTER 25 OF 25 · Practise and continue
25. Continue through the spatial evidence routes
Return to Zonal Statistics for inclusion rules, valid-cell counts, weighting and the distinction between NoData and zero. Geospatial Analysis connects target units, overlays and scale choices.
For modelled spatial structure, Geostatistics examines dependence, prediction support and model uncertainty. Geographic Information Science provides the wider geographic-data framework. Keep those modelling questions distinct from the exact accounting bounds here.
For a separate resampling issue, use why bilinear resampling can invent land-cover classes. For actual software behaviour, consult the current ArcGIS Pro tool documentation, checked on 26 October 2026; its output conventions are not assumed to implement our hypothetical inverse calculation.
Continue with Learning That Transfers into Life and the How X Works hub. All areas, scalar limits and fields in this article are original teaching constructions. Their bounds depend on the explicitly stated source-support, geometry and feasible-field assumptions.