The Gold Standard of Weather Forecasting | Comparing Displaced Rainfall Patterns with Neighbourhood Fractions

Our teaching event is at least five millimetres of rainfall during one specified hour in a specified grid cell. A cell with eight millimetres is an event; one with zero is not. The exact equality convention is part of the definition, so a cell at five millimetres also counts.

This turns a rainfall-amount field into a binary event field of ones and zeros. It deliberately discards information about how far above or below the threshold each amount lies. That simplification supports one verification question, not every rainfall question.

The forecast and reference observation must concern the same hour, spatial support and threshold. Comparing a forecast hourly accumulation with an observation from a different period changes the task. A neighbourhood score cannot repair that definition mismatch.

Record issue time as well as valid time. A forecast available the previous day and an update available just before the hour have different lead times. Their scores can be compared descriptively, but a claim about relative forecast capability needs the difference in available information to remain visible.

For this worksheet, every input is supplied on the same equal-cell teaching grid with complete observations. Later chapters deliberately remove some of those conditions. Beginning with a clear simple case makes it possible to identify which part of the method needs redesign when the event, mask or reference changes.

Contents · Next chapter

Number thirteen equal cells from one to thirteen along a one-dimensional teaching line. Each cell represents a one-kilometre-wide segment in this model. The reference has eight millimetres in cell six and zero elsewhere. The forecast has eight millimetres in cell seven and zero elsewhere.

After applying the five-millimetre threshold, the reference event field has one wet cell at six. The forecast has one at seven. The amount and wet-cell count match, but the location differs by one cell.

We will score the same nine centre locations, cells three through eleven, for neighbourhood widths one, three and five. That fixed centre set keeps the comparison population unchanged. Even the five-cell windows fit within the thirteen-cell input domain, so no padding or shortened edge window is needed.

At width one, each neighbourhood is just its centre cell. The reference event is missed at six, and an event is forecast incorrectly at seven. The other scored centres are dry in both fields. This is the strict spatial comparison we will broaden.

The line is an arithmetic teaching model, not a complete representation of rainfall geography. Actual two-dimensional windows have a shape, area and grid definition. The same score formula can apply to their fractions, but the neighbourhood geometry and observational support must be stated rather than inferred from a colourful map.

Contents · Previous chapter · Next chapter

For a width-three window, take the centre cell and its immediate neighbour on either side. Its event fraction is the number of wet cells divided by three. Do this separately for forecast and reference at every scored centre.

The reference’s wet cell six lies inside windows centred at five, six and seven. Those reference fractions are one-third. The forecast’s wet cell seven lies inside windows centred at six, seven and eight, giving forecast fractions one-third there.

The fractions are local spatial coverage values, not automatically probabilities of rain at the centre. No ensemble was introduced, and we have not calibrated a probabilistic forecast. The numerical range zero to one does not make every fraction the same kind of probability statement.

A useful checking question is why cell six contributes to three windows rather than one. The windows overlap. That overlap is part of the method and means the nine fraction records should not be treated as nine independent pieces of meteorological evidence merely because they occupy nine table rows.

Contents · Previous chapter · Next chapter

Let fᵢ and oᵢ be forecast and observed event fractions at scored centre i. Form the squared-difference sum D = Σ(fᵢ − oᵢ)². Form the reference normalisation E = Σ(fᵢ² + oᵢ²). When E is positive, FSS = 1 − D/E.

The same expression is often written using means rather than sums. If the same N centres enter both numerator and denominator, the common factor one over N cancels. Do not cancel it when the two expressions use different masks or populations.

The forecast-verification working group's technical guide presents the neighbourhood fractions method and this score definition. Our cell records and subsequent derivations are original examples. We use the formula to ask a specific spatial question rather than turn the score’s name into a general approval of a forecast.

D measures disagreement between the two fraction fields. E scales that disagreement by their squared fractions. It is not a measured uncertainty or the number of independent samples. Both sums are dimensionless because they use event fractions.

Before taking the ratio, check that every fᵢ and oᵢ uses the same threshold, window rule and valid centre. Otherwise the arithmetic may be well formed while its two inputs describe different questions. A score becomes interpretable only when the verification contract travels with its value.

Contents · Previous chapter · Next chapter

At width one in the displaced example, f and o are binary. Only two scored centres contribute to D: six has forecast zero and reference one; seven has forecast one and reference zero. Each squared difference is one, so D = 2.

The normalisation E also equals two because there is one forecast wet cell and one reference wet cell. Thus FSS = 1 − 2/2 = 0. The dry agreement at the other seven scored centres contributes zero to both sums.

This does not say that every aspect of the forecast was wrong. It says that the thresholded wet cells have no exact overlap under this width-one score. The amount, timing and nearby placement still need their own interpretation.

A different metric, such as fraction of correctly classified cells, would include correct dry cells in its numerator. Here that fraction is seven out of nine. The two numerical summaries answer different questions. A high dry-cell accuracy does not create wet-cell overlap, and a zero width-one FSS does not erase every useful property of the forecast.

Keep the statement narrow: there is no exact event overlap in the scored region. That is a defensible interpretation. Calling the entire forecasting system useless from one invented displaced patch would be an overclaim unrelated to the amount of arithmetic performed.

Contents · Previous chapter · Next chapter

Use the fraction table. Differences occur only at centres five and eight. Their magnitudes are one-third, so D = 1/9 + 1/9 = 2/9. There are three nonzero forecast fractions and three nonzero reference fractions, each one-third, giving E = 6/9.

Therefore FSS = 1 − (2/9)/(6/9) = 1 − 1/3 = 2/3. The score has risen from zero to approximately 0.667 without moving either original wet cell.

The change comes from comparing local fractional coverage instead of exact cell events. At centres six and seven, each three-cell window contains one wet cell in each field, even though the wet cells occupy different positions within the window. Those local summaries now agree.

This is the intended scale-sensitive behaviour, but it has a cost: some location detail has been removed. A user whose task requires rainfall at cell six cannot conclude that the forecast was correct there merely because nearby window fractions agree.

Ask the learner to explain the improvement without saying that the rain moved. The forecast and reference fields are unchanged. Only the representation being compared has changed. That distinction is the core of the article and remains important when real maps contain far more detail than this small table.

Contents · Previous chapter · Next chapter

A width-five window contains the centre plus two cells on either side. The reference wet cell six contributes one-fifth to centres four through eight. The forecast wet cell seven contributes one-fifth to centres five through nine.

Only centres four and nine differ, so D = 2/25. Ten squared nonzero fractions contribute to E, giving E = 10/25. The score is 1 − (2/25)/(10/25) = 0.80.

Every row scores centres three through eleven. The full input windows remain inside cells one through thirteen, and both wet patches’ contributing windows are represented. That controlled setup avoids changing the sample while changing the scale.

A broader window has improved this particular score, but it has not increased the forecast’s original spatial precision. Nor should this example be read as a theorem that every field’s score must increase monotonically under every neighbourhood or masking rule. The calculation establishes the behaviour of these specific patterns under these rules.

State physical scale too. Five one-kilometre cells form a five-kilometre-wide footprint in this one-dimensional model. The same count of cells on a different-resolution grid represents another physical neighbourhood.

Contents · Previous chapter · Next chapter

Expand D: Σ(fᵢ − oᵢ)² = Σfᵢ² + Σoᵢ² − 2Σfᵢoᵢ. Substitution gives FSS = 2Σfᵢoᵢ / Σ(fᵢ² + oᵢ²), provided the denominator is positive.

This form explains several properties directly. Fractions are nonnegative, so the score cannot be negative. The inequality 2fo ≤ f² + o² shows it cannot exceed one. Identical fraction fields give one when at least one fraction is nonzero.

If one fraction field is entirely zero and the other has an event contribution, every product fᵢoᵢ is zero, giving score zero. The zero result follows from the overlap structure rather than from an arbitrary rule that a missing event should receive a particular grade.

For our width-three example, the two overlapping centres contribute products 1/9 each. Doubling their sum gives 4/9; dividing by E = 6/9 gives 2/3 again.

The algebra offers an independent calculation and an interpretation. It does not convert the score into a probability that the forecast is correct. A dimensionless value in the unit interval can describe similarity without being a calibrated event probability. Keep that distinction visible when presenting a score alongside the probabilistic forecasts discussed in the existing Weather reliability guide.

Contents · Previous chapter · Next chapter

Consider a wider buffered teaching line containing one reference wet cell and one forecast wet cell, separated by d whole cells. Use width w equal-cell windows and a scored centre set large enough to include all windows affected by either event, with no edge truncation.

Each event contributes fraction 1/w at w centres. Thus E = 2w/w² = 2/w. If d is smaller than w, their nonzero fraction supports overlap at w − d centres. The overlap form gives FSS = [2(w − d)/w²]/[2/w] = 1 − d/w.

If d is at least w, the fraction supports do not overlap and the score is zero. Combining the cases gives max(0, 1 − d/w) for this deliberately restricted single-cell pattern.

For displacement one and widths three and five, the formula returns 2/3 and 0.80. For displacement two, it returns 1/3 and 0.60. This provides a useful changed-context check without recalculating every window by hand.

Do not apply the formula to arbitrary rain fields. Multiple patches, different wet areas, unequal cell weights and edge rules change the sums. The derivation works because we counted the exact fraction supports of one event in each field. Its value is understanding that geometry, not replacing the general score with a shortcut whose conditions have disappeared.

Contents · Previous chapter · Next chapter

Take one window containing all thirteen cells in the original example. Both fields have one wet cell, so both fractions equal 1/13. Their difference is zero and the score is one, despite the one-cell displacement.

The same would happen if their single wet cells were much farther apart within that whole-domain window. The summary retains the number of wet cells but discards their arrangement. Agreement in total event coverage cannot establish agreement in every local position.

This is a constructive explanation of the trade-off. A coarser representation can be valuable for a regional question while being insufficient for a street-level question. Neither scale is inherently correct for every user. The verification method should match the spatial information the intended task requires.

It also explains why a score should be reported with its neighbourhood, threshold and region. A bare value of one can sound like a perfect forecast when it means only perfect agreement in one broad coverage fraction.

For a learning check, move the forecast wet cell to twelve while leaving the reference at six. The whole-domain fractions remain equal, so the one-window score remains one. The strict event locations have moved farther apart. That changed example makes the information loss explicit: the score did exactly what its chosen representation allowed, and the overclaim would belong to the interpretation.

Contents · Previous chapter · Next chapter

Neighbourhood event fractions are formed from thresholded cells. Averaging rainfall amounts first and then thresholding the average creates another quantity. The two operations do not generally commute.

Consider three invented cells with amounts zero, zero and nine millimetres during the defined hour. At a five-millimetre threshold, one of three cells is wet, so the event fraction is one-third. The mean rainfall amount is three millimetres, whose threshold indicator is zero.

The answers differ because one asks what fraction of cells exceed the threshold, while the other asks whether the neighbourhood’s average amount exceeds it. Both questions can be defined, but they should not be labelled as the same event fraction.

A second example uses amounts four, four and four. The event fraction is zero and the mean is four, also below the threshold. That accidental agreement does not justify exchanging the operations in every case. A changed example is necessary to reveal the distinction.

When reading a verification product, inspect the preprocessing order. Were amounts accumulated over the correct interval, regridded under a stated method, thresholded, and then converted to fractions? Changing any of those steps can change the resulting event field. A correct score calculation cannot recover the original intended question if the input representation was constructed differently.

Contents · Previous chapter · Next chapter

Our original eight-millimetre wet cells could be changed to eighty millimetres without changing their five-millimetre event indicators. Under that threshold, the binary fields and their neighbourhood scores would be unchanged.

This does not mean the rainfall amount difference is unimportant. It means the chosen score is not retaining that information. An event-coverage comparison can be useful while leaving intensity error unresolved.

For another case, suppose reference and forecast have wet cells in exactly the same positions, but their above-threshold amounts differ substantially. Width-one FSS can be one, while an amount-based error metric reports a difference. There is no contradiction because the two metrics examine different representations.

A meaningful evaluation often uses more than one view: event coverage at specified thresholds, amount errors, timing and other task-relevant evidence. More metrics are not automatically better; each should answer a distinct question whose definition remains clear.

Do not increase a threshold after seeing the maps merely to obtain a favourable score without reporting that choice. The threshold is part of the verification design. If several thresholds are examined, show the resulting pattern rather than select one unexplained headline. The same discipline applies to choosing neighbourhood widths: a score’s interpretation depends on the scale that produced it.

Contents · Previous chapter · Next chapter

If both fraction fields are zero at every scored centre, D = 0 and E = 0. The stated formula becomes zero divided by zero and is undefined. It does not automatically return one from the algebra.

An implementation may adopt a documented convention for no-event cases, omit them from a particular aggregate or report their frequency separately. Those choices affect summaries and need to be visible. This guide does not impose a universal software convention.

If only one field is entirely zero while the other has nonzero fractions, E is positive and D equals E, so FSS is zero. That case is mathematically different from both fields having no event contribution.

For a practical record, retain how many cases were dry in both fields and how they entered the reported result. A dataset dominated by dry conditions can look different depending on whether those cases were treated as perfect, undefined or excluded. The event-verification question should determine the rule, not the desire for a convenient average.

The key learning step is to inspect the denominator before division. A familiar formula does not guarantee that every possible input yields a defined number. Explaining an undefined score can be more informative than concealing it behind a default value whose meaning nobody can reconstruct.

Contents · Previous chapter · Next chapter

Our thirteen-cell example was designed so all required windows fit and all data are present. Actual fields can contain coastlines, missing observations or analysis boundaries. The method must define which cells enter each window and which centres enter the score.

Suppose a three-cell window has two valid cells, one wet and one dry, while the third is missing. Treating the missing cell as dry gives fraction one-third. Using only valid cells gives one-half. Neither value can be selected silently while claiming the same definition.

A shortened edge window also changes the denominator of its local fraction. Zero padding assumes an event state outside the available field; it does not merely make the array easier to compute. A common interior mask can avoid some edge issues but changes the region being evaluated.

The forecast and observation rules need compatible support. If the forecast fraction uses three cells while the observation fraction uses a different unreported subset, their difference combines spatial sampling with forecast disagreement.

A useful evidence check records input validity, window validity, centre selection and weighting separately. Missing observations are not dry weather. When a design changes the valid sample, compare how much of an apparent score change came from different cases rather than a different forecast. The simple worksheet makes this easier by keeping the centre set fixed, but that is a controlled teaching choice rather than a universal solution.

Contents · Previous chapter · Next chapter

A width-five window on one-kilometre cells represents a different physical footprint from a width-five window on five-kilometre cells. Comparing only the number of cells can therefore compare different spatial questions.

In two dimensions, window shape also matters. A square, circle and elongated region can include different nearby events even when a single quoted radius or area looks similar. The physical grid, projection and weighting should be appropriate to the intended comparison.

Our one-dimensional formula uses equal cell contributions. On a grid where cells represent unequal areas, an unweighted cell fraction need not equal the fraction of physical area experiencing the event. If area coverage is the target, the weighting requires a clear definition and suitable data.

These points are not reasons to abandon a neighbourhood score. They identify the information needed to make its scale meaningful. A reader should be able to translate the reported window into the actual spatial support rather than assume that a larger integer means the same neighbourhood in every product.

For transfer, keep the physical target fixed and change grid resolution. Ask which window and weighting would represent that target under the new grid. The answer is not necessarily the same cell count. This exercise tests whether the learner understands the spatial question beneath the array operations.

Contents · Previous chapter · Next chapter

Consider two invented forecast cases, each scored at width one over one hundred cells. In case A, one wet cell is displaced with no overlap, giving D = 2, E = 2 and FSS = 0. In case B, all one hundred cells are wet in both fields, giving D = 0, E = 200 and FSS = 1.

The unweighted mean of the two case scores is 0.50. Pooling the component sums gives 1 − (2 + 0)/(2 + 200) = 100/101, approximately 0.990. These are different summaries.

The pooled score is an E-weighted mean of the individual defined scores, because Dⱼ = Eⱼ(1 − FSSⱼ). Cases with larger normalising sums have more influence. The two cases’ equal cell counts do not make their E values equal.

Which summary should be reported depends on the intended aggregation question. Do not call one the other. If means rather than raw sums were saved for unequal-sized samples, the corresponding sample weights also matter before components are pooled.

The large pooled value in this example does not erase case A’s failed wet-cell placement. It reflects the weighting. A useful report can show stratified cases or regimes alongside a pooled result so the reader sees what dominates it. The arithmetic is exact; the interpretive mistake would be presenting a pooled number as though it guaranteed similar performance on every type of weather case.

Contents · Previous chapter · Next chapter

The name Fractions Skill Score should not replace a justified reference comparison. A score of 0.80 under one threshold, window and event frequency does not by itself establish the same predictive benefit as 0.80 under another setting.

The primary research How to derive skill from the Fractions Skill Score examines how reference-score choices affect interpretation. Its arXiv record was revised in January 2025. The relevant lesson here is to state the reference and comparison design rather than treat one numerical threshold as a universal certificate of useful skill.

Sampling uncertainty matters too. Overlapping windows are not independent forecast cases. Many neighbouring fractions from one rain field do not necessarily supply the information of many independent weather situations. A reliable uncertainty assessment needs the actual sampling structure and an appropriate method.

Reference observations also have limitations. Radar-derived rainfall and gauge records differ in coverage and measurement process. The radar and rainfall-estimation guide develops that evidence chain. A neighbourhood comparison cannot make an unsuitable observation representative merely by averaging it.

Finally, user value depends on the task. Regional coverage agreement can matter to one investigation while exact local timing matters to another. Report the score with its event, scale, mask, lead time, reference and uncertainty. Then state which unresolved user question still needs another form of evidence.

Contents · Previous chapter · Next chapter

Task A moves the forecast wet cell from seven to eight while leaving the reference at six. Use a sufficiently buffered line and score every affected window. For width three, the single-cell displacement formula gives 1 − 2/3 = 1/3. For width five it gives 1 − 2/5 = 0.60.

Explain why the score fell relative to the one-cell shift at each fixed width. The fraction supports overlap at fewer centres. Do not say the broader scale has repaired the original location; it has changed how closeness is represented.

Task B keeps the original cell positions but raises the event threshold to ten millimetres while both amounts remain eight. Both fields now have no events, so the formula has a zero denominator. The old scores do not apply to the newly defined event.

Task C uses the original threshold and exactly matching wet positions but forecast amounts eighty instead of eight. The event score is unchanged, while an amount comparison is different. Name the information the threshold removed.

Task D gives only five observed cells within a larger forecast map and no masking rule. Can the full-domain score be reproduced? Not yet. The valid windows and centres have not been defined. Identifying that missing information is part of a correct solution, not a failure to apply the formula.

Contents · Previous chapter · Next chapter

Does a larger FSS mean the rain fell in the correct cell?

It means better agreement of the specified fraction fields. Our whole-domain example gives one despite displaced wet cells. Exact location needs the appropriate finer-scale evidence.

Are neighbourhood fractions probabilities of rain at the centre?

Not automatically. Here they are spatial event-coverage fractions from deterministic fields. A probability forecast needs its own definition and verification.

Can I report one score without a threshold or window?

The number would omit essential meaning. Threshold, spatial support, region, valid time and comparison sample define what was evaluated.

Is a completely dry case always perfect?

The formula is undefined when both fraction fields are zero. Any treatment in an implementation or aggregate must be stated. That is different from one field missing events that the other contains.

Does this replace official weather warnings?

No. These are fictional verification exercises. Actual conditions, warnings and instructions belong to the responsible meteorological and emergency authorities. A retrospective spatial similarity score is not a safety decision rule.

Contents · Previous chapter · Next chapter

For the production chain, return to Weather Forecasting and Weather. For the different question of probabilistic performance, use Forecast Reliability, Sharpness and Decision Thresholds.

For observational evidence and interpretation, read Weather Radar and Rainfall Estimation, Measurement and Data Literacy. These routes keep a derived reference field and a verification score connected to the process that produced them.

Primary source routes reviewed on 22 October 2026 are the forecast-verification working group's neighbourhood-method account and Antonio and Aitchison's research on interpreting FSS skill. The thirteen-cell pattern, displacement derivation, dry and threshold cases, aggregation example and practice tasks are original teaching constructions. They make no performance claim about a named forecast service.

Continue through Learning That Transfers into Life and How X Works. A useful finish is to calculate one score and explain which spatial detail it preserves and which it removes. That explanation keeps the method useful when the map becomes much larger than the worksheet.

Contents · Previous chapter · Return to Weather Forecasting