Separate a collection of plausible futures from a guaranteed range.
By the end: Define an event before counting members and question shared error.
An ensemble is a set of forecasts for the same target time and location. Members vary initial conditions and may vary model assumptions or tendencies. Their divergence describes uncertainty represented by that construction, not every possible source of uncertainty.
The twelve invented rainfall values in this lab are already-computed outcomes for one site over a fixed 24-hour period. They are not live forecasts, independent observations, or a physical rain simulation. Equal member weights are stipulated so the event fraction can be inspected.
An ensemble member is one complete forecast calculation, not a bucket containing a fraction of tomorrow's rain. In a dynamical ensemble, a member begins with a particular estimated atmospheric state and follows specified model rules. Reading several members asks how the answer changes under the represented alternatives. The rain values here retain only the final accumulation at one site. They discard the evolving winds, humidity, and rain timing that might have produced it. Different full weather paths can therefore give the same displayed total without representing identical atmospheric states.
Initial uncertainty and model uncertainty enter at different places. Uncertain starting humidity can alter how soon saturation occurs, even with unchanged model equations. A different treatment of unresolved clouds can alter rainfall despite an unchanged starting state. ECMWF's uncertainty discussion identifies both sources and notes that assimilation partly links them. A collection varying only starting conditions cannot automatically reveal every error in cloud physics. Conversely, varying one model assumption does not replace estimating the initial state. The twelve finished totals do not label either source, so they cannot separate their contributions numerically.
| Member | First 12 hours, mm | Next 12 hours, mm |
|---|---|---|
| A | 12 | 0 |
| B | 0 | 12 |
| C | 6 | 6 |
Consider a separate paper example with four invented site totals, 2, 4, 14, and 16 mm over one specified 24-hour interval. Their sum is 36 mm and their equally weighted mean is 9 mm. Yet none of the four members predicts 9 mm. Two describe relatively dry outcomes and two describe much wetter outcomes. Replacing the collection with its mean hides that separation. The mean remains a well-defined arithmetic quantity, but a statement about the average does not tell us whether moderate rainfall is the main scenario being represented.
Equal weights mean that each of those four alternatives contributes one quarter to any counted fraction. That is a construction rule, not evidence that nature chooses among four equally likely possibilities. If credible weights were instead 0.4, 0.4, 0.1, and 0.1, the weighted mean would be 5.4 mm. Multiply each total by its weight and add the products. The heavier weights on dry alternatives lower the mean. Such weights would require justification in a real product. They are stipulated here only to expose an assumption that raw counting usually leaves implicit.
The same distinction applies to an event such as at least 12 mm in that interval. Equal counting gives two qualifying alternatives out of four. Under the proposed unequal weights, the qualifying weight is 0.1 plus 0.1, or 0.2. Neither calculation changes a rainfall total. They summarize the same alternatives under different weighting assumptions. A probability interpretation needs evidence that the construction represents uncertainty well for comparable situations. Without that evidence, the calculation is a weighted scenario fraction, useful for describing the ensemble but not certified as the event's real probability.
These four paper totals refer to one site over the same 24-hour window; μ is mean rainfall. They are not the lab's twelve members.
F is qualifying scenario weight, dimensionless; neither set of weights is empirically calibrated.
| Quantity | Equal weights | Dry-weighted |
|---|---|---|
| Weights | 0.25 each | 0.4, 0.4, 0.1, 0.1 |
| Mean, mm in 24 hours | 9 | 5.4 |
| Weight at least 12 mm | 0.5 | 0.2 |
Operational ensemble systems can perturb starting states and model tendencies rather than assigning a few handpicked rain totals. Their construction tries to represent uncertainty that evolves through the forecast equations. That purpose does not make every possible outcome appear among their members. A process omitted from all members can remain omitted after averaging. The paper comparison isolates weighting, while the existing lab isolates thresholds and common shifts. Neither provides a control for generating a new atmospheric integration. Treating these finished totals as a simulation of storm formation would add a mechanism the lesson does not implement.
The selected event is rainfall at least the threshold, in millimeters over that 24-hour window. At a 10 mm threshold, three of twelve members qualify, so the raw fraction is 25%. Changing the threshold changes the event, not the members.
An ensemble mean of 7 mm is not the same statement as a 25% chance of at least 10 mm. Nor does 25% mean rain for one quarter of the day or over one quarter of the region. Always identify the location, accumulation period, and threshold.
An event definition converts a continuous quantity into a yes or no question. Rainfall total is a number in millimeters; reaching a threshold is a statement that is either satisfied or not. For a member value r and threshold T, the event here is r greater than or equal to T. The equality matters when a displayed total lands exactly on the boundary. A threshold change can move members between categories without changing any forecast calculation. It changes the question asked of the results, rather than making the underlying atmospheric forecast wetter or drier.
Location and accumulation window belong inside that question. A 24-hour total of 12 mm can arrive in one hour or in many short showers. These have identical totals but different implications for drainage. Likewise, a site forecast and an average across a catchment describe different quantities. A catchment average can exceed a threshold while some individual sites remain dry. The rain lab specifies one site and one fixed 24-hour period. Its member totals cannot answer questions about peak hourly intensity, arrival time, or the fraction of a surrounding region receiving rain.
| Member total, mm | At least 12 mm | Above 12 mm |
|---|---|---|
| 11 | No | No |
| 12 | Yes | No |
| 13 | Yes | Yes |
For the unshifted lab collection, take a new threshold of 8 mm. The qualifying values are 9, 12, 18, and 24 mm. Counting gives four members and dividing by twelve gives one third. At a threshold of 12 mm, three qualify because equality includes the 12 mm member. At 13 mm, only 18 and 24 qualify. The fraction drops by one twelfth when the boundary passes 12 mm. That jump comes from finite counting, not from a physical discontinuity in how clouds produce rainfall at that exact depth.
A threshold curve reports many such questions for one collection. Increasing the threshold cannot increase the qualifying count, because every member satisfying the higher threshold also satisfies the lower one. This set inclusion gives a useful arithmetic check independent of any probability claim. The plotted samples decrease or stay level for that reason. Between samples the actual raw fraction is a step function, with jumps just beyond member values under the inclusive convention. Straight segments in the figure merely help compare selected thresholds. They do not establish a continuous rainfall distribution between those thresholds.
The finite denominator also limits the fractions available to equal counting. Twelve members give increments of one twelfth, about 8.33 percentage points. A count of four is exactly one third even if a display rounds it to 33.3%. Printing more decimals would not add new scenarios or establish finer probability resolution. In the existing control, move the inclusive threshold from 8 to 9 mm and then to 10 mm while retaining the reference members. The first change preserves the count; the second removes the 9 mm member. These are event changes, not forecast-time advances.
Qualifying fraction, count / 12
Inclusive threshold, mm in 24 hours
A zero count at a high threshold means that none of these twelve alternatives reaches it. It does not establish impossibility outside the represented set. Similarly, twelve qualifying members mean unanimous representation, not a guarantee about the observed outcome. Establishing how often observations fall beyond an ensemble range requires repeated verification and attention to how members were constructed. The counting operation itself cannot answer that empirical question. This is why even a mathematically exact fraction should carry its denominator and event definition when communicated, rather than becoming an unexplained percentage detached from the forecast target.
Threshold changes retain all members. Shift changes retain the event threshold and replace the member values; negative rain values are floored at zero. This is a constructed sensitivity case, not a correction estimated from data.
3/12 members qualify: 25.0%. Ensemble mean: 7.00 mm. Equal weights assumed; this raw fraction is not calibrated by the lab.
| Quantity | Value |
|---|---|
| Event | At least 10 mm in the specified 24 hours |
| Qualifying members | 12, 18, 24 mm |
| Raw member fraction | 3/12 = 25% |
| Ensemble mean | 7 mm |
The existing common-shift buttons replace every rain total by the shifted value, with negative results set to zero. They do not add uncertainty independently to each member. A positive shift moves the whole collection toward wetter totals while retaining its order. A negative shift can merge several distinct dry values at zero because negative rainfall is disallowed. The threshold selection persists when the shift changes, so comparisons can hold the event fixed. Conversely, changing the threshold keeps the current shift. These retained selections make it possible to distinguish an altered question from an altered collection.
Hold the event at at least 8 mm in 24 hours at the site. With the reference collection, four members qualify. With the plus 5 mm shift, the original values 3 mm and above qualify, giving eight members. With the minus 5 mm shift, the original values must reach at least 13 mm, leaving two qualifying members. The resulting fractions are one third, two thirds, and one sixth. This comparison is deliberately constructed. It shows sensitivity to a shared offset and does not estimate a real bias from rain-gauge observations or forecast archives.
Fraction at least 8 mm, count / 12
Common shift, mm in 24 hours
The positive shift raises the mean from 7 to 12 mm because each of twelve totals gains exactly 5 mm. The negative shift is less symmetric. Subtracting 5 from the original sum of 84 would give a sum of 24 mm if negative totals were allowed. Flooring instead raises that sum to 44 mm by removing negative contributions. The resulting mean is 44 divided by 12, about 3.667 mm. A lower bound has changed the arithmetic. It would be incorrect to describe the minus button as always lowering the mean by exactly 5 mm.
The lab has twelve totals summing to 84 mm. S denotes the sum of member totals, not rainfall accumulated across twelve days.
Flooring at zero adds back 5 + 5 + 4 + 3 + 2 + 1 = 20 mm from the negative entries.
Spread and common displacement can be separated more cleanly with the four paper totals 2, 4, 14, and 16 mm. Their mean is 9 mm and their deviations from that mean are minus 7, minus 5, plus 5, and plus 7 mm. Add 3 mm to every total. The mean becomes 12 mm, while each deviation stays the same. Their range also remains 14 mm. This example has no flooring, so the shared shift changes location without changing spread. Spread alone would not disclose that all four scenarios moved together.
For a numerical measure of that spread, square the four deviations and average them with the stipulated equal weights. The squares are 49, 25, 25, and 49 square millimeters. Their mean is 37 square millimeters, and its square root is about 6.083 mm. This is the spread of the four-member distribution, using denominator four. It is not an unbiased estimate of an unknown population variance, which would use different assumptions and often a different denominator. Adding 3 mm preserves all four deviations, so it preserves this measure as well as the range.
For paper totals 2, 4, 14, 16 mm, the mean is 9 mm. V is distribution variance with denominator four, not a sample-variance estimate.
Adding 3 mm to all four totals preserves each departure from the mean and hence this spread.
| Quantity | Original totals | All plus 3 mm |
|---|---|---|
| Totals, mm | 2, 4, 14, 16 | 5, 7, 17, 19 |
| Mean, mm | 9 | 12 |
| Range, mm | 14 | 14 |
| Mean squared deviation, mm² | 37 | 37 |
A shared offset is one possible failure mode, not a complete account of forecast error. Errors can depend on rainfall amount, terrain, season, or storm type. Flooring in the lab already shows why even a simple adjustment can alter the distribution unevenly. Correcting a real forecast would require estimating and checking a correction on suitable separate cases. The shift control provides no observations and performs no such estimation. Use its paired fractions and means to understand sensitivity under a stated change, not to choose an adjustment that makes an actual weather forecast trustworthy.
If all members share a dry bias, a narrow range can be confidently wrong. The shift control changes all constructed values together to illustrate a common error. Adding copies of the same members would increase the count without adding new information.
Operational probability products may use calibration and postprocessing rather than raw fractions. Skill must be checked against observations over many relevant cases. The smallest and largest members are not guaranteed physical bounds, especially for rare extremes.
The largest member is a feature of the sampled collection rather than a physical ceiling. In the reference lab it is 24 mm over the stated 24 hours. A real outcome above 24 mm could reflect a plausible tail missed by finite sampling, shared model error, or an event definition that does not match the observation. These explanations require different investigations. Simply enlarging a plotted vertical axis does not resolve them. The ensemble's numerical range answers which totals appear in its members; coverage asks how that range relates to outcomes across forecast cases.
Dependence matters when interpreting how much information a collection contains. Members from related models can share approximations, and nearby starting states do not become independent observations merely because their integrations diverge. Imagine four constructed totals of 6, 7, 8, and 9 mm made with one cloud scheme. They form a narrow collection. If an omitted process would raise all totals, the narrow spread cannot expose it. This possibility is a model-structure concern, not evidence that a particular omitted process really exists. Diagnosing it would require comparisons with suitable observations or alternative model treatments.
| Quantity | Reference collection | What it does not establish |
|---|---|---|
| Minimum | 0 mm in 24 hours | No verified forecast-error bound |
| Maximum | 24 mm in 24 hours | No physical upper bound |
| Range width | 24 mm | No verified coverage rate |
Counting only marginal site events also loses information about joint events. Consider four hypothetical pairs of site totals. In collection A, the pairs are 0 and 0, 0 and 0, 12 and 12, and 12 and 12 mm. In collection B, they are 0 and 12, 0 and 12, 12 and 0, and 12 and 0 mm. At either site, two of four members exceed 10 mm in both collections. Yet both sites exceed together in two members of A and none of B. Identical site fractions do not determine joint risk.
The member pairing carries that joint information. To count the event that both sites exceed, inspect both totals within each same member, then count qualifying pairs. Multiplying two marginal fractions would give one quarter, but that product assumes independence between the two site events. Neither of these constructed collections satisfies that assumption. Collection A moves the sites together; collection B makes their exceedances mutually exclusive. The existing lab contains only one site, so it cannot calculate either joint event. The paired table is a separate paper example that explains what extra information a regional question needs.
Calibration requires another kind of collection, an archive of forecasts and their corresponding outcomes. Suppose a system often issues a raw member fraction near one third for a specified event. One can group comparable issued forecasts and ask how often that event actually occurs. A systematic mismatch would indicate that the raw fraction needs interpretation or correction. Twelve members for one target do not supply twelve verifying weather outcomes. There will be one observed total for that target. Confusing members with verification cases would falsely multiply the evidence supporting a probability interpretation before any repeated forecast evaluation.
| Event over the same 24 hours | Collection A | Collection B |
|---|---|---|
| Site 1 at least 10 mm | 2/4 | 2/4 |
| Site 2 at least 10 mm | 2/4 | 2/4 |
| Both at least 10 mm | 2/4 | 0/4 |
| Either at least 10 mm | 2/4 | 4/4 |
Postprocessing can use historical relationships between forecast quantities and outcomes to produce adjusted probabilities. That is different from asserting that any selected shift fixes calibration. An adjustment can fit its training cases yet fail elsewhere, especially if the event is rare or the forecast system changes. Evaluation must retain the same target definition and use cases not chosen to make the adjustment look good. These requirements explain why an ensemble product may legitimately report a probability different from its raw member fraction. Whether that reported probability is useful remains a question for verification, not a consequence of member counting alone.
For the displayed unshifted members, what fraction reaches at least 5 mm?
Include values equal to the threshold.
| Threshold | Qualifying count | Fraction |
|---|---|---|
| At least 5 mm | 6 | 50% |
| At least 10 mm | 3 | 25% |
Six members qualify: 5, 6, 9, 12, 18, and 24. The fraction is 6/12 = 50%. The event has changed from the 10 mm question, so the probability can change without any new forecast run.
Duplicate each member ten times. Does the raw 25% fraction become a better calibrated 120-member forecast?
Look for new information and verification, not just sample size.
| Representation | Qualifying | Total | Fraction |
|---|---|---|---|
| Original | 3 | 12 | 25% |
| Ten copies each | 30 | 120 | 25% |
The fraction stays 25%, and no new trajectories or uncertainty sources have been represented. Duplication does not establish calibration or justify treating all entries as independent evidence.
Selected reference passages support the definitions and modeling distinctions below. The examples, explanations, and implementations are original. Terminology and assumptions are stated within each lesson; source review is not an independent audit of this course. Optional links are not runtime dependencies.
Initial-condition and model perturbations. Operational configurations change; the twelve invented rain members are not ECMWF output.
Selected sections on reliability, resolution, Brier scores, skill, and forecast value. The eight-case sample is deliberately too small to establish real calibration.