Skew, Scale and the Shape of UK Flood Growth

Research question

Which parts of a flood growth curve are scale, dispersion, asymmetry and upper-tail growth; how much does a finite AMAX record constrain each observed target; and can finite-record sampling explain the deficient L-CV of one frozen continuous-model configuration? A parallel hypothesis asked whether rating-derived information could support transfer beyond gauged stations.

Definitions: scale, dispersion, shape and growth

The index-flood scale is the reference flow magnitude that multiplies a dimensionless growth curve. It answers “how large is the characteristic flood?” before the curve answers how that flow grows with return period. A scale error moves flows in magnitude even when the dimensionless curve is unchanged.

L-CV describes dispersion: the relative spread in the annual-maximum series. L-skew describes asymmetry: how strongly the distribution departs from a symmetric shape. GF100 is the fitted 100-year growth factor relative to the index flood. It is an output of the adopted evidence and family, not a synonym for L-skew. These roles are related, but they are not interchangeable.

Cohort and method

The growth revalidation selected 874 gauged stations. Each station had at least 10 non-rejected AMAX years and per-peak stage available for the rating analysis. The record-length experiment estimated within-station sampling variance by resampling years within each station, then recomputing L-moments and GF100. It did not substitute differences between stations for finite-record variability within a station.

The separate finite-sample experiment fixed CK1g configuration c6_t3zero across 27 basins. It used deterministic replay, 10,000 synthetic years and non-overlapping record-length windows matched to each observed series. The full synthetic record tests the model’s long-run L-CV; the matched windows test the distribution that would be seen at the observed record lengths.

Target ceilings from finite records

The estimated target ceilings were 0.857 for L-CV, 0.581 for L-skew and 0.723 for GF100 in the approximately 40-year experiment. These are estimates of how reproducible each observed target is under the frozen resampling design. They are not achieved model scores and the source does not report an uncertainty band for the ceilings.

Median within-station GF100 sampling variability was about ±13.0%. This is not universal design-flow uncertainty: it does not include index-flood error, rating error, family choice, model structure or local data-quality decisions. Finite-record sampling constrains L-CV, L-skew and GF100 differently, so the three statistics should remain separate rather than being collapsed into a monotonic rule.

Rating hypothesis and retraction

The original gauged association was strong: rating-derived features explained L-CV with R² 0.710 versus 0.420 for catchment descriptors. That comparison was self-referential for transfer because the rating evidence came from the target station’s own record—the same gauged setting in which the L-CV target was observed.

Spatial-cluster cross-validation is appropriate for neighbour and geographic transfer, but it cannot remove within-station self-reference when an evaluation station’s rating-derived feature and target L-moments come from the same observed record. The frozen held-out, train-only transfer set is therefore the relevant capability test, not the small change under spatial clustering: rating L-CV was 0.294 versus 0.248 for FEH pooling (n=40). The rating features were still extracted at each held-out target. They cannot run at an ungauged site. Any earlier reading of this work as a capability without a gauged target record is therefore withdrawn; the remaining result is a bounded gauged association.

Frozen finite-sample experiment

Observed median L-CV was 0.205, compared with 0.131 in the frozen 10,000-year model series. The model’s matched record-length windows had median 0.129 and P95 0.159. Observed L-CV exceeded that P95 in 21 of 27 basins, so the discrepancy was not confined to a small number of sites.

The finite-sample spread closed only 38% of the aggregate gap and the median record-length bias was −0.002. Short records did not inflate L-CV through upward bias. The experiment therefore closes one defence for c6_t3zero: sampling records at the observed lengths did not rescue its missing dispersion.

Implications for growth curves

A growth-curve review should begin by separating the index-flood magnitude from dimensionless growth. Within the growth evidence, low L-CV can flatten a growth curve even when L-skew is adequate. Matching asymmetry does not manufacture missing dispersion, and a plausible L-CV does not by itself validate upper-tail growth.

GF100 then records the combined consequence of the fitted L-moments and adopted distribution family at one return period. It is useful as an auditable tail summary, but it does not identify which part of a discrepancy came from scale, dispersion, asymmetry, family choice or finite-record target noise. Those evidence layers should be reported beside it.

Proposed family protocol

The proposed family protocol would compare GLO, KAP3 and GEV using the same frozen evidence record. It would report fitting validity, growth factors at common return periods and the sensitivity of the upper tail to family choice, while keeping index-flood scale and evidence custody visible.

That protocol is not yet demonstrated. No completed GLO/KAP3/GEV comparison is reported in these source artefacts, so their range is not a demonstrated uncertainty envelope and must not be presented as achieved coverage or a probability interval.

Limitations

• The growth cohort required both usable AMAX and per-peak stage, so its selection is narrower than all gauged AMAX stations.
• The target ceilings belong to one within-station resampling design and approximately 40-year experiment; no uncertainty band for those ceilings was reported.
• Rating-derived inputs require the target gauge record and remain self-referential for any use that assumes no target observations.
• The finite-sample verdict covers L-CV in one frozen configuration and 27 basins. It does not test every growth statistic, architecture or physical model.
• The proposed candidate-family comparison has not been executed as a demonstrated experiment.

What we do not conclude

• We do not turn ±13.0% GF100 variability or the target ceilings into a universal allowance, a site-specific interval or a design-flow uncertainty band.
• We do not infer a universal ordering in which one of L-CV, L-skew or GF100 must always be the noisiest finite-record target.
• We do not attribute positive L-CV bias to finite record length; the frozen experiment measured a median bias of −0.002.
• We do not claim that target-record rating features provide a route for sites without gauged observations.
• We do not generalise one frozen model failure to all continuous or physical growth-curve approaches.
• We do not present the proposed GLO/KAP3/GEV protocol as a completed family range or uncertainty envelope.

Disclosure boundary

This report discloses the research questions, broad cohort rules, within-station resampling design, frozen replay design, exact aggregate tables, failed hypotheses, claim statuses, source revisions and the scope of each conclusion. Its only evidence sources are the frozen Hydrometric growth revalidation and finite-sample experiment.

It withholds raw licensed AMAX, stage and flow records, exact feature engineering and selection, learned coefficients, per-station private evidence, model-engine internals, calibration recipes and code-level workflow logic. The public result can be audited without exposing the data or implementation needed to reconstruct the proprietary method.