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Declustering groups dependent threshold exceedances into extreme-event clusters and retains a representative, commonly the cluster maximum. It aims to make the events used in a peaks-over-threshold method approximately independent. The event rate must be estimated consistently with the retained clusters when computing return levels.

A 100-year return level is a flow threshold whose probability of being exceeded in a given year is one percent, under a stationary model for annual extremes. Equivalently, its long-run average waiting time between years with an exceedance is one hundred years. It does not imply that exceedances occur regularly one hundred years apart, or that one is guaranteed in any particular hundred-year interval. This definition uses yearly exceedance events even when the underlying observations are daily maxima.

In the original PDF's probability plot, most points lie close to the diagonal. The quantile plot likewise agrees well over most of the range, but the largest empirical flow, around $68.5$, is well above its fitted quantile, around $66$. The density plot captures the strongly decreasing right-skewed bulk of the excesses. The return level plot shows a broadly reasonable fit to most observations, with one conspicuously high extreme and increasingly wide uncertainty at long return periods. Consequently the generalized Pareto distribution is a reasonable first approximation, but its most extreme tail is less convincingly represented.

Reading the return-level curve at the 100-year tick givesThese are approximate graphical readings, not parameter-based calculations; rounding to whole units would give about $67$ with an interval roughly $65$ to $69$. The two blue curves are uncertainty bounds for the fitted return level, not prediction bounds for individual observations. The daily-data independence assumption and the stability of the fit as the threshold changes should be checked, for example by examining clusters of high flows and using declustering of extremes where necessary. A 100-year level also extrapolates beyond the 25-year record.

For the geometric distribution supported on positive integers, set $q=1-p$. The geometric series and its derivative giveEquivalently, the sum of the tail probabilities is ${\sum }_{k\ge 0}P(T>k)={\sum }_{k\ge 0}{q}^k=1/p$. For $p=0.02$, the expected wait is 50 years, so this is the 50-year return level. The exceedance probability determines the return period, but does not by itself determine a numerical flow rate.

The peaks-over-threshold method fits a generalized Pareto distribution to exceedances of a high threshold and also estimates their frequency. Together these determine tail probabilities and return levels. For dependent observations, threshold exceedances may occur in clusters; declustering of extremes can separate extreme events before fitting a model for independent events.

A return level answers the inverse question to a return period: which threshold has the specified rarity? Let $F(z)=F_{\mathrm{annual}}(z)=\mathbb{P}(M_j\le z)$ be the common cumulative distribution function of annual sample maxima in a model with stationarity. For a continuous, strictly increasing $F$ and $T>1$, the $T$-year return level isIt has the same units as $M_j$, whereas $T$ is a count of annual blocks. Independence between years is unnecessary for this one-year quantile definition; it is needed for the geometric waiting and multi-year formulas under return period. For a discrete probability distribution, use the generalized quantile function $F^{-1}(q)=\inf \{z:F(z)\ge q\}$. Then $\mathbb{P}(M_j>z_T)\le 1/T$, with equality not guaranteed at jumps; an exact requested exceedance probability may be unattainable.

For a concrete standard model, choose the maxima convention of the Gumbel distribution, with location $\mu$ and scale $b>0$:The second formula follows by taking logarithms twice and solving $F(z_T)=1-1/T$. The maxima cumulative distribution function and its inverse are given in NIST: extreme value type I distribution, maximum case. With the deliberately chosen toy values $\mu =10$ and $b=2$ in arbitrary units, $z_{10}=14.501$, $z_{100}=19.200$, and $z_{1000}=23.815$, rounded to three decimals. In particular, $19.200$ is approximately the annual quantile at cumulative probability $0.99$; it is not a prediction of the largest observation in the next 100 years. Under independence, the probability that all those annual sample maxima stay below the exact $z_{100}$ is $0.99^{100}\approx 0.366$.

The time block matters. If a toy year contains $m$ independent daily blocks, each with exceedance probability $q$ at one fixed threshold, its annual exceedance probability is $p=1-(1-q{)}^m$, giving annual return period $1/[1-(1-q{)}^m]$. The reciprocal $1/q$ counts daily blocks. Dependence among daily exceedances requires their joint law instead; a threshold crossed on several successive days still produces only one exceedance indicator for that year's sample maximum. The peaks-over-threshold method and declustering of extremes address a different choice of extreme-event sampling.

A return level plot graphs $T$ horizontally and $z_T$ vertically, often using a logarithmic scale for $T$. To read it, choose a return period, move up to the curve, and read the corresponding return level. Conversely, a threshold's horizontal coordinate gives the reciprocal of its annual exceedance probability. The usual plot of fitted quantiles against return periods, and optional pointwise confidence intervals, are documented in Mathieu Ribatet, POT package: return level plot.

The near-straight shape has a mathematical explanation: since $-\log (1-1/T)\sim 1/T$ as $T\to \infty$, the Gumbel distribution gives $z_T=\mu +b\log T+o(1)$. Multiplying a large return period by ten therefore increases the return level by approximately $b\log 10$, about $4.605$ units for this toy curve. A curve indexed by rarity puts several thresholds on one common scale while retaining their measurement units.

For a fitted curve, a pointwise confidence interval concerns uncertainty in the estimated quantile at a specified $T$; it is distinct from a prediction interval for a future annual sample maximum, and pointwise intervals need not form a simultaneous confidence band. A far-right curve value can also be an extrapolation beyond the observation record: extending the line adds no observations to support the chosen tail model. The diagram above uses fixed toy parameters, so it shows neither estimation uncertainty nor an observational record. Its interpretation throughout assumes the stated annual block and stationarity.