Fiorillo et al. (2026) Magnitude of hydrological events and extremes using the Z value

Identification

• Journal: Climate Risk Management
• Year: 2026
• Date: 2026-03-20
• Authors: Francesco Fiorillo, Guido Leone
• DOI: 10.1016/j.crm.2026.100811

Research Groups

Department of Sciences and Technologies, University of Sannio, Benevento, Italy.

Short Summary

This study introduces a statistical method using the dimensionless Z value, derived from standardizing hydrological time series, to quantify event magnitude and define extremes. It demonstrates that the Z value offers a more stable and context-invariant measure of hydrological event magnitude than the traditional return period, particularly for extreme events.

Objective

• To propose and validate a statistical approach using the Z value for measuring the magnitude of hydrological events and objectively defining extremes, offering an alternative to the return period concept.

Study Configuration

• Spatial Scale: Regional (Campania, Southern Italy) and specific river basin (Vltava River, Czech Republic) scales for application examples; the proposed method is designed to be spatially invariant.
• Temporal Scale: Long-term historical time series (ranging from 70 to 195 years) of annual, seasonal (6-month), and hourly hydrological variables. The method is applicable across various timescales (hourly to annual).

Methodology and Data

• Models used:
  • Frequency analysis of time series.
  • Indirect standardization: transformation of data probability distribution into a standard normal distribution using Z(x) = Φ−1[F(x)].
  • Direct standardization: Z(x) = (x − μ) / σ, for normally distributed data.
  • Probability distribution functions: Generalized Extreme Value (GEV), Normal, and Weibull distributions.
  • Goodness-of-fit tests: Anderson-Darling test, qq-plot.
  • Uncertainty estimation: Bootstrap for confidence intervals.
• Data sources:
  • Historical records of annual maximum discharge for the Vltava River (Prague, Czech Republic, 1827–2002).
  • Annual time series of 6-month precipitation (November to April) from a rain gauge in Campania, Italy (1920–2022).
  • Annual minimum flow of Serino karst springs (Campania, Italy, 1887–2022).
  • Annual maximum rainfall intensity (1, 3, 6, 12, and 24 hours duration) from the Cava de’ Tirreni rain gauge (Amalfi Coast, Italy, starting 1954).

Main Results

• The Z value provides a dimensionless, probability-based measure of event magnitude, invariant from variable type, units, mean, variance, distributional shape, and spatial context.
• An "extreme" event is defined as having a Z value ≥+2 (upper tail) or ≤–2 (lower tail), corresponding to a probability F(Z) ≤0.0228 or [1 – F(Z)] ≤0.0228.
• For normally distributed data (e.g., 6-month precipitation with mean μ = 0.9123 m and standard deviation σ = 0.2294 m), direct standardization is applicable. For non-normally distributed data (e.g., Vltava River annual floods, Serino springs minimum flow, Cava de’ Tirreni maximum rainfall intensity), indirect standardization using a fitted cumulative distribution function (e.g., GEV, Weibull) is required.
• The Z value scale exhibits a quasi-linear relationship with the hydrological variable, especially in the extreme tails, which contrasts with the exponential relationship observed with the return period.
• The Z value scale is more stable and less sensitive to model uncertainty and measurement errors in the tails of the distribution compared to the return period. For instance, a 10% measurement error in discharge can double the return period but only causes a quasi-linear variation in the Z value.
• Applications demonstrated the identification of extreme dry/wet periods (precipitation), extreme low flows (e.g., discharge Q ≤1.0587 m³/s for Serino springs), and extreme rainstorms (rainfall intensity) using the Z value criterion.

Contributions

• Proposes a universal, probability-based, and context-invariant method (using the Z value) for measuring hydrological event magnitude and defining extremes, extending the Standardized Precipitation Index (SPI)-like criterion to any hydrological variable.
• Offers a more stable and robust scale for communicating event magnitude, particularly for extreme events and in the presence of uncertainty, compared to the widely used return period.
• Highlights the limitations and potential misinterpretations of the return period, especially under non-stationary conditions and high uncertainty in distribution tails.
• Provides a clear, measurable, and calculable definition of "extreme event" based on statistical attributes, distinct from impact-based definitions.

Funding

• This study was funded by the Department of Sciences and Technologies of the University of Sannio (Benevento, Italy).

Citation

@article{Fiorillo2026Magnitude,
  author = {Fiorillo, Francesco and Leone, Guido},
  title = {Magnitude of hydrological events and extremes using the Z value},
  journal = {Climate Risk Management},
  year = {2026},
  doi = {10.1016/j.crm.2026.100811},
  url = {https://doi.org/10.1016/j.crm.2026.100811}
}