Two-component Extreme Value Distribution Research Articles

Goodness-of-fit, identifiability and extrapolation: Can the Two-Component Extreme Value distribution be used in at-site flood frequency analysis?

When fitting three-parameter flood frequency models to annual maximum (AM) flood series, the lack-of-fit can be mitigated by censoring potentially influential low flows (PILFs). An alternative and less-studied approach is to apply mixture probability models with four or more parameters, which trade off greater flexibility to fit AM series against the need to deal with degeneracy which arises when there is insufficient information to identify mixture components. However, the issue of degeneracy and the lack of a robust inference framework present a significant barrier to adoption. This study investigated the potential of the most parsimonious mixture model, the four-parameter Two-Component Extreme Value (TCEV) model. A Bayesian framework using Markov chain Monte Carlo sampling was developed to robustly characterize parameter uncertainty even in the presence of degeneracy. Two new posterior diagnostics based on the strength of the TCEV components were developed to aid identification of degeneracy. Armed with a robust inference framework, the study evaluated the potential of TCEV using a case study based on 31 catchments in eastern Australia with records exceeding 70 years. The evaluation used short to long records to compare TCEV and Log Pearson III fit and extrapolative uncertainty. The Log Pearson approach (LP3-PILF) censors PILFs following the approach described in Australian Rainfall and Runoff. With respect to goodness-of-fit, we found in most cases that TCEV fitted AM flood peaks well without the need to censor or stratify data, and overall, no clear difference emerged between TCEV and LP3-PILF. However, and contrary to expectation, TCEV produced high flow quantile confidence intervals consistently narrower than LP3-PILF even in the presence of degeneracy. While more case studies in different regions are needed to confirm the potential of TCEV, this study is a reminder that goodness-of-fit is a necessary but not sufficient criterion for selecting the probability model that best represents flood frequency.

Modelling multidecadal variability in flood frequency using the Two-Component Extreme Value distribution

The Two-Component Extreme Value (TCEV) distribution is traditionally known as the exact distribution of extremes arising from Poissonian occurrence of a mixture of two exponential exceedances. In some regions, flood frequency is affected by low-frequency (decadal) climate fluctuations resulting in wet and dry epochs. We extend the exact distribution of extremes approach to such regions to show that the TCEV arises as the distribution of annual maximum floods for Poissonian occurrences and (at least two) exponential exceedances. A case study using coastal basins in Queensland and New South Wales (Australia) affected by low-frequency climate variability, shows that the TCEV produces good fits to the marginal distribution over the entire range of observed values without the explicit need to resort to climate covariates and removal of potentially influential low values. Moreover, the TCEV reproduces the observed dog-leg, a key signature of different flood generation processes. A literature review shows that the assumptions underpinning the TCEV are conceptually consistent with available evidence on climate and flood mechanisms in these basins. We provide an extended domain of the TCEV distribution in the L-moment ratio diagram to account for the wider range of parameter values encountered in the case study and show that for all basins, L-skew and L-kurtosis fall within the extended domain of the TCEV.

Fit of the two-component extreme value (TCEV) distribution through of maximum likelihood

En muchas cuencas medianas y grandes de nuestro país y del mundo, su registro anual de crecientes está integrado por eventos que fueron generados por fenómenos físicamente diferentes. Por ejemplo, muchas crecientes se originan con tormentas locales y una porción pequeña es generada por lluvias ciclónicas, de amplia cobertura y duración, que generan crecientes extraordinarias. Para este tipo de registros fue propuesta la distribución TCEV (two–component extreme value) con cuatro parámetros de ajuste, la cual tiene una base teórica que permite una interpretación aproximada para dos mecanismos de generación de las crecientes y que además es capaz de reproducir la variabilidad real del coeficiente de asimetría. En este trabajo se detalla su génesis y el método de ajuste por máxima verosimilitud, según dos versiones numéricas: (1) sustitución sucesiva y (2) maximización de la función objetivo. Se procesaron seis registros de crecientes, cuya amplitud varió de 31 a 72 datos, con tres a seis valores dispersos o crecientes que se apartan de la tendencia general. Se contrastan las predicciones del modelo TCEV, ajustado con cada método numérico, contra las obtenidas con las distribuciones de aplicación establecida bajo precepto (LP3, GVE y LOG) y la Wakeby. Aceptando al error estándar de ajuste como criterio de selección, se deduce que la distribución TCEV es la mejor opción en dos de los seis registros procesados. Por último, se formulan las Conclusiones, las cuales sugieren la aplicación sistemática de la distribución TCEV con ambos métodos numéricos, en los registros que presentan dos poblaciones mezcladas.

Initial results of sediment yield measurement interpretation using a regional approach: Southern Italy case study

Abstract. The theoretical TCEV (Two Component Extreme Value) distribution was applied to interpret the sediment yield datasets available in Southern Italy. The analysis is based on hydrological data collected for twelve catchments located in Calabria and Basilicata. A hierarchical approach was used to obtain a regional parent distribution which was used to determine the return time for each event. The hierarchical approach proposed in this study includes two stages. The first stage served for calibration and made it possible to estimate the parameters of the theoretical TCEV distribution. More specifically, the hypothesis of homogeneity with regard to the skewness coefficient and the coefficient of variation was verified using the datasets related to nine catchments. The second stage consisted in verifying the goodness of the theoretical distribution on three independent datasets provided by three experimental catchments not involved in the calibration. Overall results show that, even if the TCEV distribution was conceived to estimate peak flow, its concept of “double component” can be extended to predict sediment yield on a regional scale.

Paleohydrology of extraordinary floods along the Swakop River at the margin of the Namib Desert and their paleoclimate implications

The Swakop River is a major ephemeral river (drainage basin area about 30,000 km2) crossing the western Namib Desert. Its hydrology and limited water resources depend on storms and floods. Therefore, the hydrology and flow regime in the Swakop River basin determines the availability of water for human use. In this study we present a millennia-long record of extreme floods, using paleoflood hydrological methods and optically stimulated luminescence (OSL) ages. The record includes 44 large floods ranging in date from the late Pleistocene to the present. The oldest six to eight floods are dated from the late Pleistocene to the mid-Holocene (between 14,900 ± 1700 and 5100 ± 620 years ago), with calculated peak discharges of 550–1280 m3 s−1. Eighteen floods, eleven of which were between 1000 and 1600 m3 s−1, occurred between 740 ± 90 and 190 ± 10 years ago. Additional 17 later floods occurred between about AD 1850 and 1963, with peak discharges between 100 and 1000 m3 s−1. The historical record which partly parallels the latter period recorded 17 large floods during the period 1893–1963. Flood frequency analysis (FFA) of the combined systematically gaged record (1963–2003) and the paleoflood record since AD 1270 (740 ± 90 years, the most complete record), was applied using the maximum likelihood method. A two-component extreme value distribution (TCEV) estimates the 1% annual exceedance probability magnitude, using only the systematic gaged record at 520 m3 s−1. The combined paleoflood, historical, and gage records resulted in a much larger magnitudes for the 1% annual exceedance probability flood magnitude of 1320–1350 m3 s−1. The 0.1% annual exceedance probability magnitude is estimated at 770 m3 s−1 and 2100–2150 m3 s−1, respectively.The hydroclimatic interpretation of this paleoflood record is complex, due to the discontinuity of the record. During the late Holocene, our flood data show three intervals differing in their flood regime: (1) 1280 ± 80 to 740 ± 90 years BP, with one extraordinary flood (>200 m3 s−1) about every 180 years; (2) 740 ± 90 to 200 ± 45 14C years BP, with one large flood per 80 years; and (3) 200 ± 45 14C years BP to AD1963, with one large flood per 7 years. Interval 2 represents large floods, probably related to the transition from a drier climate to the colder and probably wetter climate of the Little Ice Age (LIA). The third period relates to the end of the LIA and to a wetter period of the 19th and early 20th centuries in which more rainfall in the Kalahari and Namib Deserts and many historical floods were documented.The presence of three distinct calcretes rich in calcified rhizoliths at one of the sites, two of which were dated by OSL to 13.2 ± 1.9 ka and 8.8 ± 1.0 ka, indicates, at least locally, higher moisture availability than in the current semiarid climate; the presence of denser vegetation can be associated with increased groundwater seepage and wetter climatic conditions.

Regional Analysis of Rainfall-Depth-Duration Equation for South Italy

The two-component extreme value (TCEV) distribution, which is currently applied in Italy for the regional rainfall and flood frequency analysis, and its hierarchical parameter estimating procedure, are briefly reviewed. Then, a new rainfall-depth-duration equation, based on TCEV distribution and applicable for the hypothesis of both duration-dependent and duration-independent parameter estimates, is proposed. Finally, for the investigated Italian geographical regions (Apulia, Basilicata, Calabria, and Campania), the regionalization procedure is developed, and the rainfall-duration-frequency factor appearing in the rainfall-depth-duration equation is calibrated. The isoline maps of the mean rainfall having a duration equal to 24 h and of the exponent n, appearing in the power equation linking the mean μt of the TCEV distribution with the duration t, are also plotted.

The two-component extreme value distribution for flood frequency analysis: Derivation of a new estimation method

The two component extreme value (TCEV) distribution has recently been shown to account for most of the characteristics of the real flood experience. A new method of parameter estimation for this distribution is derived using the principle of maximum entropy (POME). This method of parameter estimation is suitable for application in both the site-specific and regional cases and appears simpler than the maximum likelihood estimation method. Statistical properties of the regionalized estimation were evaluated using a Monte Carlo approach and compared with those of the maximum likelihood regional estimators.

Regional flood frequency estimation using the two-component extreme value distribution

ABSTRACT The two-component extreme value (TCEV) distribution, suggested for modelling Italian annual flood series (AFS), is shown to account both for the presence of flow outliers and for the high variability of the skewness of historical AFSs. Using such a model, it is found that the T-year flood varies with log(T) much more rapidly for large values of T than is the case for small values of T. The presence of four parameters in the TCEV distribution introduces great uncertainty in estimating the T-year flood when the parameters are estimated from a single series. Regional analysis, which exploits basin similarities, is then needed, not only at ungauged sites but also at gauged ones. Regionalization techniques, by which all flood data over a region are combined to produce a single regional flood distribution, are described and their application to Italian data is shown.

Two‐Component Extreme Value Distribution for Flood Frequency Analysis

Theoretical considerations, supported by statistical analysis of 39 annual flood series (AFS) of Italian basins, suggest that the two‐component extreme value (TCEV) distribution can be assumed as a parent flood distribution, i.e., one closely representative of the real flood experience. This distribution belongs to the family of distributions of the annual maximum of a compound Poisson process, which is a solid theoretical basis for AFS analysis. However, the two‐parameter distribution of this family, obtained on the assumption of identically distributed floods, does not account for the high variability of both observed skewness and largest order statistics, so that a significant number of observed floods qualify as outliers under this distribution. The more general TCEV distribution assumes individual floods to arise from a mixture of two exponential components. Its four parameters can be estimated by the maximum likelihood method. A regionalized TCEV distribution, with parameters representative of a set of 39 Italian AFS's, was shown to closely reproduce the observed distribution of skewness and that of the largest order statistic.