M. D. Martínez

Return period maps of dry spells for Catalonia (northeastern Spain) based on the Weibull distribution / Périodes de retour des périodes sèches en Catalogne (nord-est de l'Espagne) à partir de la distribution de Weibull

Abstract A detailed description of the spatial distribution of dry spell lengths in Catalonia (northeastern Spain) for the years 1950–2000 is obtained from a statistical analysis. The database is derived from the daily records of 75 raingauges, and a dry spell is defined as a set of consecutive days with daily rainfall amounts below thresholds of 0.1, 1.0, 5.0 and 10.0 mm/d. The Weibull model fits the distribution of dry spell lengths well, whatever the raingauge and threshold. The Weibull parameters estimated by L-moments and the L-skewness—kurtosis diagrams allow quantification of the goodness of fit between the model and the empirical distribution. Dry spell lengths for return periods of 2, 5, 10, 25 and 50 years indicate the areas where drought phenomena might be more severe, as well as how often they might occur. A regional homogeneity method is used to test whether a single set of Weibull parameters could describe the series of dry spells for the whole region.

Two complementary stress release processes based on departures from Omori’s law

Three aftershock series in Southern California, associated with the mainshocks of Landers (1992), Northridge (1994) and Hector Mine (1999) are interpreted as the superposition of a long stress relaxation process, in agreement with the Omori’s law, and a number of short episodes with sudden stress release. These short episodes are detected as departures from the expected Omori’s law. Aftershocks following a long stress relaxation process are designed as leading aftershocks (LA) and their generation rate fits well to the modified Omori’s law (MOL). The rest of aftershocks correspond to the different episodes of sudden stress release, every one designed as a cascade (CA). Cascades are found to be characterized by four basic properties. First, although the number of aftershocks NC belonging to every cascade fluctuates greatly along the stress relaxation process, a clear positive trend of NC with the elapsed time since the mainshock is observed in the initial phase of the aftershock sequence. After a critical elapsed time, very short in comparison with the aftershock sequence length, this positive trend diminishes significantly or even becomes negative. Second, the aftershock generation rate, GR, for every cascade, is almost constant. Third, GR for the successive CAs, represented as a function of its starting time measured since the mainshock, decreases according to a power-law. Four, the validity of the Gutenberg-Richter law is preserved for the sequence of all aftershocks belonging to cascades, with values of the b-parameter quite similar to those deduced for the complete aftershock series. Additionally, some statistical and fractal (self-affine) properties of CAs are analyzed. Given that the number of aftershocks with minimum magnitudes assuring catalogue completeness ranges, for the three seismic crises, from 6,000 up to 20,000, properties concerning LA and CA can be established with a high degree of confidence. As a conclusion, a single stress relaxation process following the Omori’s law should be discarded. A physical explanation, based on the complex spatial patterns of the stress field and tectonic fractures, as well as on the two proposed relaxation processes, is qualitatively discussed.

Complexity and predictability of the monthly Western Mediterranean Oscillation index

The complexity, predictability and predictive instability of the Western Mediterranean Oscillation index (WeMOi) at monthly scale, years 1856-2000, are analysed from the viewpoint of monofractal and multifractal theories. The complex physical mechanism is quantified by: (1) the Hurst exponent, H, of the rescaled range analysis; (2) correlation and embedding dimensions, mu* and d(E), together with Kolmogorov entropy, kappa, derived from the reconstruction theorem; and (3) the critical Holder exponent, alpha(o), the spectral width, W, and the asymmetry of the multifractal spectrum, f(alpha). The predictive instability is described by the Lyapunov exponents, lambda, and the Kaplan-Yorke dimension, D-KY, while the self-affine character is characterized by the Hausdorff exponent, H-a. Relationships between the exponent beta, which describes the dependence of the power spectrum S(f) on frequency f, and the Hurst and Hausdorff exponents suggest fractional Gaussian noise (fGn) as a right simulation of empiric WeMOi. Comparisons are made with monthly North-Atlantic Oscillation and Atlantic Multidecadal Oscillation indices. The analysis is complemented with an ARIMA(p,1,0) autoregressive process, which yields a more accurate prediction of WeMOi than that derived from fGn simulations.

Fractal structure and predictive strategy of the daily extreme temperature residuals at Fabra Observatory (NE Spain, years 1917-2005)

A compilation of daily extreme temperatures recorded at the Fabra Observatory (Catalonia, NE Spain) since 1917 up to 2005 has permitted an exhaustive analysis of the fractal behaviour of the daily extreme temperature residuals, DTR, defined as the difference between the observed daily extreme temperature and the daily average value. The lacunarity characterises the lag distribution on the residual series for several thresholds. Hurst, H, and Hausdorff, Ha, exponents, together with the exponent β of the decaying power law, describing the evolution of power spectral density with frequency, permit to characterise the persistence, antipersistence or randomness of the residual series. The self-affine character of DTR series is verified, and additionally, they are simulated by means of fractional Gaussian noise, fGn. The reconstruction theorem leads to the quantification of the complexity (correlation dimension, μ*, and Kolmogorov entropy, κ) and predictive instability (Lyapunov exponents, λ, and Kaplan-Yorke dimension, DKY) of the residual series. All fractal parameters are computed for consecutive and independent segments of 5-year lengths. This strategy permits to obtain a high enough number of fractal parameter samples to estimate time trends, including their statistical significance. Comparisons are made between results of predictive algorithms based on fGn models and an autoregressive autoregressive integrated moving average (ARIMA) process, with the latter leading to slightly better results than the former. Several dynamic atmospheric mechanisms and local effects, such as local topography and vicinity to the Mediterranean coast, are proposed to explain the complex and instable predictability of DTR series. The memory of the physical system (Kolmogorov entropy) would be attributed to the interaction with the Mediterranean Sea.

Normalised monthly shortage curves: a contribution for a better understanding of monthly rain deficit in Western Europe

A new approach to the statistics of rainfall shortage at monthly scale in Western Europe is obtained from precipitation records of 115 gauges over the twentieth century. In this paper, a month is considered to have rainfall deficit when its rain amount is below the 50th percentile of the respective calendar month. The monthly shortage, MS, for every month with deficit is then computed as the absolute value of the difference between its monthly amount and the corresponding truncation level. The cumulative distributions of monthly shortage, CMS, and number of shortage months, CNM, constitute a new description of the monthly rainfall deficit. Both CMS and CNM distributions fit well to a Weibull model. Using the analogy to the normalised daily rainfall curves formulation, NRC, the relationship between CMS and CNM, named as normalised shortage curve, NSC, is modelled by the same function applied to NRCs. Similarly to NRCs, the behaviour of the NSCs strongly depends on the coefficient of variation of the monthly shortage, CVMS. Four coordinates characterising every NSC are then introduced: the CMS percentile associated with the median of CNM; the CNM percentile related to the median of CMS; and the percentiles of CMS and CNM for the average monthly shortage. In this way, the degree of asymmetric distribution of the monthly deficit is quantified. With the aim of performing a clustering process based on these four coordinates, a principal component analysis, is previously applied to remove redundancies, being obtained two uncorrelated principal components, PCs, characterising every NSC. An average linkage algorithm is then applied to these two PCs, leading to obtain spatially coherent groups of gauges with very similar NSC patterns. This clustering process permits to discard latitude and vicinity to the Atlantic Ocean or the Mediterranean Sea as main factors conditioning the monthly shortage regime.

Return period curves for extreme 5-min rainfall amounts at the Barcelona urban network

Heavy rainfall episodes are relatively common in the conurbation of Barcelona and neighbouring cities (NE Spain), usually due to storms generated by convective phenomena in summer and eastern and south-eastern advections in autumn. Prevention of local flood episodes and right design of urban drainage have to take into account the rainfall intensity spread instead of a simple evaluation of daily rainfall amounts. The database comes from 5-min rain amounts recorded by tipping buckets in the Barcelona urban network along the years 1994–2009. From these data, extreme 5-min rain amounts are selected applying the peaks-over-threshold method for thresholds derived from both 95% percentile and the mean excess plot. The return period curves are derived from their statistical distribution for every gauge, describing with detail expected extreme 5-min rain amounts across the urban network. These curves are compared with those derived from annual extreme time series. In this way, areas in Barcelona submitted to different levels of flood risk from the point of view of rainfall intensity are detected. Additionally, global time trends on extreme 5-min rain amounts are quantified for the whole network and found as not statistically significant.

Multifractality and autoregressive processes of dry spell lengths in Europe: an approach to their complexity and predictability

Dry spell lengths, DSL, defined as the number of consecutive days with daily rain amounts below a given threshold, may provide relevant information about drought regimes. Taking advantage of a daily pluviometric database covering a great extension of Europe, a detailed analysis of the multifractality of the dry spell regimes is achieved. At the same time, an autoregressive process is applied with the aim of predicting DSL. A set of parameters, namely Hurst exponent, H, estimated from multifractal spectrum, f(α), critical Hölder exponent, α0, for which f(α) reaches its maximum value, spectral width, W, and spectral asymmetry, B, permits a first clustering of European rain gauges in terms of the complexity of their DSL series. This set of parameters also allows distinguishing between time series describing fine- or smooth-structure of the DSL regime by using the complexity index, CI. Results of previous monofractal analyses also permits establishing comparisons between smooth-structures, relatively low correlation dimensions, notable predictive instability and anti-persistence of DSL for European areas, sometimes submitted to long droughts. Relationships are also found between the CI and the mean absolute deviation, MAD, and the optimum autoregressive order, OAO, of an ARIMA(p,d,0) autoregressive process applied to the DSL series. The detailed analysis of the discrepancies between empiric and predicted DSL underlines the uncertainty over predictability of long DSL, particularly for the Mediterranean region.

Complexity and predictability of the monthly Western Mediterranean Oscillation index: WeMO Predictability and Complexity

The complexity, predictability and predictive instability of the Western Mediterranean Oscillation index (WeMOi) at monthly scale, years 1856-2000, are analysed from the viewpoint of monofractal and multifractal theories. The complex physical mechanism is quantified by: (1) the Hurst exponent, H, of the rescaled range analysis; (2) correlation and embedding dimensions, mu* and d(E), together with Kolmogorov entropy, kappa, derived from the reconstruction theorem; and (3) the critical Holder exponent, alpha(o), the spectral width, W, and the asymmetry of the multifractal spectrum, f(alpha). The predictive instability is described by the Lyapunov exponents, lambda, and the Kaplan-Yorke dimension, D-KY, while the self-affine character is characterized by the Hausdorff exponent, H-a. Relationships between the exponent beta, which describes the dependence of the power spectrum S(f) on frequency f, and the Hurst and Hausdorff exponents suggest fractional Gaussian noise (fGn) as a right simulation of empiric WeMOi. Comparisons are made with monthly North-Atlantic Oscillation and Atlantic Multidecadal Oscillation indices. The analysis is complemented with an ARIMA(p,1,0) autoregressive process, which yields a more accurate prediction of WeMOi than that derived from fGn simulations.