G. Papacharalampous, H. Tyralis, A. Langousis, A. W. Jayawardena, B. Sivakumar, N. Mamassis, A. Montanari, and D. Koutsoyiannis, Probabilistic hydrological post-processing at scale: Why and how to apply machine-learning quantile regression algorithms, Water, doi:10.3390/w11102126, 2019.
We conduct a large-scale benchmark experiment aiming to advance the use of machine-learning quantile regression algorithms for probabilistic hydrological post-processing “at scale” within operational contexts. The experiment is set up using 34-year-long daily time series of precipitation, temperature, evapotranspiration and streamflow for 511 catchments over the contiguous United States. Point hydrological predictions are obtained using the Génie Rural à 4 paramètres Journalier (GR4J) hydrological model and exploited as predictor variables within quantile regression settings. Six machine-learning quantile regression algorithms and their equal-weight combiner are applied to predict conditional quantiles of the hydrological model errors. The individual algorithms are quantile regression, generalized random forests for quantile regression, generalized random forests for quantile regression emulating quantile regression forests, gradient boosting machine, model-based boosting with linear models as base learners and quantile regression neural networks. The conditional quantiles of the hydrological model errors are transformed to conditional quantiles of daily streamflow, which are finally assessed using proper performance scores and benchmarking. The assessment concerns various levels of predictive quantiles and central prediction intervals, while it is made both independently of the flow magnitude and conditional upon this magnitude. Key aspects of the developed methodological framework are highlighted, and practical recommendations are formulated. In technical hydro-meteorological applications, the algorithms should be applied preferably in a way that maximizes the benefits and reduces the risks from their use. This can be achieved by (i) combining algorithms (e.g., by averaging their predictions) and (ii) integrating algorithms within systematic frameworks (i.e., by using the algorithms according to their identified skills), as our large-scale results point out.
Full text: http://www.itia.ntua.gr/en/getfile/2001/1/documents/water-11-02126.pdf (6451 KB)
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D. Veneziano, A. Langousis, and C. Lepore, New asymptotic and pre-asymptotic results on rainfall maxima from multifractal theory, Water Resources Research, 45, doi:10.1029/2009WR008257, 2009.
Contrary to common belief, Fisher-Tippett’s extreme value (EV) theory does not typically apply to annual rainfall maxima. Similarly, Pickands’ extreme excess (EE) theory does not typically apply to rainfall excesses above thresholds on the order of the annual maximum. This is true not just for long averaging durations d, but also for short d and in the high-resolution limit as d->0. We reach these conclusions by applying large deviation theory to multiplicative rainfall models with scale-invariant structure. We derive several asymptotic results. One is that, as d->0, the annual maximum rainfall intensity in d, Iyr,d, has generalized extreme value (GEV) distribution with a shape parameter k that is significantly higher than that predicted by EV theory and is always in the EV2 range. The value of k does not depend on the upper tail of the marginal distribution, but on regions closer to the body. Under the same conditions, the excesses above levels close to the annual maximum have generalized Pareto distribution with parameter k that is always higher than that predicted by Pickands’ EE theory. For finite d, the distribution of Iyr,d is not GEV, but in accordance with empirical evidence is well approximated by a GEV distribution with shape parameter k that increases as d decreases. We propose a way to estimate k under pre-asymptotic conditions from the scaling properties of rainfall and suggest a near-universal k(d) relationship. The new estimator promises to be more accurate and robust than conventional estimators. These developments represent a significant conceptual change in the way rainfall extremes are viewed and evaluated.
Additional material:
A. Langousis, and D. Veneziano, Long-term rainfall risk from tropical cyclones in coastal areas, Water Resources Research, 45, doi:10.1029/2008WR007624, 2009.
We develop a methodology for the frequency of extreme rainfall intensities caused by tropical cyclones (TCs) in coastal areas. The model does not account for landfall effects. This makes the developed framework best suited for open-water sites and coastal areas with flat topography. The mean rainfall field associated with a TC with maximum tangential wind speed Vmax, radius of maximum winds Rmax, and translation speed Vt is obtained using a physically-based model, whereas rainfall variability at both large scales (from storm to storm) and small scales (due to rainbands and local convection) is modeled statistically. The statistical component is estimated using precipitation radar (PR) data from the TRMM mission. Taylor’s hypothesis is used to convert spatial rainfall intensity fluctuations to temporal fluctuations at a given location A. The combined physical-statistical model gives the distribution of the maximum rainfall intensity at A during an averaging period D for a TC with characteristics (Vmax, Rmax, Vt) that passes at a given distance from A. To illustrate the use of the model for long-term rainfall risk analysis, we formulate a recurrence model for tropical cyclones in the Gulf of Mexico that make landfall between longitudes 85o-95oW. We then use the rainfall and recurrence models to assess the rainfall risk for New Orleans. For return periods of 100 years or more and long averaging durations (D around 12-24 hours), tropical cyclones dominate over other rainfall event types, whereas the reverse is true for shorter return periods or shorter averaging durations.
Additional material:
A. Langousis, and D. Veneziano, Theoretical model of rainfall in tropical cyclones for the assessment of long-term risk, Journal of Geophysical Research-Atmospheres, 114, doi:10.1029/2008JD010080, 2009.
We propose a theoretical model to evaluate the rainfall intensity field due to large-scale horizontal wind convergence in tropical cyclones (TCs). The model is intended as one component of a methodology to assess the risk of extreme rainfall intensities from TCs. The other components are a recurrence relation for the model parameters and track and a statistical representation of the deviations of rainfall intensity from model predictions. The latter are primarily caused by rainbands and local convective activity and is the focus of an upcoming communication. The vertical flux of moisture and the associated surface rain rate are calculated using basic thermodynamics and a simple numerical model for the vertical winds inside the TC boundary layer. The tropical cyclone is characterized by the radial profile of the tangential wind speed at gradient level, the storm translation velocity, the surface drag coefficient, and the average temperature and saturation ratio inside the TC boundary layer. A parametric analysis shows the sensitivity of the symmetric and asymmetric components of the rainfall field to various storm characteristics.
Additional material:
A. Langousis, D. Veneziano, P. Furcolo, and C. Lepore, Multifractal rainfall extremes: Theoretical analysis and practical estimation, Chaos Solitons and Fractals, 39, 1182–1194, doi:10.1016/j.chaos.2007, 2009.
We study the extremes generated by a multifractal model of temporal rainfall and propose a practical method to estimate the Intensity-Duration-Frequency (IDF) curves. The model assumes that rainfall is a sequence of independent and identically distributed multiplicative cascades of the beta-lognormal type, with common duration D. When properly fitted to data, this simple model was found to produce accurate IDF results [Langousis A, Veneziano D. Intensity– duration–frequency curves from scaling representations of rainfall. Water Resources Research, 2007; 43: doi: 10.1029/2006WR005245]. Previous studies also showed that the IDF values from multifractal representations of rainfall scale with duration d and return period T under either d->0 or T->oo, with different scaling exponents in the two cases. We determine the regions of the (d, T)-plane in which each asymptotic scaling behavior applies in good approximation, find expressions for the IDF values in the scaling and non-scaling regimes, and quantify the bias when estimating the asymptotic power-law tail of rainfall intensity from finite-duration records, as was often done in the past. Numerically calculated exact IDF curves are compared to several analytic approximations. The approximations are found to be accurate and are used to propose a practical IDF estimation procedure.
Additional material:
D. Koutsoyiannis, C. Makropoulos, A. Langousis, S. Baki, A. Efstratiadis, A. Christofides, G. Karavokiros, and N. Mamassis, Climate, hydrology, energy, water: recognizing uncertainty and seeking sustainability, Hydrology and Earth System Sciences, 13, 247–257, doi:10.5194/hess-13-247-2009, 2009.
Since 1990 extensive funds have been spent on research in climate change. Although Earth Sciences, including climatology and hydrology, have benefited significantly, progress has proved incommensurate with the effort and funds, perhaps because these disciplines were perceived as “tools” subservient to the needs of the climate change enterprise rather than autonomous sciences. At the same time, research was misleadingly focused more on the “symptom”, i.e. the emission of greenhouse gases, than on the “illness”, i.e. the unsustainability of fossil fuel-based energy production. Unless energy saving and use of renewable resources become the norm, there is a real risk of severe socioeconomic crisis in the not-too-distant future. A framework for drastic paradigm change is needed, in which water plays a central role, due to its unique link to all forms of renewable energy, from production (hydro and wave power) to storage (for time-varying wind and solar sources), to biofuel production (irrigation). The extended role of water should be considered in parallel to its other uses, domestic, agricultural and industrial. Hydrology, the science of water on Earth, must move towards this new paradigm by radically rethinking its fundamentals, which are unjustifiably trapped in the 19th-century myths of deterministic theories and the zeal to eliminate uncertainty. Guidance is offered by modern statistical and quantum physics, which reveal the intrinsic character of uncertainty/entropy in nature, thus advancing towards a new understanding and modelling of physical processes, which is central to the effective use of renewable energy and water resources.
Remarks:
Blogs and forums that have discussed this article: Climate science; Vertical news; Outside the cube.
Update 2011-09-26: The removed video of the panel discussion of Nobelists entitled “Climate Changes and Energy Challenges” (held in the framework of the 2008 Meeting of Nobel Laureates at Lindau on Physics) which is referenced in footnote 1 of the paper, still cannot be located online. However, Larry Gould has an audio file of the discussion here.
Full text: http://www.itia.ntua.gr/en/getfile/878/17/documents/hess-13-247-2009.pdf (1476 KB)
Additional material:
See also: http://dx.doi.org/10.5194/hess-13-247-2009
Works that cite this document: View on Google Scholar or ResearchGate
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D. Veneziano, C. Lepore, A. Langousis, and P. Furcolo, Marginal methods of intensity-duration-frequency estimation in scaling and nonscaling rainfall, Water Resources Research, 43, doi:10.1029/2007WR006040, 2007.
Practical methods for the estimation of the Intensity-Duration-Frequency (IDF) curves are usually based on the observed annual maxima of the rainfall intensity I(d) in intervals of different duration d. Using these historical annual maxima, one estimates the IDF curves under the condition that the rainfall intensity in an interval of duration d with return period T is the product of a function a(T) of T and a function b(d) of d (separability condition). Various parametric or semi-parametric assumptions on a(T) and b(d) produce different specific methods. As alternatives, we develop IDF estimation procedures based on the marginal distribution of I(d). If the marginal distribution scales in a multifractal way with d, this condition can be incorporated. We also consider hybrid methods that estimate the IDF curves using both marginal and annual-maximum rainfall information. We find that the separability condition does not hold and that the marginal and hybrid methods perform better than the annual-maximum estimators in terms of accuracy and robustness relative to outlier rainfall events. This is especially true for long return periods and when the length of the available record is short. Marginal and hybrid methods produce accurate IDF estimates also when only a few years of continuous rainfall data are available
Additional material:
A. Langousis, and D. Veneziano, Intensity-duration-frequency curves from scaling representations of rainfall, Water Resources Research, 43, doi:10.1029/2006WR005245, 2007.
We develop methods to estimate the intensity-duration-frequency (IDF) curves for three rainfall models with local multifractal behavior and varying complexity. The models use the classical notion of exterior and interior process, respectively for the variation of rainfall intensity at (approximately) storm and sub-storm scales. The exterior process is non-scaling and differs in the three models, whereas the interior process is stationary multifractal in all cases. The model based IDF curves are robust, against outliers and can be obtained from only very few years of rainfall data. In an application to a 24-year rainfall record from Florence, Italy, the models closely reproduce the empirical IDF curves and make similar extrapolations for return periods longer than the historical record.
Additional material:
D. Veneziano, A. Langousis, and P. Furcolo, Multifractality and rainfall extremes: A review, Water Resources Research, 42, doi:10.1029/2005WR004716, 2006.
The multifractal representation of rainfall and its use to predict rainfall extremes have advanced significantly in recent years. This paper summarizes this body of work and points at some open questions. The need for a coherent overview comes in part from the use of different terminology, notation and analysis methods in the literature and in part from the fact that results are dispersed and not always readily available. Two important trends have marked the use of multifractals for rainfall and its extremes. One is the recent shift of focus from asymptotic scaling properties (mainly for the intensity-duration-frequency curves and the areal reduction factor) to the exact extreme distribution under non-asymptotic conditions. This shift has made the results more relevant to hydrologic applications. The second trend is a more sparing use of multifractality in modeling, reflecting the limits of scale invariance in space-time rainfall. This trend has produced models that are more consistent with observed rainfall characteristics, again making the results more suitable for application. Finally we show that rainfall extremes can be analyzed using rather rough models, provided the parameters are fitted to an appropriate range of large-deviation statistics.
Additional material:
A. Langousis, and D. Koutsoyiannis, A stochastic methodology for generation of seasonal time series reproducing overyear scaling behaviour, Journal of Hydrology, 322, 138–154, 2006.
In generating synthetic time series of hydrological processes at sub-annual scales it is important to preserve seasonal characteristics and short-term persistence. At the same time, it is equally important to preserve annual characteristics and overyear scaling behaviour. This scaling behaviour, which is equivalent to the Hurst phenomenon, has been detected in a large number of hydroclimatic series and affects seriously planning and design of hydrosystems. However, when seasonal models are used the preservation of annual characteristics and overyear scaling is a difficult task and is often ignored unless disaggregation techniques are applied, which, however, involve several difficulties (e.g. in parameter estimation) and inaccuracies. As an alternative, a new methodology is proposed that directly operates on seasonal time scale, avoiding disaggregation, and simultaneously preserves annual statistics and the scaling properties on overyear time scales. Two specific stochastic models are proposed, a simple widely used seasonal model with short memory to which long-term persistence is imposed using a linear filter, and a combination of two sub-models, a stationary one with long memory and a cyclostationary one with short memory. Both models are tested in a real world case and found to be accurate in reproducing all the desired statistical properties and virtually equivalent from an operational point of view.
Additional material:
See also: http://dx.doi.org/10.1016/j.jhydrol.2005.02.037
Works that cite this document: View on Google Scholar or ResearchGate
Other works that reference this work (this list might be obsolete):
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| 12. | Srivastav, R., K. Srinivasan, and S. P. Sudheer, Simulation-optimization framework for multi-site multi-season hybrid stochastic streamflow modeling, Journal of Hydrology, doi:10.1016/j.jhydrol.2016.09.025, 2016. |
D. Veneziano, and A. Langousis, The maximum of multifractal cascades: Exact distribution and approximations, Fractals, 13 (4), 311–324, 2005.
We study the distribution of the maximum M of multifractal measures using discrete cascade representations. For such discrete cascades, the exact distribution of M can be found numerically. We evaluate the sensitivity of the distribution of M to simplifying approximations, including independence of the measure among the cascade tiles and replacement of the dressing factor by a random variable with the same distribution type as the cascade generator. We also examine how the distribution of M varies with the dimensionality of the support and the multiplicity of the cascade. Of these factors, dependence of the measure among different cascade tiles has the highest effect on the distribution of M. This effect comes mainly from long-range dependence. We use these findings to propose a simple approximation to the distribution of M and give charts to implement the approximation for beta-lognormal cascades.
Additional material:
D. Veneziano, and A. Langousis, The areal reduction factor: A multifractal analysis, Water Resources Research, 41, doi:10.1029/2004WR003765, 2005.
The areal reduction factor (ARF) η is a key quantity in the design against hydrologic extremes. For a basin of area a and a duration d, η(a, d, T) is the ratio between the average rainfall intensity in a and d with return period T and the average rainfall intensity at a point for the same d and T. Empirical ARF charts often display scaling behavior. For example, for large ( a/d) ratios and given T, the ARF tends to behave like (√a/d)^(-α) for some α. Here we obtain scaling properties of the ARF under the condition that space-time rainfall has multifractal scale invariance. The scaling exponents of the ARF are related in a simple way to the multifractal properties of the parent rainfall process. We consider regular and highly elongated basins, quantify the effect of rainfall advection, and investigate the bias from estimating the ARF using sparse raingauge networks. We also study the effects of departure of rainfall from exact multifractality. The results explain many features of empirical ARF charts while suggesting dependencies on advection, basin shape, and return period that are difficult to quantify empirically. The theoretical scaling relations may be used to extrapolate the ARF beyond the empirical range of a, d and T.
Additional material:
D. Koutsoyiannis, and A. Langousis, Precipitation, Treatise on Water Science, edited by P. Wilderer and S. Uhlenbrook, 2, 27–78, doi:10.1016/B978-0-444-53199-5.00027-0, Academic Press, Oxford, 2011.
The study of precipitation has been closely linked to the birth of science, by the turn of the 7th century BC. Yet, it continues to be a fascinating research area, since several aspects of precipitation generation and evolution have not been understood, explained and described satisfactorily. Several problems, contradictions and even fallacies related to the perception and modelling of precipitation still exist. The huge diversity and complexity of precipitation, including its forms, extent, intermittency, intensity, and temporal and spatial distribution, do not allow easy descriptions. For example, while atmospheric thermodynamics may suffice to explain the formation of clouds, it fails to provide a solid framework for accurate deterministic predictions of the intensity and spatial extent of storms. Hence, uncertainty is prominent and its understanding and modelling unavoidably relies on probabilistic, statistical and stochastic descriptions. However, the classical statistical models and methods may not be appropriate for precipitation, which exhibits peculiar behaviours including Hurst-Kolmogorov dynamics and multifractality. This triggered the development of some of the finest stochastic methodologies to describe these behaviours. Inevitably, because deduction based on deterministic laws becomes problematic, as far as precipitation is concerned, the need for observation of precipitation becomes evident. Modern remote sensing technologies (radars and satellites) have greatly assisted the observation of precipitation over the globe, whereas modern stochastic techniques have made the utilization of traditional raingauge measurements easier and more accurate. This chapter reviews existing knowledge in the area of precipitation. Interest is in the small- and large-scale physical mechanisms that govern the process of precipitation, technologies and methods to estimate precipitation in both space and time, and stochastic approaches to model the variable character of precipitation and assess the distribution of its extremes.
Additional material:
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| 1. | Khalil, B., and J. Adamowski, Record extension for short-gauged water quality parameters using a newly proposed robust version of the line of organic correlation technique, Hydrol. Earth Syst. Sci. , 16, 2253-2266, doi: 10.5194/hess-16-2253-2012, 2012. |
| 2. | #Langousis, A. and V. Kaleris, A statistical approach to estimate spatial rainfall averages using point rainfall measurements from a single location and runoff data, Proceedings of the 2nd Joint Conference of EYE-EEDYP "Integrated Water Resources Management for Sustainable Development" (Ed.: P. Giannopoulos and A. Dimas), 75-80, Patras, Greece, 2012. |
| 3. | Langousis, A., and V. Kaleris, Theoretical framework to estimate spatial rainfall averages conditional on river discharges and point rainfall measurements from a single location: an application to western Greece, Hydrol. Earth Syst. Sci., 17, 1241-1263, 10.5194/hess-17-1241-2013, 2013. |
| 4. | #Khalil, B., J. Adamowski and A. Belayneh, Evaluation of the performance of eight record extension techniques under different levels of data contamination: A Monte Carlo study, Proceedings, Annual Conference - Canadian Society for Civil Engineering, 3, 2249-2258, 2013. |
| 5. | Langousis, A., and V. Kaleris, Statistical framework to simulate daily rainfall series conditional on upper-air predictor variables, Water Resources Research, 10.1002/2013WR014936, 2014. |
| 6. | Khalil, B., and J. Adamowski, Comparison of OLS, ANN, KTRL, KTRL2, RLOC, and MOVE as Record-extension techniques for water quality variables, Water, Air, & Soil Pollution, 10.1007/s11270-014-1966-1, 2014. |
| 7. | Khalil, B., and J. Adamowski, Evaluation of the performance of eight record-extension techniques under different levels of association, presence of outliers and different sizes of concurrent records: a Monte Carlo study, Water Resources Management, 10.1007/s11269-014-0799-4, 2014. |
| 8. | Kienzler, P., N. Andres, D. Naef-Huber and M. Zappa, Derivation of extreme precipitation and flooding in the catchment of Lake Sihl to improve flood protection in the city of Zurich, Hydrologie Und Wasserbewirtschaftung, 59 (2), 48-58, 10.5675/HyWa_2015,2_1, 2015. |
| 9. | Müller, H. and U. Haberlandt, Temporal Rainfall Disaggregation with a Cascade Model: From Single-Station Disaggregation to Spatial Rainfall, J. Hydrol. Eng., 10.1061/(ASCE)HE.1943-5584.0001195, 04015026, 2015. |
D. Veneziano, and A. Langousis, Scaling and fractals in hydrology, Advances in Data-based Approaches for Hydrologic Modeling and Forecasting, edited by B. Sivakumar and R. Berndtsson, 145 pages, World Scientific, 2010.
Virtually, all areas of hydrology have been deeply influenced by the concepts of fractality and scale invariance. The roots of scale invariance in hydrology can be traced to the pioneering work of Horton, Shreve, Hack and Hurst on the topology and metric properties of river networks and on river flow. This early work uncovered symmetries and laws that only later were recognized as manifestations of scale invariance. Le Cam, who in the early 1960s pioneered the development of multi-scale pulse models of rainfall, provided renewed impetus to the use of scale-based models. Fractal approaches in hydrology have become more rigorous and widespread since Mandelbrot systematized fractal geometry and multifractal processes were discovered. This chapter reviews the main concepts of fractality and scale invariance, the construction of scale-invariant processes, their properties, and the inference of scale invariance from data. We highlight the recent developments in four areas of hydrology: rainfall, fluvial erosion topography, river floods, and flow through porous media.
Additional material:
A. Langousis, D. Veneziano, and S. Chen, Boundary layer model for moving tropical cyclones, Hurricanes and Climate Change, edited by J. Elsner and T. Jagger, 265–286, Springer, 2008.
We propose a simple theoretical model for the boundary layer (BL) of moving tropical cyclones (TCs). The model estimates the horizontal and vertical wind velocity fields from a few TC characteristics: the maximum tangential wind speed Vmax, the radius of maximum winds Rmax, and Holland’s B parameter away from the surface boundary where gradient balance is approximately valid, in addition to the storm translation velocity Vt, the surface drag coefficient CD, and the vertical diffusion coefficient of the horizontal momentum K. The model is based on Smith’s (1968) formulation for stationary (axi-symmetric) tropical cyclones. Smith’s model is first extended to include storm motion and then solved using the momentum integral method. The scheme is computationally very efficient and is stable also for large B values and fast-moving storms. Results are compared to those from other studies (Shapiro 1983; Kepert 2001) and validated using the Fifth-Generation Pennsylvania State Univer-sity/NCAR Mesoscale Model (MM5). We find that Kepert’s (2001) BL model significantly underestimates the radial and vertical fluxes, whereas Shapiro’s (1983) slab-layer formulation produces radial and vertical winds that are a factor of about two higher than those produced by MM5. The velocity fields generated by the present model are consistent with MM5 and with tropical cyclone observations. We use the model to study how the symmetric and asymmetric components of the wind field vary with the storm parameters mentioned above. In accordance with observations, we find that larger values of B and lower values of Rmax produce horizontal and vertical wind profiles that are more picked near the radius of maximum winds. We also find that, when cy-clones in the northern hemisphere move, the vertical and storm-relative ra-dial winds intensify at the right-front quadrant of the vortex, whereas the storm-relative tangential winds are more intense in the left-front region. The asymmetry is higher for faster moving TCs and for higher surface drag coefficients CD.
Additional material:
G. Papacharalampous, H. Tyralis, A. Langousis, A. W. Jayawardena, B. Sivakumar, N. Mamassis, A. Montanari, and D. Koutsoyiannis, Large-scale comparison of machine learning regression algorithms for probabilistic hydrological modelling via post-processing of point predictions, European Geosciences Union General Assembly 2019, Geophysical Research Abstracts, Vol. 21, Vienna, EGU2019-3576, European Geosciences Union, 2019.
Quantification of predictive uncertainty in hydrological modelling is often made by post-processing point hydrological predictions using regression models. We perform an extensive comparison of machine learning algorithms in obtaining quantile predictions of daily streamflow under this specific approach. The comparison is performed using a large amount of real-world data retrieved from the Catchment Attributes and MEteorology for Large sample Studies (CAMELS) dataset. Various climate types are well-represented by the examined catchments. The point predictions are obtained using the GR4J model, a lumped conceptual hydrological model comprising of four parameters, while their post-processing is made by predicting conditional quantiles of the hydrological model's errors. The latter are transformed to conditional quantiles of daily streamflow and finally assessed by using various performance metrics. The machine learning regression algorithms are also benchmarked against the quantile regression algorithm.
Full text: http://www.itia.ntua.gr/en/getfile/1943/1/documents/EGU2019-3576.pdf (33 KB)
H. Tyralis, and A. Langousis, Modelling of rainfall maxima at different durations using max-stable processes, European Geosciences Union General Assembly 2018, Geophysical Research Abstracts, Vol. 20, Vienna, EGU2018-2299, European Geosciences Union, 2018.
The multivariate extreme value distribution (MEVD) has been used to model the dependence of rainfall block maxima at different temporal resolutions, as a means of estimating intensity-duration-frequency (IDF) curves for engineering applications. It is characterized by max-stability, which assumes that under proper renormalization, the rainfall block maxima at different temporal resolutions are extreme value distributed and the degree of their dependence remains invariant to the severity of the event. Due to these properties, and contrary to other commonly used approaches, MEVD allows for more conservative return level estimates at those durations used for model fitting. Max-stable processes are continuous extensions of MEVD, which are more flexible, and allow for extrapolation to temporal resolutions beyond those used for model fitting. Here we: 1) propose using max-stable processes to model rainfall block maxima, 2) apply the Brown-Resnick, Schlather and extremal-t models to hourly rainfall data, and 3) compare the obtained results to traditional approaches for IDF estimation. We discuss advantages and limitations regarding the use of max-stable processes in IDF estimation, and their potential use in hydrologic practice.
Full text: http://www.itia.ntua.gr/en/getfile/1805/1/documents/EGU2018-2299.pdf (32 KB)
A. Langousis, R. Deidda, and A. Carsteanu, A Simple approximation to multifractal rainfall maxima using a generalized extreme value distribution model, International Precipitation Conference (IPC10), Coimbra, Portugal, 2010.
Define Id to be the average rainfall intensity inside an interval of duration d, and denote by Iyr,d the maximum of Id in a year. Based on a standard asymptotic result from extreme value (EV) theory, assuming independence and distributional identity between the variables Id, Iyr,d is typically assumed to follow a generalized extreme value (GEV) distribution with shape parameter k(d) that depends on the extreme upper tail of the distribution of Id. Estimation of k(d) from either at-site or regional rainfall data is generally difficult for two reasons. The first is the poor knowledge of the upper tail of the distribution of Id, even for long rainfall records. The other is more theoretical and it is related to the applicability of the asymptotic EV result, when the number n = 1yr/d of the d-intervals in a year (or, equivalently, the number n of the Id variables over which the maximum Iyr,d is taken) is finite. In a recent study, Veneziano et al. (Water Resour. Res., doi:10.1029/2009WR008257) showed that for multifractal rainfall and typical values of d, 1yr/d is too small for convergence of Iyr,d to a GEV distribution. Hence, k(d) cannot be derived from asymptotic arguments and it is influenced by a region of the distribution of Id that is close to the body thereof, rather than its extreme upper tail. Here, we propose a simple method to theoretically calculate the shape parameter k(d) of a GEV distribution model fitted to Iyr,d, as a function of the averaging duration d. We do so by assuming that rainfall is stationary multifractal below some maximum temporal scale D, and estimate k(d) by fitting a GEV distribution to the maximum of n = 1yr/d independent and identically distributed Id variables. To keep the method simple and suitable for practical applications, we analytically approximate the distribution of Id and estimate k(d) by recursively solving a non-linear equation. The suggested method, to theoretically constraint the shape parameter of a GEV distribution model and then fit the model to the recorded annual rainfall maxima, is compared to the classical annual maximum (AM) approach, where all GEV parameters are calculated from data. The relative performance of the methods is evaluated by comparing the bias, variance and root mean square error (RMSE) of each approach, using the differences between the empirical annual maxima and those calculated theoretically from the fitted distribution models and the empirical exceedance probabilities.
Additional material:
R. Deidda, A. Langousis, and G. Mascaro, Intercomparison of regionalization approaches for extreme rainfall modeling, International Precipitation Conference (IPC10), Coimbra, Portugal, 2010.
We discuss the performances of two different approaches in evaluating extreme rainfall distributions from empirical rainfall records. The first approach fits a generalized extreme value (GEV) distribution to the empirical annual maxima. The second approach makes use of a larger portion of information and fits a Generalized Pareto (GP) distribution to the rainfall excesses above a range of suitable intensity thresholds u. We call this the “multiple threshold method” (MTM). The analysis is conducted using a database of 217 daily rainfall records from Sardinia (Italy) during the period 1922 to 1996, each having more than 40 years of observations. The performance of each approach is evaluated using a jacknife-type technique in order to estimate and compare the accuracy at ungauged sites. Initially, we fit the two distribution models independently to all available records, excluding one site per time. Then, we infer the distribution parameters at the excluded sites either through kriging or by assuming constant parameter values inside statistically homogeneous sub-regions. Finally, we discuss the results of the two approaches in terms of bias, variance and root mean square error, using the differences between the empirical annual maxima and those calculated theoretically from the fitted distribution models and the empirical exceedance probabilities of the ranked maxima.
D. Veneziano, A. Langousis, and C. Lepore, Annual rainfall maxima: Theoretical Estimation of the GEV shape parameter k using multifractal models, Eos Trans. AGU, 90(52), San Francisco, American Geophysical Union, 2009.
The annual maximum of the average rainfall intensity in a period of duration d, Iyear(d), is typically assumed to have generalized extreme value (GEV) distribution. The shape parameter k of that distribution is especially difficult to estimate from either at-site or regional data, making it important to constraint k using theoretical arguments. In the context of multifractal representations of rainfall, we observe that standard theoretical estimates of k from extreme value (EV) and extreme excess (EE) theories do not apply, while estimates from large deviation (LD) theory hold only for very small d. We then propose a new theoretical estimator based on fitting GEV models to the numerically calculated distribution of Iyear(d). A standard result from EV and EE theories is that k depends on the tail behavior of the average rainfall in d, I(d). This result holds if Iyear(d) is the maximum of a sufficiently large number n of variables, all distributed like I(d); therefore its applicability hinges on whether n = 1yr/d is large enough and the tail of I(d) is sufficiently well known. One typically assumes that at least for small d the former condition is met, but poor knowledge of the upper tail of I(d) remains an obstacle for all d. In fact, in the case of multifractal rainfall, also the first condition is not met because, irrespective of d, 1yr/d is too small (Veneziano et al., 2009, WRR, in press). Applying large deviation (LD) theory to this multifractal case, we find that, as d -> 0, Iyear(d) approaches a GEV distribution whose shape parameter kLD depends on a region of the distribution of I(d) well below the upper tail, is always positive (in the EV2 range), is much larger than the value predicted by EV and EE theories, and can be readily found from the scaling properties of I(d). The scaling properties of rainfall can be inferred also from short records, but the limitation remains that the result holds under d -> 0 not for finite d. Therefore, for different reasons, none of the above asymptotic theories applies to Iyear(d). In practice, one is interested in the distribution of Iyear(d) over a finite range of averaging durations d and return periods T. Using multifractal representations of rainfall, we have numerically calculated the distribution of Iyear(d) and found that, although not GEV, the distribution can be accurately approximated by a GEV model. The best-fitting parameter k depends on d, but is insensitive to the scaling properties of rainfall and the range of return periods T used for fitting. We have obtained a default expression for k(d) and compared it with estimates from historical rainfall records. The theoretical function tracks well the empirical dependence on d, although it generally overestimates the empirical k values, possibly due to deviations of rainfall from perfect scaling. This issue is under investigation.
A. Langousis, and D. Veneziano, Extreme rainfall intensities and long-term rainfall risk from tropical cyclones, European Geosciences Union General Assembly 2009, Geophysical Research Abstracts, Vol. 11, Vienna, European Geosciences Union, 2009.
We develop a methodology to estimate the rate of extreme rainfalls at coastal sites due to tropical cyclones (TCs). A basic component of the methodology is the probability distribution of ID,max, the maximum rainfall intensity at the site over a period D during the passage of a TC with given characteristics θ. The long-term rainfall risk is obtained by combining the conditional distribution of (ID,max|θ) with a recurrence model for θ. To illustrate the use of the model for long-term rainfall risk analysis, we formulate a recurrence model for tropical cyclones in the Gulf of Mexico that make landfall between longitudes 85o-95oW and compare the intensity-duration-frequency (IDF) curves for New Orleans obtained by the present model with similar curves in the literature based on continuous rainfall records. The latter include all types of rainstorms.
Additional material:
C. Lepore, D. Veneziano, and A. Langousis, Annual rainfall maxima: Practical estimation based on large-deviation results, European Geosciences Union General Assembly 2009, Geophysical Research Abstracts, Vol. 11, Vienna, European Geosciences Union, 2009.
In a separate communication (Veneziano et al., “Annual Rainfall Maxima: Large-deviation Alternative to Extreme-Value and Extreme-Excess Methods,” EGU 2009), we show that, at least for scale-invariant rainfall models, classical extreme value analysis based on Gumbel’s extreme value (EV) theory and peak-over-threshold (PoT) analysis based on Pickands’ extreme excess (EE) theory do not apply to annual rainfall maxima (AM). A more appropriate theoretical setting is provided by large-deviation (LD) theory. This paper delves with some practical implications of these findings.
Additional material:
D. Veneziano, A. Langousis, and C. Lepore, Annual rainfall maxima: Large-deviation alternative to extreme-value and extreme-excess methods, European Geosciences Union General Assembly 2009, Geophysical Research Abstracts, Vol. 11, Vienna, European Geosciences Union, 2009.
Contrary to common belief, Gumbel’s extreme value (EV) and Pickands’ extreme excess (EE) theories do not generally apply to rainfall maxima at the annual level. This is true not just for long averaging durations d, as one would expect, but also in the high-resolution limit as d->0. We reach these conclusions by studying the annual maxima of scale-invariant rainfall models with a multiplicative structure. We find that for d->0 the annual maximum rainfall intensity in d, Iyear(d), has a generalized extreme value (GEV) distribution with a shape parameter k that is significantly higher than that predicted by Gumbel’s theory and is always in the EV2 range. Under the same conditions, the excess above levels close to the annual maximum has generalized Pareto (GP) distribution with a parameter k that is always higher than that predicted by Pickands’ theory. The proper tool to obtain these results is large deviation (LD) theory, a branch of probability that has been largely ignored in stochastic hydrology.
Additional material:
A. Langousis, and D. Veneziano, Rainfall Hazard from Tropical Cyclones, European Geosciences Union General Assembly 2008, Geophysical Research Abstracts, Vol. 10, Vienna, European Geosciences Union, 2008.
The assessment of rainfall hazard from tropical cyclones (TCs) typically relies on empirical models, which estimate the rainfall field for a given TC intensity and motion category as the ensemble average over the historical storms in that category. The coarseness of the classification, the exclusion of other relevant storm parameters (most notably the radius of maximum winds), and the paucity of the historical data make the estimates rather inaccurate. Moreover, ensemble averaging suppresses the all-important fluctuations due to rainbands and local convection and severely underestimates the rainfall maxima. We have developed an alternative approach to evaluate the probability distribution of the maximum rainfall intensity in a period d at a geographical point due to passage of a tropical cyclone. This maximum rainfall intensity, Imax(θ, r, d), depends on the duration d, the distance r of the point from the storm center, and several storm parameters θ. The calculated distributions of Imax(θ, r, d) are in good agreement with maximum rainfall intensities estimated from TRMM measurements.
Additional material:
C. Lepore, M.I.P. de Lima, D. Veneziano, A. Langousis, and J.L.M.P. de Lima, Statistical characterization of extreme rainfall climate along the future high-speed rail track in Portugal, European Geosciences Union General Assembly 2008, Geophysical Research Abstracts, Vol. 10, Vienna, European Geosciences Union, 2008.
The characterization of rainfall and its extremes at different spatial and temporal scales is important for the evaluation of environmental risks along high-speed rail systems. Rainfall can induce accidents for example through the activation of landslides and affects operations by causing slowdown. We present an exploratory analysis of point rainfall data from mainland Portugal, distributed over an area of approximately 30 km by 460 km. This corridor includes the high-speed rail track that is now being designed for Portugal. This is the first step in the assessment of hydrologic risk for this infrastructure. The dataset includes rainfall time series with heterogeneous length and resolution: some of the data are from tipping bucket rain gauges with hourly resolution and an average length of 5 years, while others are from conventional gauges with daily data over periods of more than 50 years. Key statistical characteristics of the rainfall process, including extremes, are investigated to increase our understanding of the variability of the rainfall climate along the rail track. Maps showing the areal distribution of statistical characteristics of the process are drawn to better illustrate the spatial variability of the rainfall threat in the study area.
Additional material:
D. Veneziano, C. Lepore, M.I.P. de Lima, A. Langousis, and J.L.M.P. de Lima, Comparison of IDF estimation methods at selected locations of mainland Portugal, European Geosciences Union General Assembly 2008, Geophysical Research Abstracts, Vol. 10, Vienna, European Geosciences Union, 2008.
Rainfall extremes are typically represented by means of Intensity-Duration-Frequency (IDF) curves. Different methods have been developed to estimate these curves from historical rainfall records. These include methods based on annual maxima series (AMS), peaks over threshold (POT) values and marginal distribution information. While annual maximum values are most directly relevant to the IDF values, they have the drawback of reducing one year worth of data to a single value. By contrast, POT and marginal-distribution methods utilize the data more fully, but rely on simplifying assumptions to estimate the IDF curves. The AMS approach, which is widely used in hydrology, can have different complexities depending on the parameterization and estimation procedure. The method assumes a parametric or nonparametric dependence of the extremes on the return period T and the duration d. There is theoretical and empirical evidence that certain assumptions of the method (e.g. that the multiplicative effect of T is the same for different d) are incorrect. The POT approach produces results that may be sensitive to the threshold. Also this method requires assumptions, most critically on the type of distribution of the POT values. Marginal distribution methods assume independence of the rainfall process in non-overlapping d intervals. This assumption generally overestimates the IDF values but is accurate for very long return periods T. Another assumption concerns the form of the marginal distribution, in particular in the upper tail. We have significant empirical evidence that this tail has a lognormal shape. If the rainfall process displays multifractal scale invariance, then this condition can be used to regularize the dependence of the marginal distribution on d. In a “hybrid” variant of the marginal method it is possible to include information on the annual maxima. We compare the above IDF estimation methods at selected sites of mainland Portugal. The rainfall records typically include 20 or more years of high-resolution data and about 60 years of daily precipitation values.
Additional material:
C. Lepore, D. Veneziano, and A. Langousis, Lognormal upper tail of rainfall intensity and POT values: Implications on the IDF curves, European Geosciences Union General Assembly 2008, Geophysical Research Abstracts, Vol. 10, Vienna, European Geosciences Union, 2008.
Let I(d) be the average rainfall intensity in an interval of duration d and denote by I(d;i*) the peak-over-threshold (POT) value of I(d) for threshold i* and by Imax(d) the annual maximum of I(d). Hydrologic risk assessment and design depend critically on the upper tail of the distribution of Imax(d). Since the events I(d) > i in non-overlapping d-intervals become independent as i becomes large and for even moderate thresholds i* the excursions of I(d) above i* may be considered Poisson, the upper tail of the distribution of Imax(d) may be estimated from the upper tails of I(d) and I(d;i*). It has been argued that the distribution of Imax(d) should be of the GEV type, with recent propensity for EV2. However, the GEV claim follows from asymptotic arguments. We take another look at the distributions of I(d) and I(d;i*) and their implications on the upper tail of Imax(d). Specifically we show that there is empirical and theoretical evidence that for durations d ≤ 1 day the upper tail of I(d) has a lognormal behavior over a wide range of intensities and that also the distribution of I(d;i*) has a significant lognormal range. These conclusions are supported by both historical and simulated rainfall records and theoretical analysis. For the simulations we use a partly theoretical model in which storm occurrence times and durations are extracted from a historical record but the storm intensity and within-storm fluctuations are generated randomly, the latter using a beta-lognormal multifractal process. Results on Imax(d) obtained by fitting distributions of I(d) and I(d;i*) with upper lognormal tails are compared to results from directly fitting a GEV distribution to the observed annual maximum intensities and from fitting a GP distribution to exceedances above threshold i*.
Additional material:
A. Langousis, D. Veneziano, and S. Chen, Theoretical estimation of the mean rainfall intensity field in tropical cyclones: Axi-symmetric component and asymmetry due to motion, 1st International Summit on Hurricanes and Climate Change, Crete, Greece, Crete, Greece, 2007.
We develop a simple theoretical model for the mean rainfall intensity field in tropical cyclones (TCs). The model estimates the axi-symmetric rainfall profile Isym(r) as well as the asymmetric component due to storm motion Imot(r,θ), where r is the radial distance from the TC center and θ is the azimuth relative to the direction of the storm. Currently, the model does not include asymmetries due to wind shear, coastline geometry and topography, or fluctuations associated with rainbands and small-scale convection; hence its main use is to provide large-scale rainfall estimates. Rainfall intensity is estimated as the vertical outflow of water vapor at the top of the TC boundary layer (BL). The analysis combines Holland’s (1980) (or any other) tangential wind profile, an Ekman-type solution for the horizontal and vertical wind profiles inside the TC boundary layer, and moist air thermodynamics. The BL solution for wind is based on Smith’s (1968) formulation, which is modified and solved to account for the effects of storm motion. The axi-symmetric rainrate Isym(r) is zero for r = 0, increases to a maximum Imax at a distance Rrain, and then decays to zero in an approximately power-law way. Model results are compared to those from other studies (Shapiro, 1983; Kepert, 2001; Kepert and Wang, 2001). The three formulations are generally in good agreement for both horizontal and vertical fluxes, except for close to the storm center where nonlinear effects are dominant and Kepert’s (2001) solution is less accurate and for the far-field where both the Shapiro (1983) and Kepert and Wang (2001) approaches are affected by numerical instabilities. The present scheme is computationally very efficient and stable also for high storm translation velocities. In a parametric analysis, we study how the symmetric component and the motion-induced asymmetries of rainfall depend on TC characteristics such as the maximum tangential wind velocity Vmax, the radius of maximum winds Rmax, Holland’s B parameter, and the temperature T in the boundary layer. More intense cyclones have higher Imax and lower Rrain. The pickness of the tangential wind velocity profile, expressed through Holland’s B parameter, has insignificant effects on the Isym(r) profile. These theoretical findings are in agreement with observations. The model shows that when cyclones in the Northern hemisphere move, their mean rainrate intensifies in the north-east quadrant relative to the direction of motion and de-intensifies in the south-west quadrant. The asymmetry is concentrated near the TC center and is stronger for less intense and faster-moving storms.
Additional material:
D. Veneziano, C. Lepore, A. Langousis, and P. Furcolo, Scaling, partial-scaling and classical methods of IDF curve estimation, European Geosciences Union General Assembly 2007, Geophysical Research Abstracts, Vol. 9, Vienna, European Geosciences Union, 2007.
We describe a new class of IDF curve estimation methods based on the marginal distributions of rainfall intensity in intervals of different duration d and compare them to standard procedures that use historical annual maxima. First we develop and test the new methods assuming multifractal rainfall and then generalize the procedures for partial or complete lack of scaling. In the case of multifractal rainfall, a scale-invariance condition links the marginal distributions for different values of d and makes parameterization especially lean. We compare the new and existing IDF methods using historical and synthetic rainfall records. The use of marginal rather than annual-maximum information increases the stability and robustness of the new IDF estimators against outliers. Hence the new methods are applicable also to a few years of continuous rainfall data. While conventional methods frequently assume independent multiplicative effects of duration and return period on the IDF values, the new methods are capable of modeling the dependencies that are often observed in empirical IDF curves. In complexity, the new methods are comparable to the conventional ones.
Additional material:
A. Langousis, D. Veneziano, C. Lepore, and P. Furcolo, Simple IDF estimation under multifractality, European Geosciences Union General Assembly 2007, Geophysical Research Abstracts, Vol. 9, Vienna, European Geosciences Union, 2007.
We study the extremes generated by a multifractal model of temporal rainfall and propose a practical method to estimate the Intensity-Duration-Frequency (IDF) curves. The model represents rainfall as a sequence of independent and identically distributed multiplicative cascades of common duration D. This simple model was previously found to be adequate for IDF analysis. Here we show how the IDF curves produced by the model can be accurately approximated by simple formulas and use these approximations to propose a practical IDF estimation procedure. We discuss the estimation of model parameters and the calculation of return-period values and present an application to a 24-year rainfall record from Florence, Italy. The proposed approach has several advantages over conventional methods that directly fit parametric IDF models to the annual rainfall maxima. One is that fitting the multifractal model to the entire record (as opposed to using only the annual maxima) confers stability and robustness to the estimated IDF curves. For example, sensitivity to using a fraction of the historical record (down to 5 years) and to including/excluding outlier years is much smaller than in conventional methods. Another advantage is that the shape of the IDF curves needs not be externally specified but is determined by the fitted model.
Additional material:
D. Veneziano, C. Lepore, A. Langousis, and P. Furcolo, Comparison of IDF estimation methods, International Precipitation Conference (IPC09), Paris, Universite Paris Est, Ecole Nationale des Ponts et Chaussees, 2007.
We compare several estimators of the Intensity-Duration-Frequency (IDF) curves from continuous at-site rainfall records. These include parametric and semi-parametric estimators based on the historical annual maxima (AM estimators), peak-over-threshold (POT) methods, estimators based on the marginal distribution (MD) of rainfall intensity, and hybrid estimators (HY) that combine marginal and annual-maximum information. Comparison is in terms of bias, variance and RMS error. These performance measures vary with the length of the record D, the averaging duration d, and the return period T. The analysis uses subsets of actual and synthetic rainfall records of duration between 24 and 1000 years. The empirical IDF curves from each entire record are used as reference to assess the bias. Another element of comparison is the sensitivity of the estimators to outliers, defined as annual extremes whose estimated return period far exceeds the duration of the record. Broadly speaking, one would expect AM estimators to perform best for very long records, MD estimators to be superior when only a few years of data are available, and POT estimators to be competitive in the intermediate case of a few decades on record. These expectations are based on the qualitative reasoning that POT and MD methods use increasingly large samples (leading to smaller error variance), but the observed variable is increasingly removed from the annual maximum (which may increase the bias). The AM methods assume that the IDF value is a separable function of d and T. Parametric versions of these methods specify the form of these functions except for a few parameters, whereas semi-parametric versions specify only the functional dependence on d. Dependence on T in the parametric case is based on the assumption that the annual maximum has a GEV distribution. We find that the separability assumption is often violated and the IDF estimates for long return periods are highly variable and sensitive to outliers. This is especially true for the semi-parametric methods, which impose loose constraints on the tail of the annual maximum distribution. None of the other methods assumes separability. We apply the POT method assuming that the excess above the threshold has either Pareto or Generalized Pareto (GP) distribution. In the Pareto case, the model is highly constrained and results in very high bias especially for large T (because empirical distributions deviate significantly from Pareto). In the GP case the bias is small, but unless the record is very long the variance is large due to the difficulty of constraining the shape parameter of the distribution. Marginal-distribution methods assume that the rainfall intensities in separate d-intervals are independent and have a lognormal tail. These methods are statistically stable, robust against outliers, and applicable also when the rainfall record is short. Finally, hybrid methods scale the annual maximum distribution from MD analysis such that its mean value coincides with the average of the recorded annual maxima. These estimators are the best-performing ones when the continuous record has a length of a few decades or the record is short but in addition one has a long annual maximum series.
Full text: http://www.itia.ntua.gr/en/getfile/1014/1/documents/IPC9_2007.pdf (515 KB)
A. Langousis, and D. Veneziano, A simple theoretical model for the mean rainfall field of tropical cyclones, Eos Trans. AGU, 87(52), San Francisco, American Geophysical Union, 2006.
We develop a simple model for the mean rainfall intensity profile in tropical cyclones (TCs) before landfall. The model assumes that rainfall is caused primarily by condensation of the humid outflow at the top of the TC boundary layer. This upward-directed flux originates from convergence of the horizontal winds in the boundary layer. The model combines Holland’s (1980) representation of the tangential wind speed in the main vortex, an Ekman-type solution for the horizontal and vertical wind profiles inside the TC boundary layer, and moist air thermodynamics to estimate how the mean rainrate i depends on radial distance r from the low pressure center and azimuth θ relative to the direction of motion. We start by studying the axisymmetric component i(r), which is also the mean rainrate profile for zero translational velocity. i(r) depends on the maximum pressure deficit ΔPmax (or the maximum tangential wind speed Vmax), Holland’s B parameter, the radius of maximum winds Rwind, and the depth-averaged temperature in the boundary layer T. The mean rainrate is zero for r = 0, increases to a maximum imax at a distance Rrain somewhat larger than Rwind, and then decays to zero in an almost exponential way. More intense cyclones tend to have lower Rrain and higher imax. The difference Rrain-Rwind is higher for tangential wind profiles that are more picked around Rwind. Such wind profiles are generally associated with more intense cyclones and higher B values. When cyclones in the Northern hemisphere move, the mean rainrate intensifies in the north-east quadrant relative to the direction of motion and de-intesifies in the south-west quadrant. These azimuthal effects are stronger for faster-moving storms. For preliminary validation, we compare model estimates of i(r) under representative parameters with ensemble averages from 548 CAT12 and 212 CAT35 TCs extracted from the TRMM dataset (Lonfat et al., Mon. Wea. Rev., 132 (2004): 1645-1660). The model reproduces very well the shape and intensity of the empirical profiles, except near the TC center where the high gradient of the rainfall intensity and the limited resolution of the data make the empirical values less accurate.
Additional material:
D. Veneziano, and A. Langousis, Multifractality and the estimation of extreme rainfall, European Geosciences Union General Assembly 2006, Geophysical Research Abstracts, Vol. 8, Vienna, European Geosciences Union, 2006.
Two important research directions have marked the recent use of multifractals in rainfall modeling and extreme rainfall analysis. One is the shift of focus from the asymptotic scaling properties of the intensity-duration-frequency curves to exact extreme distributions under non-asymptotic conditions. The second is a more sparing use of multifractality in modeling, to reflect the limits of scale invariance in rainfall. Both trends have made the results more relevant to hydrologic applications. Here we consider three rainfall models with limited multifractal properties, which are simple, robust against outliers, and require only a few years of rainfall data. The models use the classical notions of exterior and interior process, respectively for the variation of rainfall intensity at (approximately) synoptic and sub-synoptic scales. The exterior process is non-scaling and differs in the three models, whereas the interior process is stationary multifractal in all cases. The fact that rainfall is not a simple multifractal process should increase the complexity of its representation and parameter estimation. However, focus on what is important for rainfall extremes keeps the models simple and appropriate for engineering application. We discuss the estimation of model parameters and the calculation of return-period values, and present an application to a 24-year rainfall record from Florence, Italy. All models produce IDF curves that are consistent with the empirical ones and provide similar extrapolations for return periods longer than the duration of the historical record. Sensitivity to using a fraction of the historical record (down to 5 years) is relatively small.
Additional material:
D. Veneziano, and A. Langousis, The Maximum of multifractal cascades: exact distribution and approximations, European Geosciences Union General Assembly 2005, Geophysical Research Abstracts, Vol. 7, Vienna, European Geosciences Union, 2005.
In many applications, one is interested in the maximum M of a multifractal measure at a given resolution r. Previous studies have focused on the scaling of quantiles of M with the exceedance probability and the resolution, but this falls short of characterizing the distribution FM. While calculation of FM for continuous multifractal models is difficult, in the case of discrete cascades such distribution can be obtained through an iterative numerical procedure. The distribution depends on the cascade generator Y, the integer multiplicity of the cascade m, and the resolution r = m^n at which the measure is considered. We evaluate FM for lognormal and beta-lognormal cascades and study its sensitivity to simplifying approximations, including the assumption of independence of the measure in different cascade tiles and the replacement of the dressing factor by a random variable of the same type as the generator Y. We also examine how FM varies with the multiplicity m and the Euclidean dimension of the support d. Concerning dependence among the cascade tiles, we find that: (i) at high resolutions r, dependence affects significantly the body and lower tail of FM but not the extreme upper tail; (ii) for low r, the entire distribution FM is unaffected by dependence; (iii) long-range dependence is more important for FM than short-range dependence; and (iv) for log-stable cascades with fixed index of stability α, the effect of dependence on FM varies in a simple analytic way with the co-dimension parameter C1. We also find that approximating the dressing factor with a variable having the same distribution type as Y induces little error on the distribution of M, except in the extreme upper tail region. The effect of m on FM is modest. Also the effect of the Euclidean space dimension d is small, if what is kept fixed is the volumetric (not the linear) resolution. We use these findings to propose a simple approximation to the distribution of M that includes the effect of dependence, and give charts and explicit formulas to implement the approximation for beta-lognormal cascades.
Additional material:
D. Veneziano, and A. Langousis, The rainfall areal reduction factor: A multifractal analysis, European Geosciences Union General Assembly 2004, Geophysical Research Abstracts, Vol. 6, Nice, European Geosciences Union, 2004.
The areal reduction factor (ARF) η is a key parameter in the design for hydrologic extremes. For a basin of area A, η(A, D, T) is the ratio between the area-average rainfall intensity over a duration D with return period T and the point rainfall intensity for the same D and T. Besides depending on A, D and possibly T, the ARF is affected by the shape of the basin and by a number of seasonal, climatic and topographic characteristics. Commonly used formulas and charts for η have been derived by smoothing or curve-fitting empirical area reduction factors extracted from raingage network records. We derive results on η(A, D, T) based on modeling intense space-time rainfall as a conserved multifractal field. A key parameter is the ratio r = vhyd/vatm between the hydrologic velocity vhyd= (√A)/D and the atmospheric velocity vatm= (√Arain)/Drain, where √Arain and Drain are the characteristic linear dimension and duration of rainfall “structures”. While η depends on A, D and T in a complicated way, some simple asymptotic results hold: For v small (say v < 0.2), η is close to 1 and for v large (say v > 5) η ~ A^(-0.5) for given A, D and T ->oo, whereas η ~ A^(-0.5γ1) for given D, T and A -> 0. The constant γ1 in the exponent of the latter expression is the order of singularity for which the co-dimension function c(γ) of the multifractal rainfall process equals 1; hence γ1 < 1. These analytical scaling results are in generally good agreement with empirical findings. Discrepancies for small A and D are due to the finite range of scaling behavior of physical rainfall and bias in the empirically estimated extreme area rainfall due to sparseness of the raingage network. These effects are evaluated through numerical simulation and corrections to empirical ARF formulas are suggested.
Additional material:
A. Langousis, and D. Koutsoyiannis, A stochastic methodology for generation of seasonal time series reproducing overyear scaling, Hydrofractals '03, An international conference on fractals in hydrosciences, Monte Verita, Ascona, Switzerland, doi:10.13140/RG.2.2.15242.88006, ETH Zurich, MIT, Université Pierre et Marie Curie, 2003.
In generating synthetic time series of hydrologic processes at sub-annual scale it is important to preserve seasonal characteristics and short-term persistence. At the same time, it is equally important to preserve annual characteristics and over year scaling behaviour. This scaling behaviour, which is equivalent to the Hurst phenomenon, has been detected in a large number of hydroclimatic series and affects seriously planning and design of hydrosystems. However, when seasonal models are used, the preservation of annual characteristics and overyear scaling is a difficult task and is often ignored. Disaggregation techniques are the only way to produce synthetic series that are consistent with historical series in several time scales, from seasonal to multiyear, simultaneously. Such techniques involve two or more steps, where in the first step annual series are generated, which are subsequently disaggregated to finer scales. However, disaggregation involves several difficulties (e.g. in parameter estimation), inaccuracies and is a slow procedure. As an alternative, a new methodology is proposed that directly operates on seasonal time scale, avoiding disaggregation, and simultaneously preserves annual statistics and the scaling properties on overyear time scales thus respecting the Hurst phenomenon.
Related works:
Full text:
A. Langousis, Stochastic modeling and estimation of extreme rainfalls, Department of Civil Engineering, Univ. of Patras, June 2010.
Additional material:
A. Langousis, Simple methods for extreme rainfall estimation from multifractal theory, Integrated planning of flood protection: A challenge for the future, Athens, Association of Civil Engineers of Greece, 2010.
Additional material:
A. Langousis, Assessing rainfall risk from tropical cyclones, Dep. of Env. Eng., Technical Univ. of Crete, January 2010.
We develop a methodology for the frequency of extreme rainfall intensities caused by tropical cyclones (TCs) in coastal areas. The mean rainfall field associated with a TC with maximum tangential wind speed Vmax, radius of maximum winds Rmax, and translation speed Vt is obtained using a physically-based model, whereas rainfall variability at both large scales (from storm to storm) and small scales (due to rainbands and local convection) is modeled statistically. The statistical component is estimated using precipitation radar (PR) data from the TRMM mission. Taylor’s hypothesis is used to convert spatial rainfall intensity fluctuations to temporal fluctuations at a given location A. The combined physical-statistical model gives the distribution of the maximum rainfall intensity at A during a period of duration D for a TC with characteristics (Vmax, Rmax, Vt) that passes at a given distance from A. To illustrate the use of the model for long-term rainfall risk analysis, we formulate a recurrence model for tropical cyclones in the Gulf of Mexico that make landfall between longitudes 85o- 95oW. We then use the rainfall and recurrence models to assess the rainfall risk for New Orleans. For return periods of 100 years or more and long averaging durations (D around 12-24 hours), tropical cyclones dominate over other rainfall event types, whereas the reverse is true for shorter return periods or shorter averaging durations.
Additional material:
A. Langousis, Extreme rainfall intensities and long-term rainfall risk from tropical cyclones, Dep. of Civil and Env. Eng, Nicosia, Cyprus, November 2009.
We develop a methodology for the frequency of extreme rainfall intensities caused by tropical cyclones (TCs) in coastal areas. The mean rainfall field associated with a TC with maximum tangential wind speed Vmax, radius of maximum winds Rmax, and translation speed Vt is obtained using a physically-based model, whereas rainfall variability at both large scales (from storm to storm) and small scales (due to rainbands and local convection) is modeled statistically. The statistical component is estimated using precipitation radar (PR) data from the TRMM mission. Taylor’s hypothesis is used to convert spatial rainfall intensity fluctuations to temporal fluctuations at a given location A. The combined physical-statistical model gives the distribution of the maximum rainfall intensity at A during a period of duration D for a TC with characteristics (Vmax, Rmax, Vt) that passes at a given distance from A. To illustrate the use of the model for long-term rainfall risk analysis, we formulate a recurrence model for tropical cyclones in the Gulf of Mexico that make landfall between longitudes 85o- 95oW. We then use the rainfall and recurrence models to assess the rainfall risk for New Orleans. For return periods of 100 years or more and long averaging durations (D around 12-24 hours), tropical cyclones dominate over other rainfall event types, whereas the reverse is true for shorter return periods or shorter averaging durations.
Additional material:
A. Langousis, Assessing rainfall risk from tropical cyclones, Department of Civil Engineering, Univ. of Patras, June 2009.
We develop a methodology for the frequency of extreme rainfall intensities caused by tropical cyclones (TCs) in coastal areas. The mean rainfall field associated with a TC with maximum tangential wind speed Vmax, radius of maximum winds Rmax, and translation speed Vt is obtained using a physically-based model, whereas rainfall variability at both large scales (from storm to storm) and small scales (due to rainbands and local convection) is modeled statistically. The statistical component is estimated using precipitation radar (PR) data from the TRMM mission. Taylor’s hypothesis is used to convert spatial rainfall intensity fluctuations to temporal fluctuations at a given location A. The combined physical-statistical model gives the distribution of the maximum rainfall intensity at A during a period of duration D for a TC with characteristics (Vmax, Rmax, Vt) that passes at a given distance from A. To illustrate the use of the model for long-term rainfall risk analysis, we formulate a recurrence model for tropical cyclones in the Gulf of Mexico that make landfall between longitudes 85o- 95oW. We then use the rainfall and recurrence models to assess the rainfall risk for New Orleans. For return periods of 100 years or more and long averaging durations (D around 12-24 hours), tropical cyclones dominate over other rainfall event types, whereas the reverse is true for shorter return periods or shorter averaging durations.
Additional material:
A. Langousis, Extreme rainfall intensities and long-term rainfall risk from tropical cyclones, Risk Management Solutions, London, UK, June 2009.
We develop a methodology for the frequency of extreme rainfall intensities caused by tropical cyclones (TCs) in coastal areas. The mean rainfall field associated with a TC with maximum tangential wind speed Vmax, radius of maximum winds Rmax, and translation speed Vt is obtained using a physically-based model, whereas rainfall variability at both large scales (from storm to storm) and small scales (due to rainbands and local convection) is modeled statistically. The statistical component is estimated using precipitation radar (PR) data from the TRMM mission. Taylor’s hypothesis is used to convert spatial rainfall intensity fluctuations to temporal fluctuations at a given location A. The combined physical-statistical model gives the distribution of the maximum rainfall intensity at A during a period of duration D for a TC with characteristics (Vmax, Rmax, Vt) that passes at a given distance from A. To illustrate the use of the model for long-term rainfall risk analysis, we formulate a recurrence model for tropical cyclones in the Gulf of Mexico that make landfall between longitudes 85o- 95oW. We then use the rainfall and recurrence models to assess the rainfall risk for New Orleans. For return periods of 100 years or more and long averaging durations (D around 12-24 hours), tropical cyclones dominate over other rainfall event types, whereas the reverse is true for shorter return periods or shorter averaging durations.
Additional material:
A. Langousis, Assessing rainfall risk from tropical cyclones, Travelers Insurance, Connecticut, USA, April 2009.
We develop a methodology for the frequency of extreme rainfall intensities caused by tropical cyclones (TCs) in coastal areas. The mean rainfall field associated with a TC with maximum tangential wind speed Vmax, radius of maximum winds Rmax, and translation speed Vt is obtained using a physically-based model, whereas rainfall variability at both large scales (from storm to storm) and small scales (due to rainbands and local convection) is modeled statistically. The statistical component is estimated using precipitation radar (PR) data from the TRMM mission. Taylor’s hypothesis is used to convert spatial rainfall intensity fluctuations to temporal fluctuations at a given location A. The combined physical-statistical model gives the distribution of the maximum rainfall intensity at A during a period of duration D for a TC with characteristics (Vmax, Rmax, Vt) that passes at a given distance from A. To illustrate the use of the model for long-term rainfall risk analysis, we formulate a recurrence model for tropical cyclones in the Gulf of Mexico that make landfall between longitudes 85o- 95oW. We then use the rainfall and recurrence models to assess the rainfall risk for New Orleans. For return periods of 100 years or more and long averaging durations (D around 12-24 hours), tropical cyclones dominate over other rainfall event types, whereas the reverse is true for shorter return periods or shorter averaging durations.
Additional material:
A. Langousis, Assessing rainfall risk from tropical cyclones, AIR Worldwide, Boston, MA, USA, May 2009.
We develop a methodology for the frequency of extreme rainfall intensities caused by tropical cyclones (TCs) in coastal areas. The mean rainfall field associated with a TC with maximum tangential wind speed Vmax, radius of maximum winds Rmax, and translation speed Vt is obtained using a physically-based model, whereas rainfall variability at both large scales (from storm to storm) and small scales (due to rainbands and local convection) is modeled statistically. The statistical component is estimated using precipitation radar (PR) data from the TRMM mission. Taylor’s hypothesis is used to convert spatial rainfall intensity fluctuations to temporal fluctuations at a given location A. The combined physical-statistical model gives the distribution of the maximum rainfall intensity at A during a period of duration D for a TC with characteristics (Vmax, Rmax, Vt) that passes at a given distance from A. To illustrate the use of the model for long-term rainfall risk analysis, we formulate a recurrence model for tropical cyclones in the Gulf of Mexico that make landfall between longitudes 85o- 95oW. We then use the rainfall and recurrence models to assess the rainfall risk for New Orleans. For return periods of 100 years or more and long averaging durations (D around 12-24 hours), tropical cyclones dominate over other rainfall event types, whereas the reverse is true for shorter return periods or shorter averaging durations.
Additional material:
A. Langousis, Hazards from the last economic crisis, AXIA, 9 May 2009.
A. Langousis, Development and faulty choices, ΤΑ ΝΕΑ, 15 July 2008.
A. Langousis, Society and environment under development, ΤΑ ΝΕΑ, 31 July 2007.
A. Langousis, Extreme Rainfall Intensities and Long-term Rainfall Risk from Tropical Cyclones, PhD thesis, 107 pages, MIT, Boston, USA, 2008.
We develop a methodology for the frequency of extreme rainfall intensities caused by tropical cyclones (TCs) in coastal areas. The mean rainfall field associated with a TC with maximum tangential wind speed Vmax, radius of maximum winds Rmax, and translation speed Vt is obtained using a physically-based model, whereas rainfall variability at both large scales (from storm to storm) and small scales (due to rainbands and local convection) is modeled statistically. The statistical component is estimated using precipitation radar (PR) data from the TRMM mission. Taylor’s hypothesis is used to convert spatial rainfall intensity fluctuations to temporal fluctuations at a given location A. The combined physical-statistical model gives the distribution of the maximum rainfall intensity at A during a period of duration D for a TC with characteristics (Vmax, Rmax, Vt) that passes at a given distance from A. To illustrate the use of the model for long-term rainfall risk analysis, we formulate a recurrence model for tropical cyclones in the Gulf of Mexico that make landfall between longitudes 85o- 95oW. We then use the rainfall and recurrence models to assess the rainfall risk for New Orleans. For return periods of 100 years or more and long averaging durations (D around 12-24 hours), tropical cyclones dominate over other rainfall event types, whereas the reverse is true for shorter return periods or shorter averaging durations.
A. Langousis, The Areal Reduction Factor - A Multifractal Analysis, MSc thesis, 117 pages, MIT, Boston, 2004.
The Areal Reduction Factor (ARF) η is a key parameter in the design for hydrologic extremes. For a basin of area A, η(A, D, T) is the ratio between the area-average rainfall intensity over a duration D with return period T and the point rainfall intensity for the same D and T. Besides depending on A, D and possibly T, the ARF is affected by the shape of the basin and by a number of seasonal, climatic and topographic characteristics. Another factor on which ARF depends is the advection velocity, vad, of the rainfall features. Commonly used formulas and charts for the ARF have been derived by smoothing or curve-fitting empirical ARFs extracted from raingauge network records. Here we derive some properties of the ARF under the assumption that space-time rainfall is exactly or approximately multifractal. We do so for various shapes of the rainfall collecting region and for vad = 0 and vad =/ 0. When vad = 0, a key parameter in the analysis is the ratio ures = vres/ve between the "response velocity" vres = L/D, where L is the maximum linear dimension of the region, and the "evolution velocity" ve = Le/De, where Le and De are the characteristic linear dimension and characteristic duration of organized rainfall features. The effect of vad =/ 0 depends on the shape of the region. For highly elongated basins, both the direction and magnitude of advection are influential, whereas for regular shaped regions only the magnitude vad matters. We review ways in which rainfall has been observed to deviate from exact multifractality and models that capture such deviations. We show how the ARF behaves when rainfall is a bounded cascade in space and time. We also investigate the effect of estimating areal rainfall from raingauge network measurements. We find that bounded-cascade deviations from multifractality and sparse spatial sampling distort in similar ways the scaling properties of the ARF. Finally we show how one can reproduce various features of empirical ARF charts by using multifractal and bounded cascade models and considering the effects of sparse spatial sampling.
A. Langousis, Developement of cyclostationary stochastic hydrological models preserving short-term memory and long-term persistence, Diploma thesis, 327 pages, Department of Water Resources, Hydraulic and Maritime Engineering – National Technical University of Athens, July 2003.
As an alternative, a new methodology is proposed that directly operates on seasonal time scale, avoiding disaggregation, and simultaneously preserves annual statistics and the scaling properties on overyear time scales thus respecting the Hurst phenomenon.
Full text: