Impact of wind on the spatial distribution of rain over micro‐scale topography: numerical modelling and experimental verification

The wind‐driven‐rain effect refers to the redistribution of rainfall over micro‐scale topography due to the existence of local perturbed wind‐flow patterns. Rainfall measurements reported in the literature point to the fact that the wind‐driven‐rain distribution can show large variations over micro‐scale topography. These variations should be taken into account in hillslope hydrology, in runoff and erosion studies and in the design of rainfall monitoring networks. In practice, measurements are often not suitable for determining the wind‐driven‐rain distribution. Therefore, a few researchers have employed numerical modelling. In order to provide confidence in using numerical models, experimental verification for a range of different topographic features is imperative. The objective of this study is to investigate the adequacy of a two‐dimensional Computational Fluid Dynamics (CFD) model to predict the wind‐driven‐rain distribution over small‐scale topography. The numerical model is applied to a number of topographic features, including a succession of cliffs, a small isolated hill, a small valley and a field with ridges and furrows. The numerical results are compared with the corresponding measurement results reported in the literature. It is shown that two‐dimensional numerical modelling can provide a good indication of the wind‐driven‐rain distribution over each type of micro‐scale topography that is considered in this study. It is concluded that more detailed verification procedures are currently inhibited due to the lack of available and detailed spatial and temporal rainfall data from field measurements. Copyright © 2005 John Wiley & Sons, Ltd.     

Incorporating rainfall intensity into daily rainfall records for simulating runoff and infiltration into soil profiles

Hydrological modelling often implies the use of rainfall data. Its quality and resolution directly affect the accuracy of the simulation results. This study illustrates that a simple approach of incorporating rainfall intensity information in daily rainfall records significantly improves the simulation of surface runoff and rainfall infiltration into soil profiles. The procedure is developed using a frequency analysis on rainfall data of the Royal Meteorological Institute of Belgium, collected with a resolution of 10 min and for a consecutive period of 61 years. The frequency analysis of the data allowed the incorporation of rainfall intensity information into daily rainfall records. To test the effect of this approach the surface runoff and water flow into three different soil types was simulated using the HYDRUS‐1D model for a typical dry, normal and wet year. The simulation results whereby the observed 10‐min rainfall data was used as input were considered as the reference. Comparative analysis revealed that the simulations using the 10 min rainfall data deducted from the incorporation of rainfall intensity into daily rainfall records, deviate a maximum 1·2% from the reference and produce much better results than the Soil Conservation Service (SCS) runoff curve‐number method because rainfall intensity is considered in the procedure presented. The SCS curve‐number method typical overestimates surface runoff during periods of low rainfall intensity (winter) and underestimate runoff during periods of high rainfall intensities (summer). Copyright © 2002 John Wiley & Sons, Ltd.     

Flood simulation using the gauge‐adjusted radar rainfall and physics‐based distributed hydrologic model

This paper reports the results of an investigation into flood simulation by areal rainfall estimated from the combination of gauged and radar rainfalls and a rainfall–runoff model on the Anseong‐cheon basin in the southern part of Korea. The spatial and temporal characteristics and behaviour of rainfall are analysed using various approaches combining radar and rain gauges: (1) using kriging of the rain gauge alone; (2) using radar data alone; (3) using mean field bias (MFB) of both radar and rain gauges; and (4) using conditional merging technique (CM) of both radar and rain gauges. To evaluate these methods, statistics and hyetograph for rain gauges and radar rainfalls were compared using hourly radar rainfall data from the Imjin‐river, Gangwha, rainfall radar site, Korea. Then, in order to evaluate the performance of flood estimates using different rainfall estimation methods, rainfall–runoff simulation was conducted using the physics‐based distributed hydrologic model, Vflo^?. The flood runoff hydrograph was used to compare the calculated hydrographs with the observed one. Results show that the rainfall field estimated by CM methods improved flood estimates, because it optimally combines rainfall fields representing actual spatial and temporal characteristics of rainfall. Copyright © 2008 John Wiley & Sons, Ltd.     

Geostatistical interpolation of space–time rainfall on Tamshui River basin,Taiwan

Taiwan suffers from heavy storm rainfall during the typhoon season. This usually causes large river runoff, overland flow, erosion, landslides, debris flows, loss of power, etc. In order to evaluate storm impacts on the downstream basin, a real‐time hydrological modelling is used to estimate potential hazard areas. This can be used as a decision‐support system for the Emergency Response Center, National Fire Agency Ministry, to make ‘real‐time’ responses and minimize possible damage to human life and property. This study used 34 observed events from 14 telemetered rain‐gauges in the Tamshui River basin, Taiwan, to study the spatial–temporal characteristics of typhoon rainfall. In the study, regionalized theory and cross‐semi‐variograms were used to identify the spatial‐temporal structure of typhoon rainfall. The power form and parameters of the cross‐semi‐variogram were derived through analysis of the observed data. In the end, cross‐validation was used to evaluate the performance of the interpolated rainfall on the river basin. The results show the derived rainfall interpolator represents the observed events well, which indicates the rainfall interpolator can be used as a spatial‐temporal rainfall input for real‐time hydrological modelling. Copyright © 2007 John Wiley & Sons, Ltd.     

Analytical model for a geomorphological instantaneous unit hydrograph

The rainfall‐runoff modelling of a river basin can be divided into two processes: the production function and the transfer function. The production function determines the proportion of gross rainfall actually involved in the runoff. The transfer function spreads the net rainfall over time and space in the river basin. Such a transfer function can be modelled using the approach of the geomorphological instantaneous unit hydrograph (GIUH). The effectiveness of geomorphological models is actually revealed in rainfall‐runoff modelling, where hydrologic data are desperately lacking, just as in ungauged basins. These models make it possible to forecast the hydrograph shape and runoff variation versus time at the basin outlet. This article is an introduction to a new GIUH model that proves to be simple and analytical. Its geomorphological parameters are easily available on a map or from a digital elevation model. This model is based on general hypotheses on symmetry that provide it with multiscale versatile characteristics. After having validated the model in river basins of very different nature and size, we present an application of this model for rainfall‐runoff modelling. Since parameters are determined relying on real geomorphological data, no calibration is necessary, and it is then possible to carry out rainfall‐runoff simulations in ungauged river basins. Copyright © 2006 John Wiley & Sons, Ltd.     

Rainfall‐runoff modelling using artificial neural networks technique: a Blue Nile catchment case study

A rainfall‐runoff model based on an artificial neural network (ANN) is presented for the Blue Nile catchment. The best geometry of the ANN rainfall‐runoff model in terms of number of hidden layers and nodes is identified through a sensitivity analysis. The Blue Nile catchment (about 300 000 km²) in the Nile basin is selected here as a case study. The catchment is classified into seven subcatchments, and the mean areal precipitation over those subcatchments is computed as a main input to the ANN model. The available daily data (1992–99) are divided into two sets for model calibration (1992–96) and for validation (1997–99). The results of the ANN model are compared with one of physical distributed rainfall‐runoff models that apply hydraulic and hydrologic fundamental equations in a grid base. The results over the case study area and the comparative analysis with the physically based distributed model show that the ANN technique has great potential in simulating the rainfall‐runoff process adequately. Because the available record used in the calibration of the ANN model is too short, the ANN model is biased compared with the distributed model, especially for high flows. Copyright © 2005 John Wiley & Sons, Ltd.     

A distributed monthly hydrological model for integrating spatial variations of basin topography and rainfall

Hydrological models at a monthly time‐scale are important tools for hydrological analysis, such as in impact assessment of climate change and regional water resources planning. Traditionally, monthly models adopt a conceptual, lumped‐parameter approach and cannot account for spatial variations of basin characteristics and climatic inputs. A large requirement for data often severely limits the utility of physically based, distributed‐parameter models. Based on the variable‐source‐area concept, we considered basin topography and rainfall to be two major factors whose spatial variations play a dominant role in runoff generation and developed a monthly model that is able to account for their influences in the spatial and temporal dynamics of water balance. As a hybrid of the Xinanjiang model and TOPMODEL, the new model is constructed by innovatively making use of the highly acclaimed simulation techniques in the two existing models. A major contribution of this model development study is to adopt the technique of implicit representation of soil moisture characteristics in the Xinanjiang model and use the TOPMODEL concept to integrate terrain variations into runoff simulation. Specifically, the TOPMODEL topographic index ln(a/tanβ) is converted into an index of relative difficulty in runoff generation (IRDG) and then the cumulative frequency distribution of IRDG is used to substitute the parabolic curve, which represents the spatial variation of soil storage capacity in the Xinanjiang model. Digital elevation model data play a key role in the modelling procedures on a geographical information system platform, including basin segmentation, estimation of rainfall for each sub‐basin and computation of terrain characteristics. Other monthly data for model calibration and validation are rainfall, pan evaporation and runoff. The new model has only three parameters to be estimated, i.e. watershed‐average field capacity WM, pan coefficient η and runoff generation coefficient α. Sensitivity analysis demonstrates that runoff is least sensitive to WM and, therefore, it can be determined by a prior estimation based on the climate and soil properties of the study basin. The other two parameters can be determined using optimization methods. Model testing was carried out in a number of nested sub‐basins of two watersheds (Yuanjiang River and Dongjiang River) in the humid region in central and southern China. Simulation results show that the model is capable of describing spatial and temporal variations of water balance components, including soil moisture content, evapotranspiration and runoff, over the watershed. With a minimal requirement for input data and parameterization, this terrain‐based distributed model is a valuable contribution to the ever‐advancing technology of hydrological modelling. Copyright © 2006 John Wiley & Sons, Ltd.     

Assessment of rainfall spatial variability and its influence on runoff modelling: A case study in the Brue catchment,UK

This study explores rainfall spatial variability and its influence on runoff modelling. A novel assessment scheme integrated with coefficient of variance and Moran's I is introduced to describe effective rainfall spatial variability. Coefficient of variance is widely accepted to identify rainfall variability through rainfall intensity, whereas Moran's I reflects rainfall spatial autocorrelation. This new assessment framework combines these two indicators to assess the spatial variability derived from both rainfall intensity and distribution, which are crucial in determining the time and magnitude of runoff generation. Four model structures embedded in the Variable Infiltration Capacity model are adopted for hydrological modelling in the Brue catchment of England. The models are assigned with 1, 3, 8, and 27 hydrological response units, respectively, and diverse rainfall spatial information for 236 events are extracted from 1995. This study investigates the model performance of different partitioning based on rainfall spatial variability through peak volume (Q_p) and time to peak (T_p), along with the rainfall event process. The results show that models associated with dense spatial partitioning are broadly capable of capturing more spatial information with better performance. It is unnecessary to utilize models with high spatial density for simple rainfall events, though they show distinct advantages on complex events. With additional spatial information, Q_p experiences a notable improvement over T_p. Moreover, seasonal patterns signified by the assessment scheme imply the feasibility of seasonal models.     

A new magnitude category disaggregation approach for temporal high‐resolution rainfall intensities

Simulation of quick runoff components such as surface runoff and associated soil erosion requires temporal high‐resolution rainfall intensities. However, these data are often not available because such measurements are costly and time consuming. Current rainfall disaggregation methods have shortcomings, especially in generating the distribution of storm events. The objectives of this study were to improve point rainfall disaggregation using a new magnitude category rainfall disaggregation approach. The procedure is introduced using a coupled disaggregation approach (Hyetos and cascade) for multisite rainfall disaggregation. The new procedure was tested with ten long‐term precipitation data sets of central Germany using summer and winter precipitation to determine seasonal variability. Results showed that dividing the rainfall amount into four daily rainfall magnitude categories (1–10, 11–25, 26–50, >50 mm) improves the simulation of high rainfall intensity (convective rainfall). The Hyetos model category approach (Hyetos_Cat) with seasonal variation performs representative to observed hourly rainfall compared with without categories on each month. The mean absolute percentage accuracy of standard deviation for hourly rainfall is 89.7% in winter and 95.6% in summer. The proposed magnitude category method applied with the coupled Hyetos_Cat–cascade approach reproduces successfully the statistical behaviour of local 10‐min rainfall intensities in terms of intermittency as well as variability. The root mean square error performance statistics for disaggregated 10‐min rainfall depth ranges from 0.20 to 2.38 mm for summer and from 0.12 to 2.82 mm for the winter season in all categories. The coupled stochastic approach preserves the statistical self‐similarity and intermittency at each magnitude category with a relatively low computational burden. Copyright © 2014 John Wiley & Sons, Ltd.     

Using data-derived perturbations to incorporate uncertainty in generating stochastic areal rainfall from point rainfall

Abstract

Raingauge measurements are commonly used to estimate daily areal rainfall for catchment modelling. The variation of rainfall between the gauges is usually inadequately captured and areal rainfall estimates are therefore very uncertain. A method of quantifying these uncertainties and incorporating them into ensembles of areal rainfall is demonstrated and tested. The uncertainties are imposed as perturbations based on the differences in areal rainfall that result when half of the raingauges are alternately omitted. Also included is a method of: (a) estimating the proportion rainfall that falls on areas where no gauges are located that are consequently computed as having zero rain, and (b) replacing them with plausible non-zero rainfalls. The model is tested using daily rainfall from two South African catchments and is found to exhibit the expected behaviour. One of the two parameters of the model is obtained from the rainfall data, while the other has direct physical interpretation.

Editor D. Koutsoyiannis; Associate editor C. Onof

Citation Ndiritu, J., 2013. Using data-derived perturbations to incorporate uncertainty in generating stochastic areal rainfall from point rainfall. Hydrological Sciences Journal, 58 (8), 1704–1717.     

Flood hydrograph attenuation induced by a reservoir system: analysis with a distributed rainfall‐runoff model

Spatially distributed hydrologic models can be effectively utilized for flood event simulation over basins where a complex system of reservoirs affecting the natural flow regime is present. Flood peak attenuation through mountain reservoirs can, in fact, mitigate the impact of major floods in flood‐prone areas of the lower river valley. Assessment of this effect for a complex reservoir system is performed with a spatially distributed hydrologic model where the surface runoff formation and the hydraulic routing through each reservoir and the river system are performed at a fine spatial and time resolution. The Toce River basin is presented as a case study, because of the presence of 14 active hydroelectric dams that affect the natural flow regime. A recent extreme flood event is simulated using a multi‐realization kriging method for modelling the spatial distribution of rainfall. A sensitivity analysis of the key elements of the distributed hydrologic model is also performed. The flood hydrograph attenuation is assessed. Several possible reservoir storage conditions are used to characterize the initial condition of each reservoir. The results demonstrate how a distributed hydrologic model can contribute to defining strategies for reservoir management in flood mitigation. Copyright © 2004 John Wiley & Sons, Ltd.     

Snow cover and runoff modelling in a high mountain catchment with scarce data: effects of temperature and precipitation parameters

Snowmelt is an important source of runoff in high mountain catchments. Snowmelt modelling for alpine regions remains challenging with scarce gauges. This study simulates the snowmelt in the Karuxung River catchment in the south Tibetan Plateau using an altitude zone based temperature‐index model, calibrates the snow cover area and runoff simulation during 2003–2005 and validates the model performance via snow cover area and runoff simulation in 2006. In the snowmelt and runoff modelling, temperature and precipitation are the two most important inputs. Relevant parameters, such as critical snow fall temperature, temperature lapse rate and precipitation gradient, determine the form and amount of precipitation and distribution of temperature and precipitation in hydrological modelling of the sparsely gauged catchment. Sensitivity analyses show that accurate estimation of these parameters would greatly help in improving the snowmelt simulation accuracy, better describing the snow‐hydrological behaviours and dealing with the data scarcity at higher elevations. Specifically, correlation between the critical snow fall temperature and relative humidity and seasonal patterns of both the temperature lapse rate and the precipitation gradient should be considered in the modelling studies when precipitation form is not logged and meteorological observations are only available at low elevation. More accurate simulation of runoff involving snowmelt, glacier melt and rainfall runoff will improve our understanding of hydrological processes and help assess runoff impacts from a changing climate in high mountain catchments. Copyright © 2013 John Wiley & Sons, Ltd.     

Including spatial distribution in a data‐driven rainfall‐runoff model to improve reservoir inflow forecasting in Taiwan

Multi‐step ahead inflow forecasting has a critical role to play in reservoir operation and management in Taiwan during typhoons as statutory legislation requires a minimum of 3‐h warning to be issued before any reservoir releases are made. However, the complex spatial and temporal heterogeneity of typhoon rainfall, coupled with a remote and mountainous physiographic context, makes the development of real‐time rainfall‐runoff models that can accurately predict reservoir inflow several hours ahead of time challenging. Consequently, there is an urgent, operational requirement for models that can enhance reservoir inflow prediction at forecast horizons of more than 3 h. In this paper, we develop a novel semi‐distributed, data‐driven, rainfall‐runoff model for the Shihmen catchment, north Taiwan. A suite of Adaptive Network‐based Fuzzy Inference System solutions is created using various combinations of autoregressive, spatially lumped radar and point‐based rain gauge predictors. Different levels of spatially aggregated radar‐derived rainfall data are used to generate 4, 8 and 12 sub‐catchment input drivers. In general, the semi‐distributed radar rainfall models outperform their less complex counterparts in predictions of reservoir inflow at lead times greater than 3 h. Performance is found to be optimal when spatial aggregation is restricted to four sub‐catchments, with up to 30% improvements in the performance over lumped and point‐based models being evident at 5‐h lead times. The potential benefits of applying semi‐distributed, data‐driven models in reservoir inflow modelling specifically, and hydrological modelling more generally, are thus demonstrated. Copyright © 2012 John Wiley & Sons, Ltd.     

Variation in curve numbers derived from plot runoff data for New South Wales (Australia)

The curve number method is a simple one parameter (the curve number) rainfall runoff model. While its theoretical underpinning has been questioned it remains a powerful hydrological tool in the absence of detailed data and is therefore used extensively in hydrological models. This study aims to characterize the variation in maximum retention values (S), which underlie curve numbers, for a range of agricultural treatments across a large spatial area in New South Wales (NSW), Australia. The data used for the analysis spans several decades of rainfall runoff observations. A range of different derivation methods result in variation in mean and variance of S. In particular, methods that emphasize the larger storms result in greater S and thus lower runoff. For larger spatial scales, emphasis on larger storms gives more reliable estimates of S. Systematic variation in S arises from variations in treatment, pre‐runoff soil moisture, rainfall depth, and variations in cover. On the basis of the analysis, a table of curve number values for different land uses found in NSW is presented. The resulting distributions of S and curve numbers provide guidance for rainfall runoff modelling studies in the agricultural important areas of NSW. Copyright © 2011 John Wiley & Sons, Ltd.     

Rainfall‐runoff analysis based on competing linear impulse responses: decomposition of rainfall‐runoff processes

Many recent studies have successfully used neural networks for non‐linear rainfall‐runoff modelling. Due to fundamental limitation of linear structures, approaches employing linear models have been generally considered inferior to the neural network approaches in this area. However, the authors believe that with an appropriate extension, the concept of linear impulse responses can be a viable tool since it enables one to understand underlying dynamics of rainfall‐runoff processes. In this paper, the use of competing impulse responses for rainfall‐runoff analysis is proposed. The proposed method is based on the switch over of competing linear impulse‐responses, each of which satisfies the constraints of non‐negativity and uni‐modality. The computational analyses performed for the rainfall‐runoff data in the Seolma‐Chun experimental basin, Korea showed that the proposed method can yield promising results. Considering the basin characteristics as well as the results from this study, it may be concluded that three impulse responses are enough for rainfall‐runoff analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

A Bayesian method for missing rainfall estimation using a conceptual rainfall–runoff model

The estimation of missing rainfall data is an important problem for data analysis and modelling studies in hydrology. This paper develops a Bayesian method to address missing rainfall estimation from runoff measurements based on a pre-calibrated conceptual rainfall–runoff model. The Bayesian method assigns posterior probability of rainfall estimates proportional to the likelihood function of measured runoff flows and prior rainfall information, which is presented by uniform distributions in the absence of rainfall data. The likelihood function of measured runoff can be determined via the test of different residual error models in the calibration phase. The application of this method to a French urban catchment indicates that the proposed Bayesian method is able to assess missing rainfall and its uncertainty based only on runoff measurements, which provides an alternative to the reverse model for missing rainfall estimates.     

Understanding plot‐scale hydrology of Lesser Himalayan watershed—A field study and HYDRUS‐2D modelling approach

Soil moisture dynamics have a significant effect on overland flow generation. Catchment aspect is one of the major controlling factors of overland flow and soil moisture behaviour. A few experimental studies have been carried out in the uneven topography of the Himalayas. This study presents plot‐scale experiments using portable rainfall simulator at an altitude of 1,230 m above mean sea level and modelling of overland flow using observed datasets. Two plots were selected in 2 different aspects of Aglar watershed of Lesser Himalaya; the agro‐forested (AF) plot was positioned at the north aspect whereas the degraded (DE) plot was located at the south aspect of the hillslope. HS flumes and rain gauges were installed to measure the runoff at the outlet of the plot and the rainfall depth during rainfall simulation experiments. Moreover, 10 soil moisture sensors were installed at upslope and downslope locations of both the plots at 5, 15, 25, 35, and 45 cm depth from ground level to capture the soil moisture dynamics. The tests were conducted at intensities of 79.8 and 75 mm/hr in AF plot and 82.2 and 72 mm/hr in the DE plot during Test 1 and Test 2, respectively. The observed data indicate the presence of reinfiltration process only in the AF plot. The high water holding capacity and the presence of reinfiltration process results in less runoff volume in the AF plot compared with the DE plot. The Hortonian overland flow mechanism was found to be the dominant overland flow mechanism as only a few layers of top soil get saturated during all of the rainfall–runoff experiments. The runoff, rainfall, and soil moisture data were subsequently used to calibrate the parameters of HYDRUS‐2D overland flow module to simulate the runoff hydrograph and soil moisture. The components of hydrograph were evaluated in terms of peak discharge, runoff volume and time of concentration, the results were found to be within the satisfactory range. The goodness of fit of simulated hydrographs were more than 0.85 and 0.95 for AF and DE plot, respectively. The model produced satisfactory simulation results of soil moisture for all of the rainfall–runoff experiments. The HYDRUS‐2D overland flow module was found promising to simulate the runoff hydrograph and soil moisture in plot‐scale research.     

Storm‐event rainfall–runoff modelling approach for ungauged sites in Taiwan

This work develops a top‐down modelling approach for storm‐event rainfall–runoff model calibration at unmeasured sites in Taiwan. Twenty‐six storm events occurring in seven sub‐catchments in the Kao‐Ping River provided the analytical data set. Regional formulas for three important features of a streamflow hydrograph, i.e. time to peak, peak flow, and total runoff volume, were developed via the characteristics of storm event and catchment using multivariate regression analysis. Validation of the regional formulas demonstrates that they reasonably predict the three features of a streamflow hydrograph at ungauged sites. All of the sub‐catchments in the study area were then adopted as ungauged areas, and the three streamflow hydrograph features were calculated by the regional formulas and substituted into the fuzzy multi‐objective function for rainfall–runoff model calibration. Calibration results show that the proposed approach can effectively simulate the streamflow hydrographs at the ungauged sites. The simulated hydrographs more closely resemble observed hydrographs than hydrographs synthesized using the Soil Conservation Service (SCS) dimensionless unit hydrograph method, a conventional method for hydrograph estimation at ungauged sites in Taiwan. Copyright © 2008 John Wiley & Sons, Ltd.     

Extreme value modelling of daily areal rainfall over Mediterranean catchments in a changing climate

Heavy rainfall events during the fall season are causing extended damages in Mediterranean catchments. A peaks‐over‐threshold model is developed for the extreme daily areal rainfall occurrence and magnitude in fall over six catchments in Southern France. The main driver of the heavy rainfall events observed in this region is the humidity flux (FHUM) from the Mediterranean Sea. Reanalysis data are used to compute the daily FHUM during the period 1958–2008, to be included as a covariate in the model parameters. Results indicate that the introduction of FHUM as a covariate can improve the modelling of extreme areal precipitation. The seasonal average of FHUM can improve the modelling of the seasonal occurrences of heavy rainfall events, whereas daily FHUM values can improve the modelling of the events magnitudes. In addition, an ensemble of simulations produced by five different general circulation models are considered to compute FHUM in future climate with the emission scenario A1B and hence to evaluate the effect of climate change on the heavy rainfall distribution in the selected catchments. This ensemble of climate models allows the evaluation of the uncertainties in climate projections. By comparison to the reference period 1960–1990, all models project an amplification of the mean seasonal FHUM from the Mediterranean Sea for the projection period 2070–2099, on average by +22%. This increase in FHUM leads to an increase in the number of heavy rainfall events, from an average of 2.55 events during the fall season in present climate to 3.57 events projected for the period 2070–2099. However, the projected changes have limited effects on the magnitude of extreme events, with only a 5% increase in the median of the 100‐year quantiles. Copyright © 2011 John Wiley & Sons, Ltd.