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Weather generators (WGs) are tools that create synthetic weather data, which are statistically similar to the observed data. Considering the limitations of most rainfall generators to preserve satisfactorily the fundamental statistical properties (e.g., spatial and temporal coherence) simultaneously, we propose a new framework for multisite rainfall generation. The framework consists of three main components: (1) a spatiotemporal rainfall field, described as spatial modes and their corresponding temporal evolution based on empirical orthogonal function analysis (EOFA). (2) The time series of these spatial modes, decomposed into intrinsic mode functions (IMFs) with characteristic frequencies (periods) using Hilbert-Huang transform (HHT). (3) Stochastic simulation (SS), achieved by assigning random phases for the specific IMFs. The current model, EHS ( E OFA + H HT + S S), is compared with two other typical multi-site rainfall generators, MulGETS (parametric model) and KNN (non-parametric model) for a network of 12 stations in Xiang River basin, China. These three models are assessed based on their ability to simulate sequences with statistical attributes that are similar to those observed. We compare the basic statistics (mean, standard deviation, skewness), extreme value characteristics (95th percentile and maximum), spatial dependence (spatial correlation and spatial continuity ratio), and temporal dependence statistic (autocorrelation, wet/dry spells, and low-frequency variability). The results show that EHS rainfall generator has a similar capacity as KNN model in reproducing the spatial structure of the original rainfall field, and has a greater ability than MulGETS and KNN model to preserve the historical temporal statistics, especially the autocorrelation at various time scales and low-frequency variability. Overall, EHS is a useful model for generating realistic multi-site rainfall field and can be expected to generate plausible scenarios for impact studies.

Vulnerability assessment is an essential step for urban flood risk management. Due to the objective existence of statistical errors in the variables and spatial heterogeneity, as well as the similarity of flood experience, precautionary status and environments for neighbouring areas, an urban flood vulnerability assessment essentially entails a multi-factor decision-making process that accounts for uncertainties and the local spatial dependence of neighbourhoods. To address these issues, an uncertainty-based vulnerability assessment approach embedded with an improved moving split-window (IMSW) analysis and probabilistic method is proposed. To develop an integrated urban flood vulnerability assessment model, three technical issues need to be resolved: (1) the development of damage curves of land-use types, (2) an uncertainty model and spatial dependence model of damages based on an IMSW analysis, and (3) an uncertainty-based framework to quantify vulnerability considering the uncertainties and local spatial dependence of the damage curves as well as the spatial heterogeneity of risky assets simultaneously. The proposed method is applied to an urban area in China and provides an integrated vulnerability assessment for different land-use types. Moreover, uncertainty and sensitivity analyses were performed to analyse the spatial differentiation and the dominant parameters of vulnerability. Compared with two other commonly used methods, the proposed local-correlated method (LCM) is more in line with reality, and provides more reasonable information for making a better-informed decision.

Satellite products, like all datasets, are subject to errors and uncertainties. Due to inherent biases embedded in satellite precipitation estimates, we present an error-adjustment approach based on the statistical differences between satellite precipitation products and in-situ observations (observed errors) employing two widely-used error models, namely that additive and the multiplicative error models, in an attempt to assess their suitability for the error correction of satellite-based daily precipitation estimates over northeast Austria. An error-adjustment technique based on the concept of the copula is adopted and applied to correct the supplied precipitation fields. It was found that IMERG precipitation estimates improved after error adjustment when compared to original satellite precipitation estimate (OSPE). The additive error model resulted in a better improvement by fitting the entire range of data when compared with the multiplicative error model. Moreover, the additive error model extracted the error with more accuracy and produced a better estimation of their characteristics, while the method based on the multiplicative error was less robust. However, the overall spatial dependence of the observed errors is reasonably preserved as that of the generated errors by copula. In addition, the validation results implied that the simulated realizations error-adjusted band, encompassed the observed data reasonably. Moreover, the copula-based simulations associated with the additive error model performed much better in comparison to the multiplicative error model. Overall, by using the t-copula model with an emphasis on the additive error model and imposing the simulated error fields on the OSPE, one may generate multiple realizations of precipitation fields.

Precipitation plays an important role in any hydrological analysis. The aim of this study is to empirically investigate the behaviour of spatial dependence of precipitation fields. This would determine whether the Gaussian assumption is fulfilled in regard to symmetric spatial dependence structure between low and high precipitation values. An asymmetry function that can incorporate zero precipitation amounts is introduced on the basis of empirical bivariate copulas. The asymmetry function is calculated by integrating the empirical bivariate copula density in the upper right and the lower left parts for any given quantile thresholds. Zero precipitation amounts are handled as latent variables and the thresholds, therefore, are set to be bigger than probability of zero. The Gaussian simulations based testing is applied for determining the degree of uncertainty. The empirical bivariate copulas are constructed using the concept of regionalized variables in spatial random process with a given separating distance. For any selected time interval, the precipitation over the region of interest is assumed to be a single realization of a spatially stationary random process. The investigations are conducted in Singapore and Bavaria. To take temporal characteristics and seasons into account, precipitation occurrences with different time scales for different seasons are analysed. The empirical evidence proves that the precipitation events tend to follow the positive asymmetric spatial dependence structure, particularly at a short separating distance. This implies that precipitation amounts with higher intensities tends to be more spatially correlated than that with lower intensities because precipitation occurrences tend to occur in a clustered manner. Consequently, spatial precipitation models that are based on the symmetric Gaussian dependence could result in an underestimation of the spatial extent of actual precipitation extremes. Moreover, this pattern is higher for smaller time scales (hourly to daily).

The spatial distribution of rainfall has a significant influence on catchment dynamics and the generation of streamflow time series. However, there are few stochastic models that can simulate long sequences of stochastic rainfall fields continuously in time and space. To address this issue, the first goal of this study was to present a new parsimonious stochastic model that produces daily rainfall fields across the catchment. To achieve parsimony, the model used the latent-variable approach (because this parsimoniously simulates rainfall occurrences as well as amounts) and several other assumptions (including contemporaneous and separable spatiotemporal covariance structures). The second goal was to develop a comprehensive and systematic evaluation (CASE) framework to identify model strengths and weaknesses. This included quantitative performance categorisation that provided a systematic, succinct and transparent method to assess and summarise model performance over a range of statistics, sites, scales and seasons. The model is demonstrated using a case study from the Onkaparinga catchment in South Australia. The model showed many strengths in reproducing the observed rainfall characteristics with the majority of statistics classified as either statistically indistinguishable from the observed or within 5% of the observed across the majority of sites and seasons. These included rainfall occurrences/amounts, wet/dry spell distributions, annual volumes/extremes and spatial patterns, which are important from a hydrological perspective. One of the few weaknesses of the model was that the total annual rainfall in dry years (lower 5%) was overestimated by 15% on average over all sites. An advantage of the CASE framework was that it was able to identify the source of this overestimation was poor representation of the annual variability of rainfall occurrences. Given the strengths of this continuous daily rainfall field model it has a range of potential hydrological applications, including drought and flood risk.

Understanding the spatial and temporal variability of rainfall has always been a great challenge, and the impacts of climate change further complicate this issue. The present study employs the concepts of complex networks to study the spatial connections in rainfall, with emphasis on climate change and rainfall scaling. Rainfall outputs (during 1961–1990) from a regional climate model (i.e. Weather Research and Forecasting (WRF) model that downscaled the European Centre for Medium-range Weather Forecasts, ECMWF ERA-40 reanalyses) over Southeast Asia are studied, and data corresponding to eight different temporal scales (6-hr, 12-hr, daily, 2-day, 4-day, weekly, biweekly, and monthly) are analyzed. Two network-based methods are applied to examine the connections in rainfall: clustering coefficient (a measure of the network’s local density) and degree distribution (a measure of the network’s spread). The influence of rainfall correlation threshold ( T ) on spatial connections is also investigated by considering seven different threshold levels (ranging from 0.5 to 0.8). The results indicate that: (1) rainfall networks corresponding to much coarser temporal scales exhibit properties similar to that of small-world networks, regardless of the threshold; (2) rainfall networks corresponding to much finer temporal scales may be classified as either small-world networks or scale-free networks, depending upon the threshold; and (3) rainfall spatial connections exhibit a transition phase at intermediate temporal scales, especially at high thresholds. These results suggest that the most appropriate model for studying spatial connections may often be different at different temporal scales, and that a combination of small-world and scale-free network models might be more appropriate for rainfall upscaling/downscaling across all scales, in the strict sense of scale-invariance. The results also suggest that spatial connections in the studied rainfall networks in Southeast Asia are weak, especially when more stringent conditions are imposed (i.e. when T is very high), except at the monthly scale.

Extreme low and high precipitation can cause natural disasters (e.g., drought and flood events) which have devastating effects on natural environment and the human society. In this study, we present a framework for assessing the stability of spatial dependence of extreme precipitation and their concurrence probability in a nested catchment. The Pearl River basin was used as a case study to test the applicability of the framework. This framework is threefold. This first part involves modeling spatial dependence of both summer maxima and winter minima precipitation between pairs of scattered gauges over the basin by employing the pair-copula constructions as a baseline model. The second part of this framework aims to identify the variability of spatial dependence as a result of the effects of external drivers. Specifically, variations in the spatial dependence of extreme precipitation in the Pearl River basin over different periods associated with the influence of large-scale climate signals were examined. The final part quantifies the probability of occurring extreme precipitation-related events simultaneously over broader areas related to a specific event at a given gauge and the possible spatial extents being affected. The results indicate that the presented framework is able to capture the large-scale spatial dependence structures of both summer maxima and winter minima precipitation. It also allows for explicitly estimating the pairwise concurrence probability of extreme high and low precipitation events and event-impacted areas across the basin.

Emmanuel et al. (2015) proposed rainfall variability indexes intended to summarize the influence of spatial rainfall organization on hydrograph features at the catchment outlet. The present article shows how the proposed indexes may be used in a real-world case study to analyze the influence of spatial rainfall organization on hydrograph modeling. The selected case study is located in the Cevennes Region of southeastern France. The proposed methodology is as follows: the tested flow events are split into two subsets according to the values of their rainfall variability indexes; then, a comparison is drawn between modeled and measured hydrographs separately for each subset. The results obtained suggest that, on average, modeling results based on high-resolution rainfall data are improved for the subset whose rainfall variability influence is expected to be significant according to index values. Though limited to a relatively small number of hydrographs, this case study can be viewed as a first confirmation that the proposed method, based on the rainfall variability indexes of Emmanuel et al. (2015), is pertinent to investigating the influence of spatial rainfall variability on hydrograph modeling results.

We apply the concepts of complex networks to investigate the properties of rainfall. Specifically, we examine the rainfall properties in terms of spatial connections, temporal scale, and network size. We employ the clustering coefficient method to rainfall data at six different temporal scales (daily, 2-day, 4-day, 8-day, 16-day, and monthly) from a large number of stations in the Murray-Darling basin in Australia. We consider different correlation thresholds to identify the existence of links between stations. To account for the influence of network size (i.e. number of stations) and length of data, we consider three different networks: (1) 430 stations with 30 years of daily data; (2) 383 stations with 30 years of daily data; and (3) 383 stations with 64 years of daily data. The results indicate that the nature of spatial connections changes with correlation threshold, with changes occurring at different temporal scales for different thresholds. Identification of an appropriate threshold is key to understand the rainfall connectivity properties.

Spatial geostatistical interpolation of point measurements of streambed attributes in the hyporheic zone may be constrained by the streambed anisotropy, and data density and spatial distribution may significantly impact the results. Spatial clustering and low spatial data density can be caused by bedrock outcropping at the streambed limiting installation of in-stream piezometers. This study examines parameter error variability of the geostatistical interpolation using anisotropic interpolation methods and increasing the data density by adding left censored values (i.e., data below measurement limit) to locations where measurements were limited by exposed bedrock lining the streambed. The reduction in relative standard error of the interpolation was determined for the spatial distributions of streambed attributes including hydraulic conductivity, seepage flux, and mercury solute flux measured in two different years along a study reach in East Fork Poplar Creek, Tennessee, USA. Two methods to impute the left censored values were compared including the conventional half the detection limit substitution method, and the Stochastic Approximation of Expectation-Maximization (SAEM) algorithm, which both had comparable results. Imputing left censored data increased the data density to recommended ranges, reduced data clustering, increased the spatial dependence for some attributes, and reduced the standard error for each of the three attributes. For the reach considered herein, addition of the left censored values resulted in a larger error reduction than the consideration of anisotropy within the interpolation, which confirms the benefit of data addition to increase data density within data-limited river corridors.

Stochastic rainfall generation is important for a range of hydrologic and water resources applications. Stochastic rainfall can be generated using a number of models; however, preserving relevant attributes of the observed rainfall—including rainfall occurrence, variability and the magnitude of extremes—continues to be difficult. This paper develops an approach to constrain stochastically generated rainfall with an aim of preserving the intensity-durationfrequency (IFD) relationships of the observed data. Two main steps are involved. First, the generated annual maximum rainfall is corrected recursively by matching the generated intensity-frequency relationships to the target (observed) relationships. Second, the remaining (non-annual maximum) rainfall is rescaled such that the mass balance of the generated rain before and after scaling is maintained. The recursive correction is performed at selected storm durations to minimise the dependence between annual maximum values of higher and lower durations for the same year. This ensures that the resulting sequences remain true to the observed rainfall as well as represent the design extremes that may have been developed separately and are needed for compliance reasons. The method is tested on simulated 6 min rainfall series across five Australian stations with different climatic characteristics. The results suggest that the annual maximum and the IFD relationships are well reproduced after constraining the simulated rainfall. While our presentation focusses on the representation of design rainfall attributes (IFDs), the proposed approach can also be easily extended to constrain other attributes of the generated rainfall, providing an effective platform for post-processing of stochastic rainfall generators.

Temporal rainfall distribution can often have significant influence on other hydrological processes such as runoff generation or rainfall interception. High-frequency rainfall data from 30 stations in Slovenia were analysed in order to improve the knowledge about the temporal rainfall distribution within a rainfall event. Using the pre-processed rainfall data Huff curves were determined and the binary shape code (BSC) methodology was applied. Although Slovenia covers only about 20,000 km2, results indicate large temporal and spatial variability in the precipitation pattern of the analysed stations, which is in agreement with the different Slovenian climate types: sub-Mediterranean, temperate continental, and mountain climate. Statistically significant correlation was identified between the most frequent BSC types, mean annual precipitation, and rainfall erosivity for individual rainfall stations. Moreover, different temporal rainfall distributions were observed for rainfall events with shorter duration (less than 12 h) than those with longer duration (more than 24 h). Using the analysis of the Huff curves it was shown that the variability in the Huff curves decreases with increasing rainfall duration. Thus, it seems that for shorter duration convective storms a more diverse temporal rainfall distribution can be expected than for the longer duration frontal precipitation where temporal rainfall distribution shows less variability.

Summary Impacts of global climate change on water resources systems are assessed by downscaling coarse scale climate variables into regional scale hydro-climate variables. In this study, a new multisite statistical downscaling method based on beta regression (BR) is developed for generating synthetic precipitation series, which can preserve temporal and spatial dependence along with other historical statistics. The beta regression based downscaling method includes two main steps: (1) prediction of precipitation states for the study area using classification and regression trees, and (2) generation of precipitation at different stations in the study area conditioned on the precipitation states. Daily precipitation data for 53 years from the ANUSPLIN data set is used to predict precipitation states of the study area where predictor variables are extracted from the NCEP/NCAR reanalysis data set for the same interval. The proposed model is applied to downscaling daily precipitation at ten different stations in the Campbell River basin, British Columbia, Canada. Results show that the proposed downscaling model can capture spatial and temporal variability of local precipitation very well at various locations. The performance of the model is compared with a recently developed non-parametric kernel regression based downscaling model. The BR model performs better regarding extrapolation compared to the non-parametric kernel regression model. Future precipitation changes under different GHG (greenhouse gas) emission scenarios also projected with the developed downscaling model that reveals a significant amount of changes in future seasonal precipitation and number of wet days in the river basin.

Summary Two dominant processes determine solute concentration in groundwater: vertical infiltration and horizontal advection. The goal of this paper is to incorporate both processes into a geostatistical model for spatial estimation of solute concentrations in groundwater. A multivariate copula-based methodology is demonstrated that considers infiltration via the marginal distribution and solute transport via the multivariate spatial dependence structure. The novel approach is compared to traditional methods as Ordinary- and External Drift Kriging. Leave-one-out cross-validation demonstrates that the novel approach estimates better both in concentration and in probability space, and improves the quantification and quality of uncertainty. The gain in uncertainty reduction is equivalent to at least a few hundred additional observations when Ordinary Kriging was used. Both censored and not-censored measurements are included. An ideal neighborhood size is estimated via cross-validation. The methodology is general and can incorporate other kinds of secondary information. It can be used to evaluate effects of land use changes.

The frequency and severity of extreme weather events have been noted to be changing in both time and space as a result of rising global temperatures. In this regard, the analysis of joint occurrences of extreme climate events has gained considerable importance for planning and emergency management as such events often result in system failures that extend beyond a single location in the region impacted. By their very definition, extremes cannot be sampled frequently, necessitating approaches that are robust and efficient and can take advantage of the Big Data movement we are now a part of. This study introduces a new concept for characterizing such dependence, termed relative connectivity, formulated using the theory of empirical copulas which represent a non-parametric method to describe the joint dependence of random variables. The spatial relative connectivity of extreme rainfall events across 2,708 rainfall stations in Australia is ascertained and conclusions drawn. The extreme rainfall events in Western Australia, southeast South Australia, and southwest Victoria exhibit higher relative connectivity than other regions, suggesting a stronger spatial dependence and higher possibility of concurrence when compared to other regions. The overall variation of relative connectivity is also examined under different low-frequency climatic anomalies (i.e. El Nino and La Nina events) and temperature states. We observe that spatial dependence across raingauges increases under both El Nino and La Nina periods, while it weakens at warmer temperatures, with localized lower relative connectivity at target raingauges.

Summary Design rainfall statistics are the primary inputs used to assess flood risk across river catchments. These statistics normally take the form of Intensity–Duration–Frequency (IDF) curves that are derived from extreme value probability distributions fitted to observed daily, and sub-daily, rainfall data. The design rainfall relationships are often required for catchments where there are limited rainfall records, particularly catchments in remote areas with high topographic relief and hence some form of interpolation is required to provide estimates in these areas. This paper assesses the topographic dependence of rainfall extremes by using elevation-dependent thin plate smoothing splines to interpolate the mean annual maximum rainfall, for periods from one to seven days, across Australia. The analyses confirm the important impact of topography in explaining the spatial patterns of these extreme rainfall statistics. Continent-wide residual and cross validation statistics are used to demonstrate the 100-fold impact of elevation in relation to horizontal coordinates in explaining the spatial patterns, consistent with previous rainfall scaling studies and observational evidence. The impact of the complexity of the fitted spline surfaces, as defined by the number of knots, and the impact of applying variance stabilising transformations to the data, were also assessed. It was found that a relatively large number of 3570 knots, suitably chosen from 8619 gauge locations, was required to minimise the summary error statistics. Square root and log data transformations were found to deliver marginally superior continent-wide cross validation statistics, in comparison to applying no data transformation, but detailed assessments of residuals in complex high rainfall regions with high topographic relief showed that no data transformation gave superior performance in these regions. These results are consistent with the understanding that in areas with modest topographic relief, as for most of the Australian continent, extreme rainfall is closely aligned with elevation, but in areas with high topographic relief the impacts of topography on rainfall extremes are more complex. The interpolated extreme rainfall statistics, using no data transformation, have been used by the Australian Bureau of Meteorology to produce new IDF data for the Australian continent. The comprehensive methods presented for the evaluation of gridded design rainfall statistics will be useful for similar studies, in particular the importance of balancing the need for a continentally-optimum solution that maintains sufficient definition at the local scale.

The present study investigates the use of the spatial copula approach for predicting flood quantiles at ungauged basins. Spatial copulas are the formalization of traditional geostatistics by copulas. In regional flood frequency analysis (RFFA), the regression of flood quantiles is often carried out at the logarithmic scale. Consequently, traditional interpolation methods introduce a bias and provide suboptimal predictions. In this study, the copula framework is examined for offering proper corrections in this framework. Moreover, copula techniques separate the regional distribution of flood quantiles from spatial dependence. This provides a full probabilistic model that represents a more flexible framework where proper combinations of regional distribution and dependence can be adapted to various situations that are encountered in RFFA. The adequacy of the investigated methodology is evaluated on a real world case study involving hydrometric stations from southern Quebec, Canada. Results show that the spatial copula framework is able to deal with the problem of bias, is robust to the presence of problematic stations and may improve the quality of quantile predictions while reducing the level of complexity of the models used in RFFA.

Summary Between 9th and 16th September 2013, northeast Colorado received some of its most extreme rainfall on record. The event affected 6 major rivers and their tributaries and 14 counties, breaking observed records for accumulations from sub-daily through to annual total. NOAA’s rainfall atlases indicated that this event had an anticipated return period of 1000 years. We use the rainfall that led to the 2013 Colorado floods as a case study in order to explore how a large event can affect the generalized extreme value (GEV) parameter estimates often used by designers and planners. We employ daily rainfall observations, with at least 30 years of data, from stations across Colorado’s Front Range of the Rocky Mountains to develop a spatial statistical model for annual maximum daily rainfall. We produce estimates of relatively rare events such as the 1% Annual Exceedance Probability (AEP) level and of extremely rare events such as the return period associated with Boulder’s 2013 observation. To explore sensitivity, we compare estimates including and excluding data from 2013, and both using only individual station data and our model which borrows strength across multiple stations. We compute the uncertainty associated with all of our estimates, and find large uncertainties associated with extremely rare events. Our statistical model is a spatial hierarchical model and we employ a two-stage approach for inference which can be implemented by practitioners. Additionally, the spatial model allows us to interpolate spatially and estimate the GEV parameters at unobserved locations. A further development of the model makes use of an alternatively defined space in terms of elevation and a climate variable, rather than geographical space defined by longitude and latitude, which seems to better account for orographic effects. In addition to producing AEP level and return period estimates to the annual maximum data, we investigate sensitivity to the choice of block length. We find point estimates indicate the tail to be much heavier when a longer block length is used, but the uncertainty associated with this parameter is such that one cannot say the difference is significant. To describe the spatial extent of severe storms, we also investigate the amount of data dependence between station locations. We find evidence in the record for storms with large spatial extent, although an extremal dependence parameter estimate indicates that this dependence is relatively weak.

Summary Adequate knowledge of spatial connections in rainfall is important for reliable modeling of catchment processes and water management. This study applies the ideas of network theory to examine and interpret the spatial connections in rainfall in Australian conditions. As case studies, monthly rainfall data across a network of raingages from two vastly different areas are studied: (1) Western Australia – data over a period of 67 years (1937–2003) from 57 raingages; and (2) Sydney catchment – data over a period of 114 years (1890–2003) from 47 monitoring stations. The spatial rainfall connections in the two networks are examined using clustering coefficient (CC), a popular network connectivity measure. The clustering coefficient measures the local density and quantifies the network’s tendency to cluster. Different values of rainfall correlation threshold (CT) are used to measure the strength of connections in rainfall between different stations and, hence, to calculate CC. The clustering coefficient values are interpreted in terms of topographic factors (latitude, longitude, and elevation) and rainfall properties (mean, standard deviation, and coefficient of variation).

Given a record of daily rainfall over a network of gauges, this paper describes a method of linking the Gauge Wetness Ratio (GWR) on a given day to the joint distribution of the parameters of the anisotropic correlogram defining the spatial statistics of simulated radar-rainfall fields. We generate a large number of Gaussian random fields by sampling from the correlogram parameters conditioned on the GWR and then conditionally merge these fields to the gauge observations transformed into the Gaussian domain. Availability of such a tool allows better spatially distributed hydrological modelling, because good quality ensemble spatial information is required for such work, as it yields uncertainty of the fields so generated. To achieve these ends, correlograms of many Gaussianised daily accumulations of radar images were developed using the Fast Fourier Transform to generate their sample power spectra. Empirical correlograms were fitted using a 2D exponential distribution to yield the 3 key parameters of the correlogram: the range, the anisotropy ratio and the direction of the major axis. It was found that the range follows a Gamma distribution while the anisotropy parameters follow a Loglogistic one; a t5 copula was adequate to capture the bivariate negative dependence structure between the range and ratio. The Radar Wetted Area Ratio (RWAR) drives the parameters of the correlogram, and its link with GWR is modelled by a transition probability matrix. We take each of the generated Gaussian random fields and conditionally merge it with Gaussianised rainfall values at the gauge locations using Ordinary Kriging. The method produces realistic simulated radar images, on a grid chosen to suit the data, which match the gauge observations at their locations. Ensemble simulations of 1000 samples were used to derive the median and the inter-quartile range of the fields; these were found to narrow near the control gauge locations, as expected, emphasising the value of high density gauge networks. Ongoing research is looking towards integration of the presented methodology with a stochastic daily rainfall generator for useful spatial rainfall simulation over catchments with gauged records.

Summary One of the key challenges in hydrology is to accurately measure and predict the spatial and temporal distribution of rainfall. Rain gauges measuring at point locations are often considered as the “ground truth” for grid based radar rainfall calibration. Usually, no consideration is given to the uncertainty in the measurement that varies depending on the number of rain gauges that fall within each grid cell. If this uncertainty in the rain gauge network measurements is ignored, the Z–R relationship used to convert reflectivity (Z) to rainfall (R) will be biased. We investigate the effects of point gauge rainfall uncertainty on parameter bias in the Z–R relationship. An error model is developed to compute point gauge rainfall uncertainty at the radar grid resolution. This error model has two components: (1) the error in the gauge measurement itself, and (2) the error introduced by the gauge not capturing the spatial variability within a radar pixel. The Simulation Extrapolation method (SIMEX) is used to determine the extent of parameter bias present in the rainfall–reflectivity relationship as a result of this uncertainty. When considering the point gauge rainfall uncertainty a 4% decrease in the average radar rainfall estimates is found.

The pattern-oriented memory (POM) is a novel historical rainfall interpolation method that explicitly takes into account the time dimension in order to interpolate areal rainfall maps. The method is based on the idea that rainfall patterns exist and can be identified over a certain area by means of non-linear regressions. Having been previously benchmarked with a vast array of interpolation methods using proxy satellite data under different time and space availabilities, in the scope of the present contribution POM is applied to rain gauge data in order to produce areal rainfall maps. Tested over the Zambezi River Basin for the period from 1979 to 1997 (accurate satellite rainfall estimates based on spaceborne instruments are not available for dates prior to 1998), the novel pattern-oriented memory historical interpolation method has revealed itself as a better alternative than Kriging or Inverse Distance Weighing in the light of a Monte Carlo cross-validation procedure. Superior in most metrics to the other tested interpolation methods, in terms of the Pearson correlation coefficient and bias the accuracy of POM's historical interpolation results are even comparable with that of recent satellite rainfall products. The new method holds the possibility of calculating detailed and performing daily areal rainfall estimates, even in the case of sparse rain gauging grids. Besides their performance, the similarity to satellite rainfall estimates inherent to POM interpolations can contribute to substantially extend the length of the rainfall series used in hydrological models and water availability studies in remote areas. (C) 2014 Elsevier B.V. All rights reserved.

Summary This paper maps out a methodology developed for downscaling Regional Circulation Model (RCM) estimates of rainfall to networks of gauge sites. We have produced, from an RCM, a set of gauge records which (i) captures the changes modelled by the RCM and (ii) has the statistics of the observations in the control period after suitable transformation. These statistics include the marginal distributions, the wet/dry sequences and the spatial dependence structure of rainfall. We have been careful to eliminate the RCM’s inherent bias, yet preserve its message of the past and possible future. The key issue is the ability of the method to transfer information from the RCM scale to the gauge scale (in networks), to serve planning, engineering and agricultural practitioners who are comfortable with such a product, in contrast to the raw RCM precipitation output. We exploit atmospheric Circulation Patterns (CPs) to condition the downscaling. Our main goal is to address the following issues which are germane to the downscaling process: • Scale : there is a considerable difference between frequency distribution functions (fdfs) and cross correlation coefficients (cccs) at the gauge and RCM grid-block scale; we show that precipitation distributions, as well as spatial correlations, are not reproduced well by the RCMs. • Information : A picture, or a take on the ‘future’, is available from RCMs. The problem is: how do we go about transferring the model signal to the gauge scale, concomitantly removing the bias? • CPs : We recorrelate the spatial dependence of the downscaled estimates at the gauge scale then we correct the fdfs at the gauge scale conditioned on CPs. • Check : We devise methods of transferring the RCM signal (fdfs and cccs) to gauge networks, both in a historical ten year period for verifying that the procedure is effective and importantly in a second nine year period for validating the procedure on unused data. The downscaling of RCM rainfall to gauge sites was achieved in five regions in South Africa. These gauge rainfall estimates may be used directly in design and planning scenarios, even under Climate Change, using this methodology.

Summary The coincidence of precipitation for an inter-basin water transfer project may determine the feasibility of the project and whether there is enough water to be diverted. The degree of coincidence statistically depends upon the spatial dependence of precipitation and can be measured by the multivariate probability distribution. A copula-based approach, which captures such dependence structure, is proposed to quantify the synchrony and asynchrony of precipitation for the middle route of South-to-North water transfer project in China. A test procedure is suggested for testing whether the selected copula is able to simultaneously measure the overall and tail dependencies of the observations. Goodness-of-fit tests indicate that the asymmetric trivariate Clayton copula is appropriate to model these dependencies of precipitation in different regions of the project. Combination frequencies of wet, dry and normal conditions and conditional probabilities for some extreme deficit rainfall events are calculated using the proposed procedure. The probability that is beneficial to water transfer is large enough to guarantee the amounts of water transferable, on the whole. But for some extreme deficit rainfall events, the possibility for water transfer would become very small.

Accurate estimation of spatial rainfall is crucial for modelling hydrological systems and planning and management of water resources. While spatial rainfall can be estimated either using rain gauge-based measurements or using satellite-based measurements, such estimates are subject to uncertainties due to various sources of errors in either case, including interpolation and retrieval errors. The purpose of the present study is twofold: (1) to investigate the benefit of merging rain gauge measurements and satellite rainfall data for Australian conditions and (2) to produce a database of retrospective rainfall along with a new uncertainty metric for each grid location at any timestep. The analysis involves four steps: First, a comparison of rain gauge measurements and the Tropical Rainfall Measuring Mission (TRMM) 3B42 data at such rain gauge locations is carried out. Second, gridded monthly rain gauge rainfall is determined using thin plate smoothing splines (TPSS) and modified inverse distance weight (MIDW) method. Third, the gridded rain gauge rainfall is merged with the monthly accumulated TRMM 3B42 using a linearised weighting procedure, the weights at each grid being calculated based on the error variances of each dataset. Finally, cross validation (CV) errors at rain gauge locations and standard errors at gridded locations for each timestep are estimated. The CV error statistics indicate that merging of the two datasets improves the estimation of spatial rainfall, and more so where the rain gauge network is sparse. The provision of spatio-temporal standard errors with the retrospective dataset is particularly useful for subsequent modelling applications where input error knowledge can help reduce the uncertainty associated with modelling outcomes.

Summary There is a standing challenge in obtaining long localized records of rainfall data in many large river basins of the developing world. Recent spaceborne instrumentation offers a consistent source of rainfall information, but this information covers only a relatively limited time period. In this context, and given its consistence, a question rises on the potential offered by this new wealth of information to improve our understanding of the rainfall patterns and how to use them in order to alleviate the historical problems of scarcity of observed historical records. The present research focuses on the interpolation of historical rainfall records over large spatial scales and low availability of observed point data, with distances between measurement points in the order of tenths to hundreds of kilometers and temporal scales ranging from daily to monthly. The main goals of the work are twofold: firstly, to evaluate the potential of using a novel pattern-oriented interpolation technique to learn complex spatial rainfall patterns from satellite data and applying this knowledge in the interpolation of historical rainfall maps; secondly, to assess the performance of the proposed methodology by comparing its results to those of other interpolation techniques suitable for spatially sparse datasets. The proposed pattern-oriented interpolation technique uses modern data sources to enhance the reliability of the interpolation of historical rainfall areal distributions. Results show that, under given conditions, the pattern-oriented memory class of models can considerably reduce the errors traditionally associated with historical rainfall interpolation at large spatial scales and under low availability of spatial data.

Summary Spatial rainfall variability is considered an important factor affecting the accuracy of streamflow prediction. This study evaluated the effect of spatial rainfall variability on water balance modelling by conducting a series of virtual experiments. The three-layer Variable Infiltration Capacity model (VIC-3L) was applied using both uniform and spatially variable rainfall for 60 catchments in south-eastern Australia. The spatially variable rainfall was generated from the 0.05° gridded SILO daily rainfall with different degrees of variability. The VIC-3L model was calibrated against observed daily streamflow using gridded SILO daily rainfall to generate reference parameter values. Then the model was applied using the generated spatially variable rainfall and reference parameter values to produce virtual water balance components associated with different spatially variable rainfall. Differences between the lumped and distributed modelling (i.e. virtual experiments) represent the effects of spatial rainfall variability on water balance modelling. The results showed that spatial rainfall variability interacts with catchments characteristics to influence hydrological processes and the effects are not uniform on different water balance components. For a given rainfall total, ignoring spatial rainfall variability will result in underestimation of the total streamflow volume and overestimation of evapotranspiration. Effect of spatial rainfall variability on water balance modelling is more pronounced in catchments with larger rainfall variability. In most cases, information on spatial rainfall variability will help to improve accuracy of water balance modelling. However, in some cases, lumped water balance modelling (i.e. ignoring spatial rainfall variability) may outperform distributed modelling due to catchment filtering effect. The results from this study suggest that more accurate information of rainfall spatial distribution in a catchment can help to improve water balance modelling.

Summary This paper evaluates three distinct approaches for disaggregating daily rainfall to sub-daily sequences: (1) random multiplicative cascades (microcanonical and canonical versions), (2) point process (randomized Bartlett–Lewis model – RBLM), and (3) resampling (method of fragments). These methods are used to perform disaggregation of daily rainfall to hourly rainfall at four point locations across Australia (Sydney, Perth, Cairns, and Hobart), which are associated with different climatic regimes. The methods are evaluated based on parameter estimation procedures applied (including introduction of the sequential Monte Carlo sampler in RBLM), the capability of the resulting sequences to reproduce standard validation statistics, and the representation of observed rainfall variability and intermittency, within-day wet spells, and extreme rainfall percentiles. The results generally indicate that the method of fragments outperforms the other models. While all the models are found to simulate reasonably well the commonly used statistical measures (e.g. mean and dry proportions) of rainfall at the hourly timestep, the microcanonical model is found to significantly overestimate the hourly rainfall variance. With respect to extreme value characteristics, the resampling approach is found to match well the observed intensity–frequency relationship at an hourly scale, with the cascade models underestimating (canonical) and overestimating (microcanonical) extreme rainfall. The point process model’s performance is poor in Cairns but reasonably good at other locations. An analysis of the empirical within-day wet- and dry-spell distributions further reveals that the cascade-based models are not robust for observed wet and dry spells.

Normally, rainfall data is collected on a daily, monthly or annual basis in the form of discrete observations. The aim of this study is to convert these rainfall values into a smooth curve or function which could be used to represent the continuous rainfall process at each region via a technique known as functional data analysis. Since rainfall data shows a periodic pattern in each region, the Fourier basis is introduced to capture these variations. Eleven basis functions with five harmonics are used to describe the unimodal rainfall pattern for stations in the East while five basis functions which represent two harmonics are needed to describe the rainfall pattern in the West. Based on the fitted smooth curve, the wet and dry periods as well as the maximum and minimum rainfall values could be determined. Different rainfall patterns are observed among the studied regions based on the smooth curve. Using the functional analysis of variance, the test results indicated that there exist significant differences in the functional means between each region. The largest differences in the functional means are found between the East and Northwest regions and these differences may probably be due to the effect of topography and, geographical location and are mostly influenced by the monsoons. Therefore, the same inputs or approaches might not be useful in modeling the hydrological process for different regions.

Summary Accurate rainfall input is a prerequisite for simulations that aim to generate accurate hydrographs, which are crucial for flood forecasting, particularly in regions that are prone to frequent typhoon (tropical cyclone) invasions, such as Southern Asia. Few studies have investigated the effect of spatial resolution in typhoon rainfall monitoring on modeled hydrographs. Eight typhoon cases were examined in a mountainous watershed (335 km2) featuring hourly radar-based 1.3-km resolution rainfall estimates. Radar-based hourly rainfall was subsampled at various densities in space, and then re-interpolated to full scale for modeling. The highest resolution rainfall datasets were taken as an ideal input in TOPMODEL for calibration and to derive the reference hydrographs, which were further used to examine the response of modeled hydrographs to imperfect rainfall. The correlation between rainfall similarities (compared with radar-based) and corresponding hydrograph similarities (compared with reference) were identified. The two most important findings were as follows: (1) in predicting flood peak timing in mesoscale watershed, high spatial resolution is not required because typhoon-induced rainfall is less variable in space and more concentrated in the temporal scale and (2) satisfactory hydrographs with EC > 0.8 were obtained in 96% test cases, indicating that even a totally biased rainfall (in terms of total amount and rainfall field) may produce a plausible hydrograph. Hydrologic models transfer the spatiotemporal rainfall input into time-series discharge, in which the spatial dimension is converted into travel time. Those positive and negative rainfall biases in space may compensate once allocated in the same arrival time frame in the hydrograph. This explains why in many cases sparsely gauged rainfall input also generates promising hydrographs. In other words, as discussing the effect of other distributed factors on simulated hydrographs, the highly accurate rainfall input is an essential prerequisite to prevent the compensation.

Summary This paper presents an investigation of rainfall error models used in hydrological model calibration and prediction. Traditional calibration methods assume input error to be negligible: an assumption which can lead to bias in parameter estimation and compromise model predictions. In response, a growing number of studies now specify an error model for rainfall input, usually simple in form due to both difficulties in understanding sampling errors in rainfall, and to computational constraints during parameter estimation. Such rainfall error models have not typically been validated against experimental evidence. In this study we use data from a dense gauge/radar network in the Mahurangi catchment (New Zealand) to directly evaluate the form of basic statistical rainfall error models. For this catchment, our results confirm the suitability of a multiplicative error formulation for correcting mean catchment rainfall values during high-rainfall periods (e.g., intensities over 1 mm/h); or for longer timesteps at any rainfall intensity (timestep 1 day or greater). We show that the popular lognormal multiplier distribution provides a relatively close approximation to the true error characteristics but does not capture the distribution tails, especially during heavy rainfall where input errors would have important consequences for runoff prediction. Our research highlights the dependency of rainfall error structure on the data timestep.

Summary This paper aims to clarify the dependence existing between spatial rainfall organisation, basin morphology and runoff response. This is obtained by applying a spatial rainfall metric which describes the spatial rainfall organisation in terms of concentration and dispersion statistics as a function of the flow travel time measured along the river network. The metric is based on the observation that runoff routing through branched channel networks imposes an effective averaging of spatial rainfall excess at equal travel time, in spite of the inherent spatial variability. High resolution radar rainfall fields and a distributed hydrologic model are employed to examine how effective are these statistics in describing the degree of spatial rainfall organisation which is important for runoff modelling, and in quantifying the effects of neglecting the spatial rainfall variability on flood modelling. The investigation focuses on three extreme flash flood events occurred on the Carpathian range (Romania) in the period 2005–2007. The size of the study catchments ranges between 36 and 167 km2. The analysis reported here shows that neglecting the spatial rainfall variability results in a considerable loss of simulation Nash–Sutcliffe (NS) efficiency in almost 30% of the cases (NS less than 0.8), with NS less than 0.6 in one of the cases. This provides a significant documentation of the influence of the spatial rainfall variability on runoff modelling for catchment size less than 160 km2. Moreover, it is shown that these rainfall statistics, used in combination, are able to isolate and describe the features of rainfall spatial organisation which have significant impact on runoff simulation. An alternative rainfall spatial variability index, which is computed irrespective of flow travel time structure, provides a comparatively poor description of the influence of neglecting rainfall spatial variability on flood modelling. Overall, this implies that rainfall organisation measured along the river network by using the travel time coordinate may be a significant property of rainfall spatial variability when considering flood response modelling.

Summary The variogram structure is an effective tool in order to appraise the rainfall spatial variability. In areas with disperse raingauge network, this paper suggests a 3-D estimation of the variogram, as alternative to the classical 2-D approach for spatiotemporal rainfall analysis. The context deals with the estimation of the spatial variability of maximum intensity of rainfall for a given duration δ . Hence, a 3-coordinate vector (location – rainfall duration – rainfall intensity) is associated to each monitoring location rather than the two coordinate vector, based only on the location in relation to intensity subject to duration. A set of averaging time intervals is taken into account ( δ ranging from 5 min to 2 h). The advantage of the 3-D approach is that it results on a standardized variogram which uniquely characterizes the rainfall event. On the contrary, for the 2-D approach, variograms are subject to intensity duration. The kriging with external drift is performed to make the spatial interpolations and to compute the kriging variance maps. A full comparison of the accuracy of both methods (2-D, 3-D) using cross-validation scheme, shows that the 3-D kriging leads to significantly lower prediction errors than the classical 2-D kriging. It is further suggested to quantify the effect of 3-D and 2-D kriging on the areal rainfall distribution and on the standard deviation of the kriging error SDKE. It is noticed that the 3-D SDKE field displays an empirical distribution which represents a median position among the 2-D distributions corresponding to SDKE ( δ ) fields. On the other hand, results are compared to those obtained through ordinary kriging. In the 3-D approach, cross-validation performances and SDKE maps are found to be less sensitive to the kriging method.

Summary Rainfall depth-duration-frequency (DDF) curves describe rainfall depth as a function of duration for given return periods and are important for the design of hydraulic structures. This paper focusses on the effects of dependence between the maximum rainfalls for different durations on the estimation of DDF curves and the modelling of uncertainty of these curves. For this purpose the hourly rainfall depths from 12 stations in the Netherlands are analysed. The records of these stations are concatenated to one station-year record, since no geographical variation in extreme rainfall statistics could be found and the spatial dependence between the maximum rainfalls appears to be small. A generalized extreme value (GEV) distribution is fitted to the 514 annual rainfall maxima from the station-year record for durations of 1, 2, 4, 8, 12 and 24 h. Subsequently, the estimated GEV parameters are modelled as a function of duration to construct DDF curves, using the method of generalized least squares to account for the correlation between GEV parameters for different durations. A bootstrap estimate of the covariance matrix of the estimated GEV parameters is used in the generalized least squares procedure. It turns out that the shape parameter of the GEV distribution does not vary with duration. The bootstrap is also used to obtain 95%-confidence bands of the DDF curves. The bootstrap distribution of the estimated quantiles can be described by a lognormal distribution. The parameter σ of this distribution (standard deviation of the underlying normal distribution) is modelled as a function of duration and return period.

Summary Satellite-based rainfall monitoring is widely used for climatological studies because of its full global coverage but it is also of great importance for operational purposes especially in areas such as Africa where there is a lack of ground-based rainfall data. Satellite rainfall estimates have enormous potential benefits as input to hydrological and agricultural models because of their real time availability, low cost and full spatial coverage. One issue that needs to be addressed is the uncertainty on these estimates. This is particularly important in assessing the likely errors on the output from non-linear models (rainfall-runoff or crop yield) which make use of the rainfall estimates, aggregated over an area, as input. Correct assessment of the uncertainty on the rainfall is non-trivial as it must take account of • the difference in spatial support of the satellite information and independent data used for calibration • uncertainties on the independent calibration data • the non-Gaussian distribution of rainfall amount • the spatial intermittency of rainfall • the spatial correlation of the rainfall field This paper describes a method for estimating the uncertainty on satellite-based rainfall values taking account of these factors. The method involves firstly a stochastic calibration which completely describes the probability of rainfall occurrence and the pdf of rainfall amount for a given satellite value, and secondly the generation of ensemble of rainfall fields based on the stochastic calibration but with the correct spatial correlation structure within each ensemble member. This is achieved by the use of geostatistical sequential simulation. The ensemble generated in this way may be used to estimate uncertainty at larger spatial scales. A case study of daily rainfall monitoring in the Gambia, west Africa for the purpose of crop yield forecasting is presented to illustrate the method.

A stochastic modeling framework for multisite generation of daily rainfall is developed with an aim of representing both short and higher time scale dependence in the generated rainfall sequences. The framework simulates rainfall at individual locations using separate models for rainfall occurrences and rainfall amounts on the simulated wet days. The spatial correlations in the generated occurrences and amounts are induced using spatially correlated yet serially independent random numbers. The rainfall occurrence model is based on a modification of the transition probabilities of the traditional Markov model through an analytically derived factor that represents the influence of rainfall aggregated over long time periods in an attempt to incorporate low-frequency variability in simulations. The rainfall amounts on the wet days are generated using a nonparametric conditional simulation approach. The utility of the proposed method is illustrated by applying the model on a network of 30 raingauge stations around Sydney, Australia, and comparing a range of statistics describing daily and higher time scale distribution and dependence attributes. The analyses of the results show that the method adequately captures daily as well as aggregated higher time scale rainfall characteristics at individual locations including the spatial distribution of rainfall over the region.

Summary A semi-parametric stochastic modeling framework for generation of daily rainfall at multiple locations is presented. The proposed framework represents longer-term variability and low-frequency features such as drought, while still simulating other daily key distributional and dependence attributes present in the observed rainfall record with sufficient spatial coherency. The rainfall occurrences at individual sites are simulated using a two-state, first-order Markov model. The transition probabilities of the Markov model are modified by using “aggregate” predictor variables that are indicative of how wet it has been over a period of time. The rainfall amounts on the simulated wet days are generated using a nonparametric kernel density estimation approach. Multisite spatial correlations in the rainfall occurrences and amounts series are represented by driving the single-site models with spatially correlated random numbers. The model is applied on a network of 30 raingauge stations around Sydney in eastern Australia. The analyses of results show that the model is capable of reproducing daily and higher time-scale key spatial and temporal characteristics of rainfall desired in most hydrologic applications.

Joint distributions of rainfall intensity and depth, rainfall intensity and duration, or rainfall depth and duration are important in hydrologic design and floodplain management. Multivariate rainfall frequency distributions have usually been derived using one of three fundamental assumptions: (1) Either rainfall variables (e.g., intensity, depth, and duration) have each the same type of the marginal probability distribution, (2) the variables have been assumed to have joint normal distribution or have been transformed and assumed to have joint normal distribution, or (3) they have been assumed independent-a trivial case. In reality, however, rainfall variables are dependent, do not follow, in general, the normal distribution, and do not have the same type of marginal distributions. This study aims at deriving bivariate rainfall frequency distributions using the copula method in which four Archimedean copulas were examined and compared. The advantage of the copula method is that no assumption is needed for the rainfall variables to be independent or normal or have the same type of marginal distributions. The bivariate distributions are then employed to determine joint and conditional return periods, and are tested using rainfall data from the Amite River basin in Louisiana, United States.

Summary This paper presents a comparison of three multi-site stochastic weather generators for simulation of point rainfall occurrences at a network of 30 raingauge stations around Sydney, Australia. The approaches considered include a parametric hidden Markov model (HMM), a multi-site stochastic precipitation generation model (proposed by [Wilks, D.S., 1998. Multi-site generalization of a daily stochastic precipitation model, J. Hydrol. 210, 178–191.]) and a non-parametric K-nearest neighbour (KNN) model. The HMM generates the precipitation distribution conditional on a discrete weather state representing certain identified spatial rainfall distribution patterns. The spatial dependence is maintained by assumption of a common weather state across all stations while the temporal dependence is simulated by assuming the weather state to be Markovian in nature. The Wilks model preserves serial dependence through the assumption of an order one Markov dependence at each location. The spatial dependence is simulated by prescribing a dependence pattern on the uniform random variates used to generate the rainfall occurrence at each location from the associated conditional probability distribution. The K-nearest neighbour approach simulates spatial dependence by simultaneously generating precipitation occurrence at all locations. Temporal persistence is simulated through Markovian assumptions on the rainfall occurrence process. The three methods are evaluated for their ability to model spatial and temporal dependence in the rainfall occurrence field and also the relative ease with which the assumptions of spatial and temporal dependence can be accommodated. Our results indicate that all the approaches are successful in reproducing spatial dependence in the multi-site rainfall occurrence field. However, the different orders of assumed Markovian dependence in the observed data limit their ability in representing temporal dependence at time scales longer than a few days. While each approach comes with its own advantages and disadvantages, the alternative proposed by Wilks has an overall advantage in offering a mechanism for modelling varying orders of serial dependence at each point location, while still maintaining the observed spatial dependence with sufficient accuracy.

Summary The mesoscale variability of the Sahelian rainfall is analysed from a series of 30 high time resolution rainfall series covering 13 years and a 110 × 160 km2 area in the region of Niamey. It is shown that the convective scale variability is strongly influencing the spatial pattern of rainfields at larger time and spatial scales. This means that a proper assessment of the rainfall patterns at the mesoscale in the Sahel requires raingauge networks with a sufficient density to allow sampling this convective scale variability. This is usually not the case with operational networks whose density is in the order of 1–2 gauges per 10,000 km2. Computations carried out here show that the areal rainfall estimation error increases from 3% to 16% at the annual scale and from 21% to 113% at the event scale when the number of stations over a 100 × 100 km2 area decreases from 12 to 1. While being highly variable in space, the Sahelian rainfall is also highly intermittent in time. An analysis of the series of tipping bucket times leads to compute that 50% of the annual rain falls in less than 4 h with intensities larger than 35 mm/h. Areal rainfall statistics are compared to point rainfall statistics for event accumulated rainfall. The implication of these results for studying the influence of rainfall intermittency on runoff is discussed in a final section, as an introduction to a companion paper.

Hydrological impacts of climate change are frequently assessed by off-line forcing of a hydrological model with climatic scenarios from either Global Circulation Models (GCMs) or simpler analogue models. Most hydrological models require a daily time step or smaller while observed climatology and GCM and analogue model output is generally available on a monthly time step. This study investigates and improves a rainfall disaggregation model currently used to convert monthly rainfall totals down to the daily time step. The performance of the model is evaluated using daily data from a network of raingauges covering the Nile basin and contrasted with data from a relatively dense raingauge network from the Blackwater Catchment, in the Southeast of the UK. Whilst the model preserves the mean properties of rainfall occurrence and depth, there is significant overestimation of rainfall variability. Regional calibration and better formulation of the generator improve simulation of variability as well as other aspects of rainfall properties. Hence the parameters required by the weather generator model cannot be regarded as universal. Proportional correction of daily amounts is applied to insure that monthly totals are preserved, allowing retention of interannual variability, and this was shown to have little effect on the distribution of wet day amounts. The calibration of parameter estimation equations has investigated spatial dependence of climate variables and parameters and found that (as expected) rainfall properties exhibit scale-dependence, which may be utilized to transfer data from one spatial scale to another. In order to complete the framework, a model is developed to estimate the wet fraction from monthly total when the former is not available.

We examine the hypothesis that basins characterized by (1) marked spatial variability in precipitation, and (2) less of a filtering effect of the input rainfall signal will show improved outlet simulations from distributed versus lumped models. Basin outflow response to observed spatial variability of rainfall is examined for several basins in the Distributed Model Intercomparison Project. The study basins are located in the Southern Great Plains and range in size from 795 to 1645 km2. We test our hypothesis by studying indices of rainfall spatial variability and basin filtering. Spatial variability of rainfall is measured using two indices for specific events: a general variability index and a locational index. The variability of basin response to rainfall event is measured in terms of a dampening ratio reflecting the amount of filtering performed on the input rainfall signal to produce the observed basin outflow signal. Analysis of the observed rainfall and streamflow data indicates that all basins perform a range of dampening of the input rainfall signal. All basins except one had a very limited range of rainfall location index. Concurrent time series of observed radar rainfall estimates and observed streamflow are analyzed to avoid model-specific conclusions. The results indicate that one basin contains complexities that suggest the use of distributed modeling approach. Furthermore, the analyses of observed data support the calibrated results from a distributed model.

Using the same data set as in Part I of this series (obtained from a dense rain gauge network operating with 15 s sampling over the period September 1991 to December 1993), the rainfall climatology of Norfolk Island is examined in terms of the relationship between rainfall accumulations R and fractional time raining (FTR) F . In addition, linear, cubic and power-law fits to the relationship are used as estimators and compared with actual monthly mean rainfall totals at integrations of 1 h, 1 day and 1 month. Cumulative errors in estimated island mean rainfall are compared as a function of both the monthly march and as a function of increasing F . It is shown that F may be used successfully to provide estimates of rainfall accumulation even at the shortest integration time (when the R – F relationship is non-linear) by way of the cubic fit. Temporal variations in the fitting parameters are found to be similar at each gauge site and spatial differences in fitting parameters between sites are systematic. On a year-by-year basis, the mean rainfall per FTR was found to decrease at all sites. A high correlation between the two parameters of the power-law fitting method enables R and F to be related using a single-parameter, though estimates are generally inferior to the linear or cubic fit. Using integration times of 12, 24, 48 and 96 h, the data set was sampled at intervals between 15 s (the present gauge array resolution) and 1800 s (a typical remote sensing resolution) providing R – F correlation trends and mean rainfall per unit F . Outputs from the sampling could not be normalized in terms of an integration-to-sampling time ratio.

Knowledge of the relationship between rainfall intensity and kinetic energy and its variations in time and space is important for erosion prediction. However, between studies considerable variations exist in the reported shape and coefficients of this relationship. Some differences can be explained by methods of measurement and interpretation and sample size, range and bias, while part of the variability corresponds to actual differences in rainfall generating mechanisms. The present paper critically reviews published studies of rainfall intensity and kinetic energy with a view to derive a general predictive equation of an exponential form. The performance of this general equation is compared to that of existing equations using measured rainfall intensity and kinetic energy data for a site in southeastern Australia. It appeared that the energy of individual storms could only be predicted with limited accuracy because of natural variations in rainfall characteristics. By and large, the general equation produced energy estimates that were within 10% of predictions by a range of parameterisations of the exponential model fitted to specific data-sets. Re-calculation of rainfall erosivity factors as obtained by the older and revised USLE approaches does not seem warranted for most locations. However, in regions experiencing strong oceanic influence or at high elevations, overall rainfall energy appears to be considerably lower than predicted by the general or USLE equations. Conversely, data collected at semi-arid to sub-humid locations suggest that rainfall energy may be higher than expected under those conditions. Standardised measurements are needed to evaluate rainfall intensity–kinetic energy relationships for such areas.

This paper deals with the sensitivity of distributed hydrological models to different patterns that account for the spatial distribution of rainfall: spatially averaged rainfall or rainfall field. The rainfall data come from a dense network of recording rain gauges that cover approximately 2000 km 2 around Mexico City. The reference rain sample accounts for the 50 most significant events, whose mean duration is about 10 h and maximal point depth 170 mm. Three models were tested using different runoff production models: storm-runoff coefficient, complete or partial interception. These models were then applied to four fictitious homogeneous basins, whose sizes range from 20 to 1500 km 2 . For each test, the sensitivity of the model is expressed as the relative differences between the empirical distribution of the peak flows (and runoff volumes), calculated according to the two patterns of rainfall input: uniform or non-uniform. Differences in flows range from 10 to 80%, depending on the type of runoff production model used, the size of the basin and the return period of the event. The differences are generally moderate for extreme events. In the local context, this means that uniform design rainfall combining point rainfall distribution and the probabilistic concept of the areal reduction factor could be sufficient to estimate major flood probability. Differences are more significant for more frequent events. This can generate problems in calibrating the hydrological model when spatial rainfall localization is not taken into account: a bias in the estimation of parameters makes their physical interpretation difficult and leads to overestimation of extreme flows.

A stochastic spatial rainfall generator is developed for use at the small spatial scale of urban and small hydrographic catchments. The generator is based on a spatial rainfall model of the conceptual and hierarchical type. It describes the spatial rainfall field in a macroscopic physically-based way by distinguishing rainfall entities with different scales: rain cells, cell clusters, small and large mesoscale areas (or rain storms). For applications at small spatial scales, the individual rain cells need a detailed description. Data of a dense network of rain gauges at Antwerp, enclosing 5940 rain cells in 807 rain storms are used to derive such description. For separation of the rain cells in the rainfall time series, an algorithm is developed based on the identification of increasing and decreasing rain cell flanks. The rain cells observed at different rain gauges are linked together by applying criteria for testing the similarity in rain cell properties. After separating and linking the rain cells and storms, the spatial rainfall model is calibrated to many storms by two methods (e.g. Kalman filter). The derived model structure and model parameter distributions apply to the stochastic generation of long-term time series of spatial rainfall. The model is tested by comparing intensity–duration–frequency relationships and temporal scaling properties of the generated and historical rainfall series.