Conditional simulation of remotely sensed rainfall data using a non-Gaussian v-transformed copula

Quantification of rainfall and its spatial and temporal variability is extremely important for reliable hydrological and meteorological modeling. While rain gauge measurements do not provide reasonable areal representation of rainfall, remotely sensed precipitation estimates offer much higher spatial resolution. However, uncertainties associated with remotely sensed rainfall estimates are not well quantified. This issue is important considering the fact that uncertainties in input rainfall are the main sources of error in hydrologic processes. Using an ensemble of rainfall estimates that resembles multiple realizations of possible true rainfall, one can assess uncertainties associated with remotely sensed rainfall data. In this paper, ensembles are generated by imposing rainfall error fields over remotely sensed rainfall estimates. A non-Gaussian copula-based model is introduced for simulation of rainfall error fields. The v-transformed copula is employed to describe the dependence structure of rainfall error estimates without the influence of the marginal distribution. Simulations using this model can be performed unconditionally or conditioned on ground reference measurements such that rain gauge data are honored at their locations. The presented model is implemented for simulation of rainfall ensembles across the Little Washita watershed, Oklahoma. The results indicate that the model generates rainfall fields with similar spatio-temporal characteristics and stochastic properties to those of observed rainfall data.     

Modelling bivariate extreme precipitation distribution for data‐scarce regions using Gumbel–Hougaard copula with maximum entropy estimation

A new method of parameter estimation in data scarce regions is valuable for bivariate hydrological extreme frequency analysis. This paper proposes a new method of parameter estimation (maximum entropy estimation, MEE) for both Gumbel and Gumbel–Hougaard copula in situations when insufficient data are available. MEE requires only the lower and upper bounds of two hydrological variables. To test our new method, two experiments to model the joint distribution of the maximum daily precipitation at two pairs of stations on the tributaries of Heihe and Jinghe River, respectively, were performed and compared with the method of moments, correlation index estimation, and maximum likelihood estimation, which require a large amount of data. Both experiments show that for the Ye Niugou and Qilian stations, the performance of MEE is nearly identical to those of the conventional methods. For the Xifeng and Huanxian stations, MEE can capture information indicating that the maximum daily precipitation at the Xifeng and Huanxian stations has an upper tail dependence, whereas the results generated by correlation index estimation and maximum likelihood estimation are unreasonable. Moreover, MEE is proved to be generally reliable and robust by many simulations under three different situations. The Gumbel–Hougaard copula with MEE can also be applied to the bivariate frequency analysis of other extreme events in data‐scarce regions.     

Data‐based analysis of bivariate copula tail dependence for drought duration and severity

In recent decades, copula functions have been applied in bivariate drought duration and severity frequency analysis. Among several potential copulas, Clayton has been mostly used in drought analysis. In this research, we studied the influence of the tail shape of various copula functions (i.e. Gumbel, Frank, Clayton and Gaussian) on drought bivariate frequency analysis. The appropriateness of Clayton copula for the characterization of drought characteristics is also investigated. Drought data are extracted from standardized precipitation index time series for four stations in Canada (La Tuque and Grande Prairie) and Iran (Anzali and Zahedan). Both duration and severity data sets are positively skewed. Different marginal distributions were first fitted to drought duration and severity data. The gamma and exponential distributions were selected for drought duration and severity, respectively, according to the positive skewness and Kolmogorov–Smirnov test. The results of copula modelling show that the Clayton copula function is not an appropriate choice for the used data sets in the current study and does not give more drought risk information than an independent model for which the duration and severity dependence is not significant. The reason is that the dependence of two variables in the upper tail of Clayton copula is very weak and similar to the independent case, whereas the observed data in the transformed domain of cumulative density function show high association in the upper tail. Instead, the Frank and Gumbel copula functions show better performance than Clayton function for drought bivariate frequency analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

Asymmetric copula in multivariate flood frequency analysis

The univariate flood frequency analysis is widely used in hydrological studies. Often only flood peak or flood volume is statistically analyzed. For a more complete analysis the three main characteristics of a flood event i.e. peak, volume and duration are required. To fully understand these variables and their relationships, a multivariate statistical approach is necessary. The main aim of this paper is to define the trivariate probability density and cumulative distribution functions. When the joint distribution is known, it is possible to define the bivariate distribution of volume and duration conditioned on the peak discharge. Consequently volume–duration pairs, statistically linked to peak values, become available. The authors build trivariate joint distribution of flood event variables using the fully nested or asymmetric Archimedean copula functions. They describe properties of this copula class and perform extensive simulations to highlight differences with the well-known symmetric Archimedean copulas. They apply asymmetric distributions to observed flood data and compare the results those obtained using distributions built with symmetric copula and the standard Gumbel Logistic model.     

Modelling radar-rainfall estimation uncertainties using elliptical and Archimedean copulas with different marginal distributions

Abstract

Given that radar-based rainfall has been broadly applied in hydrological studies, quantitative modelling of its uncertainty is critically important, as the error of input rainfall is the main source of error in hydrological modelling. Using an ensemble of rainfall estimates is an elegant solution to characterize the uncertainty of radar-based rainfall and its spatial and temporal variability. This paper has fully formulated an ensemble generator for radar precipitation estimation based on the copula method. Each ensemble member is a probable realization that represents the unknown true rainfall field based on the distribution of radar rainfall (RR) error and its spatial error structure. An uncertainty model consisting of a deterministic component and a random error factor is presented based on the distribution of gauge rainfall conditioned on the radar rainfall (GR|RR). Two kinds of copulas (elliptical and Archimedean copulas) are introduced to generate random errors, which are imposed by the deterministic component. The elliptical copulas (e.g. Gaussian and t-copula) generate the random errors based on the multivariate distribution, typically of decomposition of the error correlation matrix using the LU decomposition algorithm. The Archimedean copulas (e.g. Clayton and Gumbel) utilize the conditional dependence between different radar pixels to obtain random errors. Based on those, a case application is carried out in the Brue catchment located in southwest England. The results show that the simulated uncertainty bands of rainfall encompass most of the reference raingauge measurements with good agreement between the simulated and observed spatial dependences. This indicates that the proposed scheme is a statistically reliable method in ensemble radar rainfall generation and is a useful tool for describing radar rainfall uncertainty.

Editor D. Koutsoyiannis; Associate editor S. Grimaldi     

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.     

Use of a nonstationary copula to predict future bivariate low flow frequency in the Connecticut river basin

Frequency analysis of streamflow provides an essential ingredient in our understanding of hydrologic events and provides needed guidance in the design and management of water resources infrastructure. However, traditional hydrologic approaches often fail to include important external effects that can result in unpredictable or unforeseen changes in streamflow. Moreover, previous studies investigating multiple characteristics of streamflow do not address a nonstationary approach. This study explores nonstationary frequency analysis of bivariate characteristics, including occurrence and severity, of annual low flow in the Connecticut River Basin, United States. To investigate bivariate low flow frequency, copulas and their marginal distributions are constructed by using stationary and nonstationary approaches. Our study results indicate that streamflow used in this study demonstrate significant nonstationarity. Over time, the occurrence and severity of low flows are shown to be lower with the same probability based on the results of nonstationary copulas. Bivariate low flow frequencies in the years 1970, 2000, and 2030, and their joint return periods are estimated under the nonstationary copulas. Copyright © 2016 John Wiley & Sons, Ltd.     

An advanced data collection procedure in bivariate drought frequency analysis

In recent years, the bivariate frequency analysis of drought duration and severity using independent drought events and copula functions has been under extensive application. Meanwhile, emphasis on the procedure of independent drought data collection leads to the omission of the actual potential of short-term extreme droughts within a long-term independent drought. However, a long-term individual continuous drought as an Unconnected Drought Runs can be considered as a combination of short-term Connected Drought Runs. Thus, an advanced and new procedure of data collection in the bivariate drought characteristics analysis has been developed in this study. The results indicated a high relative advantage of the new proposed procedure in analysing bivariate drought characteristics (i.e., drought duration and severity frequency analysis). This advantage has been reflected in the more appropriate determination of the best copula and significant reduction in the uncertainty of bivariate drought frequency analysis.     

Spatial rainfall model using a pattern classifier for estimating missing daily rainfall data

Missing data in daily rainfall records are very common in water engineering practice. However, they must be replaced by proper estimates to be reliably used in hydrologic models. Presented herein is an effort to develop a new spatial daily rainfall model that is specifically intended to fill in gaps in a daily rainfall dataset. The proposed model is different from a convectional daily rainfall generation scheme in that it takes advantage of concurrent measurements at the nearby sites to increase the accuracy of estimation. The model is based on a two-step approach to handle the occurrence and the amount of daily rainfalls separately. This study tested four neural network classifiers for a rainfall occurrence processor, and two regression techniques for a rainfall amount processor. The test results revealed that a probabilistic neural network approach is preferred for determining the occurrence of daily rainfalls, and a stepwise regression with a log-transformation is recommended for estimating daily rainfall amounts.     

Flood prediction using parameters calibrated on limited discharge data and uncertain rainfall scenarios

ABSTRACT

Discharge observations and reliable rainfall forecasts are essential for flood prediction but their availability and accuracy are often limited. However, even scarce data may still allow adequate flood forecasts to be made. Here, we explored how far using limited discharge calibration data and uncertain forcing data would affect the performance of a bucket-type hydrological model for simulating floods in a tropical basin. Three events above thresholds with a high and a low frequency of occurrence were used in calibration and 81 rainfall scenarios with different degrees of uncertainty were used as input to assess their effects on flood predictions. Relatively similar model performance was found when using calibrated parameters based on a few events above different thresholds. Flood predictions were sensitive to rainfall errors, but those related to volume had a larger impact. The results of this study indicate that a limited number of events can be useful for predicting floods given uncertain rainfall forecasts.     

Modelling diurnal cycles in point rainfall properties

A hybrid model for point rainfall has been explored to model the diurnal cycles in rainfall properties. The hybrid model is a product of two random processes: an occurrence process and an intensity process. Two occurrence process models, first‐order Markov chain and periodic discrete autoregressive, were compared initially. Fourier series was fitted to the properties of the occurrence and intensity processes of the observed data in order to reduce the number of model parameters. The Bayesian and Akaike information criteria were used to identify the optimum number of harmonics of the Fourier series. Simulation results of the two hybrid models were similar, if not identical, and compared well with the observed. In the average sense, the introduction of diurnal cycles in the model parameters did not improve the reproduction of the observed aggregation properties of the occurrence process. However, the diurnal distributions of the aggregation statistics were significantly improved by increasing the order of the Markov chain model. Also the information criteria tend to favour higher than first‐order Markov chain models. Copyright © 2001 John Wiley & Sons, Ltd.     

联结(copula)函数在概率地震危险性分析中的应用

介绍了一种成熟的已广泛应用于金融领域的估计不确定性的方法——copula函数方法, 并推广了n重的Frankrsquo;s copulas; 在实际应用中, 本文采用了俞言祥和美国西部由Boore等人给出的两个衰减模型, 针对其中存在的模型不确定性以及它们之间的相互依赖性, 构造出概率意义上联合的copula分布函数, 并将其应用于实例分析. 结果表明, 对比于传统逻辑树中所用的线性结合方法, copula将两者带来的概率分布写成一个联合概率分布, 能够很好地考虑双方不尽相同的意见. 另外, 由于copula函数可采用各种各样的边际分布函数来获得联合概率分布, 且在金融风险投资评估已有大量的应用, 因此在现代地震危险性评估中将有着广泛的前景.      

Evaluation and comparison of interpolated gauge rainfall data and gridded rainfall data in Florida,USA

A variety of spatially continuous rainfall products are available but little evaluation of their accuracy has been published for areas with high spatial variability in rainfall. Five gridded rainfall products (PRISM, RTMA, and the interpolated Florida Automated Weather Network, FAWN, rainfall layers based on three interpolated methods) were assessed for Florida State. Point-to-pixel and pixel-to-pixel comparisons were performed to compare the five products. On average, the PRISM and RTMA products resulted in a better fit with the daily FAWN rainfall datasets, while FAWN-based interpolated products resulted in a better fit with the monthly FAWN rainfall datasets based on point-to-pixel analysis. Inverse distance weighting and ordinary kriging methods performed slightly better than the thin plate spline method in predicting daily rainfall. In general, monthly and seasonal rainfall amounts from PRISM and RTMA products were higher and lower, respectively, than reference rainfall amounts from FAWN gauge stations and FAWN-based interpolated products.     

Measuring drop size distribution and kinetic energy of rainfall using a force transducer

The relatively high cost of commercially available raindrop spectrometers and disdrometers has inhibited detailed and intensive research on drop size distribution, kinetic energy and momentum of rainfall which are important for understanding and modelling soil erosion caused by raindrop detachment. In this study, an approach to find the drop size distribution, momentum and kinetic energy of rainfall using a relatively inexpensive device that uses a piezoelectric force transducer for sensing raindrop impact response is introduced. The instrument continuously and automatically records, on a time‐scale, the amplitude of electrical pulses produced by the impact of raindrops on the surface of the transducer. The size distribution of the raindrops and their respective kinetic energy are calculated by analysing the number and amplitude of pulses recorded, and from the measured volume of total rainfall using a calibration curve. Simultaneous measurements of the instrument, a rain gauge and a dye‐stain method were used to assess the performance of the instrument. Test results from natural and simulated rainfalls are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Bivariate flood frequency analysis using the copula function: a case study of the Litija station on the Sava River

As an alternative to the commonly used univariate flood frequency analysis, copula frequency analysis can be used. In this study, 58 flood events at the Litija gauging station on the Sava River in Slovenia were analysed, selected based on annual maximum discharge values. Corresponding hydrograph volumes and durations were considered. Different bivariate copulas from three families were applied and compared using different statistical, graphical and upper tail dependence tests. The parameters of the copulas were estimated using the method of moments with the inversion of Kendall's tau. The Gumbel–Hougaard copula was selected as the most appropriate for the pair of peak discharge and hydrograph volume (Q‐V). The same copula was also selected for the pair hydrograph volume and duration (V‐D), and the Student‐t copula was selected for the pair of peak discharge and hydrograph duration (Q‐D). The differences among most of the applied copulas were not significant. Different primary, secondary and conditional return periods were calculated and compared, and some relationships among them were obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Entropy,consistency in rainfall distribution and potential water resource availability in Australia

Understanding the variability in monthly rainfall amounts is important for the management of water resources. We use entropy, a measure of variability, to quantify the rainfall variability in Australia. We define the entropy of stable rainfall (ESR) to measure the long‐term average rainfall variability across the months of the year. The stations in northern Australia observe substantially more variability in rainfall distributions and stations in southern Australia observe less variability in rainfall distribution across the months of the year. We also define the consistency index (CI) to compare the distribution of the monthly rainfall for a given year with the long‐term average monthly rainfall distribution. Higher value of the CI indicates the rainfall in the year is consistent with the overall long‐term average rainfall distribution. Areas close to the coastline in northern, southern and eastern Australia observe more consistent rainfall distribution in individual years with the long‐term average rainfall distribution. For the studied stations, we categorize the years into different potential water resource availability on the basis of annual rainfall amount and CI. For almost all Australian rainfall stations, El Niño years have a greater risk of having below median and relatively inconsistent rainfall distribution than La Niña years. The results may be helpful for developing area‐specific water usage strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

A new seasonal design flood method based on bivariate joint distribution of flood magnitude and date of occurrence

Abstract

Seasonal design floods which consider information on seasonal variation are very important for reservoir operation and management. The seasonal design flood method currently used in China is based on seasonal maximum (SM) samples and assumes that the seasonal design frequency is equal to the annual design frequency. Since the return period associated with annual maximum floods is taken as the standard in China, the current seasonal design flood cannot satisfy flood prevention standards. A new seasonal design flood method, which considers dates of flood occurrence and magnitudes of the peaks (runoff), was proposed and established based on copula function. The mixed von Mises distribution was selected as marginal distribution of flood occurrence dates. The Pearson Type III and exponential distributions were selected as the marginal distribution of flood magnitude for annual maximum flood series and peak-over-threshold samples, respectively. The proposed method was applied at the Geheyan Reservoir, China, and then compared with the currently used seasonal design flood methods. The case study results show that the proposed method can satisfy the flood prevention standard, and provide more information about the flood occurrence probabilities in each sub-season. The results of economic analysis show that the proposed design flood method can enhance the floodwater utilization rate and give economic benefits without lowering the annual flood protection standard.

Citation Chen, L., Guo, S. L., Yan, B. W., Liu, P. & Fang, B. (2010) A new seasonal design flood method based on bivariate joint distribution of flood magnitude and date of occurrence. Hydrol. Sci. J. 55(8), 1264–1280.     

Application of an optical disdrometer to characterize simulated rainfall and measure drop-size distribution

ABSTRACT

Knowledge of rainfall characteristics such as drop-size distribution is essential for the development of erosion-mitigation strategies and models. This research used an optical disdrometer to elucidate the relationships between raindrop-size distribution, median volume drop diameter (D₅₀), kinetic energy and radar reflectivity (dBz) of simulated rainfall of different intensities. The D₅₀ values were higher for the simulated rain than for natural rain at almost all rainfall intensities, perhaps due to variations in rainfall types and the turbulence in natural rain that breaks up large drops. The kinetic energy ranged from 26.67 to 5955.51 J m^?2 h ^?1, while the median volume drop diameter (D₅₀) was in the range 1.94–7.25 mm, for intensities between 1.5 and 202.6 mm h^?1. The relationship between radar reflectivity (Z) and the intensity (R) of the simulated rain was best described by a power law function (Z = aR^b), with a and b coefficients in the ranges 162–706 and 0.94–2.46, respectively, throughout the range of rainfall intensities (1.5–202.6 mm h^?1).     

Stemflow production and intrastorm rainfall intensity variation: an experimental analysis using laboratory rainfall simulation

Laboratory rainfall simulation experiments using a small artificial olive tree are used to show that the fraction of rain falling at a constant intensity that becomes stemflow rises from 9% at 2.5 mm/h to 36.1% at 35 mm/h. Natural rainfall events commonly exhibit wide fluctuations of intensity. Simulated rainfall events each having a mean intensity of 10 mm/h, but containing short intensity peaks of 20 to 100 mm/h at varying intra‐event positions, were used to explore the effect of varying intensity profiles. Results demonstrate that changes in rainfall event profile are associated with wide variation in stemflow flux, stemflow volume and stemflow fraction. When applied to an initially dry plant, rainfall events with a late intensity peak yielded an average peak stemflow flux up to 188% larger than events of contrasting profile, such as early peak events. The increase was smaller, up to 141%, when rain was applied to plants that were already partially wet, but was again found in events with a late intensity peak. Moreover, such events yielded a peak stemflow flux up to approximately seven times larger than comparable events of uniform intensity. Likewise, changing event profile with no change in rainfall depth was associated with a maximum stemflow fraction that was 31% larger than theminimum stemflow fraction, and a maximum stemflow volume that was nearly 37% larger than the minimum stemflow volume. These results suggest that rainfall event profile exerts a significant effect on all of the studied stemflow parameters. It is hypothesized that this is a consequence of the way in which intensity profile affects the rate of wetting‐up of trickle pathways on the plant, and variation in the time taken for these pathways to become fully connected. Event profile must therefore be considered along with plant architecture in seeking to understand stemflow. Copyright © 2014 John Wiley & Sons, Ltd.     

Copula-based mixed models for bivariate rainfall data: an empirical study in regression perspective

A comprehensive parametric approach to study the probability distribution of rainfall data at scales of hydrologic interest (e.g. from few minutes up to daily) requires the use of mixed distributions with a discrete part accounting for the occurrence of rain and a continuous one for the rainfall amount. In particular, when a bivariate vector (X, Y) is considered (e.g. simultaneous observations from two rainfall stations or from two instruments such as radar and rain gauge), it is necessary to resort to a bivariate mixed model. A quite flexible mixed distribution can be defined by using a 2-copula and four marginals, obtaining a bivariate copula-based mixed model. Such a distribution is able to correctly describe the intermittent nature of rainfall and the dependence structure of the variables. Furthermore, without loss of generality and with gain of parsimony this model can be simplified by some transformations of the marginals. The main goals of this work are: (1) to empirically explore the behaviour of the parameters of marginal transformations as a function of time scale and inter-gauge distance, by analysing data from a network of rain gauges; (2) to compare the properties of the regression curves associated to the copula-based mixed model with those derived from the model simplified by transformations of the marginals. The results from the investigation of transformations’ parameters are in agreement with the expected theoretical dependence on inter-gauge distance, and show dependence on time scale. The analysis on the regression curves points out that: (1) a copula-based mixed model involves regression curves quite close to some non-parametric models; (2) the performance of the parametric regression decreases in the same cases in which non-parametric regression shows some instability; (3) the copula-based mixed model and its simplified version show similar behaviour in term of regression for mid-low values of rainfall. An erratum to this article can be found at