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.     

Propagation of uncertainty from rainfall to runoff: A case study with a stochastic rainfall generator

We explore the impact of uncertainties in the spatial–temporal distribution of rainfall on the prediction of peak discharge in a typical mountain basin. To this end, we use a stochastic generator previously developed for rainfall downscaling, and we estimate the basin response by adopting a semi-distributed hydrological model. The results of the analysis provide information on the minimum rainfall resolution needed for operational flood forecasting, and confirm the sensitivity of peak discharge estimates to errors in the determination of the power spectrum of the precipitation field.     

A geostatistical approach for areal rainfall statistics assessment

Areal rainfall statistics are more relevant in flood hydrology and water resources management than point rainfall statistics when it comes to help designing dams or hydraulic structures. This paper presents a geostatistically based method to derive the areal statistics from point statistics. Assuming that the distribution models of point rainfall and areal belong to the same class of models and that the rainfall process is stationary, it is shown how the parameters of the areal distribution model can directly be computed from the parameters of the point distribution models in case of a non stationary process, an approximation is derived that yielded good results when applied to a mountainous region in Southern France. The method also allows the computation of the areal reduction factors in a very general form.     

A geostatistical approach for areal rainfall statistics assessment

Areal rainfall statistics are more relevant in flood hydrology and water resources management than point rainfall statistics when it comes to help designing dams or hydraulic structures. This paper presents a geostatistically based method to derive the areal statistics from point statistics. Assuming that the distribution models of point rainfall and areal belong to the same class of models and that the rainfall process is stationary, it is shown how the parameters of the areal distribution model can directly be computed from the parameters of the point distribution models in case of a non stationary process, an approximation is derived that yielded good results when applied to a mountainous region in Southern France. The method also allows the computation of the areal reduction factors in a very general form.     

Greatest observed one-day point and areal rainfall of India

The greatest one-day rain amounts recorded at individual stations in the country during the last 41-year period from 1940 onwards were examined for all observatories as well as State rain-gauge stations in an attempt to bring out up-to-date information on the greatest recorded point rainfall for the duration of one day. Outstanding one-day point rainfall amounts recorded prior to 1940 were also examined and have been included in this note along with their date and year of occurrence by way of comparison. A generalized chart has been prepared based on the percentage ratios of the greatest one-day rainfall to the mean annual rainfall of about 300 observatory stations distributed uniformly over the entire country. On the basis of Depth-Area-Duration (DAD) analyses of the most severe rainstorms which occurred over different plain areas of the country, it has been found that the 2 July, 1941, rainstorm gave the highest areal rain depths in the country for different areas. Comparison with similar areal rain depths of the tropical USA has shown that rain depths of the July, 1941, rainstorm were higher for all areas excepting the areas of 500 sq. miles (1295 sq. km) and 1000 sq. miles (2590 sq. km).     

基于面平均雨量误差修正的实时洪水预报修正方法

空间集总式水文模型的洪水预报精度会受到面平均雨量估计误差的严重影响.点雨量监测值的误差类型、误差大小以及流域的雨量站点密度和站点的空间分布都会影响到面平均雨量的计算.为提高实时洪水预报精度,本文提出了一种基于降雨系统响应曲线洪水预报误差修正方法.通过此方法估计降雨输入项的误差,从而提高洪水预报精度.此方法将水文模型做为输入和输出之间的响应系统,用实测流量和计算流量之间的差值做为信息,通过降雨系统响应曲线,使用最小二乘估计原理,对面平均雨量进行修正,再用修正后的面平均雨量重新计算出流过程.将此修正方法结合新安江模型使用理想案例进行检验,并应用于王家坝流域的16场历史洪水以及此流域不同雨量站密度的情况下,结果证明均有明显修正效果,且在雨量站密度较低时修正效果更加明显.该方法是一种结构简单且不增加模型参数和复杂度的实时洪水修正的新方法.     

The effect of estimating areal rainfall using self‐similarity topography method on the simulation accuracy of runoff prediction

Rainfall input for hydrologic modelling is assumed uniformly distributed over the entire catchment. This can lead to significant errors. Investigations of areal rainfall in mountain areas are typically limited by a lack of adequate meteorological and hydrogeological records. This study focuses on areal rainfall in mountain areas within the Kaidu River Basin, China, with the aim of analyzing the influence of areal rainfall on the simulation accuracy of runoff prediction. We conducted a simulation using MIKE 11/NAM rainfall‐runoff model over 92 days of the rain season and compared the simulation error in different methods. On the basis of properties of self‐similarity degree (SSD) in analyzing the detailed characteristics of terrain, areal rainfall was calculated to model the runoff. The results of the model simulations are generally consistent with observed data, indicating that the self‐similarity topography method is able to reflect the spatial change of rainfall. This indicates that the proposed methodology is applicable for the management of water resources in mountain area. The modelling and self‐similarity topography method study allowed quantification of the spatial rainfall and provided an insight into their implications in hydrological forecasting. Copyright © 2011 John Wiley & Sons, Ltd.     

Nonlinear ensemble prediction of chaotic daily rainfall

The significance of treating rainfall as a chaotic system instead of a stochastic system for a better understanding of the underlying dynamics has been taken up by various studies recently. However, an important limitation of all these approaches is the dependence on a single method for identifying the chaotic nature and the parameters involved. Many of these approaches aim at only analyzing the chaotic nature and not its prediction. In the present study, an attempt is made to identify chaos using various techniques and prediction is also done by generating ensembles in order to quantify the uncertainty involved. Daily rainfall data of three regions with contrasting characteristics (mainly in the spatial area covered), Malaprabha, Mahanadi and All-India for the period 1955–2000 are used for the study. Auto-correlation and mutual information methods are used to determine the delay time for the phase space reconstruction. Optimum embedding dimension is determined using correlation dimension, false nearest neighbour algorithm and also nonlinear prediction methods. The low embedding dimensions obtained from these methods indicate the existence of low dimensional chaos in the three rainfall series. Correlation dimension method is done on the phase randomized and first derivative of the data series to check whether the saturation of the dimension is due to the inherent linear correlation structure or due to low dimensional dynamics. Positive Lyapunov exponents obtained prove the exponential divergence of the trajectories and hence the unpredictability. Surrogate data test is also done to further confirm the nonlinear structure of the rainfall series. A range of plausible parameters is used for generating an ensemble of predictions of rainfall for each year separately for the period 1996–2000 using the data till the preceding year. For analyzing the sensitiveness to initial conditions, predictions are done from two different months in a year viz., from the beginning of January and June. The reasonably good predictions obtained indicate the efficiency of the nonlinear prediction method for predicting the rainfall series. Also, the rank probability skill score and the rank histograms show that the ensembles generated are reliable with a good spread and skill. A comparison of results of the three regions indicates that although they are chaotic in nature, the spatial averaging over a large area can increase the dimension and improve the predictability, thus destroying the chaotic nature.     

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

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

Evaluating spatio-temporal representations in daily rainfall sequences from three stochastic multi-site weather generation approaches

Many hydrological and agricultural studies require simulations of weather variables reflecting observed spatial and temporal dependence at multiple point locations. This paper assesses three multi-site daily rainfall generators for their ability to model different spatio-temporal rainfall attributes over the study area. The approaches considered consist of a multi-site modified Markov model (MMM), a reordering method for reconstructing space–time variability, and a nonparametric k-nearest neighbour (KNN) model. Our results indicate that all the approaches reproduce adequately the observed spatio-temporal pattern of the multi-site daily rainfall. However, different techniques used to signify longer time scale observed temporal and spatial dependences in the simulated sequences, reproduce these characteristics with varying successes. While each approach comes with its own advantages and disadvantages, the MMM 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. The reordering method is simple and intuitive and produces good results. However, it is primarily driven by the reshuffling of the simulated values across realisations and therefore may not be suited in applications where data length is limited or in situations where the simulation process is governed by exogenous conditioning variables. For example, in downscaling studies where KNN and MMM can be used with confidence.     

A fine-scale point process model of rainfall with dependent pulse depths within cells

Abstract

A cluster point process model is considered for the analysis of fine-scale rainfall time series. The model is based on three Poisson processes. The first is a Poisson process of storm origins, where each storm has a random (exponential) lifetime. The second is a Poisson process of cell origins that occur during the storm lifetime, terminating when the storm finishes. Each cell has a random lifetime that follows an exponential distribution (or terminates when the storm terminates, whichever occurs first). During cell lifetimes, a third Poisson process of instantaneous pulses occurs. The model is essentially an extension of the well-known Bartlett-Lewis rectangular pulses model, with the rectangular profiles replaced with a Poisson process of instantaneous pulse depths to ensure more realistic rainfall profiles for fine-scale series. Model equations, derived in Cowpertwait et al. (2007 Cowpertwait, P., Isham, V. and Onof, C. 2007. Point process models of rainfall: developments for fine-scale structure. Proceedings of the Royal Society of London, Series A, 463: 2569–2587. [Crossref], [Web of Science ®] , [Google Scholar]), are used to fit different sets of properties to a 60 year record of 5-min data taken from Kelburn, New Zealand. As in the previous work, two superposed processes are used to account for two main and distinct precipitation types (convective and stratiform). By treating the within-cell pulses as dependent random variables, it is found, by simulation, that improved fits to extreme values and the proportion of dry intervals are obtained.

Citation Cowpertwait, P. S. P., Xie, G., Isham, V., Onof, C. & Walsh, D. C. I. (2011) A fine-scale point process model of rainfall with dependent pulse depths within cells. Hydrol. Sci. J. 56(7), 1110–1117.     

A cascade approach to continuous rainfall data generation at point locations

High-resolution temporal rainfall data sequences serve as inputs for a range of applications in planning, design and management of small (especially urban) water resources systems, including continuous flow simulation and evaluation of alternate policies for environmental impact assessment. However, such data are often not available, since their measurements are costly and time-consuming. One alternative to obtain high-resolution data is to try to derive them from available low-resolution information through a disaggregation procedure. This study evaluates a random cascade approach for generation of high-resolution rainfall data at a point location. The approach is based on the concept of scaling in rainfall, or, relating the properties associated with the rainfall process at one temporal scale to a finer-resolution scale. The procedure involves two steps: (1) identification of the presence of scaling behavior in the rainfall process; and (2) generation of synthetic data possessing same/similar scaling properties of the observed rainfall data. The scaling identification is made using a statistical moment scaling function, and the log–Poisson distribution is assumed to generate the synthetic rainfall data. The effectiveness of the approach is tested on the rainfall data observed at the Sydney Observatory Hill, Sydney, Australia. Rainfall data corresponding to four different successively doubled resolutions (daily, 12, 6, and 3 h) are studied, and disaggregation of data is attempted only between these successively doubled resolutions. The results indicate the presence of multi-scaling behavior in the rainfall data. The synthetic data generated using the log–Poisson distribution are found to exhibit scaling behaviors that match very well with that for the observed data. However, the results also indicate that fitting the scaling function alone does not necessarily mean reproducing the broader attributes that characterize the data. This observation clearly points out the extreme caution needed in the application of the existing methods for identification of scaling in rainfall, especially since such methods are also prevalent in studies of the emerging satellite observations and thus in the broader spectrum of hydrologic modeling.     

A spatial–temporal point process model of rainfall for the Thames catchment, UK

A spatial–temporal point process model of rainfall is fitted to data taken from 23 sites across the Thames catchment, London. For the period January 1970 to December 1988 the sites have daily data, whilst for the period January 1989 to November 2003 the sites have hourly data. In addition, the records contain missing values. The fitted model is used to infill the missing values, to disaggregate the daily data to hourly data and to generate spatially consistent series at a further 25 sites across the catchment. The model is validated by considering statistical properties that are not used in the fitting procedure, which include extreme values, the proportion of dry intervals, and annual totals. Overall the performance of the model is good, with the exception of an over-estimation in cross-correlation between pairs of sites for the disaggregated series.     

A two steps disaggregation method for highly seasonal monthly rainfall

 The need for high resolution rainfall data at temporal scales varying from daily to hourly or even minutes is a very important problem in hydrology. For many locations of the world, rainfall data quality is very poor and reliable measurements are only available at a coarse time resolution such as monthly. The purpose of this work is to apply a stochastic disaggregation method of monthly to daily precipitation in two steps: 1. Initialization of the daily rainfall series by using the truncated normal model as a reference distribution. 2.␣Restructuring of the series according to various time series statistics (autocorrelation function, scaling properties, seasonality) by using a Markov chain Monte Carlo based algorithm. The method was applied to a data set from a rainfall network of the central plains of Venezuela, in where rainfall is highly seasonal and data availability at a daily time scale or even higher temporal resolution is very limited. A detailed analysis was carried out to study the seasonal and spatial variability of many properties of the daily rainfall as scaling properties and autocorrelation function in order to incorporate the selected statistics and their annual cycle into an objective function to be minimized in the simulation procedure. Comparisons between the observed and simulated data suggest the adequacy of the technique in providing rainfall sequences with consistent statistical properties at a daily time scale given the monthly totals. The methodology, although highly computationally intensive, needs a moderate number of statistical properties of the daily rainfall. Regionalization of these statistical properties is an important next step for the application of this technique to regions in where daily data is not available.     

A class of hidden Markov models for regional average rainfall

Abstract

Basic hidden Markov models are very useful in stochastic environmental research but their ability to accommodate sufficient dependence between observations is somewhat limited. However, they can be modified in several ways to form a rich class of flexible models that are useful in many environmental applications. We consider a class of hidden Markov models that incorporate additional dependence among observations to model average regional rainfall time series. The focus of the study is on models that introduce additional dependence between the state level and the observation level of the process and also on models that incorporate dependence at observation level. Construction of the likelihood function of the models is described along with the usual second-order properties of the process. The maximum likelihood method is used to estimate the parameters of the models. Application of the proposed class of models is illustrated in an analysis of daily regional average rainfall time series from southeast and southwest England for the winter season during 1931 to 2010. Models incorporating additional dependence between the state level and the observation level of the process captured the distributional properties of the daily rainfall well, while the models that incorporate dependence at the observation level showed their ability to reproduce the autocorrelation structure. Changes in some of the regional rainfall properties during the time period are also studied.

Editor D. Koutsoyiannis     

Stemflow on the woody parts of plants: dependence on rainfall intensity and event profile from laboratory simulations

This paper presents the first experimental study of how rainfall intensity and event profile affects stemflow behaviour on the rigid branches and stems of leafless, woody plants. Constant intensity rainfall simulation experiments showed that stemflow fraction rises with intensity. Varying intensity experiments showed that the stemflow fraction and stemflow flux vary with the rainfall event intensity profile and peak intensity. Stemflow fraction tends to be larger when intensity peaks occur early in the rainfall event, and variable intensity events exhibited peak stemflow fluxes >3 times those seen in constant intensity events. Moreover, experiments in which incident drop energy was reduced by a mesh screen suspended above the test plant commonly showed increases of >100% (and exceeding 300% under particular intensity profiles) in stemflow fraction, depth and peak stemflow flux. The results suggest that the development of trickle pathways along woody branches is facilitated by rain of moderate intensity and that splash dislodgement of attached water progressively reduces the adhesion of drops during intense rainfall. Thus, in plants with extensive woody branches, it is not merely rainfall intensity that determines stemflow fraction but the temporal variations in rainfall intensity. This offers a new explanation for increased stemflow production when trees are leafless, than when foliage is present, in terms of the reduced intensity peaks during rain in the dormant season. Copyright © 2013 John Wiley & Sons, Ltd.     

A stochastic model of low flows

A model is developed for annual low flow hydrographs. Its two primary components reflect the fact that hydrologic processes during streamflow rise (function of water input) and recession (function of basin storage) are different. Durations of periods of rise (wet intervals) and recession (dry intervals) are modelled by discrete probability distributions — negative binomial for dry intervals and negative binomial or modified logarithmic series for wet intervals depending on goodness of fit. During wet intervals, the total inflow is modelled by the lognormal distribution and daily amounts are allocated according to a pattern-averaged model. During dry intervals, the flow recedes according to a deterministic-stochastic recession model. The model was applied to three Canadian basins with drainage area ranging from 2210 to 22000 km² to generate 50 realizations of low flow hydrographs. The resulting two standard-error confidence band for the simulated probability distribution of annual minimum 7-day flows enclosed the probability distribution estimated from the observed record. A sensitivity analysis for the three basins revealed that in addition to the recession submodel, the most important submodel is that describing seasonality. The state of the basin at the beginning of the low flow period is of marginal importance and the daily distribution of input is unimportant.     

A stochastic model of low flows

A model is developed for annual low flow hydrographs. Its two primary components reflect the fact that hydrologic processes during streamflow rise (function of water input) and recession (function of basin storage) are different. Durations of periods of rise (wet intervals) and recession (dry intervals) are modelled by discrete probability distributions — negative binomial for dry intervals and negative binomial or modified logarithmic series for wet intervals depending on goodness of fit. During wet intervals, the total inflow is modelled by the lognormal distribution and daily amounts are allocated according to a pattern-averaged model. During dry intervals, the flow recedes according to a deterministic-stochastic recession model. The model was applied to three Canadian basins with drainage area ranging from 2210 to 22000 km² to generate 50 realizations of low flow hydrographs. The resulting two standard-error confidence band for the simulated probability distribution of annual minimum 7-day flows enclosed the probability distribution estimated from the observed record. A sensitivity analysis for the three basins revealed that in addition to the recession submodel, the most important submodel is that describing seasonality. The state of the basin at the beginning of the low flow period is of marginal importance and the daily distribution of input is unimportant.