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.     

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.     

Relationship between point rainfall, average sampled rainfall and ground truth at the event scale in the Sahel

Geostatistical techniques are used to quantify the reference mean areal rainfall (ground truth) from sparse raingaugenetworks. Based on the EPSAT-Niger event cumulative rainfall, a linear relationship between the ground truth considered as the mean area rainfall estimated from the densely available raingauge network and the area rainfall estimated from sparse network are derived. Also, a linear relationship between the ground truth and point rainfall is established. As it was reported experimentally by some authors, the slope of these relationships is less than one. Based on the geostatistical framework, the slope and the ordinate at the origin can be estimated as a function of the spatial structure of the rainfall process. It is shown that the slope is smaller than one. For the special case of one gauge inside a fixed area or a Field Of View (FOV), an areal reduction factor is derived. It has a limit value which depends only on the size of the area and the spatial structure of the rainfall process. The relative variance error of estimating the FOV cumulative rainfall from point rainfall is also given.     

Relationship between point rainfall, average sampled rainfall and ground truth at the event scale in the Sahel

Geostatistical techniques are used to quantify the reference mean areal rainfall (ground truth) from sparse raingaugenetworks. Based on the EPSAT-Niger event cumulative rainfall, a linear relationship between the ground truth considered as the mean area rainfall estimated from the densely available raingauge network and the area rainfall estimated from sparse network are derived. Also, a linear relationship between the ground truth and point rainfall is established. As it was reported experimentally by some authors, the slope of these relationships is less than one. Based on the geostatistical framework, the slope and the ordinate at the origin can be estimated as a function of the spatial structure of the rainfall process. It is shown that the slope is smaller than one. For the special case of one gauge inside a fixed area or a Field Of View (FOV), an areal reduction factor is derived. It has a limit value which depends only on the size of the area and the spatial structure of the rainfall process. The relative variance error of estimating the FOV cumulative rainfall from point rainfall is also given.     

Influence of rainfall spatial variability on flood prediction

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 km² 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 km². 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.     

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.     

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.     

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.     

Classification of rainfall into small- and large-scale events by statistical pattern recognition

Scale properties of daily areal rainfall in south-eastern Norway are investigated, and the events are classified into small- and large-scale events by statistical pattern recognition. The coefficient of variation and the standard deviation, computed from all precipitation gauges for one event, are considered to be mutually independent and are assigned gamma densities. The two statistical parameters form a feature vector which is used as the discriminatory variable in the classification algorithm. Empirical and theoretical semivariograms were fitted to the classified sets of events, and the parameters of the semivariogram (nugget, sill and range) were investigated for each set. Different values for all parameters were found, and the differences in range, describing the scale of the phenomena, were further investigated by the use of cross validation with the MSIE (mean squared interpolation error) criterion. Optimal range of daily small-scale precipitation was found to be 16 km (19 km without nugget effect), and 33 km (70 km without nugget effect) for daily large-scale precipitation. These optimal values for range, however, proved to be of small consequence for the estimated areal precipitation using kriging interpolation. Areal precipitation was computed for catchments of size 50 km² and 1200 km², and insignificant differences in the computed areal precipitation were found when applying the class-specific semivariograms instead of the mean semivariogram calculated from all events regardless of class. However, when areal reduction factors (ARFs) were derived from the analytical expressions of the class-specific semivariograms, considerable differences were found for the ARF curves of the two precipitation types. The difference became more marked for precipitation events of low probability of occurrence.     

Modelling point rainfall occurrences with the modified Bartlett-Lewis rectangular pulses model

A six parameter stochastic point process model, known as the modified Bartlett-Lewis Rectangular Pulses Model, is applied to fairly long hourly rainfall data recorded at Valentia (relatively a wet location) and Shannon Airport (relatively a dry location), Ireland. Five different sets of statistics of the rainfall data of each month, assuming local stationarity within the month, are used to estimate the parameters and to simulate model output. The problems of parameter stability/sensitivity and identification are discussed and it has been shown that the sensitivity of the model parameters to the choice of six statistics can be avoided by estimating the six parameters by optimization from 16 statistics namely mean, variance, lag-1 autocorrelation corfficient and proportion dry of hourly, 6-hourly, 12-hourly, and 24-hourly rainfalls. Some useful properties of the rainfall depth process are analysed using the notion of event-based statistics. The conditional distributions of rainfall depth and maximum intensity, mean event profiles, and various other features of the rainfall depth process obtained from the model simulated samples compare favourably with the historical ones.     

A watershed rainfall data recovery approach with application to distributed hydrological models

Watershed areal rainfall estimation, which is one of the most important and fundamental aspects in hydrological forecasting and various kinds of catchment‐scale hydrological models, is widely used in the analysis of hydrological regime change, and its precision has a direct influence on the accuracy of hydrological forecasting and hydrological simulation. In China, it is difficult to obtain the watershed areal rainfall estimate with reliable precision and avoid the phenomenon of ‘the same effect of different parameters’ because of the low density of the rain gauge network. Therefore, a watershed rainfall data recovery approach of improving the precision of watershed areal rainfall estimation is proposed here. This approach is to build new observatories, establish the time–space relations of rainfall between newly built observatories and previously built observatories in a relatively short interval and then recover the rainfall data of newly built observatories prior to their construction through simulating the relations over a longer time. As a result, watershed rainfall information could be elaborated to improve the precision of watershed areal rainfall estimate and avoid the phenomenon of ‘the same effect of different parameters’ to a certain degree in the process of hydrological simulation. The approach is used in the hydrological simulation of Hali River catchment. In combination with the Soil Water Assessment Tool model, a better result can be obtained in the hydrological simulation. Therefore, the approach can be used in other similar catchments. Copyright © 2011 John Wiley & Sons, Ltd.     

Estimation and spatial interpolation of rainfall intensity distribution from the effective rate of precipitation

Great emphasis is being placed on the use of rainfall intensity data at short time intervals to accurately model the dynamics of modern cropping systems, runoff, erosion and pollutant transport. However, rainfall data are often readily available at more aggregated level of time scale and measurements of rainfall intensity at higher resolution are available only at limited stations. A distribution approach is a good compromise between fine-scale (e.g. sub-daily) models and coarse-scale (e.g. daily) rainfall data, because the use of rainfall intensity distribution could substantially improve hydrological models. In the distribution approach, the cumulative distribution function of rainfall intensity is employed to represent the effect of the within-day temporal variability of rainfall and a disaggregation model (i.e. a model disaggregates time series into sets of higher solution) is used to estimate distribution parameters from the daily average effective precipitation. Scaling problems in hydrologic applications often occur at both space and time dimensions and temporal scaling effects on hydrologic responses may exhibit great spatial variability. Transferring disaggregation model parameter values from one station to an arbitrary position is prone to error, thus a satisfactory alternative is to employ spatial interpolation between stations. This study investigates the spatial interpolation of the probability-based disaggregation model. Rainfall intensity observations are represented as a two-parameter lognormal distribution and methods are developed to estimate distribution parameters from either high-resolution rainfall data or coarse-scale precipitation information such as effective intensity rates. Model parameters are spatially interpolated by kriging to obtain the rainfall intensity distribution when only daily totals are available. The method was applied to 56 pluviometer stations in Western Australia. Two goodness-of-fit statistics were used to evaluate the skill—daily and quantile coefficient of efficiency between simulations and observations. Simulations based on cross-validation show that kriging performed better than other two spatial interpolation approaches (B-splines and thin-plate splines).     

Effect of the inter-annual variability of rainfall statistics on stochastically generated rainfall time series: part 2. Impact on watershed response variables

This study analyzes how the stochastically generated rainfall time series accounting for the inter-annual variability of rainfall statistics can improve the prediction of watershed response variables such as peak flow and runoff depth. The modified Bartlett–Lewis rectangular pulse (MBLRP) rainfall generation model was improved such that it can account for the inter-annual variability of the observed rainfall statistics. Then, the synthetic rainfall time series was generated using the MBLRP model, which was used as input rainfall data for SCS hydrologic models to produce runoff depth and peak flow in a virtual watershed. These values were compared to the ones derived from the synthetic rainfall time series that is generated from the traditional MBLRP rainfall modeling. The result of the comparison indicates that the rainfall time series reflecting the inter-annual variability of rainfall statistics reduces the biasness residing in the predicted peak flow values derived from the synthetic rainfall time series generated using the traditional MBLRP approach by 26–47 %. In addition, it was observed that the overall variability of the peak flow and run off depth distribution was better represented when the inter-annual variability of rainfall statistics are considered.     

A stochastic model describing the impact of daily rainfall depth distribution on the soil water balance

Simplified, vertically-averaged soil moisture models have been widely used to describe and study eco-hydrological processes in water-limited ecosystems. The principal aim of these models is to understand how the main physical and biological processes linking soil, vegetation, and climate impact on the statistical properties of soil moisture. A key component of these models is the stochastic nature of daily rainfall, which is mathematically described as a compound Poisson process with daily rainfall amounts drawn from an exponential distribution. Since measurements show that the exponential distribution is often not the best candidate to fit daily rainfall, we compare the soil moisture probability density functions obtained from a soil water balance model with daily rainfall depths assumed to be distributed as exponential, mixed-exponential, and gamma. This model with different daily rainfall distributions is applied to a catchment in New South Wales, Australia, in order to show that the estimation of the seasonal statistics of soil moisture might be improved when using the distribution that better fits daily rainfall data. This study also shows that the choice of the daily rainfall distributions might considerably affect the estimation of vegetation water-stress, leakage and runoff occurrence, and the whole water balance.     

Effect of rainfall uncertainty on the performance of physically based rainfall–runoff models

This paper analyses the effect of rain data uncertainty on the performance of two hydrological models with different spatial structures: a semidistributed and a fully distributed model. The study is performed on a small catchment of 19.6 km² located in the north‐west of Spain, where the arrival of low pressure fronts from the Atlantic Ocean causes highly variable rainfall events. The rainfall fields in this catchment during a series of storm events are estimated using rainfall point measurements. The uncertainty of the estimated fields is quantified using a conditional simulation technique. Discharge and rain data, including the uncertainty of the estimated rainfall fields, are then used to calibrate and validate both hydrological models following the generalized likelihood uncertainty estimation (GLUE) methodology. In the storm events analysed, the two models show similar performance. In all cases, results show that the calibrated distribution of the input parameters narrows when the rain uncertainty is included in the analysis. Otherwise, when rain uncertainty is not considered, the calibration of the input parameters must account for all uncertainty in the rainfall–runoff transformation process. Also, in both models, the uncertainty of the predicted discharges increase in similar magnitude when the uncertainty of rainfall input increase.     

A multiplier-based method of generating stochastic areal rainfall from point rainfalls

Catchment modelling for water resources assessment is still mainly based on rain gauge measurements as these are more easily available and cover longer periods than radar and satellite-based measurements. Rain gauges however measure the rain falling on an extremely small proportion of the catchment and the areal rainfall obtained from these point measurements are consequently substantially uncertain. These uncertainties in areal rainfall estimation are generally ignored and the need to assess their impact on catchment modelling and water resources assessment is therefore imperative. A method that stochastically generates daily areal rainfall from point rainfall using multiplicative perturbations as a means of dealing with these uncertainties is developed and tested on the Berg catchment in the Western Cape of South Africa. The differences in areal rainfall obtained by alternately omitting some of the rain gauges are used to obtain a population of plausible multiplicative perturbations. Upper bounds on the applicable perturbations are set to prevent the generation of unrealistically large rainfall and to obtain unbiased stochastic rainfall. The perturbations within the set bounds are then fitted into probability density functions to stochastically generate the perturbations to impose on areal rainfall. By using 100 randomly-initialized calibrations of the AWBM catchment model and Sequent Peak Analysis, the effects of incorporating areal rainfall uncertainties on storage-yield-reliability analysis are assessed. Incorporating rainfall uncertainty is found to reduce the required storage by up to 20%. Rainfall uncertainty also increases flow-duration variability considerably and reduces the median flow-duration values by an average of about 20%.     

Effect of the duration and direction of storm movement on planar flow with full and partial areal coverage

Using kinematic wave equations, analytical solutions are derived for flow resulting from storms moving either up or down the plane and covering it fully or partially. By comparing the flow resulting from a moving storm with that from a stationary storm of the same duration and areal coverage, the influence of storm duration, direction and areal coverage is investigated. It is found that the direction, duration and areal coverage of storm movement have a pronounced effect on the discharge hydrograph. The runoff hydrographs resulting from storms moving downstream are quite different from those from storms moving upstream. Likewise, the areal coverage of the storm has a pronounced effect on the runoff hydrograph. Copyright © 2002 John Wiley & Sons, Ltd.     

Stochastic simulation of droughts. Part 2: regional droughts

In Part 1 we demonstrated the applicability of stochastic models to predicting the characteristics of point drought events within any planning period by means of a case study (Mohan S, Sahoo PK (2007) Hydrological Processes 21 : this issue). In addition, studies on regional droughts are important in the context of regional level planning and evolving management strategies. The small number of drought events from a particular streamflow or rainfall series, when subjected to statistical analysis in order to predict future occurrences, produces results that are not very reliable. To overcome this difficulty, we propose using a long sequence of synthetically generated annual rainfall series at various rain‐gauge stations of a region, and multiyear regional droughts were derived from both historic and generated series. The key parameters for a successful regional multiyear drought study are the critical area ratio and the critical level, and the area affected by the drought can be ascertained using these parameters. The important regional drought parameters were determined and their suitable probability distributions were arrived at by studying a total of nine possible probability models; these models can be used in predicting the longest regional drought duration and the greatest regional drought severity with a given return period. The effect of change of critical parameters on the regional drought parameters is also studied and reported. Copyright © 2007 John Wiley & Sons, Ltd.     

D) UNITED Nations

Synopsis

In areas of accentuated relief, some of the basic assumptions made in the use of standard methods of assessing areal mean rainfall are often untenable. It is shown in this paper, that, not only does topography affect the actual rainfall distribution, but that the areal variability, measured as the correlation between any two points, is also dependent on the relief. Two methods are used to show this. Once method compares the areal variability of a flat area to one of accentuated relief, while the second method relates areal variability to topographic factors using a multiple regression technique.

The conclusions reached are then used for three purposed. The first is to develop a method of ascribing objectively areas or points to a particular raingauge, taking into account the nature of the terrain. The second is to establish a procedure for estimating the rainfall at ungauged points, by taking into account the rainfall at a selected nearby rainguage and the topographic situation of the points, and the third purpose is to provide means of establishing a correction factor to be applied to a raingauge reading in order that the reading may more accurately represent the area ascribed to it.     

Quantization analysis of weather radar data with synthetic rainfall

Quantization is a process by where continuous signals are transformed into discrete values. It is an important part of the signal processing involved in using weather radar. Technological advances have made it easier to increase the number of quantization levels, as witnessed by the replacement of a 3 bit system by an 8 bit system by the UK Meteorological Office. Research has been conducted in the past demonstrating the error statistics of quantized rainfall, although these studies have used real radar data. The novelty of this study is in using synthetic rain, generated with a Poisson cluster model to represent hourly rainfall, and subsequently disaggregated using a fractal cascade to a fine 5 min time scale. The advantage of this approach is the length of time series that can be generated far outweighs the limited duration of historical rainfall series, especially at such fine time scales. This provides sufficient rainfall data, especially high intensity rainfall, to say something statistically significant about the error statistics. The models are parameterised for different months and also for a non-seasonal set. Rainfall is then generated for a summer case, a winter case, and for the non-seasonal case. It is discovered that the error distribution varies significantly as the parameters change for 3 bit rainfall. This error distribution is relatively constant for 8 bit data, within its working range (up to 126 mm/h). At a fine time scale, such high intensity events are not uncommon. This knowledge is useful when investigating historical radar data at lower quantization levels, for the purpose of flood frequency analysis, and remains relevant, especially, if as some studies have shown, the occurrence of high intensity storms is likely to increase.