Rainfall estimation using raingages and radar — A Bayesian approach: 2. An application

In Seo and Smith (this issue), a set of estimators was built in a Bayesian framework to estimate rainfall depth at an ungaged location using raingage measurements and radar rainfall data. The estimators are equivalent to lognormal co-kriging (simple co-kriging in the Gaussian domain) with uncertain mean and variance of gage rainfall. In this paper, the estimators are evaluated via cross-validation using hourly radar rainfall data and simulated hourly raingage data. Generation of raingage data is based on sample statistics of actual raingage measurements and radar rainfall data. The estimators are compared with lognormal co-kriging and nonparametric estimators. The Bayesian estimators are shown to provide some improvement over lognormal co-kriging under the criteria of mean error, root mean square error, and standardized mean square error. It is shown that, if the prior could be assessed more accurately, the margin of improvement in predicting estimation variance could be larger. In updating the uncertain mean and variance of gage rainfall, inclusion of radar rainfall data is seen to provide little improvement over using raingage data only.     

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

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

Propagation of uncertainties in interpolated rainfields to runoff errors

Conditional daily rainfields were generated using collocated raingauge radar data by a kriging interpolation method, and disaggregated into hourly rainfields using variants of the method of fragments. A geographic information system (GIS)-based distributed rainfall–runoff model was used to convert the hourly rainfields into hydrographs. Using the complete radar rainfall as input, the rainfall–runoff model was calibrated based on storm events taken from nested catchments. Performance statistics were estimated by comparing the observed and the complete radar rainfall simulated hydrographs. Degradation in the hydrograph performance statistics by the simulated hourly rainfields was used to identify runoff error propagation. Uncertainty in daily rainfall amounts alone caused higher errors in runoff (depth, peak, and time to peak) than those caused by uncertainties in the hourly proportions alone. However, the degradation, which reduced with runoff depth, caused by the combined uncertainties was not significantly different from that caused by the uncertainty of amounts alone.     

On the potential of Bartlett Lewis rectangular pulse models for simulating rainfall in Peninsular Malaysia

Abstract

The applicability of two versions of the Bartlett Lewis rectangular pulse model, the original and the modified model, is discussed for describing the temporal and spatial variation of rainfall patterns observed at 15 raingauge stations in Peninsular Malaysia over the period 1971–2008; 17 different sets of moment combinations are fitted to these models based on the generalized method of moments approach. The common statistics included in all sets are the mean, variance, lag-1 autocorrelation and the probability of dry based on the hourly rainfall data. The analysis was carried out on hourly rainfall data from all 15 stations for all months of the year. Two stations, Petaling Jaya and Kemaman, located on the west and east coasts of the Peninsula, respectively, are considered for illustration of the results, taking the months of July and November, which correspond to the driest and wettest months, corresponding to the southwest monsoon (May–August) and northeast monsoon (November–February), respectively. The best moment combination found for the illustrative results is based on the common statistics, as well as the mean and variance based on 24-h aggregated rainfall data, the inclusion of which successfully improved the model performance; the errors were significantly reduced. It was also found that the performance of the fitted models based on the mean absolute deviate error varies according to the type of Bartlett Lewis model applied: errors are much smaller for the fitted model based on the modified model as compared to the original model. In addition, the fitted statistics: mean, lag-1 autocorrelation and probability of dry are quite well fitted for several aggregated time scales; however, the variances are underestimated in both models for all aggregated time scales, particularly in the case of the original model. The results of extreme value analysis indicate that the modified model failed to reproduce the annual hourly and daily rainfall extremes satisfactorily.

Editor D. Koutsoyiannis; Associate editor C. Onof

Citation Hanaish, I.S., Ibrahim, K., and Jemain, A.A., 2013. On the potential of Bartlett Lewis rectangular pulse models for simulating rainfall in Peninsular Malaysia. Hydrological Sciences Journal, 58 (8), 1690–1703.     

The relationship between runoff rate and lag time and the effects of surface treatments at the plot scale

Abstract

The manner in which both the seasonal and regional variations in storm duration, intensity and inter-storm period manifest in the runoff response of agricultural water supply catchments is investigated. High-resolution rainfall data were analysed for a network of 17 raingauges located across the semiarid (200–500 mm year^?1) agricultural districts of southwest Western Australia. Seasonal variations in mean storm duration, mean rainfall intensity and mean inter-storm period were modelled using simple periodic functions whose parameters were then also regressed with geographic and climatic indices to create spatial fields for each of these statistics. Based on these mean values, a continuous rainfall time series can be synthesized for any location within the region, with the rainfall depth within each storm being downscaled to 5-min time steps using a bounded random cascade model. Runoff from six different catchment surface treatments (“engineered” catchments) was simulated using a conceptual water-balance model, validated using rainfall—runoff data from an experimental field site. The expected yield of the various catchment types at any other location within the study region is then simulated using the above rainfall—runoff model and synthetic rainfall and potential evaporation time series under a range of climatic settings representative of regional climate variation. The resulting coupled model can be used to estimate the catchment area required to yield an acceptable volume of runoff for any location and dam capacity, at a specified reliability level, thus providing a tool for water resource managers to design engineered catchments for water supply. Although the model presented is specific for Western Australia's southwest region, the methodology itself is applicable to other locations.     

Critical rainfall statistics for predicting watershed flood responses: rethinking the design storm concept

Recent advances have been made to modernize estimates of probable precipitation scenarios; however, researchers and engineers often continue to assume that rainfall events can be described by a small set of event statistics, typically average intensity and event duration. Given the easy availability of precipitation data and advances in desk‐top computational tools, we suggest that it is time to rethink the ‘design storm’ concept. Design storms should include more holistic characteristics of flood‐inducing rain events, which, in addition to describing specific hydrologic responses, may also be watershed or regionally specific. We present a sensitivity analysis of nine precipitation event statistics from observed precipitation events within a 60‐year record for Tompkins County, NY, USA. We perform a two‐sample Kolmogorov–Smirnov (KS) test to objectively identify precipitation event statistics of importance for two related hydrologic responses: (1) peak outflow from the Six Mile Creek watershed and (2) peak depth within the reservoir behind the Six Mile Creek Dam. We identify the total precipitation depth, peak hourly intensity, average intensity, event duration, interevent duration, and several statistics defining the temporal distribution of precipitation events to be important rainfall statistics to consider for predicting the watershed flood responses. We found that the two hydrologic responses had different sets of statistically significant parameters. We demonstrate through a stochastic precipitation generation analysis the effects of starting from a constrained parameter set (intensity and duration) when predicting hydrologic responses as opposed to utilizing an expanded suite of rainfall statistics. In particular, we note that the reduced precipitation parameter set may underestimate the probability of high stream flows and therefore underestimate flood hazard. Copyright © 2016 John Wiley & Sons, Ltd.     

Data time-step dependency of conceptual rainfall—streamflow model parameters: an empirical study with implications for regionalisation

Abstract

Employing 1-, 2-, 4-, 6-, 12- and 24-hourly data sets for two catchments (10.6 and 298 km²) in Wales, the calibrated parameters of a unit hydrograph-based model are shown to change substantially over that range of data time steps. For the smaller basin, each model parameter reaches, or approaches, a stable value as the data time step decreases, providing a straightforward method of estimating time-step independent model parameter values. For the larger basin, the model parameters also reach, or approach, stable values using hourly data, but, for reasons given in the paper, interpretation of the results is more difficult. Model parameter sensitivity analyses are presented that give insights into the relative precision on the parameters for both catchments. The paper discusses the importance of accounting for model parameter data time-step dependency in pursuit of a reduction in the uncertainty associated with estimates of flow in ungauged basins, and suggests that further work along these lines be undertaken using different catchments and models.     

Sensitivity of soil moisture field evolution to rainfall forcing

In this paper the temporal behaviour of soil moisture is modelled and statistically characterized by use of the zero‐dimensional model for soil moisture dynamics and the rectangular pulses Poisson process model for rainfall forcing. The mean, covariance and spectral density function of soil moisture (both instantaneous and locally averaged cases) are analytically derived to evaluate its sensitivity to the model parameters. Finally, the probability density function of soil moisture is derived to evaluate the effect of rainfall forcing. All the model parameters used have been tuned to the Monsoon '90 data. Results can be summarized as follows. (1) Only the soil moisture model parameters (η and nZ_r) are found to affect the autocorrelation function in a distinguishable manner. On the other hand, both the rainfall model parameter (θ) and the effective soil depth (nZ_r) are found to be of impact to the soil moisture spectrum. However, as the smoothing (or damping) effect of soil is so dominant, about ±20% variation of one parameter seems not to affect significantly the second‐order statistics of soil moisture. (2) More difference can be found by applying a longer averaging time, which is found to obviously decrease the variance but increase the correlation even though no overlapping between neighbouring soil moisture data was allowed. (3) Among rainfall model parameters, the arrival rate (λ) was found to be most important for the soil moisture evolution. When increasing the arrival rate of rainfall, the histogram of soil moisture shifts its peak to a certain value as well as becomes more concentrated around the peak. However, by decreasing the arrival rate of rainfall, a much smaller (almost to zero) mean value of soil moisture was estimated, even though the total volume of rainfall remained constant. This indicates that desertification may take place without decreasing the total volume of rainfall. Copyright © 2005 John Wiley & Sons, Ltd.     

Rainfall derivatives pricing with an underlying semi-Markov model for precipitation occurrences

Weather derivatives represent a new and particular kind of contingent claim which shares a specific underlying weather index. These derivatives are written for different temperature indices, hurricanes, frost, snowfall and rainfall, and they are available for several cities. Our paper focuses on rainfall derivatives. In order to price this kind of derivatives, we have to model daily rainfall sequences at a specific location. For this purpose, we adopt a non-homogeneous parametric semi-Markov model to describe the rainfall occurrences, and a mixture of exponential distributions for rainfall amounts. The underlying Markov process has the obvious two states: dry and wet. In addition, dry and wet sequences are estimated by using best-fitting techniques. The model parameters are determined thanks to classical log-likelihood maximization. We finally price some rainfall contracts issued by the Chicago Mercantile Exchange through Monte Carlo simulation. The numerical applications and the parameter estimations are carried out using real data.     

A doubly stochastic rainfall model with exponentially decaying pulses

We develop a doubly stochastic point process model with exponentially decaying pulses to describe the statistical properties of the rainfall intensity process. Mathematical formulation of the point process model is described along with second-order moment characteristics of the rainfall depth and aggregated processes. The derived second-order properties of the accumulated rainfall at different aggregation levels are used in model assessment. A data analysis using 15 years of sub-hourly rainfall data from England is presented. Models with fixed and variable pulse lifetime are explored. The performance of the model is compared with that of a doubly stochastic rectangular pulse model. The proposed model fits most of the empirical rainfall properties well at sub-hourly, hourly and daily aggregation levels.     

Spatial,inter and intra‐annual variability of the Upper Blue Nile Basin rainfall

In this study, monthly and annual Upper Blue Nile Basin rainfall data were analyzed to learn the rainfall statistics and its temporal and spatial distribution. Frequency analysis and spatial characterization of rainfall in the Upper Blue Nile Basin are presented. Frequency analysis was performed on monthly basin rainfall. Monthly basin average rainfall data were computed from a network of 32 gauges with varying lengths of records. Monthly rainfall probability distribution varies from month to month fitting Gamma‐2, Normal, Weibull and Log‐Normal distributions. The January, July, October and November basin rainfall fit the Gamma‐2 probability distribution. The February, June and December ones fit Weibull distribution. The March, April, May and August rainfall fit Normal distribution. The September rainfall fits Log‐Normal distribution. Upper Blue Nile Basin is relatively wet with a mean annual rainfall of 1423 mm (1960–2002) with a standard deviation of 125 mm. The annual rainfall has a Normal probability distribution. The 100‐year‐drought basin annual rainfall is 1132 mm and the 100‐year‐wet basin annual rainfall is 1745 mm. The dry season is from November through April. The wet season runs from June through September with 74% of the annual rainfall. October and May are transition months. Monthly and annual rainfalls for return periods 2‐, 5‐, 10‐, 25‐, 50‐ and 100‐year dry and wet patterns are presented. Spatial distribution of annual rainfall over the basin is mapped and shows high variation with the southern tip receiving as high as 2049 mm and the northeastern tip as low as 794 mm annual average rainfall. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimation of instantaneous peak flows from maximum mean daily flows using the HBV hydrological model

The record length and quality of instantaneous peak flows (IPFs) have a great influence on flood design, but these high resolution flow data are not always available. The primary aim of this study is to compare different strategies to derive frequency distributions of IPFs using the Hydrologiska Byråns Vattenbalansavdelning (HBV) hydrologic model. The model is operated on a daily and an hourly time step for 18 catchments in the Aller‐Leine basin, Germany. Subsequently, general extreme value (GEV) distributions are fitted to the simulated annual series of daily and hourly extreme flows. The resulting maximum mean daily flow (MDF) quantiles from daily simulations are transferred into IPF quantiles using a multiple regression model, which enables a direct comparison with the simulated hourly quantiles. As long climate records with a high temporal resolution are not available, the hourly simulations require a disaggregation of the daily rainfall. Additionally, two calibrations strategies are applied: (1) a calibration on flow statistics; (2) a calibration on hydrographs. The results show that: (1) the multiple regression model is capable of predicting IPFs with the simulated MDFs; (2) both daily simulations with post‐correction of flows and hourly simulations with pre‐processing of precipitation enable a reasonable estimation of IPFs; (3) the best results are achieved using disaggregated rainfall for hourly modelling with calibration on flow statistics; and (4) if the IPF observations are not sufficient for model calibration on flow statistics, the transfer of MDFs via multiple regressions is a good alternative for estimating IPFs. Copyright © 2015 John Wiley & Sons, Ltd.     

Spatial-temporal rainfall modelling for flood risk estimation

Some recent developments in the stochastic modelling of single site and spatial rainfall are summarised. Alternative single site models based on Poisson cluster processes are introduced, fitting methods are discussed, and performance is compared for representative UK hourly data. The representation of sub-hourly rainfall is discussed, and results from a temporal disaggregation scheme are presented. Extension of the Poisson process methods to spatial-temporal rainfall, using radar data, is reported. Current methods assume spatial and temporal stationarity; work in progress seeks to relax these restrictions. Unlike radar data, long sequences of daily raingauge data are commonly available, and the use of generalized linear models (GLMs) (which can represent both temporal and spatial non-stationarity) to represent the spatial structure of daily rainfall based on raingauge data is illustrated for a network in the North of England. For flood simulation, disaggregation of daily rainfall is required. A relatively simple methodology is described, in which a single site Poisson process model provides hourly sequences, conditioned on the observed or GLM-simulated daily data. As a first step, complete spatial dependence is assumed. Results from the River Lee catchment, near London, are promising. A relatively comprehensive set of methodologies is thus provided for hydrological application.     

Let-It-Rain: a web application for stochastic point rainfall generation at ungaged basins and its applicability in runoff and flood modeling

We present a web application named Let-It-Rain that is able to generate a 1-h temporal resolution synthetic rainfall time series using the modified Bartlett–Lewis rectangular pulse (MBLRP) model, a type of Poisson stochastic rainfall generator. Let-It-Rain, which can be accessed through the web address http://www.LetItRain.info, adopts a web-based framework combining ArcGIS Server from server side for parameter value dissemination and JavaScript from client side to implement the MBLRP model. This enables any desktop and mobile end users with internet access and web browser to obtain the synthetic rainfall time series at any given location at which the parameter regionalization work has been completed (currently the contiguous United States and Republic of Korea) with only a few mouse clicks. Let-It-Rain shows satisfactory performance in its ability to reproduce observed rainfall mean, variance, auto-correlation, and probability of zero rainfall at hourly through daily accumulation levels. It also shows a reasonably good performance in reproducing watershed runoff depth and peak flow. We expect that Let-It-Rain can stimulate the uncertainty analysis of hydrologic variables across the world.     

Stochastic generation of hourly rainstorm events

Occurrence of rainstorm events can be characterized by the number of events, storm duration, rainfall depth, inter-event time and temporal variation of rainfall within a rainstorm event. This paper presents a Monte-Carlo based stochastic hourly rainfall generation model considering correlated non-normal random rainstorm characteristics, as well as dependence of various rainstorm patterns on rainfall depth, duration, and season. The proposed model was verified by comparing the derived rainfall depth–duration–frequency relations from the simulated rainfall sequences with those from observed annual maximum rainfalls based on the hourly rainfall data at the Hong Kong Observatory over the period of 1884–1990. Through numerical experiments, the proposed model was found to be capable of capturing the essential statistical features of rainstorm characteristics and those of annual extreme rainstorm events according to the available data.     

An empirical method for approximating canopy throughfall

Rainfall replenishes surface and subsurface water but is partially intercepted by a canopy. However, it is challenging to quantify the rainfall passing through the canopy (i.e. throughfall). This study derives simple‐to‐use empirical equations relating throughfall to canopy and rainfall characteristics. Monthly throughfall is calculated by applying a mass balance model on weather data from Singapore; Vancouver, Canada; and Stanford, USA. Regression analysis is then performed on the calculated throughfall with three dependent variables (i.e. maximum canopy storage, average rainfall depth and time interval between two consecutive rainfall in a month) to derive the empirical equations. One local equation is derived for each location using data from that particular location, and one global equation is derived using data from all three locations. The equations are further verified with calculated monthly throughfall from other weather data and actual throughfall field measurements, giving an accuracy of about 80–90%. The global equation is relatively less accurate but is applicable worldwide. Overall, this study provides a global equation through which one can quickly estimate throughfall with only information on the three variables. When additional weather data are available, one can follow the proposed methodology to derive their own equations for better estimates. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Application of a data‐based mechanistic modelling (DBM) approach for predicting runoff generation in semi‐arid regions

This paper addresses the application of a data‐based mechanistic (DBM) modelling approach using transfer function models (TFMs) with non‐linear rainfall filtering to predict runoff generation from a semi‐arid catchment (795 km²) in Tanzania. With DBM modelling, time series of rainfall and streamflow were allowed to suggest an appropriate model structure compatible with the data available. The model structures were evaluated by looking at how well the model fitted the data, and how well the parameters of the model were estimated. The results indicated that a parallel model structure is appropriate with a proportion of the runoff being routed through a fast flow pathway and the remainder through a slow flow pathway. Finally, the study employed a Generalized Likelihood Uncertainty Estimation (GLUE) methodology to evaluate the parameter sensitivity and predictive uncertainty based on the feasible parameter ranges chosen from the initial analysis of recession curves and calibration of the TFM. Results showed that parameters that control the slow flow pathway are relatively more sensitive than those that control the fast flow pathway of the hydrograph. Within the GLUE framework, it was found that multiple acceptable parameter sets give a range of predictions. This was found to be an advantage, since it allows the possibility of assessing the uncertainty in predictions as conditioned on the calibration data and then using that uncertainty as part of the decision‐making process arising from any rainfall‐runoff modelling project. Copyright © 2001 John Wiley & Sons, Ltd.     

THE RHONE GLACIER SURVEYS

Abstract

This paper describes a stochastic rainfall model which has been developed to generate synthetic sequences of hourly rainfalls at a point. The model has been calibrated using data from Farnborough in Hampshire, England. This rainfall data series was divided into wet and dry spells; analysis of the durations of these spells suggests that they may be represented by exponential and generalized Pareto distributions respectively. The total volume of rainfall in wet spells was adequately fitted by a conditional gamma distribution. Random sampling from a beta distribution, defining the average shape of all rainfall profiles, is used in the model to obtain the rainfall profile for a given wet spell. Results obtained from the model compare favourably with observed monthly and annual rainfall totals and with annual maximum frequency distributions of 1, 2, 6, 12, 24 and 48 hours duration at Farnborough. The model has a total of 22 parameters, some of which are specific to winter or summer seasons.     

Effects of temporal resolution on hydrological model parameters and its impact on prediction of river discharge / Effets de la résolution temporelle sur les paramètres d'un modèle hydrologique et impact sur la prévision de l'écoulement en rivière

Abstract

Temporal resolution of rainfall plays an important role in determining the hydrological response of river basins. Rainfall temporal variability can be considered as one of the most critical elements when dealing with input data of rainfall—runoff models. In this paper, a typical lumped rainfall—runoff model is applied to long- and short-term runoff prediction using rainfall data sets with different temporal resolution, including daily, hourly and 10-min interval data, and the dependency of model performance on the time interval of the rainfall data is discussed. Furthermore, the effect of temporal resolution on model parameter values is analysed. As results, rainfall data with shorter temporal resolution provide better performance in short-term river discharge estimation, especially for storm discharge estimation. The most accurate results are obtained on the peak discharge and recession part of the hydrograph by using 10-min interval rainfall data. It is concluded that model parameter values are influenced not only by the temporal resolution of calculation but also by the rainfall intensity—duration relationship. This study provides useful information about determination of hydrological model parameters using data of different temporal resolutions.