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).     

Fractal analysis of rainfall observed in two different climatic regions

Abstract

The importance of high-resolution rainfall data to understand the intricacies of the dynamics of hydrological processes and describe them in a sophisticated and accurate way has been increasingly realized. The present study investigates the general suitability of fractal (or scaling) theory for understanding the rainfall behaviour and transforming rainfall data from one time scale to another. The study, employing a multi-fractal approach, follows the research undertaken earlier by the author (Sivakumar, 2000) employing a mono-fractal approach in which some preliminary indication as to the possibility of existence of (multi-) fractals was obtained. Rainfall data of three different resolutions, six-hourly, daily, and weekly, observed over a period of 25 years in two different climatic regions: a subtropical climatic region (Leaf River basin, Mississippi, USA); and an equatorial climatic region (Singapore) are analysed. The existence of multi-fractal behaviour in the rainfall data is investigated using (a) the power spectrum method; (b) the empirical probability distribution function (PDF) method; (c) the statistical moment scaling method; and (d) the probability distribution multiple scaling (PDMS) method. The results achieved from all these methods for the six different rainfall data sets considered indicate the existence of multi-fractal behaviour of rainfall observed in Leaf River basin and Singapore, providing further support to the results obtained using the mono-fractal approach (Sivakumar, 2000). The suitability of a multi-fractal framework to characterize the behaviour of rainfall observed in the above two significantly different climatic regions, subtropical and equatorial, seems to suggest the general suitability of the fractal theory for transforming rainfall from one time scale to another. Investigations with rainfall data from several other climatic regions are underway with a view to strengthening the above conclusions.     

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.     

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.     

Analysis and modelling of rainfall fields at different resolutions in southern Italy

Abstract

The spatial and temporal variability of the scaling properties and correlation structure of a data set of rainfall time series, aggregated over different temporal resolutions, and observed in 70 raingauges across the Basilicata and Calabria regions of southern Italy, is investigated. Two types of random cascade model, namely canonical and microcanonical models, were used for each raingauge and selected season. For both models, different hypotheses concerning dependency of parameters on time scale and rainfall height can be adopted. In particular, a new approach is proposed which consists of several combinations of models with a different scale dependence of parameters for different temporal resolutions. The goal is to improve the modelling of the main features of rainfall time series, especially for cases where the variability of rainfall changes irregularly with temporal aggregation. The results obtained with the new methodology showed good agreement with the observed data, in particular, for the summer months. In fact, during this season, rainfall heights aggregated at fine temporal resolutions (from 5 to 20 min) are more similar (relative to the winter season) to the values cumulated on 1 or 3 h (due to convective phenomena) and, consequently, the process of rainfall breakdown is nearly stationary for a range of finer temporal resolutions.

Editor D. Koutsoyiannis; Associate editor A. Montanari     

Doubly stochastic Poisson pulse model for fine-scale rainfall

Stochastic rainfall models are widely used in hydrological studies because they provide a framework not only for deriving information about the characteristics of rainfall but also for generating precipitation inputs to simulation models whenever data are not available. A stochastic point process model based on a class of doubly stochastic Poisson processes is proposed to analyse fine-scale point rainfall time series. In this model, rain cells arrive according to a doubly stochastic Poisson process whose arrival rate is determined by a finite-state Markov chain. Each rain cell has a random lifetime. During the lifetime of each rain cell, instantaneous random depths of rainfall bursts (pulses) occur according to a Poisson process. The covariance structure of the point process of pulse occurrences is studied. Moment properties of the time series of accumulated rainfall in discrete time intervals are derived to model 5-min rainfall data, over a period of 69 years, from Germany. Second-moment as well as third-moment properties of the rainfall are considered. The results show that the proposed model is capable of reproducing rainfall properties well at various sub-hourly resolutions. Incorporation of third-order moment properties in estimation showed a clear improvement in fitting. A good fit to the extremes is found at larger resolutions, both at 12-h and 24-h levels, despite underestimation at 5-min aggregation. The proportion of dry intervals is studied by comparing the proportion of time intervals, from the observed and simulated data, with rainfall depth below small thresholds. A good agreement was found at 5-min aggregation and for larger aggregation levels a closer fit was obtained when the threshold was increased. A simulation study is presented to assess the performance of the estimation method.     

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 preliminary investigation on the scaling behaviour of rainfall observed in two different climates

Abstract

The problem of transformation of rainfall data from one scale to another has been gaining considerable importance in recent years. Though the application of the concept of fractal theory, in the studies conducted thus far, nearly unanimously points at the possibility of such a transformation, the suitability of the theory to the highly variable rainfall in time and space has very often been questioned. A preliminary attempt is made herein to address this issue by investigating the existence of temporal scaling behaviour in rainfall data observed in two different climatic regions: (a) a subtropical climatic region (Leaf River basin, Mississippi, USA) and (b) an equatorial climatic region (Singapore). Rainfall data of three different resolutions, six-hourly, daily, and weekly, observed over a period of 25 years, are investigated. A mono- or simple-scaling method (box dimension method) is employed. The results achieved for the different data sets clearly indicate the existence of temporal scaling in rainfall observed in the two regions, an encouraging news on the suitability of fractal theory in understanding and modelling the rainfall process. However, the insufficiency of a single dimension to characterize the rainfall behaviour is realized, as the dimension depends on the rainfall intensity level, which, in turn, may be related to the rainfall generating mechanisms. A comparison of the box-dimension results obtained for data of different resolutions, from each of the regions, seems to indicate a possible connection between them, a prospect of tremendous practical importance. Another interesting observation is the similarity between the box dimension results obtained for rainfall data from Leaf River basin and Singapore, but this is also clearly related to the intensity level. The dependence of the dimension on the intensity threshold suggests the use of a multi-dimensional fractal approach, where the process is characterized by more than one dimension (or a dimension function) instead of one single dimension. On the basis of the present results, some potential areas for further study are identified.     

Reproduction of temporal scaling by a rectangular pulses rainfall model

The presence of scaling statistical properties in temporal rainfall has been well established in many empirical investigations during the latest decade. These properties have more and more come to be regarded as a fundamental feature of the rainfall process. How to best use the scaling properties for applied modelling remains to be assessed, however, particularly in the case of continuous rainfall time‐series. One therefore is forced to use conventional time‐series modelling, e.g. based on point process theory, which does not explicitly take scaling into account. In light of this, there is a need to investigate the degree to which point‐process models are able to ‘unintentionally’ reproduce the empirical scaling properties. In the present study, four 25‐year series of 20‐min rainfall intensities observed in Arno River basin, Italy, were investigated. A Neyman–Scott rectangular pulses (NSRP) model was fitted to these series, so enabling the generation of synthetic time‐series suitable for investigation. A multifractal scaling behaviour was found to characterize the raw data within a range of time‐scales between approximately 20 min and 1 week. The main features of this behaviour were surprisingly well reproduced in the simulated data, although some differences were observed, particularly at small scales below the typical duration of a rain cell. This suggests the possibility of a combined use of the NSRP model and a scaling approach, in order to extend the NSRP range of applicability for simulation purposes. Copyright © 2002 John Wiley & Sons, Ltd.     

A hybrid stochastic-weather-generation method for temporal disaggregation of precipitation with consideration of seasonality and within-month variations

Linking atmospheric and hydrological models is challenging because of a mismatch of spatial and temporal resolutions in which the models operate: dynamic hydrological models need input at relatively fine temporal (daily) scale, but the outputs from general circulation models are usually not realistic at the same scale, even though fine scale outputs are available. Temporal dimension downscaling methods called disaggregation are designed to produce finer temporal-scale data from reliable larger temporal-scale data. Here, we investigate a hybrid stochastic weather-generation method to simulate a high-frequency (daily) precipitation sequence based on lower frequency (monthly) amounts. To deal with many small precipitation amounts and capture large amounts, we divide the precipitation amounts on rainy days (with non-zero precipitation amounts) into two states (named moist and wet states, respectively) by a pre-defined threshold and propose a multi-state Markov chain model for the occurrences of different states (also including non-rain days called dry state). The truncated Gamma and censored extended Burr XII distributions are then employed to model the precipitation amounts in the moist and wet states, respectively. This approach avoids the need to deal with discontinuity in the distribution, and ensures that the states (dry, moist and wet) and corresponding amounts in rainy days are well matched. The method also considers seasonality by constructing individual models for different months, and monthly variation by incorporating the low-frequency amounts as a model predictor. The proposed method is compared with existing models using typical catchment data in Australia with different climate conditions (non-seasonal rainfall, summer rainfall and winter rainfall patterns) and demonstrates better performances under several evaluation criteria which are important in hydrological studies.     

Australian rainfall trends and their relation to the southern oscillation index

Rainfall is the key climate variable that governs the spatial and temporal availability of water. In this study we identified monthly rainfall trends and their relation to the southern oscillation index (SOI) at ten rainfall stations across Australia covering all state capital cities. The nonparametric Mann–Kendall (MK) test was used for identifying significant trends. The trend free pre‐whitening approach (TFPW) was used to remove the effects of serial correlation in the dataset. The trend beginning year was approximated using the cumulative summation (CUSUM) technique and the influence of the SOI was identified using graphical representations of the wavelet power spectrum (WPS). Decreasing trends of rainfall depth were observed at two stations, namely Perth airport for June and July rainfall starting in the 1970s and Sydney Observatory Hill for July rainfall starting in the 1930s. No significant trends were found in the Melbourne, Alice Springs and Townsville rainfall data. The remaining five stations showed increasing trends of monthly rainfall depth. The SOI was found to explain the increasing trends for the Adelaide (June) and Cairns (April) rainfall data and the decreasing trends for Sydney (July) rainfall. Other possible climatic factors affecting Australian rainfall are also discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

Simulation and spatio-temporal disaggregation of multi-site rainfall data for urban drainage applications

Abstract

For urban drainage and urban flood modelling applications, fine spatial and temporal rainfall resolution is required. Simulation methods are developed to overcome the problem of data limitations. Although temporal resolution higher than 10–20 minutes is not well suited for detailed rainfall—runoff modelling for urban drainage networks, in the absence of monitored data, longer time intervals can be used for master planning or similar purposes. A methodology is presented for temporal disaggregation and spatial distribution of hourly rainfall fields, tested on observations for a 10-year period at 16 raingauges in the urban catchment of Dalmuir (UK). Daily rainfall time series are simulated with a generalized linear model (GLM). Next, using a single-site disaggregation model, the daily data of the central gauge in the catchment are downscaled to an hourly time scale. This hourly pattern is then applied linearly in space to disaggregate the daily data into hourly rainfall at all sites. Finally, the spatial rainfall field is obtained using inverse distance weighting (IDW) to interpolate the data over the whole catchment. Results are satisfactory: at individual sites within the region the simulated data preserve properties that match the observed statistics to an acceptable level for practical purposes.     

Depth Estimation from Gravity Data Using the Maximum Entropy Method (MEM) and the Multi Taper Method (MTM)

The depth determination from the gravity data in frequency domain is carried out using the classical fast Fourier transform (FFT) method utilizing scaling properties of ensemble of anomalous source. The problem of calculating power spectrum from the FFT is well described in the literature. Here, the application of other high-resolution methods of power spectrum calculation, such as maximum entropy method (MEM) and multi-taper method (MTM) are explored to estimate depth to anomalous sources. At the outset, the FFT, the MEM and the MTM are tested on synthetic gravity data, generated for different types of synthetic models and then all these methods are applied to the field gravity data of the Bengal basin. The MTM with scaling is found to be superior for providing the detailed subsurface information rather than the MEM and the FFT methods in the case of synthetic as well as field examples.     

Spatial disaggregation and intensity correction of TRMM-based rainfall time series for hydrological applications in dryland catchments

Abstract

A novel approach is presented for combining spatial and temporal detail from newly available TRMM-based data sets to derive hourly rainfall intensities at 1-km spatial resolution for hydrological modelling applications. Time series of rainfall intensities derived from 3-hourly 0.25° TRMM 3B42 data are merged with a 1-km gridded rainfall climatology based on TRMM 2B31 data to account for the sub-grid spatial distribution of rainfall intensities within coarse-scale 0.25° grid cells. The method is implemented for two dryland catchments in Tunisia and Senegal, and validated against gauge data. The outcomes of the validation show that the spatially disaggregated and intensity corrected TRMM time series more closely approximate ground-based measurements than non-corrected data. The method introduced here enables the generation of rainfall intensity time series with realistic temporal and spatial detail for dynamic modelling of runoff and infiltration processes that are especially important to water resource management in arid regions.

Editor D. Koutsoyiannis

Citation Tarnavsky, E., Mulligan, M. and Husak, G., 2012. Spatial disaggregation and intensity correction of TRMM-based rainfall time series for hydrological applications in dryland catchments. Hydrological Sciences Journal, 57 (2), 248–264.     

Multi-hydro hydrological modelling of a complex peri-urban catchment with storage basins comparing C-band and X-band radar rainfall data

ABSTRACT

The spread of impervious surfaces in urban areas combined with the rise in the intensity of rainfall events as a result of climate change has led to dangerous increases in storm water flows. This paper discusses a new implementation of the fully distributed hydrological model Multi-Hydro (developed at École des Ponts ParisTech), when operating storage basins, and its ability to deal with high-resolution radar rainfall data. The peri-urban area of Massy (south of Paris, France) was selected as a case study for having six of these drainage facilities, used extensively in flood control. Two radar rainfall datasets with different spatiotemporal resolutions were used: Météo-France’s PANTHER rainfall product (C-band) and ENPC’s X-band DPSRI. The rainfall spatiotemporal variability was analysed statistically using Universal Multifractals (UM). Finally, to validate the application, the water level simulations were compared with local measurements in the Cora storage basin located next to the catchment’s single outlet.     

Improved fitting of the Bartlett-Lewis Rectangular Pulse Model for hourly rainfall

Abstract

This paper examines the applicability of the Bartlett-Lewis Rectangular Pulse Model to rainfall taken from a site in Elmdon, Birmingham, UK. The approach used is to assess the performance of the model in terms of characteristics of the precipitation process, incorporating monthly seasonality. Analytical expressions are derived to complement those presented in Rodriguez-Iturbe et al. (1987). As in that paper, the shortcomings of the simple Poisson process are reduced by the use of a Bartlett-Lewis process. Different methods of parameter estimation are examined. The characteristic features of the time distribution of rainfall events, however, can be well approximated only by optimization and this enables an improved identification of the model parameters.     

A stochastic model for the analysis of the temporal change of dry spells

In the present paper a stochastic approach which considers the arrival of rainfall events as a Poisson process is proposed to analyse the sequences of no rainy days. Particularly, among the different Poisson models, a non-homogeneous Poisson model was selected and then applied to the daily rainfall series registered at the Cosenza rain gauge (Calabria, southern Italy), as test series. The aim was to evaluate the different behaviour of the dry spells observed in two different 30-year periods, i.e. 1951–1980 and 1981–2010. The analyses performed through Monte Carlo simulations assessed the statistical significance of the variation of the mean expected values of dry spells observed at annual scale in the second period with respect to those observed in the first. The model has then been verified by comparing the results of the test series with the ones obtained from other three rain gauges of the same region. Moreover, greater occurrence probabilities for long dry spells in 1981–2010 than in 1951–1980 were detected for the test series. Analogously, the return periods evaluated for fixed long dry spells through the synthetic data of the period 1981–2010 resulted less than half of the corresponding ones evaluated with the data generated for the previous 30-year period.     

Is a chaotic multi‐fractal approach for rainfall possible?

An Erratum has been published for this article in Hydrological Processes 15 (12) 2001, 2381–2382. Applications of the ideas gained from fractal theory to characterize rainfall have been one of the most exciting areas of research in recent times. The studies conducted thus far have nearly unanimously yielded positive evidence regarding the existence of fractal behaviour in rainfall. The studies also revealed the insufficiency of the mono‐fractal approaches to characterizing the rainfall process in time and space and, hence, the necessity for multi‐fractal approaches. The assumption behind multi‐fractal approaches for rainfall is that the variability of the rainfall process could be directly modelled as a stochastic (or random) turbulent cascade process, since such stochastic cascade processes were found to generically yield multi‐fractals. However, it has been observed recently that multi‐fractal approaches might provide positive evidence of a multi‐fractal nature not only in stochastic processes but also in, for example, chaotic processes. The purpose of the present study is to investigate the presence of both chaotic and fractal behaviours in the rainfall process to consider the possibility of using a chaotic multi‐fractal approach for rainfall characterization. For this purpose, daily rainfall data observed at the Leaf River basin in Mississippi are studied, and only temporal analysis is carried out. The autocorrelation function, the power spectrum, the empirical probability distribution function, and the statistical moment scaling function are used as indicators to investigate the presence of fractal, whereas the presence of chaos is investigated by employing the correlation dimension method. The results from the fractal identification methods indicate that the rainfall data exhibit multi‐fractal behaviour. The correlation dimension method yields a low dimension, suggesting the presence of chaotic behaviour. The existence of both multi‐fractal and chaotic behaviours in the rainfall data suggests the possibility of a chaotic multi‐fractal approach for rainfall characterization. Copyright © 2001 John Wiley & Sons, Ltd.     

Effect of the inter-annual variability of rainfall statistics on stochastically generated rainfall time series: part 1. Impact on peak and extreme rainfall values

A noble approach of stochastic rainfall generation that can account for inter-annual variability of the observed rainfall is proposed. Firstly, we show that the monthly rainfall statistics that is typically used as the basis of the calibration of the parameters of the Poisson cluster rainfall generators has significant inter-annual variability and that lumping them into a single value could be an oversimplification. Then, we propose a noble approach that incorporates the inter-annual variability to the traditional approach of Poisson cluster rainfall modeling by adding the process of simulating rainfall statistics of individual months. Among 132 gage-months used for the model verification, the proportion that the suggested approach successfully reproduces the observed design rainfall values within 20 % error varied between 0.67 and 0.83 while the same value corresponding to the traditional approach varied between 0.21 and 0.60. This result suggests that the performance of the rainfall generation models can be largely improved not only by refining the model structure but also by incorporating more information about the observed rainfall, especially the inter-annual variability of the rainfall statistics.     

Estimating Flood Quantiles on the Basis of Multi-Event Rainfall Simulation — Case Study

This paper presents an approach to estimating the probability distribution of annual discharges Q based on rainfall-runoff modelling using multiple rainfall events. The approach is based on the prior knowledge about the probability distribution of annual maximum daily totals of rainfall P in a natural catchment, random disaggregation of the totals into hourly values, and rainfall-runoff modelling. The presented Multi-Event Simulation of Extreme Flood method (MESEF) combines design event method based on single-rainfall event modelling, and continuous simulation method used for estimating the maximum discharges of a given exceedance probability using rainfall-runoff models. In the paper, the flood quantiles were estimated using the MESEF method, and then compared to the flood quantiles estimated using classical statistical method based on observed data.