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.     

Further developments of a transient Poisson-cluster model for rainfall

Stochastic point processes for rainfall are known to be able to preserve the temporal variability of rainfall on several levels of aggregation (e.g. hourly, daily), especially when the cluster approach is used. One major assumption in most of the applications todate is the stationarity of the rainfall properties in time, which must be reconsidered under a climate change hypothesis. Here, we propose new theoretical developments of a Poisson-based model with cluster, namely the Neyman–Scott Rectangular Pulses Model, which treats storm frequency as a nonstationary function. In this paper, storm frequency is modelled as a linear function of time in order to reproduce an increase (or decrease) of the mean annual precipitation. The basic theory is reconsidered and the moments are derived up to the third order. Then, a calibration method based on the generalized method of moments is proposed and discussed. An application to a rainfall time series from Uccle illustrates how this model can reproduce a trend for the average rainfall. This work opens new avenues for future developments on transient stochastic rainfall models and highlights the major challenges linked to this approach.     

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.     

Stochastic modelling of seasonal and yearly rainfalls with low-frequency variability

Stochastic rainfall models are important for many hydrological applications due to their appealing ability to simulate synthetic series that resemble the statistical characteristics of the observed series for a location of interest. However, an important limitation of stochastic rainfall models is their inability to preserve the low-frequency variability of rainfall. Accordingly, this study presents a simple yet efficient stochastic rainfall model for a tropical area that attempts to incorporate seasonal and inter-annual variabilities in simulations. The performance of the proposed stochastic rainfall model, the tropical climate rainfall generator (TCRG), was compared with a stochastic multivariable weather generator (MV-WG) in various aspects. Both models were applied on 17 rainfall stations at the Kelantan River Basin, Malaysia, with tropical climate. The validations were carried out on seasonal (monsoon and inter-monsoon) and annual basis. The third-order Markov chain of the TCRG was found to perform better in simulating the rainfall occurrence and preserving the low-frequency variability of the wet spells. The log-normal distribution of the TCRG was consistently better in modelling the rainfall amounts. Both models tend to underestimate the skewness and kurtosis coefficient of the rainfall. The spectral correction approach adopted in the TCRG successfully preserved the seasonal and inter-annual variabilities of rainfall amounts, whereas the MV-WG tends to underestimate the variability bias of rainfall amounts. Overall, the TCRG performed reasonably well in the Kelantan River Basin, as it can represent the key statistics of rainfall occurrence and amounts successfully, as well as the low-frequency variability.     

Dissecting the effect of rainfall variability on the statistical structure of peak flows

This study examines the role of rainfall variability on the spatial scaling structure of peak flows using the Whitewater River basin in Kansas as an illustration. Specifically, we investigate the effect of rainfall on the scatter, the scale break and the power law (peak flows vs. upstream areas) regression exponent. We illustrate why considering individual hydrographs at the outlet of a basin can lead to misleading interpretations of the effects of rainfall variability. We begin with the simple scenario of a basin receiving spatially uniform rainfall of varying intensities and durations and subsequently investigate the role of storm advection velocity, storm variability characterized by variance, spatial correlation and intermittency. Finally, we use a realistic space–time rainfall field obtained from a popular rainfall model that combines the aforementioned features. For each of these scenarios, we employ a recent formulation of flow velocity for a network of channels, assume idealized conditions of runoff generation and flow dynamics and calculate peak flow scaling exponents, which are then compared to the scaling exponent of the width function maxima. Our results show that the peak flow scaling exponent is always larger than the width function scaling exponent. The simulation scenarios are used to identify the smaller scale basins, whose response is dominated by the rainfall variability and the larger scale basins, which are driven by rainfall volume, river network aggregation and flow dynamics. The rainfall variability has a greater impact on peak flows at smaller scales. The effect of rainfall variability is reduced for larger scale basins as the river network aggregates and smoothes out the storm variability. The results obtained from simple scenarios are used to make rigorous interpretations of the peak flow scaling structure that is obtained from rainfall generated with the space–time rainfall model and realistic rainfall fields derived from NEXRAD radar data.     

Comparison of two approaches for generation of daily rainfall data

There has been extensive research on the problem of stochastically generating daily rainfall sequences for use in water management applications. Srikanthan and McMahon [Australia Water Resources Council, Canberra, 1985] proposed a transition probability matrix (TPM) model that performs better for Australian rainfall than many alternative models, particularly where long records (say 100 years) are available. Boughton [Report 99/9, CRC for Catchment Hydrology, Monash University, Melbourne, 21pp, 1999] incorporated an empirical adjustment into the TPM model that allows the model to reproduce the observed variability in the annual rainfall. More recently, Harrold et al. [Water Resour Res 39(10, 12):1300, 1343, 2003a,b] proposed nonparametric models for the generation of daily rainfall occurrences and rainfall amounts on wet days. By conditioning on short, medium and long-term characteristics, this approach is also able to preserve the variability in annual rainfall. In this study, the above two approaches were used to generate daily rainfall data for Sydney and Melbourne, and the results evaluated. Both approaches preserved most of the daily, monthly and annual characteristics that were compared, with the nonparametric approach providing marginally better performance at the cost of greater model complexity. The nonparametric approach was also able to preserve the variability and persistence in the annual number of wet days.     

Prediction of daily rainfall state in a river basin using statistical downscaling from GCM output

Conventional statistical downscaling techniques for prediction of multi-site rainfall in a river basin fail to capture the correlation between multiple sites and thus are inadequate to model the variability of rainfall. The present study addresses this problem through representation of the pattern of multi-site rainfall using rainfall state in a river basin. A model based on K-means clustering technique coupled with a supervised data classification technique, namely Classification And Regression Tree (CART), is used for generation of rainfall states from large-scale atmospheric variables in a river basin. The K-means clustering is used to derive the daily rainfall state from the historical daily multi-site rainfall data. The optimum number of clusters in the observed rainfall data is obtained after application of various cluster validity measures to the clustered data. The CART model is then trained to establish relationship between the daily rainfall state of the river basin and the standardized, dimensionally-reduced National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis climatic data set. The relationship thus developed is applied to the General Circulation Model (GCM)-simulated, standardized, bias free large-scale climate variables for prediction of rainfall states in future. Comparisons of the number of days falling under different rainfall states for the observed period and the future give the change expected in the river basin due to global warming. The methodology is tested for the Mahanadi river basin in India.     

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.     

A cascade approach to continuous rainfall data generation at point locations

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

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).     

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 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.     

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

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

Annual and monthly rainfall data generation schemes

Synthetic annual and monthly rainfall data series are generated by using autoregressive (AR) processes, Thomas-Fiering (TF) model, method of fragments (F) and its modified version (MF), two-tier (TT) model, and a newly developed wavelet (W) approach. It is seen that the W approach is as well in preserving the statistical behavior of the observed data series as the classical annual and monthly hydrological data generation schemes used in this study. The W approach is found even better in replacing some particular characteristics such as the mean of the sequence and correlation between the successive months in the series. It is, therefore, proposed as a new annual and monthly hydrological data generation scheme.     

The influence of antecedent rainfall on Sahelian surface evaporation

This study quantifies the influence of rainfall on surface evaporation in the Sahel. A numerical model of the surface is used to extend the observations taken during the HAPEX–Sahel project, and is forced by 2 years of rain‐gauge data. The model is applied to the Southern Super Site (SSS), which covers an area of less than 100 km². The effects of rainfall variability (spatial and temporal) on soil moisture, vegetation growth and evaporation are explored. Contrasting rainfall conditions between the two years produce observed differences in the timing of the seasonal growth cycle. This correlates well with modelled root‐zone moisture deficits, and exerts a modest influence on transpiration rates. The evolution of surface evaporation is dominated, however, by the bare soil contribution in the day or two after a storm. This component also exerts a strong influence on the spatial variability of fluxes across the SSS, particularly when rain falls only in part of the area. In these cases, differences in evaporation between recently wetted and dry areas can reach 3\5 mm day⁻¹. Observations indicate that during a period of persistent rainfall gradients across the SSS, the lower atmosphere maintained a ‘memory’ of past rainfall patterns through humidity contrasts. These were the result of gradients in surface soil moisture, and therefore evaporation. The model results therefore support the possibility of a positive surface feedback mechanism affecting rainfall patterns in the region. Copyright © 2000 John Wiley & Sons, Ltd.     

The impact of different rainfall products on landscape modelling simulations

Rainfall products can contain significantly different spatiotemporal estimates, depending on their underlying data and final constructed resolution. Commonly used products, such as rain gauges, rain gauge networks, and weather radar, differ in their information content regarding intensities, spatial variability, and natural climatic variability, therefore producing different estimates. Landscape evolution models (LEMs) simulate the geomorphic changes in landscapes, and current models can simulate timeframes from event level to millions of years and some use rainfall inputs to drive them. However, the impact of different rainfall products on LEM outputs has never been considered. This study uses the STREAP rainfall generator, calibrated using commonly used rainfall observation products, to produce longer rainfall records than the observations to drive the CAESAR-Lisflood LEM to examine how differences in rainfall products affect simulated landscapes. The results show that the simulation of changes to basin geomorphology is sensitive to the differences between rainfall products, with these differences expressed linearly in discharges but non-linearly in sediment yields. Furthermore, when applied over a 1500-year period, large differences in the simulated long profiles were observed, with the simulations producing greater sediment yields showing erosion extending further downstream. This suggests that the choice of rainfall product to drive LEMs has a large impact on the final simulated landscapes. The combination of rainfall generator model and LEMs represents a potentially powerful method for assessing the impacts of rainfall product differences on landscapes and their short- and long-term evolution. © 2020 The Authors. Earth Surface Processes and Landforms published by John Wiley & Sons Ltd     

A model to consider the spatial variability of rainfall partitioning within deciduous canopy. I. Model description

In this first paper of two, a numerical simulation model capable of simulating the spatial variability of rainfall partitioning within a canopy is presented. The second paper details how the model is parameterized, and some testing of its capabilities. The first stage of the model is to derive the mature canopy structure. This is achieved through simplified individual tree structures and a random placement routine based on a modified Poisson distribution. Following this the spatial discretization for throughfall is attained as a series of layers of triangles with a tree trunk at every apex. The number of layers is derived from the leaf area index through a modified Poisson distribution. Seasonal variation in the deciduous canopy is simulated through a time vector routine. The rainfall partitioning section of the model is based upon the Rutter model which has been modified to each individual triangle layer. The main feature of this model is that it offers a method of simulating rainfall partitioning at a scale that is a function of the size of the elementary physical components (tree and leaf scale). This can be used to investigate soil moisture variations under a canopy, or to study the variations within the forest hydrological processes themselves. © 1997 John Wiley & Sons, Ltd.     

An analysis of the impact of spatial variability in rainfall on runoff and sediment predictions from a distributed model

This paper investigates the effect of introducing spatially varying rainfall fields to a hydrological model simulating runoff and erosion. Pairs of model simulations were run using either spatially uniform (i.e. spatially averaged) or spatially varying rainfall fields on a 500‐m grid. The hydrological model used was a simplified version of Thales which enabled runoff generation processes to be isolated from hillslope averaging processes. Both saturation excess and infiltration excess generation mechanisms were considered, as simplifications of actual hillslope processes. A 5‐year average recurrence interval synthetic rainfall event typical of temperate climates (Melbourne, Australia) was used. The erosion model was based on the WEPP interrill equation, modified to allow nonlinear terms relating the erosion rate to rainfall or runoff‐squared. The model results were extracted at different scales to investigate whether the effects of spatially varying rainfall were scale dependent. A series of statistical metrics were developed to assess the variability due to introducing the spatially varying rainfall field. At the catchment (approximately 150 km²) scale, it was found that particularly for saturation excess runoff, model predictions of runoff were insensitive to the spatial resolution of the rainfall data. Generally, erosion processes at smaller sub‐catchment scales, particularly when the sediment generation equation had non linearity, were more sensitive to spatial rainfall variability. Introducing runon infiltration reduced the total runoff and sediment yield at all scales, and this process was also most sensitive to the rainfall resolution. Copyright © 2011 John Wiley & Sons, Ltd.     

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.