Hydrological Sciences–Journal–des Sciences Hydrologiques, 52(5) October 2007
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Simulation and spatio-temporal disaggregation of multi-site rainfall data for urban drainage applications MARIE-LAURE SEGOND1, NATASA NEOKLEOUS1, CHRISTOS MAKROPOULOS2, CHRISTIAN ONOF1 & CEDO MAKSIMOVIC1 1 Department of Civil and Environmental Engineering, Imperial College, London SW7 2AZ, UK
[email protected] 2 School of Engineering, University of Exeter, Exeter EX4 4QF, UK
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. Key words rainfall; disaggregation; generalized linear models; Poisson-cluster processes; climate change; interpolation
Simulation et désagrégation spatio-temporelle de données de précipitation multisites pour des applications d’assainissement urbain Résumé L’étude de l’assainissement et la modélisation des crues en milieu urbain nécessitent des données de précipitations à fine résolution spatio-temporelle. Des méthodes de simulation sont développées pour surmonter le problème des limitations dues aux données. Bien qu’une résolution temporelle supérieure à 10–20 minutes ne soit pas pertinente pour des modélisations pluie–débit détaillées en milieu urbain, en l’absence de données observées, de plus longs intervalles de temps peuvent être utilisés pour des schémas d’aménagements ou des objectifs similaires. Cet article présente une méthode de désagrégation temporelle et de distribution spatiale de champs de pluie horaires, testée avec des observations sur une période de 10 ans par 16 pluviomètres dans le bassin versant urbain de Dalmuir (Royaume-Uni). Des séries de pluie journalière sont générées par un modèle linéaire généralisé. La chronique journalière du pluviomètre central du bassin versant est ensuite désagrégée en données horaires par un modèle ponctuel et ce profil horaire est appliqué linéairement dans l’espace afin d’obtenir des données horaires sur tous les sites. Finalement, la méthode de pondération par distance inverse permet de générer des champs de pluies sur tout le bassin versant. Les résultats sont satisfaisants: les données simulées aux sites ponctuels de la région présentent des propriétés qui correspondent aux statistiques observées avec une qualité acceptable pour des applications pratiques. Mots clefs pluie; désagrégation; modèles linéaires généralisés; processus de Poisson; changement climatique; interpolation
INTRODUCTION Urban hydrology is an area of hydrology focusing on cities and regions that have very high levels of human interference with natural processes. The role of rainfall is essential for urban hydrology as other input variables such as hail, snow, groundwater, wind, temperature, etc. have much less influence on urban runoff (Schilling, 1991). In urban areas, a high proportion of the rain becomes effective and produces runoff and Open for discussion until 1 April 2008
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the spatial and temporal variability of rainfall constitutes a significant source of uncertainty for hydrological modelling. The small size and the rapid response of urban catchments require rainfall to be considered at small scales. Berne et al. (2004) suggest a resolution of 3–6 min in time and 2–4 km in space for urban catchments of 1– 10 km2. Most of the time, typical national raingauge networks do not provide the high resolutions requested for urban drainage, and the record length of the collected data is inadequate for performing statistical analysis. In order to overcome the problem of data limitations, rainfall simulations are used as an alternative. In addition, these can incorporate climate change scenarios to assess the potential impacts of climate change. The AUDACIOUS (Adaptable Urban Drainage Addressing Change in Intensity, Occurrence and Uncertainty of Stormwater) project in the UK, for example, aims at investigating key aspects of the effects of climate change on existing drainage in urban areas, and at providing tools for drainage managers and operators to adapt to uncertain future climate change scenarios. Indeed, in order to generate optimum solutions in urban drainage, there is a need for new enhanced urban drainage models capable of being adapted to meet the needs of changing environmental factors, such as groundwater levels, vegetation, changing design standards and rainfall. These models will have to be capable of simulating the interactions between surface and pipe flows and will be used in conjunction with whole life cost assessment of solutions within a risk-based approach. Climate change and variability is seen as one of the biggest uncertainties hindering reliable modelling. The greatest influence on the performance of urban drainage systems is the anticipated change in the precipitation patterns in both space and time, and its impact on flood risk. By physically-based modelling, an attempt can be made to understand how changes in all aspects of precipitation (intensity, duration, frequency, location and clustering) impact flooding within the urban environment. Information on rainfall characteristics, such as presented herein, provide a contribution in this direction. Several studies have been carried out that examined the change in precipitation patterns due to climate change. Comprehensive studies include UKCIP (2002) and the UKWIR (2002). These studies do not use hourly and sub-hourly data; therefore, they are not adequate to address the needs of the urban environment. New rainfall series had to be produced as part of Building Knowledge for Climate Change (BKCC), and these have been disaggregated for use in urban drainage and roof drainage modelling (Ashley et al., 2007). The aim of this paper is to present the results of sub-daily rainfall data simulation for several raingauges in a catchment, coupled with a spatial distribution procedure to obtain a rainfall resolution, which would be as close as possible to what is required for urban hydrology and water management in urban systems. Using a generalized linear model (GLM) (Chandler & Wheater, 2002), daily rainfall data are simulated for a long period of time, and a temporal disaggregation method, based on the Poisson cluster process, is used in order to achieve finer temporal resolution. For finer spatial resolution, inverse distance weighting (IDW) is used to spatially interpolate the data. Hence, simulations of long sequences of sub-daily spatial rainfall data are obtained, with the possibility of incorporating climate change scenarios using GLMs (Chandler & Wheater, 2002).
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METHODOLOGY There is clearly an ongoing discussion in the literature on the issue of processing rainfall data to make them appropriate for a wide range of applications (for relevant reviews, see, for example, Waymire & Gupta, 1981; Onof et al., 2000). For the design of stormwater sewerage systems in particular, there is a need for short-duration precipitation data. A procedure combining a statistical multi-site model (Chandler & Wheater, 2002), which can readily represent spatial nonstationarity and temporal trends in rainfall sequences at a daily time step, and a continuous temporal stochastic model based on Poisson-cluster processes (Koutsoyiannis & Onof, 2001), was presented by Segond et al. (2006) and applied with relative success on rural catchments (Segond, 2006). Further validation of the methodology on urban catchments and an extension accounting for rainfall spatial interpolation is described here. The main steps of the procedure can be summarized as follows: 1. Generate sequences of spatially correlated daily rainfall across a network of sites using a GLM. 2. Taking a cluster of wet days, defined as a series of consecutive wet days delimited by at least one dry day, use the single-site model to simulate hourly rainfall at one raingauge, defined as the master gauge, and disaggregate using an adjusting procedure, so that the hourly rainfalls sum up to the daily totals. 3. Use the hourly pattern generated at the master gauge to disaggregate the daily information to hourly rainfall at the other gauges. If the master gauge records zero rainfall, the previous profile at a site is used. 4. Spatially interpolate the rainfall for the whole catchment using inverse distance weighting. The methodology presented above is tested using data from a network of 16 raingauges provided by Scottish Water. The raingauges, located in the urban 13 × 29 km2 Dalmuir catchment in the Glasgow region, are shown in Fig. 1. The area is relatively flat with the elevation of gauges ranging from 3 to 81 m a.s.l. The rainfall intensities provided at a 2-min temporal resolution are converted into rainfall depths at a daily and hourly time resolution to test the methodology presented herein. Data were available for the period March 2003–June 2005, which, although useful in the development of a methodology, should be considered a very short data set for rigorous applications. Ideally, the GLM software should be fitted to a longer time period (e.g. 15 years), in order to identify possible trends and capture the characteristics of the time series. Daily data for the whole raingauge network and hourly data from the master gauge are used to develop the model. The simulated hourly rainfall fields are then compared with the observed hourly data from the other gauges to test the model. The first part of this paper provides an overview and evaluates the use of GLMs for simulation of multi-site daily rainfall. The fitting of the single-site disaggregation model run at the master gauge (Site RG007) is then described before introducing the spatially-uniform temporal disaggregation applied to the satellite stations. The results of the combined scheme are then presented at a typical gauge on the catchment (RG003) before using IDW to spatially interpolate the data. A summary and discussion conclude the paper.
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Fig. 1 Map of the Glasgow catchment showing the location of the raingauges.
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RAINFALL SIMULATION AND INTERPOLATION Generalized linear model Generalized linear models extend the classical linear regression case and are reviewed in Nelder & Wedderburn (1972). Coe & Stern (1982) first used GLMs for rainfall simulation using a two-stage approach. Chandler & Wheater (2002) extended that work, proposing a GLM-based framework for interpreting the spatio-temporal structure for rainfall. Firstly, an occurrence model predicts the probability distribution of wet and dry days at a site using logistic regression according to: ⎛ p ⎞ ln⎜⎜ i ⎟⎟ = xiT β ⎝ 1 − pi ⎠
(1)
where pi is the probability of rain for the ith case in the data set, conditional on a predictor vector xi, and β is a vector of coefficients. The superscript T denotes the transpose of the vector. Next, the rainfall amount for the ith wet day in the database is taken conditional on a predictor vector ξi, to have a gamma distribution with mean μi, such that for some coefficient vector γ:
ln μi = ξ iT γ
(2)
All gamma distributions are assumed to have a common shape defined by the dispersion parameter ν (Yang et al., 2005). In this study, the application of GLMs is undertaken using the GLIMCLIM software package (Chandler, 2002). The aim is to find suitable predictors and estimate their coefficient vectors, which is carried out using likelihood methods. In order to remove apparent inconsistencies, mainly related to the recording of small amounts of rainfall (trace values), a 0.45 mm cut-off is applied to the data before modelling. Providing the threshold is small enough, this is acceptable for practical applications (Yang et al., 2005). Occurrence model To represent the rainfall occurrence, a logistic model is fitted to daily data at all sites. Eleven predictors are used and a summary of the covariates composing the occurrence model can be found in Table 1. Site altitude and a nonlinear transformation along the eastings and northings (using Legendre polynomials of degree 3) are included to account for regional variation (components 1–3 of Table 1). eastings and northings are expressed in km. Seasonality is represented via sine and cosine functions (components 4–5); the temporal dependence on the previous time steps is taken into account using indicators for the three previous days’ rainfall (components 6–9). In addition, seasonal variation in coefficients associated with previous days’ rainfall is also represented using interaction terms (components 10–11). Generating rainfall occurrence sequences with the correct spatial structure is more difficult than in the case of continuous variables. For the catchment area considered, it is assumed that all sites tend to be influenced by the same weather systems on particular days. It can thus be inferred that inter-site dependence is strong and sites tend to be either mostly wet or mostly dry. The dependence is introduced by specifying a beta-binominal distribution for the number of wet sites on any day. Parameterizing Copyright © 2007 IAHS Press
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Table 1 Covariates used in occurrence model for daily rainfall.
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Component Value Covariate number Constant –1.179 Legendre polynomial 3 for eastings (km) 0.2858 1 Legendre polynomial 3 for northings (km) 0.1691 2 Altitude (km) –2.8366 3 Daily seasonal effect, cosine component 0.1072 4 Daily seasonal effect, sine component –0.0648 5 1.7927 6 I(Y[t –1]) > 0) a I(Y[t – 2]) > 0) 0.3921 7 I(Y[t – 3]) > 0) 0.7735 8 –0.3077 9 Mean of I(Y[t – k]) > 0; k = 1 to 2) b 2-way interaction: covariates 6 and 8 –0.5925 10 2-way interaction: covariates 7 and 8 –0.2563 11 “Soft” threshold for positive values 0.45 Shape parameter of beta-binomial distribution 0.5419 a I is an indicator that takes the value of 1 if the previous day’s rainfall was wet, 0 otherwise. b Similar to above, but averaged over all sites with available data.
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Fig. 2 Observed (white) distribution of number of wet sites in the Dalmuir catchment, together with distribution simulated by GLMs (grey) using a beta-binomial distribution model for spatial dependence in rainfall occurrence.
this distribution involves two terms: θt, which can be seen as a mean parameter, is calculated every day as the mean of the individual site probabilities; and φ, which controls the shape of the distribution, is assumed constant for every day and is estimated from the method of moments (Yang et al., 2005). In this case the shape parameter is 0.5419 (see Table 1). In simulation mode, a joint distribution specifies the marginal probabilities that a site is wet and of the total number of wet sites for every day (see Chandler, 2002, for a description of the mathematical process). The histogram in Fig. 2 shows the number of wet sites, observed and simulated, for the period March 2003–June 2005. The shape of the distribution is relatively well Copyright © 2007 IAHS Press
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Table 2 Covariates used in amounts model for daily rainfall. Component Constant Legendre polynomial 3 for eastings (km) Legendre polynomial 8 for northings (km) Daily seasonal effect, cosine component Daily seasonal effect, sine component ln(Y[t – 1]) > 0) “Soft” threshold for positive values Dispersion parameter Observed residual correlation structure
Value 1.3387 0.0871 0.1458 0.01 –0.0627 0.1827 0.45 0.756 -
Covariate number 1 2 3 4 5
represented up to 12 gauges. Given that there is a considerable amount of missing values in the data set, the distribution is not so well represented for a higher number of gauges, but, in all, should be considered satisfactory. Amounts model The amounts model is fitted to the same data set in order to model the rainfall amount using gamma distributions. A summary of the covariates can be found in Table 2. The predictors are similar to those in the occurrence model, but fewer in number (five covariates). Altitude and interaction terms are not considered for this model. Additionally, the amounts model contains only one term for temporal persistence (component 5 of Table 2). In order to specify a joint distribution for the amounts model, a transformation to marginal normality is applied so that spatial dependence is characterized by the intersite correlation structure of the transformed values. If Yi is the observed amount for the ith wet day in the database, and μi is the mean of the gamma distribution fitted to the observations, then the quantities: ⎛Y ri = ⎜⎜ i ⎝ μi A
⎞ ⎟⎟ ⎠
1
3
(3)
called Anscombe residuals, are approximately normally distributed and this can be verified using a normal probability plot of Anscombe residuals (Chandler & Wheater, 2002). The plot in Fig. 3 shows a good fit to a straight line, except in the lower tail of the distribution which corresponds to small rainfall values. The spatial dependence structure in the rainfall amounts model is then specified through a model for the intersite correlations between Anscombe residuals. Model checks In addition to the Anscombe residuals, the fit of either model can be assessed by plotting mean Pearson residuals by month, site and year. The Pearson residual for an observation Y is proportional to:
Y−μ (4) σ where μ and σ are respectively the mean and standard deviation of Y under the fitted model (Yang et al., 2005). If the occurrence or amounts model is correct, the Pearson residuals all come from distributions with zero mean and constant variance (Yang et al., 2005). Figure 4 shows the daily residual plots calculated for each month for the Copyright © 2007 IAHS Press
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0.4 0.2 −0.4
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Fig. 3 Ascombe residuals from amounts model.
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Fig. 4 Continuous lines represent the mean Pearson residuals from (a) occurrence and (b) amounts models by averaging over all sites. Dashed lines indicate 95% confidence limits under the assumption that the model is correct.
occurrence and amounts models by averaging over all sites. The mean Pearson residuals lie within the 95% confidence interval under the assumption that the model is correct. Simulation The calibrated occurrence and amounts models are used simultaneously to produce rainfall sequences at multiple sites. Fifty sets of simulations are generated for the period March 2003–June 2005. The envelope of simulated averaged daily rainfall over all sites for each month is compared with the corresponding Copyright © 2007 IAHS Press
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Fig. 5 Observed and simulated monthly statistics by averaging over all sites at a daily time step. The continuous line represents the observed series and the dashed lines the envelope obtained from 50 GLM simulations.
observed average series in Fig. 5. The results are encouraging as the observed series lies within the envelope generated from the simulated series. Further daily statistical checks are presented in Fig. 6 at a representative site (RG010), which has no missing values. Ten sets of simulations are generated for this site for the period March 2003–June 2005. Summary statistics include the mean, standard deviation, proportion of dry periods and lag 1 autocorrelation. The statistics of the observed series are consistent with the simulated values. Overall the fitted models give satisfactory results and are then used to generate, at each site, 10 years of daily data for the period March 2003–February 2013. Single-site rainfall temporal disaggregation
The Hyetos software (Koutsoyiannis & Onof, 2000) is used to disaggregate the simulated daily rainfall at the master gauge into hourly rainfall. The simulated daily series is divided into clusters of wet days and several rainfall simulations from the Bartlett-Lewis model (discussed below) are performed separately for each cluster of wet days. The runs continue until the sequence of simulated daily depths matches the sequence of daily totals from the GLM within a tolerance distance. A correction procedure referred to as “adjustment” (Koutsoyiannis & Onof, 2001) is then performed so that the hourly synthetic series generated by the stochastic model is modified for consistency with the given daily series. Koutsoyiannis & Onof (2001) used a simple proportional adjustment. Gauge RG007 was selected as the master gauge to run the disaggregation scheme because of its central location in the Dalmuir catchment and its continuous record length. The Bartlett-Lewis rectangular pulse model (BLRPM) The rainfall model used is the Bartlett-Lewis rectangular pulse model (BLRPM) (Rodriguez-Iturbe et al., 1987, 1988). The BLRPM can represent rainfall at a point in continuous time. Therefore it is particularly useful in a disaggregation framework where it may be used at a time step Copyright © 2007 IAHS Press
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Standard deviation
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Fig. 6 Observed and simulated monthly statistics for RG010 at a daily time step. In each plot, the solid represents the observed series and the dashed lines the envelope obtained from 10 GLM simulations.
different from that at which it is fitted. It has been applied to several climates and is able to reproduce important features of the rainfall field from the hourly to the daily scale (Onof & Wheater 1993, 1994). In the BLRPM, storm origins ti occur following a Poisson process (rate λ). Cell origins tij arrive following Poisson processes (rate β) starting at each ti; cell arrival process terminates after an exponentially distributed time vi (parameter γ). Each cell has an exponentially distributed duration wij (parameter η), and a uniform intensity Xij with a specified distribution. In this case, the distribution is assumed exponential (parameter 1/μx). Rodriguez-Iturbe et al. (1988) observed that this basic model does not provide a satisfactory reproduction of dry periods. A greater diversity of the internal wet–dry structures of storms can thus be introduced. This is achieved by randomly varying η from storm to storm; η is sampled from a gamma distribution with shape parameter α and scale parameter ν. Parameters β and γ also vary proportionally (κ = β/η and φ = γ/η are constant). A six-parameter model is thus obtained, characterized by the following set of parameters: {λ, μ x , α, ν, κ , φ} . Copyright © 2007 IAHS Press
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Fitting of the BLRPM The BLRPM model parameters are estimated using the software of Chandler & Lourmas (2004). Fitting is carried out for every month using a method of moments to minimize the following objective function: k
S (θ ) = ∑ wi [Ti − τ i (θ )]2
(5)
i =1
where θ = (θ1, …, θk) is the set of unknown parameters, wi represents the weights, Ti is the historical statistics of the data and τi the theoretical model properties as a function of θ. The statistics that are included in the objective function are the mean, 1-hour variance, 6-hour variance, 24-hour variance, 1-hour autocovariance, 1-hour proportion of dry periods and 24-hour proportion of dry periods. Initially, equal weights were adopted, but an improvement was observed when using the inverse of the squared of the historical values (averaged for all years) as weights. In order to obtain realistic storms, boundaries constraining the range of model parameters were defined as shown in Table 3. The monthly parameters are given in Table 4. Once the single-site parameters are obtained they can be input to the HYETOS disaggregation model (Koutsoyiannis & Onof, 2000). Figure 7 compares the observed and disaggregated hourly data at site RG007. It can be observed that the mean is preserved and a close agreement between the observed and disaggregated series is obtained for the standard deviation. A good fit is obtained for the lag 1 autocorrelation Table 3 Boundary constraints of parameters. Parameter λ (h-1) μ (mm/h) σ/μ Α α/ν (h-1) κ φ
Lower bound 0.000001 0.000001 1.0001 0.000001 0.000001 0.000001 0.03
Upper bound ∞ ∞ ∞ ∞ ∞ ∞ ∞
Table 4 Estimated BLRPM parameters for the master gauge (site RG007). Month
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μx (mm/h)
α
ν (h)
κ
φ
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0,03045 0,02812 0,02571 0,02889 0,03844 0,03898 0,03529 0,04490 0,05540 0,03441 0,03902 0,03480
1,34918 1,10419 1,14201 1,28064 1,66215 1,45045 3,53722 8,14571 0,89458 1,55293 0,84768 1,01000
14,09377 1,31313 1,46697 3,11727 3,14591 3,95664 5,68044 1,61167 2,06675 8,08475 6,31663 3,73000
11,56643 0,44682 0,29135 0,71515 0,96223 1,38787 0,64636 0,03259 0,49748 10,71062 3,61485 2,10000
0,18001 0,00004 0,29989 0,77340 0,27395 0,65606 0,13194 0,29271 0,48163 0,25600 0,77673 0,23900
0,03267 0,14517 0,14808 0,11841 0,13104 0,21698 0,03851 0,12844 0,13352 0,55413 0,25359 0,05830
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Fig. 7 Impact of the HYETOS single-site disaggregation scheme on properties of hourly rainfall at the master gauge.
and the skewness. It is expected that, given a longer record length, a better fit could be obtained at the calibration stage for the proportion of dry periods. Copyright © 2007 IAHS Press
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Multi-site spatially-uniform temporal disaggregation
Once the disaggregation model has been run to obtain sub-daily data at a master gauge, a standard profile is defined as the hourly rainfall at the master gauge divided by the daily total rainfall. The temporal profile is then applied to each gauge by multiplying the standard profile by the total daily rainfall at that gauge according to: H Mj H Sj = j = 1, …, 24 (6) YS YM where HSj is the hourly profile at hour j for gauge S, HMj is the master gauge hourly rainfall depth at hour j, YM and YS are the daily totals for the master gauge and a satellite site, respectively. If the master gauge records zero rainfall while other gauges record non-zero daily intensity, the previous profile is used to disaggregate at other gauges. This scheme can be applied to mostly stratiform situation. However limitations are observed for rapidly moving storms and highly variable rainfall as in the case of convective summer events. Hence the profile of hourly data of RG007 is applied at all sites to 10 years of simulated daily data generated from GLMs. Results in terms of the reproduction of standard statistics are presented at a representative site, RG003, in Fig. 8. The 10 years of simulation are divided into four periods of two and a half years to mach the historical data length. Each two and a half years are considered as one simulation. A close agreement between the historical and the modelled series is obtained for the mean, the lag 1 autocorrelation, the skewness and the standard deviation. However the model overestimates the proportion of dry periods. The main limitation is in the spatial correlation structure at an hourly level which is the same as the daily correlation since a single hourly profile is applied uniformly in space. Spatial distribution of rainfall with IDW
So far, the methodology has focused on the modelling of sequences of hourly rainfall at several locations in space. However for hydrological applications, there is a need to infer areal rainfall over a grid or a catchment from these point measurements via the use of interpolation techniques. Several techniques have been developed such as Thiessen polygons (Thiessen, 1911), inverse distance weighting (IDW) (Dirks et al. 1998), spline techniques (Naoum & Tsanis, 2004), and more sophisticated ones such as kriging (Cressie, 1991). Although much has been written, there is little or no agreement among authors on the superiority of any spatial interpolation techniques over others (Tabios & Salas, 1985; Ball & Luk, 1998; Yao & Creed, 2005). The suitability of a particular method depends on various factors including the spatial and temporal resolution of the data, in addition to the geographical location and topographic effects of the area (Naoum & Tsanis, 2004). In this investigation, the IDW method was chosen. This is because, given its degree of complexity, the use of kriging does not appear to be justified for the relatively high resolution raingauge network available for this project. In contrast to simpler schemes such as Thiessen polygons and the areal mean method, IDW produces rain fields that appear more realistic (Dirks et al., 1998; Su et al., 2005). This is Copyright © 2007 IAHS Press
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Fig. 8 Single-site disaggregation at the master gauge conditioned on simulated daily rainfall from GLMs and sub-daily temporal profile applied at a typical site on the catchment (RG010). The plots compare the properties of hourly rainfall of the observed series with those obtained from four simulations. Copyright © 2007 IAHS Press
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perceived as an advantage because the initial values (resulting from the GLM), incorporate the spatial dependence between the gauges and thus this dependence is preserved during the interpolation. A brief description of the methodology is given below. An IDW weighting factor is defined as:
1 (7) dp where w(d) is the weighting factor applied to a known value, d is the distance from the known value to the unknown value, and p is a user-selected power factor. Here weight decreases as distance increases from the interpolated points. Greater values of p assign greater influence to values closest to the interpolated point. A general form for interpolating a value using IDW is: w(d ) =
Z=
N
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n =1 N
i
∑d
p
1 ∑ p n =1 d i
(8)
where Z is the value of the interpolated point, Zi is a known value, N is the total number of known points used in interpolation. The sum of all weights for an interpolated point must be unity. The choice of power p may affect the accuracy of the resulting interpolated field. Dirks et al. (1998) showed however that the exact choice of the numerical value of the power has minimal effect on the resulting errors providing the value is within the range of about 1.5 to 4, and suggested a typical value of 2 to reasonably fit temporal scales from hourly to yearly. In this work the user is allowed to select a power (see following paragraph) but all results presented are for a standard power of 2. Still, it is suggested that further development of this work could attempt to incorporate in the IDW process (for example through a spatially variable choice of power) information derived from radar data where available (similar to Haberlandt, 2007). This could be achieved, for example, by introducing a spatial “bias” on the weighting process (Makropoulos & Butler, 2006). Such a method has not been however developed in this work, due to lack of reliable radar data. IDW implementation
The IDW was implemented through a custom-written algorithm requiring the origin of the catchment (arbitrarily defined) and the position of the raingauges used for the interpolation relative to the origin to be entered as input. In addition, the length and width of the catchment, the cell size and the power of the weighting function should also be provided by the user. In this case, the Dalmuir catchment is defined as a rectangle of 28.6 km length and 12.5 km width. The origin of the catchment is taken at 246 km and 665.7 km for eastings and northings, respectively. IDW is used to spatially interpolate the 10 years of simulated hourly rainfall sequences at the 16 raingauges to produce areal rainfall for a grid size of 500 m at each hourly time step. Copyright © 2007 IAHS Press
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Figures 9 and 10 illustrate the methodology. Figure 9 shows the hourly rainfall time series for June 2009 at a random point 6 km away from the origin of the catchment in both the Eastings and Northings directions. Figure 10 displays the interpolated field of 30 June 2009 at 23:00 h in the case where the power of the
Fig. 9 Hourly rainfall time series obtained from the IDW interpolation scheme applied to simulated rainfall for June 2009 at a random point in space.
Fig. 10 Interpolated hourly rainfall field over the Dalmuir catchment obtained from simulated rainfall sequences at the raingauge locations on 30 June 2009 at 23:00 h. Precipitation is in mm and distance in km. Copyright © 2007 IAHS Press
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weighting function is 2. Peaks and troughs can be seen in the vicinity of the raingauges in Fig. 10. Further work is necessary to derive a more realistic rainfall field. However the given data set is short in record length and there are questions of the quality of the data. This may explain some of the inconsistencies in the simulations and interpolation and additional investigation of the given historical record is necessary. Overall the methodology presented here is a powerful tool to recreate fine time and spatial resolution data. CONCLUSIONS
Simulations of fine spatio-temporal resolution rainfall can be used as input to rainfall– runoff models to support flood risk management. A disaggregation method based on Poisson-cluster processes conditioned on daily data simulated from a GLM and a spatially uniform temporal profile of hourly rainfall applied to all raingauges has been described. Spatial interpolation using inverse distance weighting is then applied to obtain a gridded rainfall field over the catchment. The results from a network of 16 raingauges in the Dalmuir catchment are promising. The simulated rainfall preserves properties that match the empirical statistics to an acceptable level for practical purposes. The simulation of 10 years of daily rainfall at 16 sites was achieved using a GLM. The stochastic nature of this procedure must be emphasized although these simulated time series are not the actual rainfall depths that will fall to the ground in the future, their statistics are consistent with the actual time series and can be used for engineering design. A disaggregation procedure was then applied at a single-site first to obtain hourly rainfall time series. The statistics of the disaggregated time series were compared with the observed values. The mean was preserved. A good agreement was obtained for most statistics but some bias was observed for the proportion of dry periods. Further improvement can be achieved at the calibration stage, however because of the short length of the recorded data set, the derived optimal parameter set was selected to carry on the investigation. Next the hourly temporal rainfall profile of the master gauge was used to downscale the daily totals at the satellite sites into hourly rainfall time series. This method can be applied provided that the gauges are relatively close in space but there is a need to assess at what distance spatial heterogeneities need to be included. Overall, the hourly time series obtained from the combined scheme were compared with the observed time series and most statistics showed good agreement. Spatial interpolation with IDW was then applied to the data to obtain some hourly rainfall field at 500 m spatial resolution. Based on subdaily rainfall data for a period of two and a half years from a network of sixteen raingauges, 10 years of hourly data interpolated to the whole catchment were obtained. The methodology is straightforward and makes use of existing models and techniques. In the context of urban flood management, possible extensions include combining the rainfall simulation methodology with a cascade model to disaggregate the hourly data to five minutes (or less) data as well as testing the modelled rainfall with the incorporation of climate change considerations to the rainfall–runoff models. A possible way to achieve this, which is compatible with the methodological frameCopyright © 2007 IAHS Press
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work described in this paper, was presented by Frost et al. (2006), in which temperature, pressure and humidity were derived by the Hadley Centre, for future and control periods under scenarios of climate change. These were used as covariates for the GLMs and the single-site disaggregation procedure was then applied to the daily simulations yielding encouraging results. A similar approach could easily be incurporated in the methodology discussed here allowing for daily simulations to be driven by climate change scenarios through GLMs covariates. Acknowledgements The authors would like to thank A. M. Ireson for his assistance in the implementation of the models. Provision of data by Scottish Water is much appreciated. In addition, we would like to thank two anonymous reviewers and J. Leitao for their useful comments, which helped to improve the manuscript. REFERENCES Ashley, R. M., Blanksby, J. R., Cashman, A., Jack, L., Wright, G., Packman, J., Fewtrell, L. & Poole, A. (2007) Adaptable Urban Drainage—Addressing Change in Intensity, Occurrence and Uncertainty of Stormwater (AUDACIOUS). J. Built Environment 33(1), 70–84. Ball, J. E. & Luk, K. C. (1998) Modelling spatial variability of rainfall over a catchment. J. Hydrol. Engng 3(2), 122–130. Berne, A., G. Delrieu, J.–D., Creutin & Obled, C. (2004) Temporal and spatial resolution of rainfall measurements required for urban hydrology. J. Hydrol. 299(3/4), 166–179. Chandler, R. E. (2002) GLIMCLIM: Generalized linear modelling for daily climate time series: software and user guide. Department of Statistical Science, University College London, http://www.homepages.ucl.ac.uk/~ucakarc/work/rain_glm.html. Chandler, R. E. & Lourmas, G. (2004) Software for single–site stochastic rainfall model fitting, User guide. DEFRA Project FD2105, Department of Statistical Science, University College London, UK. Chandler, R. E. & Wheater, H. S. (2002) Analysis of rainfall variability using generalised linear models: a case study from the west of Ireland. Water Resour. Res. doi:10.1029/2001WR000906. Coe, R. & Stern, R. D. (1982) Fitting models to daily rainfall data. J. Appl. Met. 21, 1024–1031. Cressie, N. (1991) Statistics for Spatial Data. John Wiley & Sons Inc., New York, USA. Dirks, K. N., Hay, J. E., Stow, C. D. & Harris, D. (1998) High–resolution studies of rainfall on Norfolk Island. Part II: Interpolation of rainfall data. J. Hydrol. 208(3/4), 187–193. Frost, A., Chandler, R. E. & Segond, M.-L. (2006) Multi-site downscaling for the Blackwater Catchment. University College London and Imperial College London Tech. Report, http://www.ucl.ac.uk/stats/research/Rainfall/FD2113_rpt4.pdf. Haberlandt, U. (2007) Geostatistical interpolation of hourly precipitation from rain gauges and radar for a large-scale extreme rainfall event. J. Hydrol. 332(1/2), 144–157. Koutsoyiannis, D. & Onof, C. (2000) HYETOS—a computer program for stochastic disaggregation of fine-scale rainfall. Available from http://www.itia.ntua.gr/e/softinfo/3/. Koutsoyiannis, D. & C. Onof (2001) Rainfall disaggregation using adjusting procedures on a Poisson cluster model. J. Hydrol. 246(1/4), 109. Makropoulos, C. & Butler, D. (2006) Spatial ordered weighted averaging: incorporating spatially variable attitude towards risk in spatial multicriteria decision-making. Environ. Modell. Software 21(1), 69–84. Naoum, S. & Tsanis, I. K. (2004) A hydroinformatic approach to assess interpolation techniques in high spatial and temporal resolution. Can. Water Resour. J. 29(1), 23–46. Nelder, J. A. & Wedderburn, R. W. M. (1972) Generalized linear models. J. Roy. Statist. Soc. A135(3), 370–384. Onof, C. & Wheater, H. S. (1993) Modelling of British rainfall using a random parameter Bartlett-Lewis rectangular pulse model. J. Hydrol. 149, 177–195. Onof, C. & Wheater, H. S. (1994) Improvements to the modelling of British rainfall using a random parameter BartlettLewis rectangular pulse model. J. Hydrol. 157, 177–195. Onof, C., Chandler, R. E., Kakou, A., Northrop, P., Wheater, H. S. & Isham, V. (2000) Rainfall modelling using Poisoncluster processes: a review of developments. Stochast. Environ. Res. Risk Assess. 14, 384–411. Rodriguez-Iturbe, I., Cox, D. R. & Isham, V. (1987) Some models for rainfall based on stochastic point processes. Proc. Roy. Soc. London A417, 269–298. Rodriguez-Iturbe, I., Cox, D. R. & Isham, V. S. (1988) A point process model: further developments. Proc. Roy. Soc. London A147, 283–298. Schilling, W. (1991) Rainfall data for urban hydrology: what do we need? Atmos. Res. 27(1/3), 5–21. Segond, M.-L. (2006) Stochastic modelling of space–time rainfall and the significance of spatial data for flood runoff generation. PhD Thesis, University of London, UK.
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Received 2 October 2006; accepted 1 August 2007
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