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A multiplier-based method of generating stochastic areal rainfall from point rainfalls
Physics and Chemistry of the Earth xxx (2013) xxx–xxx
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A multiplier-based method of generating stochastic areal rainfall from point rainfalls J.G. Ndiritu ⇑ School of Civil and Environmental Engineering, University of the Witwatersrand, Private Bag 3, WITS 2050, South Africa
a r t i c l e
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Article history: Available online xxxx Keywords: Areal rainfall Multipliers Uncertainty Storage Yield Reliability
a b s t r a c t Catchment modelling for water resources assessment is still mainly based on rain gauge measurements as these are more easily available and cover longer periods than radar and satellite-based measurements. Rain gauges however measure the rain falling on an extremely small proportion of the catchment and the areal rainfall obtained from these point measurements are consequently substantially uncertain. These uncertainties in areal rainfall estimation are generally ignored and the need to assess their impact on catchment modelling and water resources assessment is therefore imperative. A method that stochastically generates daily areal rainfall from point rainfall using multiplicative perturbations as a means of dealing with these uncertainties is developed and tested on the Berg catchment in the Western Cape of South Africa. The differences in areal rainfall obtained by alternately omitting some of the rain gauges are used to obtain a population of plausible multiplicative perturbations. Upper bounds on the applicable perturbations are set to prevent the generation of unrealistically large rainfall and to obtain unbiased stochastic rainfall. The perturbations within the set bounds are then fitted into probability density functions to stochastically generate the perturbations to impose on areal rainfall. By using 100 randomly-initialized calibrations of the AWBM catchment model and Sequent Peak Analysis, the effects of incorporating areal rainfall uncertainties on storage-yield-reliability analysis are assessed. Incorporating rainfall uncertainty is found to reduce the required storage by up to 20%. Rainfall uncertainty also increases flow-duration variability considerably and reduces the median flow-duration values by an average of about 20%. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Rainfall is a major driver of hydrological processes and therefore a major component of models of these processes. Although rainfall measurement by remote sensing provides better spatial coverage than rain gauge measurements, these data are generally not as readily available as rain gauge measurements and typically span over shorter periods. Rain gauge measurements are therefore still a major source of rainfall data for practical hydrological analysis including areal rainfall determination. They are also vital for verifying remotely-sensed rainfall measurements. Rain gauges are usually sparsely located and a rain gauge measures rainfall over a very small area of the catchment. The resulting areal rainfall estimates are therefore substantially uncertain irrespective of the areal rainfall estimation method used. Although areal rainfall uncertainty has been the focus of many studies (Seed and Austin, 1990; Andreassian et al., 2001; Vrugt et al., 2008; Volkmann et al., 2010), there is still a lack of methods that are sufficiently robust yet simple enough for routine practical application. The aim here was to develop such a method and assess it using ⇑ Tel.: +27 011 717 7134; fax: +27 011 717 7045. E-mail address: [email protected]
storage-yield-reliability and flow-duration analysis – both typical water resources assessment problems. It was also the intention to keep the formulation simple and robust so as to make it easily applicable in practice. The main basis of the approach is the understanding that the uncertainty in areal rainfall determination can be obtained from the differences in areal rainfall obtained if half (or about half) of the rainfall stations are alternately omitted whilst trying to maintain as uniform a catchment-wide spatial coverage as possible. Since the best estimate of areal rainfall is that obtained when all the rain gauges are used, the uncertainties obtained at half the rain gauge density would need to be scaled down before being applied at the actual density. An approach for scaling down the perturbations is therefore included in the formulation. The approach proposed has the advantage of naturally adapting to rain gauge data availability and any unique spatial rainfall patterns of the area because it uses the actual data that has been recorded which implicitly incorporates these patterns or characteristics. The approach is non-parametric therefore has the advantage of ‘‘letting the data speak for themselves’’ (Wand and Jones, 1995) which parametric method typically lack. The proposed method also allows for any method of estimating areal rainfall as the uncertainties are based on areal rainfall estimates. Furthermore, the
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Please cite this article in press as: Ndiritu, J.G. A multiplier-based method of generating stochastic areal rainfall from point rainfalls. J. Phys. Chem. Earth (2013), http://dx.doi.org/10.1016/j.pce.2013.09.010
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approach enables the inaccuracies of the areal rainfall estimation method to get naturally incorporated in the uncertainty framework. The assessment of areal rainfall uncertainty by varying rain gauge density has been studied previously (Cheng et al., 2012, 2007; Anctil et al., 2006; Andreassian et al., 2001; Duncan et al., 1993) but these studies did not specifically aim at formulating simple methods for incorporating uncertainties on areal rainfall estimates. Hydrologic uncertainty has been expressed multiplicatively in several other studies (e.g. Kavetski et al., 2006; Vrugt et al., 2008; Kuczera et al., 2010; McMillan et al., 2011) and this approach is adopted here. The differences in areal rainfall are expressed as ratios (here after called multipliers).
Table 1 Basic statistics of 8 rain gauge stations in the study catchment. Rain gauge station
MAR (mm/year)
% of rain days per year
0041713 W 0041417 W 0041060 W 0021548 W 0021130 A 0021105 W 0021230 W 0021621 W
643 411 578 564 367 456 615 620
9 16 20 31 15 22 21 26
M2,t that are reciprocals of each other can be obtained for period t as:
2. Case study catchment
M1;4 ¼ L1;4 =L2;t Since the modelling approach is data-based, it was decided to search for a reliable daily rainfall record that required very little patching. The catchment selected also needed to have a reliable natural streamflow record to enable rainfall–runoff modelling for assessing the effect of incorporating rainfall uncertainties on storage-yield-reliability and flow-duration relationships. Considering these factors, the 1280 km2 Berg catchment up to river gauging station G2H014 in South Africa (Fig. 1) was selected for the analysis. Daily rainfall data were obtained from Lynch (2003) and required an average of 3% of patching while streamflow and evaporation data were obtained from the South African Department of Water Affairs webpage (www.dwa.gov.za). Four of the rain gauge stations (0041713 W, 0021130 W, 0041417 W and 0021230 W) had daily rainfall data available over a 42 year long period (1939–1981) while the remaining four had data for the 4 year period between 1967 and 1971. Table 1 shows the basic statistics of the 8 rainfall stations for the 4 year period (1967–1971) for which all 8 had rainfall daily data available. Daily streamflow data at G2H014 (Fig. 1) were available between 1967 and 1981 although there were considerable periods with missing flows. A 4-year record of daily evaporation from 1980 to 1984 was available from station H1E002. 3. Development of stochastic areal rainfall generator 3.1. Multipliers and their characteristics If L1,t and L2,t are non-zero areal rainfall estimates obtained using half the rain gauges by alternately omitting half of them whilst trying to maintain as uniform a catchment-wide spatial coverage as possible, two multipliers of areal rainfall M1,t and
and M 2;t ¼ L2;t =L1;t
ð1Þ
The relevant characteristics of these multipliers need to be identified for them to be applied effectively for incorporating uncertainty. An expected feature of the multipliers is their variation with rainfall magnitude because higher rainfall cover larger areas and the relative differences recorded by the rain gauges would be lower in comparison to lower rainfall. There is also the expectation that the multipliers would be less varied and closer to unity as the rain gauge density increases as the computed values of L1,t and L2,t would be closer if they are determined from a more closely spaced rainfall stations. These two aspects; the variation of the multipliers with rainfall magnitude and with rain gauge density were investigated using the rainfall data for the 8 stations for the period April 15 1967 to January 5 1971. To do this, multipliers were computed for the following 3 cases: Case A: Areal rainfall from all 8 stations related to multipliers obtained using 0041713 W, 0041060 W, 0021548 W and 0021230 W as one group and 0041417 W, 0021130 A, 0021105 W and 0021621 W as the other. Case B: Areal rainfall from 4 stations; 0041417 W, 0021105 W, 0021130 A and 0021621 W related to multipliers obtained using 0041417 W and 0021105 W as one group and 0021130 A and 0021621 W as the other. Case C: Areal rainfall from 4 stations; 0041713 W, 0021230 W, 0041060 W and 0021548 W related to multipliers obtained using 0041713 W and 0021230 W as one group and 0041060 W and 0021548 W as the other. Fig. 2 shows the variation of multiplier values with areal rainfall. A few multipliers exceeded a value of 5 but these were ignored and considered too large to be applied. Fig. 2 shows the expected
Legend 0021548W Rain gauging station
0041713W
G2H014 River gauging station
0041060W
0021548
10 km
G2H014 0021230W
0021621W
Fig. 1. The Berg River catchment to G2H014.
Please cite this article in press as: Ndiritu, J.G. A multiplier-based method of generating stochastic areal rainfall from point rainfalls. J. Phys. Chem. Earth (2013), http://dx.doi.org/10.1016/j.pce.2013.09.010
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Fig. 3b. Cumulative density plots of multipliers with scaling for Cases B and C.
Fig. 2. The variation of rainfall with the multiplier values.
Fig. 3a. Cumulative density plots of multipliers for rainfall P 3 mm/day.
tendency for the multipliers to converge towards unity as rainfall increases although multiplier variability is not symmetrical about unity. Because Case A is applied at twice the rain gauge density used for Cases B and C, the variability of multipliers would be expected to be smaller for Case A but this is not apparent from Fig. 2. Cumulative density plots of the multipliers however revealed that for rainfall estimates exceeding 3 mm/day, the multiplier variability for Case A was lower than that of Cases B and C as seen in Fig. 3a. The best areal rainfall estimates are obtained at the existing (highest) rain gauge density while the perturbations can only be obtained at half this density (by alternately omitting half of the rainfall stations) or at lower densities. A method of quantifying the change in variability of perturbations as rain gauge density changes can therefore help to obtain the perturbations that need to be applied on the areal rainfall estimates obtained at the existing (highest) rain gauge density. A practical approach to this was to find out how the cumulative density plots at lower rain gauge densities can be scaled to those of higher rain gauge densities. Using this approach, it was found that scaling the lower density plots of Cases B and C by a factor of 0.68 about the median reduced the variability to that of higher density plot of Case A (Fig. 3b). The multiplier characteristics observed in Figs. 2, 3a and 3b revealed that areal rainfall is subject to large uncertainties and ignoring them is therefore hardly justifiable. The observations of
Figs. 2, 3a and 3b and trial modelling runs helped to broadly conceptualize the approach needed to incorporate uncertainties on areal rainfall estimates using multipliers. This is now described in points i to iv. (i) The specific uncertainty that needs to be assigned to any given areal rainfall is unknown but plausible probability distributions of these multipliers can be identified. A reasonable approach to incorporate uncertainties is therefore to identify these probability distributions and to use them to stochastically generate ensembles of multipliers to impose on the areal rainfall estimates. This would then generate ensembles of plausible areal rainfall estimates in place of the single value generated if uncertainties are ignored. (ii) The generation of multipliers needs to account for the variation of multiplier variability with rainfall magnitude. This can be done by grouping the rainfall into classes based on the rainfall magnitude and any rainfall-dependent shifts in multiplier behaviour. After some visual observations and trial modelling runs for the study problem, the rainfall classes applied were 0–2, 2–4, 4–6, 6–8, 8–10, 10–15, 15–20 and 20–40 mm/day. (iii) The multipliers obtained by alternately omitting of half the available rain gauges need to be reduced (scaled down) when applied on areal rainfall estimates obtained using all the available rainfall stations and the required scaling factor can be reasonably estimated. For example, for the problem studied here, if we wanted to generate multipliers to impose on areal rainfall estimates obtained at a rain gauge density of 4 stations using the probability distributions obtained by alternately omitting two of the stations a scaling factor of 0.68 would be applied for the multipliers to impose on all rainfall greater than 3 mm/day. Since there are only 8 (and not 16) rain gauge stations available, it is not possible to compute a cumulative density plot of multipliers for determining scaling factors in shifting from a density of 4 stations to 8 stations. Consequently, the scaling factor obtained for the lower rain gauge density would need to be assumed for the actual rain gauge density. The scaling factor of 0.68 for obtaining the cumulative density plot of 4 stations from that of 2 stations would be applied to scale down the cumulative density plot of 4 stations to obtain the one for 8 stations. (iv) As observed in Fig. 2, the multipliers are not symmetrically spaced about unity 1.0 because for every pair of multipliers obtained as in equation 1, the larger one will locate further from unity than the lower one as deduced below.
Please cite this article in press as: Ndiritu, J.G. A multiplier-based method of generating stochastic areal rainfall from point rainfalls. J. Phys. Chem. Earth (2013), http://dx.doi.org/10.1016/j.pce.2013.09.010
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If, in Eq. (1), rainfall L1,t is the greater of the two rainfall estimates (L1,t > L2,t), then, M1,t will be greater than 1.0 while M2,t will be less than 1.0. The differences d1,t and d2,t between each of the two multipliers and 1.0 are then
d1;t ¼ M 1;t 1 ¼ L1;t =L2;t 1 and d2;t ¼ 1 M2;t ¼ 1 L2;t =L1;t ð2Þ The difference between d1,t and d2,t is then
d1;t d2;t ¼ ðL1;t =L2;t 1Þ ðL2;t =L1;t 1Þ ¼ ½ðL1;t Þ2 ðL2;t Þ2 =L1;t L2;t ð3Þ Since L1,t > L2,t, the difference will always be greater than zero and the larger of the two multipliers will therefore always be spaced further from unity (1.0) than the lower one. A purely random selection of multipliers would select M1,t and M2,t in equal proportion and would therefore lead to over-estimation of the overall rainfall. Some modelling trial runs also revealed that there was a tendency to generate unrealistically high rainfall if very large multiplier values were used. To deal with the two problems, it was decided to apply an iterative approach for determining the upper limit on multiplier values that leads to an overall areal rainfall that is unbiased. The theoretical lower bound of multipliers is 0 and this was set as the lower limit after trial runs showed this as reasonable. It also enables a more variable sampling of multiplier than lower bounds set at higher values. After some trial runs, upper bounds ranging from 2.5 to 3.5 were found to be good starting points. Taking into account these factors, a stochastic generator of areal rainfall was formulated as follows. 3.2. Steps in stochastic areal rainfall generation (1) The rainfall is grouped into classes that reflect the variation of multipliers with rainfall magnitude (see point ii of Section 3.1). (2) A starting upper bound of the multipliers is subjectively selected within the range 2.5–3.5. (3) The rainfall class that the areal rainfall Ht obtained using all rain gauges for period t belongs to is determined. For example, if Ht = 16 mm/day, it falls in class 15–20 mm/day. (4) A multiplier within this class (i.e. any multiplier that falls in the range 15–20 mm/day on Fig. 2) is randomly obtained and scaled using an uncertainty factor u that varies between 0 and 1. For the case study catchment, u was set to 1.0 for Ht < 3 mm/day and 0.68 for Ht P 3 mm/day based on the scaling factor obtained in Fig. 3b. For a multiplier M and an uncertainty factor u, the scaled multiplier Ms is obtained as:
Ms ¼ 1:0 þ ðM 1:0Þu
ð4Þ
Eq. (4) gives a multiplier equal to 1 if u = 0 and an unscaled multiplier equal to M if u = 1. The stochastic areal rainfall for period t is then obtained as HtMs. (5) Steps 2–4 are repeated for the complete time series and for as many ensembles as required. (6) The bias in overall average rainfall after generating the required number of ensembles is determined by comparing the average of the stochastic rainfall with that of the unperturbed rainfall. If the bias is significant, another upper multiplier bound is iteratively selected and Steps 2–6 are repeated until the overall bias is negligible. Using this approach, an upper bound of 2.88 was obtained for the Berg River as illustrated in Fig. 4.
Fig. 4. The variation of bias in rainfall generation with upper multiplier bound.
4. Assessing the effect of uncertainty on storage-yield and flowduration relationships In order to assess the effect of incorporating areal rainfall uncertainty on storage-yield and flow-duration analysis, the AWBM model (Boughton, 2004) was selected for rainfall–runoff modelling. The AWBM has a parsimonious model structure and was found to perform better than a daily version of the widely used monthly Pitman model (Pitman, 1973) in trial modelling runs. Calibration was carried out by a hybrid manual-automatic approach using the SCE-UA method (Duan et al., 1992) and setting realistic initial ranges and ultimate bounds of the AWBM parameters. The computationally efficient Sequent Peak Algorithm was selected for determining storage-yield relationships. The stream flow data between 1967 and 1974 was used for validation while 1975–1981 was used for calibrating the AWBM model. The validation runs however included the previous period starting in 1939 and the 36 year-long period from 1939 to 1974 was used for storage-yield and flow-duration analysis. The assessment also recognizes that ideal incorporation of areal rainfall uncertainty should not degrade the validation performance of the rainfall–runoff modelling because the uncertainty-imposed rainfall should be as equally likely to occur as those derived from the actual measurements. The assessment followed the following steps: (1) One hundred randomly initialized calibration–validation runs were carried out using the single areal rainfall series that had not been subjected to uncertainty and the validation performance of each run was obtained. This was done using a composite measure defined as ðCE þ jBIASjþ CMAD þ RMCCÞ=2 where CE;jBIASj; CMAD and RMCC is the coefficient of efficiency, the absolute bias, the coefficient of mean absolute deviation and the residual mass curve coefficient respectively. The RMCC quantifies systematic bias as described by Ndiritu (2009). (2) A cumulative probability density plot of the 100 validation performances from step 1 was generated. (3) 100 randomly initialized calibration–validation runs with each run using perturbed areal rainfall estimates were carried out and validation performance were obtained as in step 1. (4) A cumulative probability density plot of the validation performance obtained in step 3 was generated and compared with that obtained with unperturbed rainfall (step 2). If
Please cite this article in press as: Ndiritu, J.G. A multiplier-based method of generating stochastic areal rainfall from point rainfalls. J. Phys. Chem. Earth (2013), http://dx.doi.org/10.1016/j.pce.2013.09.010
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Fig. 5. Cumulative density plots of validation performance.
Fig. 7a. Effect of rainfall uncertainty on 30 May daily flows with validation performance matched.
Fig. 6a. Effect of rainfall uncertainty on storage with performance matched.
Fig. 7b. Effect of rainfall uncertainty on 30 May daily flows with validation performance not matched.
Fig. 6b. Effect of rainfall uncertainty on storage with performance not matched.
the central measures (the median or the mean) of the validation performance with uncertainty were close enough to those without uncertainty, step 6 followed and if not, step 5 followed.
(5) Additional calibration–validation runs were made with perturbed areal rainfall estimates and 100 runs that gave a cumulative density plot of validation performance that had central measures (mean and median) adequately close to the measures obtained without uncertainty (step 2) were sampled out. (6) The 100 daily flow sequences for the period 1939–1974 using the undisturbed areal rainfall (step 2) and the 100 sequences from perturbed rainfall (step 4 or 5) were used to carry out storage-yield analysis. This was done for 9 demand levels ranging from a low level of 10% to a high level of 90% of the mean of a specified representative flow and varying in steps of 10%. The flow sequence used to set the demands was the median of the 100 flows generated using unperturbed rainfall (step 1). The storages were compared to assess the effect of uncertainties on storage-yield relationships. (7) Flow-duration analysis for the 365 days of the year using the 100 daily flow sequences used in step 6 were carried out and used to assess the effect of uncertainties on flow-duration relationships.
Please cite this article in press as: Ndiritu, J.G. A multiplier-based method of generating stochastic areal rainfall from point rainfalls. J. Phys. Chem. Earth (2013), http://dx.doi.org/10.1016/j.pce.2013.09.010
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5. Results and discussion Fig. 5 shows the cumulative density plots of validation performance and reveals that the validation performance with uncertainty (perturbations) incorporated (green dotted1 curve in Fig. 5) was substantially lower than the performance with unperturbed rainfall (dark blue spiked curve on Fig. 5). Step 5 as described in Section 4) was therefore taken and an additional 200 runs were made. From these and the initial 100 (making a total of 300), 100 ensembles that obtained a cumulative density plot that is practically identical to that obtained with unperturbed rainfall were sampled out. The resulting distribution is the red curve shown in Fig. 5. To assess the effect of uncertainty on storage, the 100 storages obtained for each level of demand were first ranked in order of magnitude. For each rank, the percentage change in required storage was obtained as 100(Su–Snu)/Snu where Su and Snu is the storage with and without uncertainty respectively. Fig. 6a is a plot of these percentages for selected demand levels. The average percentages obtained from all 9 demand levels (10–90% in steps of 10%) are also shown. Most of the changes are found to be less than zero with the exception of the top ranked storages at 10% demand and some lower ranked storages at 90% demand. The average percentage changes are all lower than zero with the reduction increasing to 20% at the lower ranks. No reason is found for this finding that would imply overdesign of storage if areal rainfall uncertainty is ignored. For completeness, a similar analysis using the original 100 ensembles obtained without sampling (step 3) was done and presented graphically in Fig. 6b. The curve for the average percentages (of all 9 demand levels) in Fig. 6b points to increases in required storage for ranks of up to 40 and a reduction in required storage for lower ranks. The percentage increases for the higher ranks at 10% demand are again observed to be much higher than for the larger demands. The effect of areal rainfall uncertainty on flow-duration relationships was assessed using 12 representative days of the year that are uniformly spread within the year. To achieve this, any day of each month could be used and the 28th of February and 30th for the other 11 months were subjectively selected. Out of the 100 flow-duration curves obtained with each flow ensemble, the 5th, 50th and 95th percentile values were determined. A comparison of these values with and without rainfall uncertainty was carried out by computing the percentage change in flow that uncertainty brings about on flows of the same exceedance probability. Figs. 7a and 7b show plots of the percentage change in flow with and without validation performance matching respectively for 30th May – a representative high flow day. The changes in flow are mostly negative and generally greater at the extremes of the flow-duration relationship. The expected increase in flow variability as a result of incorporating uncertainties is observed on Figs. 7a and 7b towards the extremes (<20 and >70% exceedance) of the flow-duration relationship. No reasons are found for the observed reduction in the 50th percentile flows and the substantially lower variabilities for the middle portions of the flow-duration relationship. A thorough search of the literature did not find any other study that has investigated the impact of incorporating rainfall uncertainty on storage-yield and flow-duration relationships although a number of studies on the effect of rainfall uncertainty on catchment modelling performance have been conducted. Andreassian et al., 2001 tested three rainfall models on three catchments of varying size and found the prediction performance of the models to improve as the quality of the rainfall input improved. Duncan et al.,
1 For interpretation of color in Fig. 5, the reader is referred to the web version of this article.
1993 obtained similar findings on tests on the effect of rain fall adequacy on the quality of stream hydrograph prediction.
6. Conclusions A method of quantifying areal rainfall uncertainty using multiplicative perturbations (multipliers) has been formulated and used to assess the impact of the uncertainties on storage-yield and flowduration relationships for a 36 year long period of the Berg River catchment in South Africa. The formulation is simple and its application mainly requires (i) the specification of the uncertainty to impose on the areal rainfall estimates, (ii) the lower and upper bounds of the multipliers to apply, and (iii) the grouping of rainfall to effect the variation of multipliers with rainfall magnitude. The analysis reveals that incorporating uncertainties using this method leads to substantial lower validation performance in rainfall–runoff modelling while an ideal incorporation of uncertainties would be expected to maintain this validation performance. To counter this, sampling is used to obtain a population of ensembles that give a similar cumulative density plot of validation performance as flows obtained without incorporation of areal rainfall uncertainties. Using these ensembles, uncertainty is found to reduce the required storage by up to 20%. Areal rainfall uncertainty is also found to reduce the flow-duration ordinates by an average of about 20%. These results may seem contradictory as a reduction in required storage can reasonably be considered to result from increased flows. Reservoir capacity however depends on variability and more detailed analysis would help explain the observations. These observations probably also reveal that the simplistic approach of sampling from perturbed runs in order to obtain the same validation performance as for the unperturbed case (step 5 of Section 4) may not be sufficiently valid. The selection of the lower and upper bounds of multipliers and of the rainfall zones to apply in the method was fairly subjective and not rigorous and this could have lead to the observed reduction in validation performance that prompted sampling out runs to achieve the required validation performance. This is particularly relevant as multiplicative perturbations could easily lead to unrealistically high rainfall if all the parameters are not selected judiciously. McMillan et al. (2011) studied multipliers for a catchment with a dense rain gauge network and high-resolution radar rainfall estimates and reported that the multipliers did not fit at the tails of lognormal distribution and this could lead to large uncertainties in predicting heavy rainfall. An alternative approach using linear perturbations that has been recently formulated (Ndiritu, 2013) is free from this limitation and the requirement to subjectively set bounds on the multiplier values to apply. Additional analysis using the multiplicative and linear perturbations is recommended. Ideally, this needs to span across different climatic regions and apply several rainfall–runoff models and storage-yield approaches. Tests on the approaches using high-density rain gauge data are also proposed. The combined use of radar and rainfall data (Sinclair and Pegram, 2005) and rainfall and satellite rainfall (Grimes et al., 1999) may however also yield robust and realistic approaches for dealing with areal rainfall uncertainties and further work will therefore also take this direction. References Anctil, F., Lauzon, N., Andreassian, V., Oudin, L., Perrin, C., 2006. Improvement of rainfall–runoff forecasts through mean areal rainfall optimization. J. Hydrol. 328, 717–725. Andreassian, V., Perrin, C., Michel, C., Usart-Sanchez, I., Lavabre, J., 2001. Impact of imperfect rainfall knowledge on the efficiency and the parameters of watershed models. J. Hydrol. 250, 206–223.
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A multiplier-based method of generating stochastic areal rainfall from point rainfalls J.G. Ndiritu ⇑ School of Civil and Environmental Engineering, University of the Witwatersrand, Private Bag 3, WITS 2050, South Africa
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i n f o
Article history: Available online xxxx Keywords: Areal rainfall Multipliers Uncertainty Storage Yield Reliability
a b s t r a c t Catchment modelling for water resources assessment is still mainly based on rain gauge measurements as these are more easily available and cover longer periods than radar and satellite-based measurements. Rain gauges however measure the rain falling on an extremely small proportion of the catchment and the areal rainfall obtained from these point measurements are consequently substantially uncertain. These uncertainties in areal rainfall estimation are generally ignored and the need to assess their impact on catchment modelling and water resources assessment is therefore imperative. A method that stochastically generates daily areal rainfall from point rainfall using multiplicative perturbations as a means of dealing with these uncertainties is developed and tested on the Berg catchment in the Western Cape of South Africa. The differences in areal rainfall obtained by alternately omitting some of the rain gauges are used to obtain a population of plausible multiplicative perturbations. Upper bounds on the applicable perturbations are set to prevent the generation of unrealistically large rainfall and to obtain unbiased stochastic rainfall. The perturbations within the set bounds are then fitted into probability density functions to stochastically generate the perturbations to impose on areal rainfall. By using 100 randomly-initialized calibrations of the AWBM catchment model and Sequent Peak Analysis, the effects of incorporating areal rainfall uncertainties on storage-yield-reliability analysis are assessed. Incorporating rainfall uncertainty is found to reduce the required storage by up to 20%. Rainfall uncertainty also increases flow-duration variability considerably and reduces the median flow-duration values by an average of about 20%. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Rainfall is a major driver of hydrological processes and therefore a major component of models of these processes. Although rainfall measurement by remote sensing provides better spatial coverage than rain gauge measurements, these data are generally not as readily available as rain gauge measurements and typically span over shorter periods. Rain gauge measurements are therefore still a major source of rainfall data for practical hydrological analysis including areal rainfall determination. They are also vital for verifying remotely-sensed rainfall measurements. Rain gauges are usually sparsely located and a rain gauge measures rainfall over a very small area of the catchment. The resulting areal rainfall estimates are therefore substantially uncertain irrespective of the areal rainfall estimation method used. Although areal rainfall uncertainty has been the focus of many studies (Seed and Austin, 1990; Andreassian et al., 2001; Vrugt et al., 2008; Volkmann et al., 2010), there is still a lack of methods that are sufficiently robust yet simple enough for routine practical application. The aim here was to develop such a method and assess it using ⇑ Tel.: +27 011 717 7134; fax: +27 011 717 7045. E-mail address: [email protected]
storage-yield-reliability and flow-duration analysis – both typical water resources assessment problems. It was also the intention to keep the formulation simple and robust so as to make it easily applicable in practice. The main basis of the approach is the understanding that the uncertainty in areal rainfall determination can be obtained from the differences in areal rainfall obtained if half (or about half) of the rainfall stations are alternately omitted whilst trying to maintain as uniform a catchment-wide spatial coverage as possible. Since the best estimate of areal rainfall is that obtained when all the rain gauges are used, the uncertainties obtained at half the rain gauge density would need to be scaled down before being applied at the actual density. An approach for scaling down the perturbations is therefore included in the formulation. The approach proposed has the advantage of naturally adapting to rain gauge data availability and any unique spatial rainfall patterns of the area because it uses the actual data that has been recorded which implicitly incorporates these patterns or characteristics. The approach is non-parametric therefore has the advantage of ‘‘letting the data speak for themselves’’ (Wand and Jones, 1995) which parametric method typically lack. The proposed method also allows for any method of estimating areal rainfall as the uncertainties are based on areal rainfall estimates. Furthermore, the
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Please cite this article in press as: Ndiritu, J.G. A multiplier-based method of generating stochastic areal rainfall from point rainfalls. J. Phys. Chem. Earth (2013), http://dx.doi.org/10.1016/j.pce.2013.09.010
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approach enables the inaccuracies of the areal rainfall estimation method to get naturally incorporated in the uncertainty framework. The assessment of areal rainfall uncertainty by varying rain gauge density has been studied previously (Cheng et al., 2012, 2007; Anctil et al., 2006; Andreassian et al., 2001; Duncan et al., 1993) but these studies did not specifically aim at formulating simple methods for incorporating uncertainties on areal rainfall estimates. Hydrologic uncertainty has been expressed multiplicatively in several other studies (e.g. Kavetski et al., 2006; Vrugt et al., 2008; Kuczera et al., 2010; McMillan et al., 2011) and this approach is adopted here. The differences in areal rainfall are expressed as ratios (here after called multipliers).
Table 1 Basic statistics of 8 rain gauge stations in the study catchment. Rain gauge station
MAR (mm/year)
% of rain days per year
0041713 W 0041417 W 0041060 W 0021548 W 0021130 A 0021105 W 0021230 W 0021621 W
643 411 578 564 367 456 615 620
9 16 20 31 15 22 21 26
M2,t that are reciprocals of each other can be obtained for period t as:
2. Case study catchment
M1;4 ¼ L1;4 =L2;t Since the modelling approach is data-based, it was decided to search for a reliable daily rainfall record that required very little patching. The catchment selected also needed to have a reliable natural streamflow record to enable rainfall–runoff modelling for assessing the effect of incorporating rainfall uncertainties on storage-yield-reliability and flow-duration relationships. Considering these factors, the 1280 km2 Berg catchment up to river gauging station G2H014 in South Africa (Fig. 1) was selected for the analysis. Daily rainfall data were obtained from Lynch (2003) and required an average of 3% of patching while streamflow and evaporation data were obtained from the South African Department of Water Affairs webpage (www.dwa.gov.za). Four of the rain gauge stations (0041713 W, 0021130 W, 0041417 W and 0021230 W) had daily rainfall data available over a 42 year long period (1939–1981) while the remaining four had data for the 4 year period between 1967 and 1971. Table 1 shows the basic statistics of the 8 rainfall stations for the 4 year period (1967–1971) for which all 8 had rainfall daily data available. Daily streamflow data at G2H014 (Fig. 1) were available between 1967 and 1981 although there were considerable periods with missing flows. A 4-year record of daily evaporation from 1980 to 1984 was available from station H1E002. 3. Development of stochastic areal rainfall generator 3.1. Multipliers and their characteristics If L1,t and L2,t are non-zero areal rainfall estimates obtained using half the rain gauges by alternately omitting half of them whilst trying to maintain as uniform a catchment-wide spatial coverage as possible, two multipliers of areal rainfall M1,t and
and M 2;t ¼ L2;t =L1;t
ð1Þ
The relevant characteristics of these multipliers need to be identified for them to be applied effectively for incorporating uncertainty. An expected feature of the multipliers is their variation with rainfall magnitude because higher rainfall cover larger areas and the relative differences recorded by the rain gauges would be lower in comparison to lower rainfall. There is also the expectation that the multipliers would be less varied and closer to unity as the rain gauge density increases as the computed values of L1,t and L2,t would be closer if they are determined from a more closely spaced rainfall stations. These two aspects; the variation of the multipliers with rainfall magnitude and with rain gauge density were investigated using the rainfall data for the 8 stations for the period April 15 1967 to January 5 1971. To do this, multipliers were computed for the following 3 cases: Case A: Areal rainfall from all 8 stations related to multipliers obtained using 0041713 W, 0041060 W, 0021548 W and 0021230 W as one group and 0041417 W, 0021130 A, 0021105 W and 0021621 W as the other. Case B: Areal rainfall from 4 stations; 0041417 W, 0021105 W, 0021130 A and 0021621 W related to multipliers obtained using 0041417 W and 0021105 W as one group and 0021130 A and 0021621 W as the other. Case C: Areal rainfall from 4 stations; 0041713 W, 0021230 W, 0041060 W and 0021548 W related to multipliers obtained using 0041713 W and 0021230 W as one group and 0041060 W and 0021548 W as the other. Fig. 2 shows the variation of multiplier values with areal rainfall. A few multipliers exceeded a value of 5 but these were ignored and considered too large to be applied. Fig. 2 shows the expected
Legend 0021548W Rain gauging station
0041713W
G2H014 River gauging station
0041060W
0021548
10 km
G2H014 0021230W
0021621W
Fig. 1. The Berg River catchment to G2H014.
Please cite this article in press as: Ndiritu, J.G. A multiplier-based method of generating stochastic areal rainfall from point rainfalls. J. Phys. Chem. Earth (2013), http://dx.doi.org/10.1016/j.pce.2013.09.010
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Fig. 3b. Cumulative density plots of multipliers with scaling for Cases B and C.
Fig. 2. The variation of rainfall with the multiplier values.
Fig. 3a. Cumulative density plots of multipliers for rainfall P 3 mm/day.
tendency for the multipliers to converge towards unity as rainfall increases although multiplier variability is not symmetrical about unity. Because Case A is applied at twice the rain gauge density used for Cases B and C, the variability of multipliers would be expected to be smaller for Case A but this is not apparent from Fig. 2. Cumulative density plots of the multipliers however revealed that for rainfall estimates exceeding 3 mm/day, the multiplier variability for Case A was lower than that of Cases B and C as seen in Fig. 3a. The best areal rainfall estimates are obtained at the existing (highest) rain gauge density while the perturbations can only be obtained at half this density (by alternately omitting half of the rainfall stations) or at lower densities. A method of quantifying the change in variability of perturbations as rain gauge density changes can therefore help to obtain the perturbations that need to be applied on the areal rainfall estimates obtained at the existing (highest) rain gauge density. A practical approach to this was to find out how the cumulative density plots at lower rain gauge densities can be scaled to those of higher rain gauge densities. Using this approach, it was found that scaling the lower density plots of Cases B and C by a factor of 0.68 about the median reduced the variability to that of higher density plot of Case A (Fig. 3b). The multiplier characteristics observed in Figs. 2, 3a and 3b revealed that areal rainfall is subject to large uncertainties and ignoring them is therefore hardly justifiable. The observations of
Figs. 2, 3a and 3b and trial modelling runs helped to broadly conceptualize the approach needed to incorporate uncertainties on areal rainfall estimates using multipliers. This is now described in points i to iv. (i) The specific uncertainty that needs to be assigned to any given areal rainfall is unknown but plausible probability distributions of these multipliers can be identified. A reasonable approach to incorporate uncertainties is therefore to identify these probability distributions and to use them to stochastically generate ensembles of multipliers to impose on the areal rainfall estimates. This would then generate ensembles of plausible areal rainfall estimates in place of the single value generated if uncertainties are ignored. (ii) The generation of multipliers needs to account for the variation of multiplier variability with rainfall magnitude. This can be done by grouping the rainfall into classes based on the rainfall magnitude and any rainfall-dependent shifts in multiplier behaviour. After some visual observations and trial modelling runs for the study problem, the rainfall classes applied were 0–2, 2–4, 4–6, 6–8, 8–10, 10–15, 15–20 and 20–40 mm/day. (iii) The multipliers obtained by alternately omitting of half the available rain gauges need to be reduced (scaled down) when applied on areal rainfall estimates obtained using all the available rainfall stations and the required scaling factor can be reasonably estimated. For example, for the problem studied here, if we wanted to generate multipliers to impose on areal rainfall estimates obtained at a rain gauge density of 4 stations using the probability distributions obtained by alternately omitting two of the stations a scaling factor of 0.68 would be applied for the multipliers to impose on all rainfall greater than 3 mm/day. Since there are only 8 (and not 16) rain gauge stations available, it is not possible to compute a cumulative density plot of multipliers for determining scaling factors in shifting from a density of 4 stations to 8 stations. Consequently, the scaling factor obtained for the lower rain gauge density would need to be assumed for the actual rain gauge density. The scaling factor of 0.68 for obtaining the cumulative density plot of 4 stations from that of 2 stations would be applied to scale down the cumulative density plot of 4 stations to obtain the one for 8 stations. (iv) As observed in Fig. 2, the multipliers are not symmetrically spaced about unity 1.0 because for every pair of multipliers obtained as in equation 1, the larger one will locate further from unity than the lower one as deduced below.
Please cite this article in press as: Ndiritu, J.G. A multiplier-based method of generating stochastic areal rainfall from point rainfalls. J. Phys. Chem. Earth (2013), http://dx.doi.org/10.1016/j.pce.2013.09.010
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If, in Eq. (1), rainfall L1,t is the greater of the two rainfall estimates (L1,t > L2,t), then, M1,t will be greater than 1.0 while M2,t will be less than 1.0. The differences d1,t and d2,t between each of the two multipliers and 1.0 are then
d1;t ¼ M 1;t 1 ¼ L1;t =L2;t 1 and d2;t ¼ 1 M2;t ¼ 1 L2;t =L1;t ð2Þ The difference between d1,t and d2,t is then
d1;t d2;t ¼ ðL1;t =L2;t 1Þ ðL2;t =L1;t 1Þ ¼ ½ðL1;t Þ2 ðL2;t Þ2 =L1;t L2;t ð3Þ Since L1,t > L2,t, the difference will always be greater than zero and the larger of the two multipliers will therefore always be spaced further from unity (1.0) than the lower one. A purely random selection of multipliers would select M1,t and M2,t in equal proportion and would therefore lead to over-estimation of the overall rainfall. Some modelling trial runs also revealed that there was a tendency to generate unrealistically high rainfall if very large multiplier values were used. To deal with the two problems, it was decided to apply an iterative approach for determining the upper limit on multiplier values that leads to an overall areal rainfall that is unbiased. The theoretical lower bound of multipliers is 0 and this was set as the lower limit after trial runs showed this as reasonable. It also enables a more variable sampling of multiplier than lower bounds set at higher values. After some trial runs, upper bounds ranging from 2.5 to 3.5 were found to be good starting points. Taking into account these factors, a stochastic generator of areal rainfall was formulated as follows. 3.2. Steps in stochastic areal rainfall generation (1) The rainfall is grouped into classes that reflect the variation of multipliers with rainfall magnitude (see point ii of Section 3.1). (2) A starting upper bound of the multipliers is subjectively selected within the range 2.5–3.5. (3) The rainfall class that the areal rainfall Ht obtained using all rain gauges for period t belongs to is determined. For example, if Ht = 16 mm/day, it falls in class 15–20 mm/day. (4) A multiplier within this class (i.e. any multiplier that falls in the range 15–20 mm/day on Fig. 2) is randomly obtained and scaled using an uncertainty factor u that varies between 0 and 1. For the case study catchment, u was set to 1.0 for Ht < 3 mm/day and 0.68 for Ht P 3 mm/day based on the scaling factor obtained in Fig. 3b. For a multiplier M and an uncertainty factor u, the scaled multiplier Ms is obtained as:
Ms ¼ 1:0 þ ðM 1:0Þu
ð4Þ
Eq. (4) gives a multiplier equal to 1 if u = 0 and an unscaled multiplier equal to M if u = 1. The stochastic areal rainfall for period t is then obtained as HtMs. (5) Steps 2–4 are repeated for the complete time series and for as many ensembles as required. (6) The bias in overall average rainfall after generating the required number of ensembles is determined by comparing the average of the stochastic rainfall with that of the unperturbed rainfall. If the bias is significant, another upper multiplier bound is iteratively selected and Steps 2–6 are repeated until the overall bias is negligible. Using this approach, an upper bound of 2.88 was obtained for the Berg River as illustrated in Fig. 4.
Fig. 4. The variation of bias in rainfall generation with upper multiplier bound.
4. Assessing the effect of uncertainty on storage-yield and flowduration relationships In order to assess the effect of incorporating areal rainfall uncertainty on storage-yield and flow-duration analysis, the AWBM model (Boughton, 2004) was selected for rainfall–runoff modelling. The AWBM has a parsimonious model structure and was found to perform better than a daily version of the widely used monthly Pitman model (Pitman, 1973) in trial modelling runs. Calibration was carried out by a hybrid manual-automatic approach using the SCE-UA method (Duan et al., 1992) and setting realistic initial ranges and ultimate bounds of the AWBM parameters. The computationally efficient Sequent Peak Algorithm was selected for determining storage-yield relationships. The stream flow data between 1967 and 1974 was used for validation while 1975–1981 was used for calibrating the AWBM model. The validation runs however included the previous period starting in 1939 and the 36 year-long period from 1939 to 1974 was used for storage-yield and flow-duration analysis. The assessment also recognizes that ideal incorporation of areal rainfall uncertainty should not degrade the validation performance of the rainfall–runoff modelling because the uncertainty-imposed rainfall should be as equally likely to occur as those derived from the actual measurements. The assessment followed the following steps: (1) One hundred randomly initialized calibration–validation runs were carried out using the single areal rainfall series that had not been subjected to uncertainty and the validation performance of each run was obtained. This was done using a composite measure defined as ðCE þ jBIASjþ CMAD þ RMCCÞ=2 where CE;jBIASj; CMAD and RMCC is the coefficient of efficiency, the absolute bias, the coefficient of mean absolute deviation and the residual mass curve coefficient respectively. The RMCC quantifies systematic bias as described by Ndiritu (2009). (2) A cumulative probability density plot of the 100 validation performances from step 1 was generated. (3) 100 randomly initialized calibration–validation runs with each run using perturbed areal rainfall estimates were carried out and validation performance were obtained as in step 1. (4) A cumulative probability density plot of the validation performance obtained in step 3 was generated and compared with that obtained with unperturbed rainfall (step 2). If
Please cite this article in press as: Ndiritu, J.G. A multiplier-based method of generating stochastic areal rainfall from point rainfalls. J. Phys. Chem. Earth (2013), http://dx.doi.org/10.1016/j.pce.2013.09.010
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Fig. 5. Cumulative density plots of validation performance.
Fig. 7a. Effect of rainfall uncertainty on 30 May daily flows with validation performance matched.
Fig. 6a. Effect of rainfall uncertainty on storage with performance matched.
Fig. 7b. Effect of rainfall uncertainty on 30 May daily flows with validation performance not matched.
Fig. 6b. Effect of rainfall uncertainty on storage with performance not matched.
the central measures (the median or the mean) of the validation performance with uncertainty were close enough to those without uncertainty, step 6 followed and if not, step 5 followed.
(5) Additional calibration–validation runs were made with perturbed areal rainfall estimates and 100 runs that gave a cumulative density plot of validation performance that had central measures (mean and median) adequately close to the measures obtained without uncertainty (step 2) were sampled out. (6) The 100 daily flow sequences for the period 1939–1974 using the undisturbed areal rainfall (step 2) and the 100 sequences from perturbed rainfall (step 4 or 5) were used to carry out storage-yield analysis. This was done for 9 demand levels ranging from a low level of 10% to a high level of 90% of the mean of a specified representative flow and varying in steps of 10%. The flow sequence used to set the demands was the median of the 100 flows generated using unperturbed rainfall (step 1). The storages were compared to assess the effect of uncertainties on storage-yield relationships. (7) Flow-duration analysis for the 365 days of the year using the 100 daily flow sequences used in step 6 were carried out and used to assess the effect of uncertainties on flow-duration relationships.
Please cite this article in press as: Ndiritu, J.G. A multiplier-based method of generating stochastic areal rainfall from point rainfalls. J. Phys. Chem. Earth (2013), http://dx.doi.org/10.1016/j.pce.2013.09.010
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5. Results and discussion Fig. 5 shows the cumulative density plots of validation performance and reveals that the validation performance with uncertainty (perturbations) incorporated (green dotted1 curve in Fig. 5) was substantially lower than the performance with unperturbed rainfall (dark blue spiked curve on Fig. 5). Step 5 as described in Section 4) was therefore taken and an additional 200 runs were made. From these and the initial 100 (making a total of 300), 100 ensembles that obtained a cumulative density plot that is practically identical to that obtained with unperturbed rainfall were sampled out. The resulting distribution is the red curve shown in Fig. 5. To assess the effect of uncertainty on storage, the 100 storages obtained for each level of demand were first ranked in order of magnitude. For each rank, the percentage change in required storage was obtained as 100(Su–Snu)/Snu where Su and Snu is the storage with and without uncertainty respectively. Fig. 6a is a plot of these percentages for selected demand levels. The average percentages obtained from all 9 demand levels (10–90% in steps of 10%) are also shown. Most of the changes are found to be less than zero with the exception of the top ranked storages at 10% demand and some lower ranked storages at 90% demand. The average percentage changes are all lower than zero with the reduction increasing to 20% at the lower ranks. No reason is found for this finding that would imply overdesign of storage if areal rainfall uncertainty is ignored. For completeness, a similar analysis using the original 100 ensembles obtained without sampling (step 3) was done and presented graphically in Fig. 6b. The curve for the average percentages (of all 9 demand levels) in Fig. 6b points to increases in required storage for ranks of up to 40 and a reduction in required storage for lower ranks. The percentage increases for the higher ranks at 10% demand are again observed to be much higher than for the larger demands. The effect of areal rainfall uncertainty on flow-duration relationships was assessed using 12 representative days of the year that are uniformly spread within the year. To achieve this, any day of each month could be used and the 28th of February and 30th for the other 11 months were subjectively selected. Out of the 100 flow-duration curves obtained with each flow ensemble, the 5th, 50th and 95th percentile values were determined. A comparison of these values with and without rainfall uncertainty was carried out by computing the percentage change in flow that uncertainty brings about on flows of the same exceedance probability. Figs. 7a and 7b show plots of the percentage change in flow with and without validation performance matching respectively for 30th May – a representative high flow day. The changes in flow are mostly negative and generally greater at the extremes of the flow-duration relationship. The expected increase in flow variability as a result of incorporating uncertainties is observed on Figs. 7a and 7b towards the extremes (<20 and >70% exceedance) of the flow-duration relationship. No reasons are found for the observed reduction in the 50th percentile flows and the substantially lower variabilities for the middle portions of the flow-duration relationship. A thorough search of the literature did not find any other study that has investigated the impact of incorporating rainfall uncertainty on storage-yield and flow-duration relationships although a number of studies on the effect of rainfall uncertainty on catchment modelling performance have been conducted. Andreassian et al., 2001 tested three rainfall models on three catchments of varying size and found the prediction performance of the models to improve as the quality of the rainfall input improved. Duncan et al.,
1 For interpretation of color in Fig. 5, the reader is referred to the web version of this article.
1993 obtained similar findings on tests on the effect of rain fall adequacy on the quality of stream hydrograph prediction.
6. Conclusions A method of quantifying areal rainfall uncertainty using multiplicative perturbations (multipliers) has been formulated and used to assess the impact of the uncertainties on storage-yield and flowduration relationships for a 36 year long period of the Berg River catchment in South Africa. The formulation is simple and its application mainly requires (i) the specification of the uncertainty to impose on the areal rainfall estimates, (ii) the lower and upper bounds of the multipliers to apply, and (iii) the grouping of rainfall to effect the variation of multipliers with rainfall magnitude. The analysis reveals that incorporating uncertainties using this method leads to substantial lower validation performance in rainfall–runoff modelling while an ideal incorporation of uncertainties would be expected to maintain this validation performance. To counter this, sampling is used to obtain a population of ensembles that give a similar cumulative density plot of validation performance as flows obtained without incorporation of areal rainfall uncertainties. Using these ensembles, uncertainty is found to reduce the required storage by up to 20%. Areal rainfall uncertainty is also found to reduce the flow-duration ordinates by an average of about 20%. These results may seem contradictory as a reduction in required storage can reasonably be considered to result from increased flows. Reservoir capacity however depends on variability and more detailed analysis would help explain the observations. These observations probably also reveal that the simplistic approach of sampling from perturbed runs in order to obtain the same validation performance as for the unperturbed case (step 5 of Section 4) may not be sufficiently valid. The selection of the lower and upper bounds of multipliers and of the rainfall zones to apply in the method was fairly subjective and not rigorous and this could have lead to the observed reduction in validation performance that prompted sampling out runs to achieve the required validation performance. This is particularly relevant as multiplicative perturbations could easily lead to unrealistically high rainfall if all the parameters are not selected judiciously. McMillan et al. (2011) studied multipliers for a catchment with a dense rain gauge network and high-resolution radar rainfall estimates and reported that the multipliers did not fit at the tails of lognormal distribution and this could lead to large uncertainties in predicting heavy rainfall. An alternative approach using linear perturbations that has been recently formulated (Ndiritu, 2013) is free from this limitation and the requirement to subjectively set bounds on the multiplier values to apply. Additional analysis using the multiplicative and linear perturbations is recommended. Ideally, this needs to span across different climatic regions and apply several rainfall–runoff models and storage-yield approaches. Tests on the approaches using high-density rain gauge data are also proposed. The combined use of radar and rainfall data (Sinclair and Pegram, 2005) and rainfall and satellite rainfall (Grimes et al., 1999) may however also yield robust and realistic approaches for dealing with areal rainfall uncertainties and further work will therefore also take this direction. References Anctil, F., Lauzon, N., Andreassian, V., Oudin, L., Perrin, C., 2006. Improvement of rainfall–runoff forecasts through mean areal rainfall optimization. J. Hydrol. 328, 717–725. Andreassian, V., Perrin, C., Michel, C., Usart-Sanchez, I., Lavabre, J., 2001. Impact of imperfect rainfall knowledge on the efficiency and the parameters of watershed models. J. Hydrol. 250, 206–223.
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Please cite this article in press as: Ndiritu, J.G. A multiplier-based method of generating stochastic areal rainfall from point rainfalls. J. Phys. Chem. Earth (2013), http://dx.doi.org/10.1016/j.pce.2013.09.010