The study area is located in Bradford, a city in West Yorkshire, England. Bradford has a maritime climate, with an average yearly rainfall of 873 mm recorded from 1981–2010 (MetOffice UK, 2016). The rain gauge network, used in this study was located at the premises of Bradford University (Fig. 1) and rainfall data were collected from paired tipping bucket rain gauges placed at eight locations covering an area of 400 m × 200 m. Data used in this study were collected from April to August 2012 and from April to August 2013. These stations were located on selected roofs of the university buildings, thereby providing controlled, secure and obstruction-free measurement locations. Each station consists of two tipping bucket type rain gauges mounted 1 m apart. On each roof the paired gauges were placed such that the height of the nearest obstruction is less than two times the distance between the gauges and the obstruction. The rim of each rain gauge was set up around 0.5 m above the surrounding ground level following UK standard practice (MetOffice UK, 2016). An example of the measurement setup (station 6) is also shown in Fig. 1. A histogram of the inter-station distances of the rain gauge network is presented in Fig. 2. Lag distances covered in this network are distributed between 21 m (stations 4–5) and 399 m (stations 1–3).

All rain gauges are ARG100 tipping bucket type with an orifice diameter of 254 mm and a resolution of 0.2 mm. Dynamic calibration was carried out for each individual gauge before deployment and visual checks were carried out every 4–5 weeks during the measurement period to ensure that the instruments were free of dirt and debris. Data loggers were reset every 4–5 weeks during data collection to avoid any significant time drift. Measurements (number of tips) were taken every minute and recorded on TinyTag data loggers mounted in each rain gauge.

Quality control procedures were performed prior to statistical analysis, taking advantage of the paired gauge setup to detect gross measurement errors. The paired gauge design provides efficient quality control of the rain gauge data records as it helps to identify the instances when one of the gauges fails, and to clearly identify periods of missing or incorrect data (Ciach and Krajewski, 2006). During the dynamic calibration of all rain gauges in the laboratory before deployment, it was identified that the highest and lowest values of the calibration factors for the tipping bucket size are 0.196 and 0.204 mm. The gauges were recalibrated in the laboratory after the first period of measurement and it was found that the largest change in calibration factor for any gauge was a maximum of 4 % of the original calibration factor. Therefore a maximum difference of 4 % in volume per tip was assumed to be caused by inherent instrument error. It was therefore decided that this is the maximum acceptable difference between any pair of gauges. Sets of cumulative rainfall data corresponding to specific events from the paired gauges were checked against each other and if the (absolute) difference in cumulative rainfall was greater than 4 %, that complete set was identified as unreliable and removed from further analysis.

The total average network rainfall depth for the summer seasons of 2012 and 2013 are 538 and 207 mm, respectively. Figure 3 shows time series of daily rainfall averaged over the network for 2012 and 2013. There is a significant difference in cumulative rainfall between 2012 and 2013. This is because 2012 was the wettest year recorded in 100 years in the UK (MetOffice UK, 2016) and 558 mm of rainfall during 2012 summer was unusually high. An average rainfall of only 360 mm was recorded during April to September over the 1981–2010 period at the nearest operational rain gauge station at Bingley, which is around 8 km from the study site with a similar ground elevation (MetOffice UK, 2016).

The data set for 2012 and 2013 contains 13 events yielding more than 10 mm network average rainfall depth each and lasting for more than 20 min. A summary of these events is presented in Table 1. Note that this event separation is only used for the presentation of results in Sect. 4.2. Hence it does not leave out any data from the development and calibration of the geostatistical model as presented in Sect. 3. Table 1 shows that the total event duration ranges from 1.5 to 11.4 h while the event network average rainfall intensity varies from 1.79 to 7.96 mm h-1. Table 1 also includes summary statistics of peaks of events (temporal averaging interval of 5 min) for the eight stations within the network. Although the spatial extent of the area is only 400 m × 200 m, it is clear that there is a considerable difference in rainfall intensity measurements indicated by the standard deviation and range of peaks observed in the individual events. The maximum standard deviation between peaks of individual events is 9.27 mm h-1 for event 8, which is around 12.5 % of the mean network peak intensity of 74.4 mm h-1. This variation provides evidence of the potential importance of analysing uncertainty in the estimation of AARI even in such a small urban catchment.

The rain gauge network contains eight measurement locations. These eight measurement locations give 28 spatial pairs at a given time instant which yields too few spatial lags than would normally be used in geostatistical modelling. For example, Webster and Oliver (2007) recommend around 100 measurement points to calibrate a geostatistical model. The procedure adapted in this study increases the number of pairs by pooling sample variograms for time instants with similar rainfall characteristics. With n measurement locations and measurements taken at t time instants, the pooling over t time instants creates t × 1/2 × n × (n-1) spatial pairs. Although this procedure increases the number of spatial pairs by a factor t, the spatial separation distances for which information is available will be limited to the original configuration of the n measurement locations.

The underlying assumption of this pooling procedure is that the spatial variability over the pooled time instants is the same. Therefore it is important to pool sample variograms of rainfall measurements with similar rainfall characteristics. Since the spatial rainfall variability is often intensity dependent (Ciach and Krajewski, 2006), the characteristics of a less intense rainfall event may not be the same as that of a high-intensity rainfall event. Hence to make the assumption of consistency of spatial variability, the range of rainfall intensity over the pooled time instants should be reasonably small. On the other hand, one should also make sure that there are enough time instants within a pooled subset to meet the data requirement to calibrate the geostatistical model. Based on the above two criteria, three rainfall intensity classes were selected. The maximum threshold value was limited to 10 mm h-1 to have enough time instants for the highest range (i.e. > 10 mm h-1) in order to produce stable variograms even at 30 min temporal averaging interval. It was then decided to divide the 0–10 mm h-1 class to two equal subclasses (i.e. < 5 and 5–10 mm h-1). This resulted in three subclasses, which is a reasonable number given the size of the data set and computational demand. The number of time instants (t) within each rainfall intensity class is presented for three temporal averaging intervals in Fig. 5. The natural characteristic of rainfall data results in the dominance of lower intensity rainfall (0.1–5.0 mm h-1) over the recording period. In addition, the number of time instants t obviously reduces with increasing temporal averaging intervals due to the aggregation process. As a consequence there are only seven time instants for the intensity range > 10 mm h-1 at the 30 min temporal averaging interval. This limits the maximum temporal averaging interval to 30 min for our analyses. For a catchment of this size (400 m × 200 m) it is very unlikely to have a response time of more than 30 min. Hence, from a hydrological point of view consideration of temporal averaging intervals longer than 30 min would not be sensible. Note that although there are only seven time instants, the pooling procedure will produce 196 (= 7 × 28) points to calculate and calibrate the geostatistical model for that intensity class.

Having chosen the rainfall intensity classes to create pooled time instants, there can still be inconsistency in spatial variability between time instants within a class and therefore assuming a single geostatistical model for the whole subset may not be realistic. To reduce this effect to a certain extent, all observations within an intensity class were standardised using the mean and standard deviation of each time instant as follows: r̃ix=rix-miSDi, where i= 1 … t, x= 1 … n; r̃ix is standardised rainfall intensity at a time instant i and location x; rix is rainfall intensity at time instant i and location x; and mi, SDi are mean and standard deviation of rainfall intensities at time instant i, respectively. Further steps were carried out on the standardised rainfall intensity.

The upper part of Fig. 6 shows the distribution of standardised rainfall intensity for a temporal averaging interval of 5 min derived using Eq. (1). From the figure it is clear that the data are not normally distributed. Distributions for other temporal averaging intervals (i.e. 2, 15, and 30 min) show a similar behaviour. But the geostatistical upscaling method to be used is based on the normal distribution. This requires the rainfall data to be normally distributed prior to the calibration of the geostatistical model. The normal score transformation (NST, also known as normal quantile transformation; Van der Waerden, 1952) is a widely used method to transform a variable distribution to the Gaussian distribution. It has widely been applied in many hydrological applications (Bogner et al., 2012; Montanari and Brath, 2004; Todini, 2008; Weerts et al., 2011). The concept of NST is to match the p quantile of the data distribution with the p quantile of the standard normal distribution. Consider a standardised rainfall intensity r̃ with cumulative distribution FR̃r̃. It is transformed to a rN value with a Gaussian cumulative distribution FRNrN as follows: rN=FRN-1FR̃r̃. Detailed description of NST including the steps involved can be found in Bogner et al. (2012), Van der Waerden (1952) and Weerts et al. (2011). The lower part of Fig. 6 shows the transformed standardised intensity for the temporal averaging interval of 5 min.

A geostatistical model of (normalised) rainfall intensity rN (derived from Sect. 3.3) at any location x can be written as rNx=px+εx, where px is the trend (explanatory part) and ε(x) is the stochastic residual (unexplanatory part). Considering the availability of data, small catchment size, and scope of this study, it was assumed that the trend is constant and does not depend on explanatory variables (e.g. topography of the area, wind direction). The stochastic term ε is spatially correlated and characterised by a variogram model. A variogram model typically consists of three parameters; nugget, sill, and range (Isaaks and Srivastava, 1989). The nugget is the value of the semi-variance at near-zero distance. It is often greater than zero because of random measurement error and micro-scale spatial variation. The range is the distance beyond which the data are no longer spatially correlated. The sill is the maximum variogram value and equal to the variance of the variable of interest (Isaaks and Srivastava, 1989)

The assumption of a constant trend makes that the spatial interpolation can be solved using an ordinary kriging system (Isaaks and Srivastava, 1989): ∑y=1nwyγxy-μ=γxz∀x=1,…,n∑x=1nwx=1, where wx, x=1,…,n are ordinary kriging weights, γxy is the semivariance between rainfall intensities at locations x and y, γxz is the semivariance between rainfall intensities at location y and prediction location z, and μ is a Lagrange parameter. Once the ordinary kriging weights are calculated using Eqs. (4) and (5), point rainfall intensities can be predicted using point kriging at any given point by taking the weighted average of the observed rainfall intensities, using the wx as weights. In this case we need a change of support from point to block as our intention is to predict the average rainfall intensity over the catchment. This is usually done by predicting at all points inside the catchment and integrating these over the catchment. This procedure is known as block kriging (Isaaks and Srivastava, 1989), which also has provisions for calculating the prediction error variance of the catchment average. But the procedure of NST as explained in Sect. 3.3 also involves back-transformation of kriging predictions to the original domain at the end (step 6). Since this transformation is typically non-linear, the back-transform of the spatial average of the transformed variable that is obtained from block kriging is not the same as the spatial average of the back-transformed variable; we need the latter and not the former. In principle, we could predict at all points within the block, back-transform all and next calculate the spatial average, but standard block kriging software implementations do not support this and neither is it possible to compute the associated prediction error variance. Hence block kriging cannot be applied. The alternative used in this study is to apply a computationally more demanding spatial stochastic simulation approach, which involves generation of a larger number of realisations and spatial averaging of these realisations. Unlike kriging, spatial stochastic simulation does not aim to minimise the prediction error variance but focuses on the reproduction of the statistics such as the histogram and variogram model (Goovaerts, 2000). The output from spatial stochastic simulation is a set of alternative rainfall realisations (“possible realities”). The mean of a large set of realisations approximates the kriging prediction, while their standard deviation approximates the kriging standard deviation. We used the sequential Gaussian simulation algorithm which involves the following steps (Goovaerts, 2000): i.

Define a prediction grid (a 25 m × 25 m regular grid in this case).

The grid size and number of simulations (i.e., the sample size) were selected considering the spatial resolution of available measurements and computational demand. It was observed that neither a finer grid nor more simulations improved the results significantly. Increasing the resolution to 10 m × 10 m only reduces the standard deviation of the prediction by less than 5 % in most cases while making the computational time six times higher (a summary on computation power is presented as Supplement).

Once the realisations have been prepared these are back-transformed by applying the inverse of Eq. (2) to all grid cells (step 6). Some values derived from spatial stochastic simulation were outside the transformed data range. Hence during back transformation (step 6) of these values linear extrapolation was used. These linear models were derived using a selected number of head and tail portion of normal Q–Q plot. This is one of the simplest and most commonly used solutions for NST back-transformation (Bogner et al., 2012; Weerts et al., 2011). Considering the scope of this study and the relatively small number of data which had to be extrapolated, other extrapolation methods were not explored. After step 6, the back-transformed realisations are spatially averaged one by one (step 7). This yields as many spatially averages as the number of realisations that had been generated in step 5. This set of values is a simple random sample from the probability distribution of the catchment average rainfall. Thus, the sample mean and standard deviation provide estimates of the mean and standard deviation of the distribution, respectively (step 8). Finally, by doing the inverse standardisation of the mean and standard deviation of the distribution to account for step 2, the AARI and associated uncertainty measure (standard deviation) were derived (step 9).

As explained in Sect. 3.4, the geostatistical model of transformed rainfall data were calibrated using variograms for three different intensity ranges. This procedure was repeated for temporal averaging intervals of 2, 5, 15, and 30 min. Exponential models were fitted to empirical variograms. The resulting variograms are presented in Fig. 7.

The variograms illustrate two properties of the collected rainfall measurements: spatial variability of rainfall and measurement error. One of the main parameters which characterises these properties is the nugget. Theoretically at zero lag distance the variance should be zero. However, most of the variograms exhibit a positive nugget effect (generally presented as nugget-to-sill ratio) at zero lag distance. This nugget effect can be due to two reasons: random measurement error and microscale spatial variability of rainfall. Unfortunately we cannot quantify these causes individually using the variograms. But there is a consistent pattern of nugget against both rainfall intensity class and temporal averaging interval which helps to interpret the variograms.

Considering the behaviour of nugget-to-sill ratio against rainfall intensity class, it can be observed that the smaller the intensity the higher the nugget-to-sill ratio, regardless of temporal averaging interval. For example, at 2 min averaging interval the nugget-to-sill ratio increases from zero to almost one (nugget variogram) as the rainfall intensity class changes from > 10 to < 5 mm h-1. The pure nugget variogram at < 5 mm h-1 means that either there is no spatial correlation at the regarded distance, or the spatial correlation of the field cannot be detected by the measurements because of the measurement error. Looking at the behaviour of nugget-to-sill ratio against temporal averaging interval, Fig. 7 shows that the smaller the averaging interval the higher the nugget-to-sill ratio, regardless of rainfall intensity class. For example, for rainfall intensity class 5.0–10.0 mm h-1 the nugget-to-sill ratio decreases from almost one to zero as the temporal averaging interval increases from 2 to 30 min. Overall these observations show that the combined effect of random measurement error and microscale special variability of rainfall characterised by nugget-to-sill ratio decreases with increasing (a) rainfall intensity class and (b) averaging interval.

Regarding the behaviour of the nugget-to-sill ratio against averaging interval, it is expected that with the averaging interval the (microscale) spatial correlation of rainfall would increase, which partly explains the observed pattern. The increase in spatial correlation of rainfall intensity with increasing temporal averaging interval agrees with other similar studies (e.g. Ciach and Krajewski, 2006; Fiener and Auerswald, 2009; Krajewski et al., 2003; Peleg et al., 2013; Villarini et al., 2008). For example, Krajewski et al. (2003) observed in their study on analysis of spatial correlation structure of small-scale rainfall in central Oklahoma a similar behaviour using correlogram functions for different temporal averaging intervals. But commenting on the decreasing trend of the nugget-to-sill ratio against intensity class, it cannot be attributed to improvement in microscale spatial correlation as it is neither natural nor proven. In fact, in Fig. 7 the behaviour of spatial correlation against rainfall intensity class does not show a distinctive trend except at the origin, i.e. the nugget effect. The absence of any consistent trend of spatial variability against intensity class was also observed in Ciach and Krajewski (2006). Meanwhile this decreasing trend of nugget-to-sill ratio against rainfall intensity corresponds well with measurement errors of tipping bucket type rain gauges caused by its sampling mechanism (hereafter referred to as TB error). This is due to the rain gauges' inability to capture small temporal variability of the rainfall time series. The behaviour of TB error against rainfall intensity as seen from Fig. 7 complements results from previous studies (Habib et al., 2001; Villarini et al., 2008). These studies also show that the TB error decreases with temporal averaging interval. Habib et al. (2001) found similar behaviour of TB error with increasing intensity (0–100 mm h-1) and also with increasing averaging interval (1, 5, and 15 min). Although the bucket size used in their study (0.254 mm) is slightly different from our rain gauge bucket size of 0.2 mm, the characteristic of the TB error against rainfall intensity for different averaging interval is consistent in both cases. In summary, the behaviour of nugget-to-sill ratio of the variograms against temporal averaging interval can be explained by the combined effect of microscale spatial variability of rainfall and TB error, while the behaviour of nugget-to-sill ratio against intensity range can mainly be attributed to the latter.

In addition to the nugget-to-sill ratio, another parameter that characterises the variograms is the range, i.e. the distance up to which there is spatial correlation. At lower temporal averaging intervals (≤ 5 min) the variograms for all rainfall intensity classes reach the variogram range very quickly (< 100 m). But at averaging intervals ≥ 15 min, the range has not been reached even at a maximum separation distance, showing the improvement in spatial correlation. High spatial variability of rainfall at shorter temporal averaging interval (≤ 5 min) is an important observation in the context of urban drainage runoff modelling, as the time step used in such models is generally around 2 min for small catchments.

The fact that the data set covers only 10 months of data from 2 years with varying climatology is something that needs to be acknowledged. However, for previous studies using such a dense network the duration of data collection is similar (e.g. 15 months – Ciach and Krajewski, 2006; 16 months – Jaffrain and Berne, 2012). These time periods are reflection of the practical and funding issues to maintain such dense networks operating accurately for extended periods. The characteristics of our data are comparable with Ciach and Krajewski (2006) and Fiener and Auerswald (2009) as these studies also used rainfall data from warm months to investigate the spatial correlation structure. Despite the fact that the data cover only 10 months all derived variogram models are stable and reliable. Webster and Oliver (2007) suggested around 100 samples to reliably estimate a variogram model. Even in the case of 30 min temporal averaging interval and > 10 mm h-1 (where we had the fewest number of observations) we had a total of 196 spatial lags to calculate the variogram. Furthermore, we demonstrated that all derived variogram models are stable and reliable by examining sub-sets of the data. We randomly selected 80 % of the data from each intensity class and reproduced the variograms to compare them with the variograms presented in Fig. 7. We had to limit the subclass percentage to 80 % to give enough time instants to reproduce variograms for all subclasses. We repeated this procedure a few times. Comparing these variograms with Fig. 7 shows that these variograms are very similar. One set of the variograms computed from 80 % of the data are presented in the Supplement. This analysis supports our claim that the variograms shown in Fig. 7 are stable and an adequate representation of the rainfall spatial variation for each intensity class and temporal averaging interval.

One of the assumptions we made during the pooling procedure is that the spatial variability is reasonably consistent within a pooled intensity class. We acknowledge that with narrower intervals the assumption of consistency in spatial variability would be more realistic. But with the available data we had to find a compromise with the number of time instants. We believe that using three intensity subclasses is a reasonable compromise. Further we also introduced step 2 (Sect. 3.2) which standardises the rainfall for each time instant within a subset. Although variograms are derived only for the whole subset, step 2 (before geostatistical upscaling) and step 9 (after geostatistical upscaling) ensure that the probabilistic model is adjusted for each time instant separately. Effectively, we assume the same correlogram for time instants of the same subclass, not the same variogram. Although this does not justify the assumption of similar spatial correlation structure within the pooled classes, it at least relaxes the assumption of the same variogram within subclasses. To compare the behaviour of variogram models for a narrower intensity interval, we produced variograms for narrower intensity classes ranging from 0 to 14 mm h-1 for the 5 min averaging interval. The highest intensity class is limited to ≥ 12 to < 14 mm h-1 as for further narrower ranges (i.e ≥ 14 to < 16 mm h-1 and so on) there are not enough sample points to produce a meaningful variogram. Narrower intensity classes means that the assumption of similar spatial variability within a pooled subset is more realistic. Comparing Figs. 7 and 8, we conclude that the variograms shown in Fig. 7 are accurate representations of the average spatial variability conditions for corresponding intensity classes.

Having calculated all variograms, the next step is to apply spatial stochastic simulation for the time instants of interest followed by steps 6 to 9 in Fig. 4 to calculate the AARI together with associated uncertainty. This procedure was carried out for all events presented in Table 1. The following sections present and discuss the predicted AARI and associated uncertainty levels derived from step 9.