1. Introduction

Regular estimates of rainfall in space and time are crucial to earth sciences because spatially and temporally complete datasets are required in ecosystem and hydrological modeling (Haberlandt and Kite 1998) or are used for validation exercises of numerical weather models (Mladek et al. 2000) or regional and global climate models (e.g., Jones et al. 1995; Murphy 1999; Fowler et al. 2005). Commonly, point measurements of rainfall are regionalized in space using some form of spatial interpolation. Various techniques are available and, depending on the purpose of the interpolated product, different methods may be more or less suitable. For example, methods that rely on optimization are likely to perform well if mean conditions, that is, monthly and annual data, are the focus. However, these methods generally produce estimated maps that are too smooth when the actual data display much more variability, a characteristic typical of daily rainfall data (Shen et al. 2001). The smoothing tends not to be uniform as it depends on the local data configuration (Goovaerts 1997). For example, maps of rainfall kriging estimates appear more variable in densely sampled areas than in sparsely sampled areas (e.g., Chappell and Ekström 2005). In contrast, conditional stochastic simulation of rainfall fields can sometimes better represent the inherent variability in the rainfall patterns as the simulations can be constrained to reproduce a particular histogram and the spatial-dependence structure. Furthermore, using simulation, several realistic rainfall fields can be generated, providing a population of simulated values at each grid cell location. This population can be queried with respect to its distribution properties and used to visualize a more realistic picture of spatial uncertainty than that obtained using an interpolated rainfall field. In addition, the simulated rainfall fields can be used as inputs to physically based models, for example rainfall–runoff simulators or river flow estimates for ensemble forecasting (Jones et al. 2005). In this paper, a stochastic simulation method is used within a space–time framework to hindcast monthly rainfall patterns for the United Kingdom over a period of 8 yr. The aim of the study is to apply a novel method within climatology that uses spatial, ancillary, and temporal information to estimate rainfall over a large region and for a period that includes large differences in rainfall amounts as well as spatial characteristics.

Generally, there are two central issues to consider when choosing an estimation method for rainfall: (i) temporal resolution and (ii) size and density of the rain gauge network used for the interpolation. Higher temporal resolution rainfall data often imply less regular behavior in the spatial domain. To resolve spatial rainfall patterns on a minute or hourly basis the estimation method must be able to handle intermittent rainfall fields with large variance; see, for example, the method of Lin and Chen (2004), which involves a combination of spatial dependence models and radial basis functions for interpolating hourly rainfall, or Barancourt et al. (1992) and Pardo-Igúzquiza et al. (2006), who used a binary random function to first identify areas of rainfall and then a continuous random function to estimate the rainfall inside the wet areas. Other approaches rely on physical principles or temporal dependence rather than spatial dependence, such as the combination of a limited area model and a downscaling scheme (Bindlish and Barros 2000), and the use of a Lagrangian coordinate system rather than the traditional Eulerian coordinate system (Amani and Lebel 1997). In most climatological applications, however, these methods are not of primary interest, as data of monthly to annual resolution are generally the focus. For these larger time steps there is sufficient time for rainfall to conform to a regular behavior, and typical rainfall patterns are likely to appear due to the predominant circulation and environmental influences, such as orography and distance to large water bodies. Hence, many climatological studies that interpolate spatial properties relying solely on spatial dependence, for example, thin-plate splines (Hutchinson 1995, 1998a, b; New et al. 1999), polynomial regression (Goodale et al. 1998; Rajagopalan and Lall 1998), distance weighting (Frei and Schär 1998; Shepard 1984; Tomczak 1998), and kriging (Atkinson and Lloyd 1998; Bacchi and Kottegoda 1995), are likely to be successful.

The second issue relates to the rainfall observation network. When data samples are dense and are believed to represent the general rainfall population, most methods will perform well. However, obvious problems occur when the spatial structure of rainfall is larger than the sample area (Dirks et al. 1998) or when the spatial structure of rainfall varies over a smaller scale than that resolved by the station network, for example, rainfall in conjunction with thunderstorms. Furthermore, when rainfall fields are estimated for several time steps, the completeness and length of rainfall records also become important as inhomogeneities can be introduced due to intermittently operating recording devices.

When data are sparse or unevenly distributed, it is sometimes possible to use ancillary variables to which rainfall is related and that are available on a dense network, such as elevation (Goovaerts 2000; Kyriakidis et al. 2001; Pardo-Igúzquiza 1998). Although rainfall is generally linked to elevation via orographic enhancement, there may be other factors with an additional significant effect on rainfall, such as “aspect.” For example, in regions with high and complex topography such as the European Alps (Frei and Schär 1998) topographic variables such as “obstruction presented by relief,” or “roughness of the relief” may prove more useful than elevation itself (Prudhomme and Reed 1998). Furthermore, the effect of topography on rainfall is often difficult to quantify, as that effect may itself be related to the prevailing weather conditions. Sometimes there is relatively little dependence between environmental variables and rainfall. For example, in some tropical regions elevation plays a secondary role relative to the more important convection (Basist et al. 1994; Sansó and Guenni 1999).

The information available for the estimation of rainfall may be increased by considering the, often more frequently sampled, temporal domain in conjunction with the spatial domain. Although the information in the temporal domain is often used in climatological work, it is rarely considered jointly with the spatial domain. For example, the temporal variability in a set of predictor variables, often circulation related, is used in weather generators to produce synthetic time series of rainfall (Wilby 1998) or, when used in combination with an interpolation scheme, rainfall series at several locations (Semenov and Brooks 1999). A joint space–time framework, where space and time variation are considered simultaneously is available in geostatistics (Kyriakidis and Journel 1999). Although not yet commonly applied within a climatological context, this framework has proven itself in many other earth science applications, for example, for mapping sediment transport (Chappell et al. 2003), deposition of atmospheric pollutants, or soil moisture. Kyriakidis and Journel (1999) provide a comprehensive review on the topic and examples of applications.

In this paper, a space–time hindcast approach is applied to monthly rainfall totals for the period 1980–87. The simulated rainfall fields are compared with observed rain gauge rainfall and the ensemble members available for each month are then used to visualize spatiotemporal uncertainty for the study period. The method applied here follows the geostatistical framework proposed by Kyriakidis and Journel (2001) and later Kyriakidis et al. (2004). It builds on the concept that a rainfall series may be broken down into a trend component and a residual component, similar to the climatological way of breaking down a series into its climatology and its accompanying anomaly (Jones et al. 1986). In a climatological approach, the anomaly would typically be interpolated onto a grid and then the climatology is added back to the anomaly field. Here, the trend component is deterministically estimated while stochastic simulation is used to derive the residual term. The stochastic component of this method is particularly attractive, as it provides a framework of assessing spatial uncertainty in the rainfall estimates. Such an assessment is valuable for several hydrological or water management applications, as it provides a level of confidence that represents the ability to estimate rainfall for that particular time and location.

2. Rainfall series and digital elevation data

Daily observations of rainfall totals for the period 1960 to 1990 were extracted from the Met Office database, the Met Office Integrated Data Archive System (MIDAS), via the British Atmospheric Data Centre (BADC) Web site (http://www.badc.rl.ac.uk). The data contained in MIDAS have been subjected to a series of quality control measures including basic range checks, to ensure that meteorological values do not lie outside long-term climatological extremes, and automatic algorithms, to ensure consistency with neighbors.

Rainfall totals were calculated for all months with a maximum of two missing daily observations. To ensure complete spatial coverage at all times, the monthly dataset was screened for spatial gaps. Unfortunately, multiple gaps in the data were found during several different years. A visual inspection of the density of stations over the United Kingdom showed the largest decrease of stations in the northern parts of the United Kingdom, particularly over much of Scotland from 1984 onward (not shown). Controls of the database indicate that this might be mostly due to observations omitted from the database rather than actual gaps in the observation network. Since it was not possible to retrieve any more data for this study, the extensive and multiple spatial gaps in the data restricted the study period to 1980–87, which is the longest period during 1960–90 with no large such gaps in the station network. For this period, all stations with at least 12 consecutive months were kept for further analysis and provided 4316 rain gauge records for the United Kingdom. To validate the simulated estimates, 10% (432 rain gauges) of the total number of rain gauges was kept from the simulation, leaving 3884 rain gauges for the simulation itself. The validation rain gauges were selected at random, and a visual inspection of their locations ensured that the entire region was represented by the validation dataset.

To regionalize temporal trend model parameters, the relationship between the model parameters and terrain elevation was used (see section 3a). The digital elevation data for this purpose were extracted from the Global Land 1-km Base Elevation (GLOBE) dataset (GLOBE Task Team et al. 1999). The horizontal coordinate system is seconds of latitude and longitude referenced to World Geodetic System 84 (WGS84). The vertical units represent elevation in meters above mean sea level (Hastings and Dunbar 1999). The elevation data were averaged into ∼5 km squares, transformed to an easting and northing grid reference system [meters], and then subjected to a 15-km smoothing window. Elevation data were averaged because, although the grid spacing is 1 km, the actual spatial resolution (i.e., the minimum distance between two objects that can be separated in the image) of the elevation is approximately three times the grid spacing (Hastings and Dunbar 1999). The conversion of the latitude and longitude coordinate system to the easting and northing grid reference system was necessary due to the latitudinal distortion of distance in the degree coordinates, and the 15-km smoothing window was imposed to give large-scale elevation, shown to have stronger correlation with rainfall compared to the immediate rain gauge elevation (Prudhomme and Reed 1998).

3. Methodology

In this paper, a space–time simulation approach based on the works of Kyriakidis et al. (2004) and Kyriakidis and Journel (2001) is used to estimate monthly rainfall totals on a 5 km by 5 km regular grid. In principal, the method builds on the concept that rainfall can be described by two components, a deterministic trend component and a stationary, zero mean residual. While the deterministic trend component describes some general smooth variability of the rainfall process (typically linked to ancillary information), the residual component describes higher frequency fluctuations around the deterministic component, that is, all variability that is not explained by the trend component.

The method can be broken down into three main stages of analysis: (i) estimation of temporal trend models at the rain gauges (section 3a); (ii) regionalization of temporal trend coefficients (see section 3b); and (iii) generation of multiple rainfall realizations for each location and time step (see section 3c). Each stage is described below.

a. Estimation of temporal trend models at rain gauges

Let us assume that the rainfall profile z_α = [z_a(t_i), i = 1, . . . , T_a]′ at the αth rain gauge can be described by the sum of two profiles: a deterministic trend component m_α = [m_a(t_i), i = 1, . . . , T_a]′ and a stationary, zero mean, residual component r_α = [r_a(t_i), i = 1, . . . , T_a]′, where t_i denotes the time instant and primes denotes transposition. Hence, for the ath rain gauge with coordinate vector u_α, the observed profile z_α = z(u_α) can be written as

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We call the deterministic trend a temporal trend, as it describes some average smooth behavior of rainfall over time. The temporal trend should represent the average behavior of rainfall at each location, which implies that the temporal trend differs from location to location within the study area.

There are different ways of defining the smoothly varying temporal trend for each rain gauge location. Here, we consider K elementary temporal profiles f_k = [ f_k(t_i), i = 1, . . . , T]′, where f_k(t_i) denotes the value of the kth elementary profile at time t_i, which can be arranged into a matrix 𝗙 = [f₁ · · · f_k · · · f_K]. By convention f₁(t_i) = 1 for all t_i; that is, f₁ is a vector of unit entries. These elementary profile(s) should ideally have a physical interpretation that is pertinent to the entire study region. In this paper, we used a spatial average rainfall series for the entire study region to derive the elementary temporal profile [see section 4 and Fig. 1, as well as Kyriakidis et al. (2004)]. Another option is to use empirical orthogonal function (EOF) analysis to derive the elementary temporal profiles (Sauquet et al. 2000).

The temporal trend component at any rain gauge is then assumed to be a weighted linear combination of such elementary profiles:

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where b_k(u_α) is a weight quantifying the contribution of the kth elementary profile f_k to the trend component m_α at the αth rain gauge.

The vector b_α of weights, or temporal trend coefficients, can be estimated using least squares regression with the individual rain gauge series. Here, regression coefficients were established for all rainfall series with at least 12 consecutive months (3884 series), hence increasing the number of rainfall observations used for the rainfall estimation. The inclusion of the additional series is important, as restricting analysis to only those time series that are temporally complete could have detrimental effects to precipitation climatologies, as shown by Willmott et al. (1996) for the United States.

b. Regionalization of temporal trend models

At this stage, temporal trend coefficients are available at all rain gauge locations. For the estimation of rainfall at the nodes of a regular grid the temporal trend coefficients must first be defined at all regular grid nodes. In this process it is necessary to consider the auto- and cross correlation of the temporal trend coefficients in space, as they are derived from the same rainfall data. A two-stage process proposed by Kyriakidis et al. (2004), which considers spatial auto- and cross correlation between the coefficients, was used for this task. First, the temporal trend coefficients were regionalized in space using a spatial regression relationship with a more densely available variable (here terrain elevation); second, the respective residuals of the temporal trend coefficients (calculated for all rain gauge locations) were subjected to co-kriging, in which residuals were estimated for all grid nodes taking into consideration their auto- and cross correlation. Finally, the regressed-derived temporal trend coefficients and the co-kriging-derived residuals were combined into regionalized temporal trend coefficients on a regular grid with the same resolution as the elevation data (see section 4 and Fig. 2). For details on co-kriging, the reader is referred to Deutsch and Journel (1998).

With the temporal trend coefficients available on a regular grid, the next step involves reconstructing the temporal trend model onto the same grid using Eq. (2):

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where b**_k(u_n) denotes the regionalized temporal trend coefficient at the nth grid node u_n; the double asterisk superscript is used to explain that such coefficients are derived from a two-step regionalization procedure (see below), and the single asterisk superscript is used to denote that the variable (here the temporal trend model) is an estimate.

c. Simulating rainfall realizations

The last stage of the analysis involves generating multiple simulations of the residual component [r, from Eq. (1)] for all grid nodes and time intervals. This task can be accomplished using geostatistical sequential Gaussian simulation (SGS: Deutsch and Journel 1998).

Gaussian simulation requires that the data have a normal distribution. For this reason, the residuals are calculated as differences between the rain gauge data and those of the estimated temporal trend:

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where the subscript G denotes that the variables are transformed to a normal distribution. In computing the normal scores of the temporal trend, the data are transformed to a normal distribution using the cumulative distribution function of the rainfall data and not that of the temporal trend data itself. This approach ensures that the temporal trend represents some average spatiotemporal variability of the precipitation process (Kyriakidis et al. 2004).

Stochastic simulation in this setting requires a model of spatiotemporal correlation structure for the Gaussian residuals, that is, a model of spatial and temporal dependence in r_G. Such a model is encapsulated in this work by a space–time variogram model (see section 4), where time is treated as a third dimension. Sequential simulation proceeds by visiting at random each grid node and generating a simulated residual Gaussian deviate at that node via Monte Carlo drawing from a local conditional distribution. At each grid node and for a predefined number of neighboring grid nodes comprising both previously simulated residuals and original residuals at rain gauges, kriging is used to estimate the mean m*_RG(u_n, t) and standard deviation σ*_RG(u_n, t) of that residual local distribution. The kriging weights are based on the space–time variogram model of the Gaussian residuals. These weights are recalculated for each grid node and each realization because the random visiting sequence (and hence the particular residuals considered in the neighborhood of each grid node) changes from one realization to another. The kriging-derived residual mean m*_RG(u_n, t) is subsequently added to the reconstructed temporal trend value m*_G(u_n, t) for that grid node so that the mean and standard deviation for the Gaussian-transformed rainfall at the nth grid node and ith time instant is now defined as m*_G(u_n, t) + m*_RG(u_n, t) and σ*_RG(u_n, t), respectively. A simulated Gaussian deviate is then generated from this local distribution and added to the conditional Gaussian-transformed rainfall dataset. When all nodes are visited, the simulated Gaussian deviates (z_G) are back-transformed to a set of values (z) that have a non-Gaussian distribution conditioned by the histogram of the original rain gauge data.

Because the normal score transformation is applied to the entire rain gauge dataset (across all locations and time instants), it is necessary to transform such simulated values again if one wishes to reproduce the histogram of a specific time or location (Kyriakidis et al. 2004). The postprocessing of each simulated rainfall field was made using a quantile transformation, where the simulated distribution is transformed to a specific rainfall distribution (Deutsch and Journel 1998).

4. Application of stochastic space–time simulation to monthly rainfall totals

The following section details how space–time stochastic simulation was applied to the U.K. monthly rainfall data and provides a validation study of the simulated rainfall fields, as well as a visualization of uncertainty in the simulated rainfall values.

First, an elementary temporal profile for the United Kingdom was derived by calculating a spatially averaged rainfall series using rain gauges with complete data coverage for the study period, in total 736 series (Fig. 1). The elementary temporal profile, which consists of T = 96 average rainfall values, was then locally modified at each rain gauge location (using all rain gauge series kept for the analysis) via a regression relationship with the local rain gauge series, Eq. (2). The temporal trend coefficients (weights) b_α were estimated using ordinary least squares as

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For this study, 𝗙 is a matrix with T = 96 rows and K = 2 columns; its first column is a vector with 96 unit entries and its second column is the spatially averaged rainfall profile. Vector b_α comprises two coefficients b₁(u_α) and b₂(u_α), which represent the intercept and slope of the regression relationship between the elementary profile and the rainfall data at the αth rain gauge.

These temporal trend coefficients were then regionalized in space using a spatial regression relationship with elevation data for the United Kingdom, followed by co-Kriging of the resulting residuals. More precisely, the kth temporal trend coefficient b_k(u_α) at the αth rain gauge is assumed to be comprised of two components: a component b*_k(u_α) explained by L auxiliary spatial variables (here elevation only) and a resulting residual component e_k(u_α) = b_k(u_α) − b*_k(u_α). The values of the first component at all A rain gauges are again derived using ordinary least squares as

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where 𝗤 is a matrix with A = 3884 rows and L = 2 columns; its first column contains A unit entries, and its second column contains A elevation values at the corresponding rain gauges. Here b̃_k = [b_k(u_α), α = 1, . . . , A]′ is the vector of the kth temporal trend coefficients at all A rain gauges (note the different notation from the vector b_α of K coefficients at the αth rain gauge), and b̃*_k = [b*_k(u_α), α = 1, . . . , A]′ is the component of b̃_k explained by the auxiliary spatial variables. In this study, the R-squared values from the spatial regression with respect to the elevation were 7.5% for b₁ and 41% for b₂, both significant at the 95% level. The correlation is low for b₁; however, the level of explanation is not crucial to the regionalization of the temporal trend coefficients, as the spatial autocorrelation of e₁ and e₂ plays in this case a more significant role in estimating the temporal trend parameters. The reverse would be true if the ancillary variables had a stronger predictive ability.

Subsequently, simple co-kriging was used to estimate the residuals (e₁ and e₂) at all grid nodes of the 261 × 160 grid with cell size 5 km². Unlike other forms of kriging, co-kriging considers the spatial dependence between multiple variables (Goovaerts 1997) and uses a linear model of coregionalization (LMC) that defines spatial dependence for e₁ and e₂ individually and the spatial dependence between them. The isotropic LMC for the residuals was estimated using an exponential variogram model in the software Gstat (Pebesma 2004; Pebesma and Wesseling 1998) (Table 1). The final estimate b**_k(u_n) of the kth temporal trend coefficient at the nth grid node is then computed by adding the co-kriging-derived residual e*_k(u_n) to the spatial regression-derived estimate b*_k(u_n):

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where b*_k(u_n) = [1 q(u_n)](𝗤′𝗤)⁻¹𝗤′b̃_k with q(u_n) denoting the elevation value at grid node u_n.

The geographical distributions of the temporal trend parameters are shown in Fig. 2. Regions with intercepts (b₁) close to 0 and slopes (b₂) close to 1 are where the elementary temporal profile is similar to the local rain gauge profile. The main spatial features in the geographical patterns of the coefficients are decreasing intercept values (Fig. 2a) and increasing slope values (Fig. 2b) with elevation. This spatial pattern indicates that areas of high elevation are generally associated with higher levels of rainfall compared to that of the elementary temporal profile.

Having defined the temporal trend coefficients (b₁ and b₂), the temporal trend (m) itself was estimated for all grid nodes, using Eq. (3), and Gaussian-distributed residuals (r_G) were then calculated for all rain gauge locations kept for the analysis, Eq. (4).

The process of generating several stochastic simulations requires a model of space–time variability of the residual (r_G) given by a space–time variogram. Because the monthly spatial rainfall pattern is likely to be conditioned mainly on regional characteristics, which are assumed to be captured by the temporal trend component, the residual component could be assumed isotropic. The residual variogram was constructed as a 3D model, where the space and time dimensions were defined by the horizontal plane and time axis, respectively. Scale differences between the two dimensions were accounted for by calculating the space–time separation distance using a generalized distance metric (Kyriakidis and Journel 1999):

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where a₁ and a₂ denote (unequal) spatial and temporal correlation length parameters and h and τ are the spatial and temporal lag distances. The correlation length parameters were estimated from variogram models fitted to the space and time dimensions separately, where the correlation length was taken as the range (i.e., the extent of spatial/temporal correlation). For this case study, a₁ and a₂ were estimated to be approximately 468 km and 4.5 months, respectively.

The space–time variogram model adopted, scaled here to a unit sill, consists of three structures: a nugget and two exponential functions:

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where h and τ are the separation distances in space and time, respectively. The variogram model was fitted to the spherical space–time variogram using the software Gstat. The parameter δ_hτ is defined as 1 when h = τ = 0, 0 if not, and ε is a very small number, ensuring that the variance contribution from the third structure is seen in the space directions but acts as a nugget effect in the time direction.

The above space–time variogram model was used for SGS (Deutsch and Journel 1998). In total 40 realizations were generated and each simulation was back-transformed from the Gaussian space to a distribution represented by all the rain gauge data. A secondary transformation was then applied to each month separately because the original transformation of rain gauge data to Gaussian deviates considers all such data (across all locations and time instants). The simulated Gaussian rainfall values for all 96 months were transformed to the rainfall values with the distribution of the rain gauge data for the corresponding months using an inverse quantile transformation (Deutsch and Journel 1998).

5. Assessment of the method

In this case study, space–time simulation was applied to monthly rainfall totals from around 3900 rainfall series across the United Kingdom over an 8-yr period to provide point estimates of rainfall on a regular grid of 5-km resolution. Estimating punctual rainfall rather than an area average enables a direct comparison with observed rain gauge data. A note of caution should be applied to the comparison between the simulations and data such as regional climate models estimates, which report grid cell area averages. The area average rate of rainfall is likely to be less than that for a point.

The results of the simulations were validated using 10% of the total number of rain gauge stations available for the simulation, giving in total 432 stations to be used for validation exercises. An RMSE was calculated for each validation station, where each rain gauge was matched with the closest located grid cell center. The RMSE values indicated reasonably accurate estimates for much of the United Kingdom, with the exception of northwest and north Scotland and northwest England (Fig. 4a). An investigation of validation stations with large RMSE showed that the large values were caused by systematic underestimation of rainfall (e.g., Fig. 5a). The cause of the discrepancy between observed and simulated rainfall was linked to a weak relationship between the elementary temporal profile and the local rain gauge rainfall series problems (Fig. 7), leading to rather inadequate temporal trend models in these regions. The weak relationship appeared to be due to rainfall in these regions being conditioned on different factors compared to other parts of the United Kingdom. For example, it is likely that topographic factors, such as elevation, shielding, etc., play a strong role in rainfall rates in these regions. However, the weak relationship could also be due to the relatively few rain gauges included from these regions in the calculation of the elementary temporal profile.

The impact of an inadequate temporal trend model on the simulation is largest during the autumn and winter seasons, when the northwest regions are associated with large rainfall amounts. This pattern explains the clear underestimation of rainfall rates between 350 and 450 mm month⁻¹ (Fig. 3). The seasonal impact on the temporal trend models is likely to influence the residuals in these regions and introduce an element of variability over time in the spatial dependence pattern. However, because the space–time variogram is the same for every time step, differences in spatial dependence over time cannot be taken into account. Hence, the only way to make the assumption of stationary residuals in space and time more plausible is to improve the temporal trend models.

There are ways in which the estimates of the temporal trends can be improved. Additional elementary profiles could be used and other ancillary variables could be included in the regionalization of the temporal trend coefficients. More precisely, given the weak correlation between the spatial averaged time series for the United Kingdom and many local rain gauge series in northwest Scotland and England, it may be worth considering the inclusion of other elementary profiles in the estimation of the temporal trends. Such additional elementary profiles could be circulation-dependent variables, similar to the predictor variables used by weather generators to generate synthetic rainfall series (Wilby 1998). For example, U.K. daily rainfall shows a strong relationship with daily vorticity (Jones et al. 1993) and some moderate improvement might be expected if seasonal indices of the North Atlantic Oscillation index (Hurrell 1995) and North Atlantic sea surface temperatures (Folland and Parker 1995) were considered.

It is also possible that other topographical variables than the 15-km smoothed elevation, for example, slope, terrain roughness, etc. (Prudhomme and Reed 1998), could replace the smoothed 15-km elevation or be used in combination with it, when regionalizing the temporal trend coefficients.

Nevertheless, despite some regional discrepancies between simulated and observed rainfall, the simulated rainfall showed reasonable accuracy for the majority of the United Kingdom. The 40 realizations that were generated provided a measure of accuracy in estimating the residual, and examples of the realizations for January 1981 were displayed in Fig. 12. The ensemble variability represents the uncertainty in estimating each grid cell rainfall estimate and can be large if the number of original residuals is sparse and the range of the correlation is short. In situations when the spread of the grid cell realizations indicates a wide distribution, it is important not to use mean statistics to represent the simulations, but rather consider each simulation in its own right. From the user’s perspective, this quality must be seen as one of the main benefits of the method. It enables the user to identify areas where estimates are less reliable due to the performance of the estimation method and offers an uncertainty range for each estimate. The grid cell distributions, however, only account for the uncertainty in the predictions of the residual component, as the uncertainty in the interpolated trend coefficients was not accounted for. The proportion of uncertainty provided by the ensemble can be assessed by testing how many of the validation observations falls within one and two standard deviations of the estimated ensemble mean, or within a specified ensemble confidence interval (Pardo-Igúzquiza et al. 2006). If the estimates for one grid cell follow a Gaussian distribution, the expected proportions would be 68% and 95%, respectively. Using observations from the 432 rain gauge stations the test showed that approximately 49% and 69% fell within the range of one and two standard deviations from the ensemble mean. For this reason, although the ensemble of simulations for each time step shows local uncertainty in the estimates, they should be considered to be a conservative measure of uncertainty. A fully stochastic approach can be achieved by combining a stochastic trend component with a stochastic residual component (Kyriakidis and Journel 1999), but this approach was outside the scope of this study. Nevertheless, estimate uncertainty as well as other sources of uncertainty, for example, grid cell resolution errors (Willmott and Johnson 2005), are highly relevant for users of gridded rainfall data in order to assess the quality and reliability of the data at hand.

6. Conclusions

In this study, space–time stochastic simulation was used to estimate monthly rainfall totals for the United Kingdom over a period of 8 yr. Attractive aspects of the methodology were the consideration of dependence of rainfall in both the time and space domain and its stochastic component. Owing to a dense station network for most of the United Kingdom, the largest information source is found in the spatial domain. In regions where rain gauge data are sparse, such as in the mountainous areas in the northwest, temporal dependence may become more important. The synthetic conditional simulations of rainfall provide a model of uncertainty regarding unknown monthly levels of rainfall totals in space and time. Such a model, albeit conservative in this work due to only the residual being considered as a stochastic variable, can be used for risk analysis within a water management framework, for example, river flow estimation.

The main problem encountered in this study concerned the estimation of the temporal trend model. This problem revealed itself as an underestimation of rainfall in specific regions. In this case, the overestimation of the temporal trend could be related back to the relationship between the local rain gauge series and the elementary profile, that is, the spatial U.K. average rainfall series. Owing to a weak relationship between the stations and the elementary profile in these regions, the resulting temporal trend model became inadequate and did not well represent the average smooth variability of rainfall in the northwest and north of the United Kingdom. There are, however, several options for improving the accuracy of the temporal trend estimates, for example, by including additional elementary temporal profiles that better represent the regional rainfall variability in these regions as elementary profiles, such as time series of indices that represent certain key mesoscale features of the atmosphere and ocean that are known to influence rainfall processes in the United Kingdom.