1. Introduction
West Africa is frequently hit by heavy rainfall events during the wet summer monsoon season, which cause widespread devastation in both urban and rural areas (Paeth et al. 2011; Engel et al. 2017; Maranan et al. 2019; Atiah et al. 2023). For instance, the recent Nigeria floods of 2022 caused over 603 deaths and displaced more than 1.4 million people (Tunde 2022). Accurate observational data describing these extreme rainfall events are crucial for investigating their hydrological impacts (Beck et al. 2017). Although ground observations of rainfall are scarce in West Africa, the density of rainfall networks in many West African countries is approximately one magnitude lower than that of the observation network operated by National Weather Services in industrialized countries like the German Weather Service (Bliefernicht et al. 2022b). Moreover, many stations struggle to transmit data to the global transmission system due to their location in conflict zones and other challenges (Salack et al. 2019).
To accurately represent the spatial variability of a river basin, distributed hydrological models rely on gridded data inputs. In West Africa, common spatial interpolation methods such as the nearest neighbor approach or kriging are commonly employed to provide spatial rainfall information based on ground observations for hydrological models (Ascott et al. 2020). However, most of these approaches suffer from certain limitations. They often fail to maintain the variability of the target variable, leading to significant underestimations (Lantuéjoul 2002, p. 1 ff.). Additionally, many interpolation methods (e.g., kriging) do not adequately address the uncertainty of the regionalization process (Webster and Oliver 2007, p. 267 ff.). To address these fundamental problems, it is essential to improve rainfall databases in data-scarce regions and develop sophisticated regionalization methods that fulfill the specific requirements of hydrological applications. In this context, geostatistical simulations can better handle biases and spatial uncertainties (Chilès and Delfiner 2012), therefore leading to more reliable regionalization of the target variable under data scarcity.
In geostatistics, simulation refers to the process of generating two- or three-dimensional variables as realizations of a stochastic process, ensuring that their statistical characteristics match those observed in the actual world (Webster and Oliver 2007). A theoretically infinite number of realizations can be created, all of which are equally probable. The most commonly applied simulation techniques use Gaussian random fields (GRFs) to characterize variables of interests. For rainfall and other meteorological variables, these techniques are based on meta-Gaussian frameworks that are transformed and converted into “real world” fields using statistical transfer functions to account for the non-Gaussian characteristics of the variable of interest. They can be conditioned on point observations and are numerically efficient (Lauzon and Marcotte 2022). There are several geostatistical methods often referred to in the literature for simulating Gaussian random fields, e.g., sequential Gaussian simulation (Deutsch and Journel 1997), fast Fourier transform (FFT) (Shinozuka and Deodatis 1991, 1996; Le Ravalec et al. 2000), or the spectral turning bands method (STBM) (Matheron 1973; Mantoglou and Wilson 1982; Emery et al. 2016; Lauzon and Marcotte 2020). A detailed discussion of geostatistical rainfall simulation can be found in Benoit and Mariethoz (2017).
The simulation of rainfall fields was the subject of many studies (Mellor 1996; Mellor et al. 2000; Haberlandt and Gattke 2004; Seo et al. 2014; Leblois and Creutin 2013; Caseri et al. 2016; Haese et al. 2017; Vaittinada Ayar et al. 2020; Yan et al. 2020, 2021; Blettner et al. 2022; Bárdossy et al. 2022). So far, only a limited number of research studies examined the potential of GRF-based geostatistical simulation for rainfall in the West African monsoon (WAM) region. These studies focused on the development of stochastic simulations methods for subhourly rainfall events, which were tested for mesoscale rainfall networks located in the Sudan–Sahel zone in West Africa. For instance, Guillot (1999) and Guillot and Lebel (1999) simulate rainfall fields with GRFs using 30 rain gauges from the Estimation of Precipitation by Satellite-Niger (EPSAT-Niger) network for an area with 16 000 km2. It appears that the meta-GRFs are able to produce acceptable rainfall simulations across the Sahel region, taking into account the statistical features of the observational data (anisotropy and intermittency). Vischel et al. (2009) used a number of different simulation methods to simulate the rainfall fields that were observed by the African Monsoon Multidisciplinary Analysis–Couplage de l’ Atmosphère Tropicale et du Cycle Hydrologique (AMMA-CATCH) network in Niger. Their geostatistical frameworks were able to generate relatively realistic event rainfall fields for different types of conditioning modalities. In a subsequent study, Wilcox et al. (2021) further developed the methodology of Vischel et al. (2009) for the generation of the rainfall fields in this challenging region.
The main objective of this study is to perform a comprehensive quality assessment of a state-of-the-art geostatistical simulation approach under data-scarce conditions. In contrast to the aforementioned rainfall studies, the geostatistical simulation method is used to generate high-resolution daily rainfall fields with a focus on rainfall extremes. As simulation approach, the STBM (Shinozuka 1971; Emery et al. 2016; Lauzon and Marcotte 2020) was chosen in this study because of its accuracy and computational efficiency (Lauzon and Marcotte 2022). The simulation approach is tested for the highest 50 extremes, based on the daily areal rainfall amount, in a mesoscale catchment (∼50 000 km2), the Ouémé River basin in Benin, covering two climatic zones (Guinea and Savannah zones) in West Africa. Moreover, the geostatistical approach is compared to various interpolation methods using a cross-validation framework and sophisticated performance measures like the continuous ranked probability score and other important evaluation measures (e.g., summary statistics and empirical cumulative distribution function). In addition, the simulation methodology is tested for conventional and novel variogram estimation techniques. For a better comparison of the quality of the STBM approaches, we also propose a low-skill reference (pure nugget effect) simulation method in this work. By conducting this comprehensive quality assessment, we aim to enhance our understanding of the performance of geostatistical simulation methods in data-scarce regions. The findings of this study will contribute to the advancement of rainfall regionalization techniques, with a focus on accurately representing rainfall extremes. This knowledge can support effective hydrological modeling and improve the management and mitigation of hydrological impacts in West Africa and other similar regions worldwide.
This article is structured as follows: section 2 provides an overview of the study area and rainfall data. Section 3 outlines the methodological workflow. This includes explanation of the simulation approach employed, the transformation of the data to Gaussian space, the estimation techniques used for variogram analysis, the interpolation methods employed for comparison, and the evaluation measure. Section 4 presents the results, including the reproduction of observations, comparison with other techniques, and evaluation of spatial structures. Section 5 offers a discussion of the results, and section 6 provides key conclusions.
2. Study region and data
a. Study area
This study focuses on extreme rainfall in the Ouémé River basin that is mostly located in Benin (West Africa, Fig. 1). The study region extends in the south to the discharge station at Bonou (Benin), before the river turns into a delta (Biao 2017). It is bounded by Nigeria to the east and by Togo to the west. The studied region encompasses two climatic zones: the Guinea zone (coast–8°N) and the Savannah zone (8°–11°N) (Omotosho and Abiodun 2007; Akinsanola et al. 2017), which have distinct seasonal rainfall patterns. The Guinea zone has a bimodal rainfall regime with maxima in June and September, while the Savannah zone has a unimodal rainfall regime with a peak in August. In West Africa, mesoscale convective systems (MCSs) are the predominant rain-producing systems (Fink et al. 2016; Maranan et al. 2018). MCSs can appear in the form of rapidly eastward-moving organized convective systems (OCSs) with intense rainfall amounts of more than 20 mm h−1, potentially reaching 100 mm h−1 for the most intense systems [see Fink et al. (2006), Fink (2010) or Maranan et al. (2018) for further information]. The rain-type MCSs are therefore likely the most significant for our investigation. Indeed, through visual inspection of the events past the year 2000 (19 events) by leveraging satellite-based rainfall loops of GSMaP (Kubota et al. 2020), 12 of these cases are linked to intense eastward-moving MCSs, some in the form of squall lines. Using the categorization by Fink et al. (2006), they can be considered as “advective MCSs/OCSs.” The remaining cases are likely “local MCSs” which partly exhibited extensive residence time (>6 h) over the region, causing enhanced daily rainfall amounts at some stations. From September to November 2021, several intense rainfall events occurred in the region of study with widespread flooding of several municipalities. At least 13 people were killed, more than 50 primary schools and 12 institutions were flooded, a number of homes were destroyed, and 608 families were displaced [International Federation of Red Cross and Red Crescent Societies (IFRC) 2022]. Figure 1 displays the study region overlaid with a digital elevation model to provide additional insights regarding the topography of the catchment.
The target region Ouémé River basin (black line), the selected 89 rainfall stations (black diamonds), the Ouémé river (gray line), the tributaries (small gray line), and the national borders to Togo and Nigeria (dashed black). The elevation model is from the Shuttle Radar Topography Mission (SRTM) (Farr et al. 2007).
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
b. Rainfall data
The daily, quality-controlled rainfall data are obtained from the Karlsruhe Surface Station Database (KASS-D) and the West African Historical Precipitation Database (WAHPD). A description of the WAHPD, with data sources and applied algorithms for quality control, is given in Bliefernicht et al. (2022b). Both databases were used as source for several other studies [see e.g., Bliefernicht et al. (2022a) and Ascott et al. (2020) for WAHPD and Schlueter et al. (2019) and Vogel et al. (2018) for KASS-D]. Combining the two datasets yields a subset of 89 stations with daily data records from 1960 to 2016 for the Ouémé River basin. The chosen stations are not evenly distributed across the basin of the Ouémé river (black diamonds in Fig. 1). It is especially lacking in the eastern part of the basin, where there are no observations. The establishment of the AMMA-CATCH observatory site (Galle et al. 2018) is responsible for the increased density of rainfall stations in the northwest corner of the basin and the subsequent rise in available stations after 1997. The mean temporal data availability for this subset is 27.21% with 4 almost complete time series (>90%) and 21 stations that contain more than 50% of the data records. As a result, strong changes over time can be observed in the daily count of stations having records (Fig. 2). The significant fluctuations observed in the data availability are primarily caused by the absence of rainfall measurements during the dry season. On average, there are 24.43 stations per day available within a basin that spans an area of ∼48 322 km2. Due to this data situation, running a geostatistical simulation to generate daily rainfall fields is challenging.
Daily distribution of daily rainfall observations with absolute number of valid records for the Ouémé River basin.
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
The purpose of this study is to simulate rainfall extremes. Therefore, we compute the daily areal rainfall amount RA (mm day−1) based on the stations’ average rainfall for each day as follows:
3. Methodology
The components of the applied geostatistical simulation work flow are shown in Fig. 3 as an example for the simulation of one high-resolution daily rainfall field. The STBM is the most important component of the methodology, as it is used to draw the GRFs. Daily rainfall is highly skewed and therefore not normally distributed by nature; the transformation to Gaussian space is used to approximate it. Adjustable parameters for the spatial continuity model include nugget C0, sill C1, range a, and the anisotropy parameters ζ, θ. The conditional simulation using STBM (CS-STBM) is implemented to ensure that simulated values at rain gauge locations match the observed values. In the end, the normal distributed fields are transformed back into the original space, and a rainfall field is created. In the following, the different steps are explained in more detail.
Flowchart of the approach followed in this study to simulate daily rainfall extremes. Abbreviations: x0, observation locations; C0, nugget; C1, sill; a, range; ζ, anisotropy stretch parameter; θ, anisotropy rotation angle; Zcs(xi), the conditioned rainfall field for the gridded output locations xi.
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
a. STBM
CS-STBM
The STBM described above generates unconditional Gaussian values that are not influenced by additional input data, such as rainfall. However, to incorporate the observed rainfall data, a geostatistical technique called conditional simulation is applied (e.g., Journel 1974). This method involves two additional interpolation steps. While simple kriging is traditionally the method of choice, its effectiveness is limited under certain conditions. Ordinary kriging (OK) provides a more robust framework, particularly in the context of sparse data. This is attributed to its capability in accounting for uncertainties associated with local mean estimates (Emery 2007). In this context, x0 represents the locations of the rainfall gauges, while xi denotes the intended output positions, typically a uniform grid over the geographical region of interest. The first step is to interpolate the computed Gaussian values at the observation locations P(x0) using OK, resulting in the interpolated values
Flowchart of the CS-STBM. The flowchart outlines the process of conditional simulation for generating rainfall fields. Abbreviations: x0, the observation location; xi, the intended output locations; Zuc(xi), the unconditional simulation;
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
b. Transformation to Gaussian space
The purpose of the transformation to the Gaussian space is to enable the application of statistical methods that assume normality, even if the data are not normally distributed. Since rainfall is not normal distributed by nature, this method has to be used to approximate it. Numerous methods exist for data transformation in meta-Gaussian models (Journel 1974; Vischel et al. 2009; Haese et al. 2017). This implementation follows the relationship between the empirical cumulative distribution functions (ECDFs), the probability density functions (PDFs) of the observation, and the normal distribution
c. Spatial correlation
Variograms are important in geostatistical research because they provide a measure of spatial correlation. Fitted theoretical variogram functions serve as the foundation for modeling in most geostatistical tools. The variogram is essential for characterizing the spatial variability and dependence of a variable. Accurate characterization of underlying spatial structures is essential for predicting values at unmeasured locations and performing spatial interpolation/simulation. Rainfall may have fundamentally diverse spatial structures, such as isotropic (same structure in all directions) or anisotropic (different structure in different directions) (Wilcox et al. 2021). To detect and analyze these structures, the traditional geostatistical workflow involves calculating the semivariogram and fitting a theoretical function. Often, modeling the variogram requires geostatistical and variable-specific expertise. Furthermore, on some days (especially with low station density), it is impossible to identify a clear variogram and an average variogram must be used. Therefore, this study introduces three techniques (S1, S2, St) designed to handle data-scarce conditions more effectively.
1) Mathematical implementation: Exponential function and geometric anisotropy
Illustrative example of geometric anisotropy: The black ellipse represents a conceptual illustration of geometric anisotropy, with a major axis (a = 4) indicating the direction of maximum correlation length and a minor axis (aζ, where ζ = 0.5) representing the direction of minimum correlation. The ellipse is rotated by an angle of θ = 45°. The dashed circle illustrates the isotropic case, where the correlation structure is the same in all directions. Adapted from Cao et al. (2014).
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
2) Semivariogram (S1)
Comparison of empirical semivariogram values
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
3) Automatic variogram detection (St)
To handle data scarcity and account for the input data, we have developed an alternative technique for automatic variogram modeling of rainfall data. In this approach, different combinations of parameter settings for C0, C1, a, θ, and ζ are tested for each day using leave-one-out cross-validation. The evaluation of these combinations is performed using the continuous ranked probability score (CRPS) (section 3). The input values for the different parameters can be found in Table 1 which are initially based on S1. However, they are subject to further refinement. Here, only lower (higher) nugget values C0 (sill values C1) are allowed. The range parameter a can vary within lower (10.0) and higher (50.0) values. Regarding direction angles θ, four options are considered: west to east (0°), southeast to northwest (45°), south to north (90°), and southwest to northeast (135°). The stretch parameter ζ is uniformly distributed from 0.25 to 1.0, where a value of 1.0 corresponds to isotropy. This approach is a combinatorial optimization. Given the assumption that C0 + C1 = 1.00, resulting in 320 × n (where n represents the number of available stations) combinations per day. The variogram model with the lowest CRPS is selected as the best model. The fitted parameters (C0, C1, a, θ, ζ) for every single extreme (St) can be seen in Table A2 in the appendix.
Parameters for the automatic variogram modeling, St.
4) Low-skill reference (S2)
For comparison and evaluation, this study introduces another variogram structure S2, which is completely random (C0 = 1.0) as low-skill reference. The idea of using a random or low-skill reference is inspired by forecast verification skill scores (e.g., Heidke skill score) which measure the proportion of accurate forecasts beyond what would be expected by random chance alone.
In this approach, unconditional simulations Zuc(xi) are back-transformed to the rainfall space using the available data for the specified day, which allows testing the methodology against randomness and provides a basis for assessing its performance. The S2 is visualized as dashed black line in Fig. 6.
d. Reference interpolation methods
The selection of standard spatial interpolation methods, such as the nearest neighbor approach, inverse distance weighting, or OK, as reference methods are due to their widespread use in providing spatial rainfall information based on ground observations for hydrological models.
1) Ordinary kriging
2) Nongeostatistical interpolation methods
We use two straightforward deterministic interpolation algorithms to test their capability to regionalize extreme rainfall events. The nearest neighbor (NN) algorithm chooses the value of the nearest point while disregarding the values of neighboring points, resulting in a piecewise-constant interpolation. Inverse distance weighted (IDW) interpolation provides more weight to points closer to the forecast location and less as distance increases. In this case, the weights are inversely proportional to the distance between the observation values to the target values. The two methods are often used to regionalize rainfall (Bárdossy et al. 2021; Mohanty et al. 2018; Chen and Liu 2012).
e. Spatial declustering
Spatial datasets can exhibit sampling biases, resulting in data points clustering in specific areas over space and time. This clustering can significantly influence geostatistical techniques like interpolation and simulation. A notable example is the increased density of rainfall stations in the northwest corner of the study area, as a result of the AMMA-CATCH observatory in northern Benin [Galle et al. (2018), Fig. 1]. Spatial declustering techniques address this issue by assigning weights to data points based on their local data density (Chilès and Delfiner 2012; Pyrcz and Deutsch 2014).
Within geostatistics, cell-based declustering is a popular approach (Deutsch 1989; Deutsch and Journel 1997; Chappell and Ekström 2005; Pyrcz and Deutsch 2014). This method involves overlaying a grid or mesh onto the study area. Each data point in a cell receives a weight inversely proportional to the number of data points within that cell. To improve the accuracy and reduce the influence of the starting point of the grid, adjustments (cell offsets) are applied to the distribution of the grid’s cells. The resulting weights are standardized to ensure their sum equals the total number of data points. A weight greater than 1.0 indicates that a sample is underweighted in the original distribution. Conversely, a weight less than 1.0 suggests the sample is being overweighted often due to clustering with other samples. In practice, various cell sizes are evaluated to find the one that minimizes the mean of the declustered distribution.
f. CRPS
4. Results
In the following, the quality assessment of the CS-STBM is discussed in terms of reproducing the observation, comparing to reference methods, and improving performance by changing spatial correlation functions.
a. Reproduction of the observation
To assess the algorithm’s capability to reproduce the observations, a total of 100 realizations were generated for each of the 50 extreme events using the CS-STBM approach. The simulations were performed at a resolution of 0.05° × 0.05° (1576 grid points) within the basin using the variogram structure S1.
The performance results of the simulation framework reproducing the ECDFs are demonstrated in Fig. 7. The observations closely align with the ensemble mean and generally fall within the 10th and 90th quantiles, indicating a satisfactory reproduction of the observed rainfall CDF. However, it should be noted that in cases where there are only a few observations available for a specific day, there may be instances where the observation deviates from the range of the ensemble (e.g., extreme event number 5 with n = 11). Consequently, constructing the ECDFs for the Gaussian transformation becomes challenging in such scenarios. Furthermore, the simulation framework effectively captures the intermittent nature of rainfall in space. This is evident in extreme event no. 8, which exhibits a proportion of zero values (p0) of 0.19. A p0 value of 0.19 indicates the occurrence of dry areas within the simulated rainfall field, highlighting the spatial variability and intermittent nature of rainfall patterns. This capability to replicate intermittent rainfall patterns is a valuable characteristic of the simulation framework. Additionally, the evaluation reveals that the integrated 10% excess of the maximum rainfall amount, when transformed back to the original space, operates as expected.
Comparison of ECDF with desired ECDF of observations (blue), ensemble mean (red), and Q10/Q90 (gray) ECDF of simulations for the highest 10 extremes × 100 realizations.
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
Regarding the summary statistics, the comparison between the simulated fields and the observed mean and variance of the rain gauges (Figs. 8 and 9) reveals a generally good agreement. The mean and variance of the simulated fields closely match the corresponding observed values. However, there are some minor inconsistencies, such as extreme event number 36 falling outside the range of the ensemble. One notable finding is that as the observation variance increases, the ensemble spread of both variance and mean in the simulations also increases. This can be observed, for example, in extreme event number 3. This suggests that with higher observation variance, the simulations capture a wider range of possible outcomes, thereby addressing the overall uncertainty.
Comparison of simulated field’s mean and rain gauges’ observed mean (raw): Boxplot illustrating the mean of 100 simulated realizations for each rain event. The box represents the interquartile range (IQR), with the bottom edge indicating the 25th percentile (Q1) and the top edge representing the 75th percentile (Q3). Data points beyond the whiskers are shown as outliers (black points). The red point represents the mean of rainfall observed by rain gauges, while the boxplot represents the mean of the simulated fields.
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
As in Fig. 8, but for comparison of simulated field’s variance and rain gauges’ observed variance.
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
Overall, the simulation framework demonstrates its proficiency in reproducing observed rainfall distributions and capturing key characteristics such as intermittency. While there may be some minor discrepancies between the simulated and observed values, the simulation framework demonstrates a reasonable ability to replicate the mean and variance of the rain gauges. The increased ensemble spread with rising observation variance indicates a potential improvement in addressing widespread uncertainty in the simulations.
b. Spatial declustering: Extreme event number 36
Figure 8 (Fig. 9) shows a significant mismatch between the simulated field’s mean (variance) and the observed mean (variance) from rain gauges for extreme event number 36 (30 August 2003). This anomaly is likely due to the clustered and centralized pattern of rainfall in the northwestern part of the basin.
Figure 10a illustrates the markedly nonuniform spatial distribution of observational points. This nonuniformity becomes particularly evident with the clustering of data points in the aforementioned part being overemphasized, introducing an inherent bias into the dataset. Cell declustering effectively mitigates this issue by assigning lower weights (less than 1) to the northwestern stations, reducing their influence. Conversely, stations in other areas of the basin receive weights greater than 1, reflecting their increased significance in the overall data representation (Fig. 10b). This adjustment in weighting leads to a significant change in the overall distribution, with a reduced mean and variance, which provides a statistically more accurate reflection of the rainfall distribution over the basin (Fig. 11).
Analysis of the observed spatial rainfall distribution and the respective weights for declustering from extreme number 36 (30 Aug 2003). (a) The observed rainfall measurements with their spatial distribution across the basin, using a color gradient to indicate the amount of rainfall (mm). (b) The declustered weights assigned to each observation station, where the size and color of the circles correspond to the weight magnitude. The declustering process adjusts the representation of each station’s data based on the nonuniformity of the spatial distribution, leading to a more balanced dataset that minimizes the bias caused by overemphasized clustering in the northwestern corner.
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
Comparison of rainfall distributions before and after declustering from extreme event number 36 (30 Aug 2003). (left) The frequency of rainfall amounts prior to any statistical adjustment, with a mean (μ) of 31.25 mm and a variance (σ2) of 1248.12 mm2. (right) The frequency of rainfall amounts after declustering has been applied to address spatial clustering bias, resulting in a revised mean (μ) of 15.23 mm and a variance (σ2) of 302.98 mm2. The histograms illustrate the effect of declustering on the representation of rainfall data, notably reducing both the average and variability of the measured values.
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
We rerun the CS-STBM using the declustered data in the same configuration as described in the previous section. This leads to a much more accurate reproduction of the mean and variance of the respective data compared to the clustered (“raw”) data (Fig. 12). This demonstrates how clustered data impacts the performance of geostatistical simulations.
As in Fig. 8, but for comparison of the (a) mean and (b) variance of the simulated field with rain gauge data (red points) for extreme event number 36 (30 Aug 2003). The “raw” label on the left represents the application without cell declustering, while “declustered” on the right denotes the results after applying cell declustering.
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
c. Comparison to other regionalization techniques
Most regionalization strategies, such as OK, tend to produce smoothed outcomes by assuming minimizing estimation variance, resulting in reduced variability. However, variograms calculated from kriged values differ from the desired ones, as the variances related to various distances are significantly reduced (Olea 1999, p. 142). In contrast, geostatistical simulation stands out as a method capable of accurately reproducing spatial correlation.
To illustrate the effectiveness of different regionalization strategies, Fig. 13 provides a direct comparison of spatial regionalization techniques for the highest recorded rainfall amount (RA). It is evident that only the simulation realizations (CS-STBM1 and CS-STBM2) are capable of generating realistic rainfall field patterns. In contrast, the smoothed models (OK and IDW) and piecewise constant model (NN) appear implausible, as they result in an even distribution of rainfall across space, which is highly unlikely in reality. Furthermore, the impact of different regionalization techniques on the spatial distribution of mean variance, considering the 50 extreme events, is illustrated in Fig. 14. Particularly, it is evident that IDW and OK significantly underestimate the variance values at unobserved locations. From this perspective, NN exhibits the best interpolation results. Interestingly, even with their Q10, the CS-STBM realizations display higher variances than OK and IDW. The ensemble mean of CS-STBM exhibits a well-distributed variance across the entire basin area. These findings highlight CS-STBM as a valuable tool for modeling more realistic rainfall fields and capturing their variability in terms of variance. Nevertheless, similar to interpolation techniques but not as pronounced, artifacts occur at the rainfall station locations. The observed peak in variance in the midstreams of the basin is very likely not the results of orographic lifting (Fig. 1). It might be partially caused by higher data availability at these stations for the analyzed 50 extreme events. This could increase their influence on the overall variance compared to stations with fewer observations, potentially introducing artifacts.
Comparison of daily rainfall fields for extreme event number 1 (2 Sep 1969): The figure displays the daily rainfall field including OBS as well as using various interpolation methods, NN, OK, IDW, and two realizations of the simulation (CS-STBM1 and CS-STBM2). Each method’s representation of the extreme rainfall event is shown to illustrate the differences and similarities between the approaches.
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
Comparison of the variance over 50 extremes: The figure presents the mean variance of the OBS (for stations that occur in at least 25 extremes) and the rainfall estimates for four interpolation methods, including NN, IDW, OK, and CS-STBM at three stages: Q10 (10th percentile), mean, and Q90 (90th percentile). The analysis highlights the performance differences of these methods in capturing the variability of extreme rainfall events.
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
In summary, Figs. 13 and 14 show the limitations of conventional regionalization strategies in terms of variability and emphasizes the unique capabilities of geostatistical simulation, specifically CS-STBM, in producing realistic rainfall patterns and preserving their variance.
d. Performance assessment of S1, S2, and St
To assess the performance of different spatial structures (S1, S2, and St) from a probabilistic perspective, the CRPS is computed using a leave-one-out cross-validation approach as described in section 3. Here, S1 represents the isotropic model, S2 is the low-skill reference, and St stands for the daily-specific model incorporating anisotropy. A total of 100 realizations were generated in a resolution of 0.05° × 0.05°. In Fig. 15, the 50 extreme values are sorted from bottom to top, and each bar represents the corresponding CRPS value. The red bars indicate the lowest CRPS value across the three spatial structures for each extreme event. Low CRPS values indicate a good match between the observations and the narrow probability density function of the simulations. This demonstrates that incorporating anisotropy significantly improves the model performance. In 45 out of 50 instances, the results were notably enhanced. The average CRPS values are 15.39 for St, 17.00 for S1, and 19.39 for S2. Comparing the performance of St with a basic average isotropic variogram S1 and a random method S2, it is evident that St outperforms both. Specifically, St yields a 9.5% improvement over S1 and a 20.63% improvement over S2. These results underscore the importance of incorporating geometric anisotropy and estimating daily specific variograms instead of relying on average variograms.
Performance assessment of S1, S2, and St: CRPS values for three variogram structures in relation to 50 extreme rainfall events. The highlighted bar in red represents the best-performing method among the three.
Citation: Journal of Hydrometeorology 25, 10; 10.1175/JHM-D-23-0123.1
5. Discussion
Geostatistical rainfall simulation is widely applied in several regions, but rarely used in the context of data-scarce regions like West Africa. In this study, we performed a quality assessment of a state-of-the-art conditional geostatistical simulation approach (CS-STBM) for generating daily rainfall fields with a focus on extremes. Our assessment revealed that the simulation approach can much better reproduce the variability of the target variable compared to interpolation techniques, leading therefore to less underestimations of rainfall extremes at ungauged sites. This result is expected as IDW and OK produce smooth results, while NN performs a block regionalization (Fig. 13). Therefore, our study is in line with other geostatistical studies and shows the importance of using simulation methods for rainfall (e.g., Haberlandt and Gattke 2004; Vischel et al. 2009; Haese et al. 2017). The high potential of stochastic rainfall simulation was also shown by other studies for the West African region, such as Guillot (1999), Guillot and Lebel (1999), Vischel et al. (2009), and Wilcox et al. (2021). Since these studies differ in their model settings (e.g., time resolution and spatial domain) and performance measures, a direct comparison with our study is difficult.
However, in contrast to the abovementioned studies, we simplified the simulation process and replaced traditional variogram estimation with developing a semiautomatic workflow that optimizes the CRPS to match the observation data. This automatic variogram estimation St is compared to CS-STBM simulation using a mean variogram (S1). We also include a low-skill reference simulation (S2) to better assess the quality of the simulation approach. The results demonstrate an improvement in 45 out of 50 extreme cases, but the overall enhancement from S2 to St is only moderate (20.63%). This finding highlights the significant challenges in achieving precise rainfall fields in our study area, primarily due to the stochastic nature of rainfall events due to convection, low sample size, and poor quality of rainfall measurements. Furthermore, it is important to note that the combination of parameters leading to the lowest CRPS offers the optimal variogram model considering the available data in terms of cross-validation. This event-based automatic variogram estimation can be only used for conditional simulations, e.g., for monitoring of heavy rainfall events. In the case of unconditional simulations, average seasonal or weather-dependent variograms are required. This method involves combinatorial optimization, which can be computationally intensive, particularly for large datasets with numerous stations. This computational complexity may limit the feasibility of using the methodology for large datasets or in real-time applications.
The research findings presented herein provide evidence of the effectiveness of the CS-STBM in capturing the spatial variability of rainfall in the Ouémé River basin. These findings underscore the value and potential of the CS-STBM as a reliable and robust simulation method for further application in hydrological modeling, risk assessment, climate impact studies, and related fields. Although the performance improvements from event-specific tuning (St) may appear marginal based on the CRPS alone (9.5%), they can be of significant consequence for hydrological simulations in data-scarce regions. Since even slight enhancements could lead to much better decisions in water resource management. In hydrological modeling, this technique can be used to generate rainfall fields for distributed hydrologic modeling (Vischel et al. 2009; Grundmann et al. 2019; Bárdossy et al. 2022), thus enabling to analyze the effect of different spatial rainfall patterns on resulting runoff responses (Nicótina et al. 2008) and to address rainfall errors and uncertainties in hydrologic modeling (McMillan et al. 2011). Furthermore, the STBM can be also valuable for climate impact assessments. It can be used as statistical downscaling approach conditioned on circulation patterns (Bárdossy and Plate 1992; Wetterhall et al. 2009; Haberlandt et al. 2015; Vaittinada Ayar et al. 2020; Bliefernicht et al. 2022a) enabling the simulation of future rainfall scenarios under various climate change projections to evaluate the potential impact on regional and local rainfall patterns. For these reasons, stochastic geostatistical simulations are acknowledged as important tools in a variety of environmental disciplines where rainfall has a significant impact. Although this approach is not limited to precipitation, the methodology can also be applied to other variables such as groundwater (Karami et al. 2022), solar radiation (Bechini et al. 2000), or soil water content (Castrignanò and Buttafuoco 2004). Overall, the presented framework can be an important tool in a wide range of environmental disciplines where rainfall and other variables exhibit significant impacts.
The limitations exposed by the applied normal score transformation highlight the importance of developing more sophisticated and targeted methods, especially for handling zero-rainfall data. In this research, converting all zero values to a single, consistent value artificially reduces the variance of the normal distribution. Additionally, this can lead to unrealistic outcomes, as zero observations close to wet stations should not be treated identically to those further away. Alternative approaches include generating a perfectly Gaussian variable (Journel 1974) or performing simulations using a truncated normal distribution with equality constraints for positive values and inequality constraints for zeros (Haese et al. 2017). Such simulations can be implemented using the S-STBM framework (Lauzon and Marcotte 2020).
Furthermore, the overall mean variogram is affected by zero observations, as transforming all zeros to a single value lowers both the variance and the variogram of that variable. A more accurate and statistically valid estimate could be derived from using indicator variograms. Setting a consistent threshold for every time step—a level significantly above the zero probability—allows the transformation of the indicator variogram into the variogram of a Gaussian equivalent (Journel and Isaaks 1984; Kyriakidis et al. 1999; Maleki et al. 2016). This method ensures that assigning the same value to zero observations does not affect the variogram calculation.
The first condition of the intrinsic hypothesis demands that the target variable must be spatially stationary, which is essential for achieving an unbiased estimation error across the entire domain (Bárdossy 1997, p. 15 ff.). However, the strong latitudinal dependence of rainfall in West Africa could indicate nonstationary behavior (Benoit et al. 2018), highlighting the spatial complexity of the region’s climate. To address the challenge of spatial nonstationarity, one strategy involves assuming a deterministic drift in the rainfall model, influenced by external variables like elevation and geographical coordinates (external drift kriging) (Haberlandt 2007; Bárdossy et al. 2021). Alternatively, using a quantile-based interpolation method (Lebrenz and Bárdossy 2019) provides a robust solution for overcoming the issues posed by nonstationarity by converting the variable of interest into quantiles.
The performance of the CS-STBM is highly dependent on the density and distribution of rain gauge stations. Spatially unevenly distributed or clustered data can lead to less reliable simulation results, highlighting the importance of a well-distributed network of rain gauges. Due to its ability to handle the nonuniform spatial distribution of observational points and thus enhance performance for extreme event number 36, cell declustering holds promise for improving results and uncertainty quantification across all extremes. Future studies could perform a more in-depth analysis of CS-STBM with and without declustering, e.g., by using more extremes. This could provide valuable additional insights not only for practitioners.
Further improvements can be made to enhance the simulation method. These include incorporating third-order statistics, such as directional asymmetry, to capture non-Gaussian spatial correlations more effectively (Bárdossy and Hörning 2017; Hörning and Bárdossy 2018; Yan et al. 2020), incorporating information from satellite retrievals (e.g., GPM IMERG or CHIRPS) or reanalysis products [e.g., ERA5, NCEP–NCAR Reanalysis Project (NNRP), and JRA-55] to enhance the accuracy of simulated rainfall fields, or considering alternative variogram estimation techniques to supplement spatial continuity, for example, by incorporating locally varying anisotropy (Boisvert and Deutsch 2011). These advancements could contribute to improving the realism and reliability of stochastic geostatistical simulations of rainfall.
6. Conclusions
In this study, a geostatistical simulation framework was presented to simulate extreme daily rainfall fields. The developed methodology was applied for 50 extremes (1960–2016) within the Ouémé River basin (Benin). Due to data scarcity in this region, running a geostatistical simulation is challenging. The study provides a simulation technique for rainfall fields to enable straightforward and robust parameter estimation and a simple simulation approach due to the widespread lack of sufficient data for rainfall inputs to hydrological models. The novelty of the model developed in this study resides in the combination of traditional method accounting for the creation of rainfall fields and intensive verification framework. Based on tests and evaluations for multiple extremes, our results demonstrate that the simulation framework serves as a valuable tool for modeling more realistic rainfall fields. This is due to its superior capability to accurately replicate both the spatial distribution and intensity of rainfall events, compared to established interpolation methods (OK, NN, and IDW). Additionally, the framework demonstrates skill in reproducing variability, as quantified by improved performance metrics such as the CRPS for each event, indicating a substantially closer match between simulated and observed rainfall distributions. Furthermore, the simulation framework can reproduce the empirical cumulative distribution function of the observation samples and outperforms classical interpolation techniques (e.g., ordinary kriging) in terms of spatial continuity and rainfall variability. We have developed a methodology that maps a reliable variogram even at low station density. It shows the overall importance for every individual time step variogram determination. The performance can be increased by another 9.5% in terms of CRPS. It also shows the importance of addressing the uncertainty in the observational data. In summary, geostatistical simulation is a powerful tool for estimating and predicting the spatial distribution of extreme daily rainfall fields. By using these techniques, it is possible to produce more accurate and reliable predictions that can be used to better understand and mitigate the impacts of extreme rainfall events. The simulations are an important input in hydrological modeling to reconstruct flood events and produce hazard maps. Overall, the application of this technique contributes to a better understanding of the impact that past and present extreme rainfall fields in West Africa have on the risks posed to both humans and the environment.
Acknowledgments.
This study is part of the FURIFLOOD (Current and future risks of urban and rural flooding in West Africa) project, which is funded by the Federal Ministry of Education and Research in Germany (Grant: 01LG2086D). We thank our partners involved in the FURIFLOOD project for their support. We also thank the editor, Yu Zhang, two anonymous reviewers, and András Bárdossy for their helpful comments that improved the quality of the manuscript.
Data availability statement.
The rain gauge data cannot be shared due to data provider restrictions. The created rainfall fields are available upon request from the corresponding author. The Python codes for the STBM and CS-STBM can be accessed on GitHub: https://github.com/rauchman1/furiflood_tools.
APPENDIX
Additional Information
Table A1 provides a descriptive overview of the 50 extreme events, while Table A2 presents the optimized variogram structures for these events.
Overview of the selected 50 extremes with respect to date, mean daily rainfall amount RA (mm day−1), daily areal rainfall variance RV (mm2 day−1), maximum observed rainfall amount per day RMAX (mm day−1), available measurements per day n (—), and fraction of zero values per day p0 (—).
Overview of the optimized variogram structures for the 50 extremes, including the nugget (C0), sill (C1), range (a), anisotropy stretch parameter (ζ), anisotropy rotation angle (θ), and the CRPS.
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