A spatially based precipitation bias correction is introduced that generalizes existing approaches. The method consists of projecting observed precipitation anomalies on to the model's modes of variability for a large set of model hindcasts to produce artificial mapped empirical orthogonal functions, which are used to bias correct forecasts. Similar to previous spatially based methods, the scheme can shift displaced anomalies, associated with the West African monsoon progression for example, to their correct location, and by construction produces a corrected field with a zero-mean bias with respect to the observations. The new method has the advantage that it only applies corrections to modes of variability for which the model has proven skill, and does not rely on a one-to-one direct correspondence between the observational and model modes, a restriction of previous methods. By processing the precipitation fields in sequences of seven pentad averages, it is also possible to including variability on shorter than monthly time-scales, important if the end product is to be used for end-user impact-focused research. The method is tested for various empirical orthogonal function-defined climate macro regions within Africa and is shown to reduce biases while also improving threat skill scores over a range of thresholds and forecast lead times.
Bias correction techniques can be broadly divided into two categories: point-wise and spatial techniques. Examples of the former include simple moment correction (Baigorria et al., 2007), cumulative distribution function (CDF) matching (Ines and Hansen, 2006; Piani et al., 2009) and quantile regression-based correction (Wood etal., 2004; Li et al., 2010; Hopson and Webster, 2010). Point-wise methods are simple to implement and can be used successfully to remove systematic model deficiencies in, say, the representation of the hydrological cycle, but do not account for spatially coherent signals; the correction applied at each point is by definition independent of neighbouring locations. This is quite critical in the forecast context if the aim is to maximize the model's predictive skill. A hypothetical model system may possess good El Niño–Southern Oscillation predictive skill, but poorly represent the teleconnections to a remote region, such as eastern Africa, such that the remote precipitation anomaly, while correctly predicted, is shifted in space. In this case, a point-wise approach is unable to correct for this and reveal the model skill.
Spatial-based techniques instead address bias correction through the mapping of dominant modes of variability in the model parameter to the equivalent (correlated) mode observed in the observational field, using singular value decomposition analysis (SVDA), canonical correlation analysis (CCA) or empirical orthogonal functions (EOF) to identify the modes of variability. Examples of this approach applied on a global scale are described by Ward and Navarra (1997), Feddersen et al. (1999), Kang et al. (2004) and Yun et al. (2005), while Feudale and Tompkins (2011) applied this technique regionally to western African monsoon precipitation. In each case, the method was applied to a single variable and operated by identifying which model/observational pairs of the leading order modes of variability were significantly correlated. The model field was then reconstructed using the model predicted principal components to scale the equivalent observational mode.
Spatially based approaches have been demonstrated to improve global model bias and skill measures. For example, the European Centre for Medium-Range Weather Forecasts (ECMWF) system 3 seasonal forecast system was demonstrated by Tompkins and Feudale (2010) to suffer from a systematic bias whereby monsoon rains were shifted to the south. Reasons for this bias were identified in sensitivity tests, but correcting the identified model deficiencies is a difficult and long-term task. Until such model improvements are achieved, Feudale and Tompkins (2011) showed that the EOF-based bias correction approach could successfully correct this bias and improve model skill scores. Nevertheless, one drawback of the spatial-based approaches as implemented until now is that they rely on a straightforward one-to-one mapping of model and observational modes of variability. The correction is in fact achieved by a simple one-to-one EOF swapping between the model and the observation. If the leading observational mode of variability in observations corresponds to the second order or lower mode in the model, or possibly a combination of modes, this will not be accounted for. A generalized mapping method is thus presented in this work that attempts to address this concern.
Another concern is the time-scale of the bias correction technique when applied to precipitation. While the point-wise approaches discussed above may be applied to daily precipitation, the same is not true of the spatial techniques, which would be confounded by the high spatial variability of daily rainfall. Thus to date these techniques have only been applied to monthly rainfall average anomalies which are spatially smoother and also Gaussian in distribution. The drawback with this is that dynamical (in contrast to statistical) applications models for disease, crop and water modelling are often integrated on a daily time step (for respective examples in each of these sectors, see Challinor et al., 2004; Hoshen and Morse, 2004; Hopson and Webster, 2010). The integrating effect of soil moisture and surface hydrology could imply that pentad or dekad rainfall may be adequate as input for some of these applications; the malaria model of Hoshen and Morse (2004) uses dekad rainfall, for example, while the study of Li and Tompkins (2012) revealed little difference in crop yield when pentad rainfall was substituted for daily rainfall, while using monthly rainfall amount was inadequate. The second aim of this work was therefore to investigate whether a spatial correction technique may be usefully applied to dekad or pentad precipitation data.
2. Data and methodology
2.1. Model data
The ECMWF forecast system currently consists of three operational components, namely a 10-day deterministic forecast, a 15-day variational ensemble prediction system (EPS) forecast that is extended to 32 days once each week (Vitart et al., 2008),* and an ensemble seasonal forecast system (System-4 since November 2011) that once a month issues a 7-month (extended to 13 months four times a year) prediction. In this work we focus on the EPS monthly forecast system for a number of reasons. Firstly, due to the frequent updates that are made to the model physics and data assimilation system, each EPS monthly forecast ensemble integration also runs a five-member hindcast suite of integrations for the same months over the previous 18 years. This hindcast set, by providing identical integrations (i.e. no observations are assimilated) as the forecast but for a historical period for which observations are available, can be employed for calibration purposes. Thus, every week a set of 19 years of month-long prediction ensembles is available using an identical forecast system.
While the System-4 seasonal forecast system also has an extensive hindcast suite, the lower-resolution and less recent model cycle (the seasonal system atmospheric model is updated less frequently, every 3–5 years) implies that the skill is lower relative to the monthly system in the overlapping first 4 weeks. Further motivation for emphasizing the monthly system is that it is rarely used operationally in Africa, the products not being presently available via the EUMETCAST or RETIM distribution services, and not distributed for consideration in the consensus forecast process that occur in western, eastern and Southern Africa each year. The aim is thus to demonstrate the usefulness and skill of a bias-corrected product for sector-relevant research and operational planning in the region.
It should be emphasized that the much more frequent updates to the monthly system compared to the seasonal system in ECMWF imply that the bias characteristics of the two systems diverge over time, only reconverging when the seasonal system is updated. To illustrate this disparity, the rainfall bias of the most recent monthly system (model cycle 37R2) and of System-4 (model cycle 36R3) compared to Global Precipitation Climatology Project (GPCP) v2.1 rainfall data are shown in (Figure 1). It is no that System-4 underestimated precipitation, with the rain band lying too lower down to the south, in stark contrast to the more recently updated monthly EPS system.
A suitable precipitation observational dataset is required in order to correct model precipitation on a temporal resolution of pentads, which is the minimum requirement for a precipitation product to be employed to drive sectorial application (Li and Tompkins, 2012). The dataset must cover the 18 years hindcast period for which the model hindcast ensembles are available. Moreover, it is desirable not to degrade the horizontal resolution of the model used, which at the moment is 50 km with a very coarse dataset.
A lack of daily station data for the continent of Africa necessitates the use of satellite-derived precipitation datasets. GPCP v1.1 daily precipitation data are available on a 1° resolution, but do not cover the complete hindcast period, being available only from late 1996. GPCP v1.0 pentad data are available for the earlier period from 1979, but on a coarser grid of 2.5° resolution. The decision is therefore taken to use the higher-resolution daily dataset, GPCP v1.1, averaged over 5 days from late 1996 onwards supplemented for the initial period of the 1990s with the GPCP v1.0 pentad dataset that is sharpened to the same resolution using a simple algorithm, outlined in Appendix A.
African rainfall follows seasonal cycles which differs across the continent. Therefore, the bias correction method is tested for eight subregions in Africa that are homogeneous in terms of their precipitation climatology defined using an EOF-based clustering detailed in Appendix B. While the bias correction method is evaluated in all subregions, there is a special focus here on the regions that contain the countries of Senegal, Ghana and Malawi, for which the corrected forecasts will be used to drive a prototype malaria prediction system within the project ‘Quantifying the impact of weather and climate on health in developing countries’ (QWeCI).
2.3. Correction methodology
Similar to previous spatial bias correction methods, the approach used here is based on an EOF decomposition of the forecast spatial precipitation anomalies with respect to the hindcast (H) mean. The hindcast anomaly field, ℋ, available for a time series of T time slices, can be decomposed into the dot product of the temporally evolving principal components and their respective spatial pattern, the empirical orthogonal function , where i represents the ith mode. Thus
In practice, only a finite number M < T of leading modes are retained in the analysis below, implying that ℋ is approximate, and hereafter for ease of notation the superscript i is assumed to range over the M modes as i = 1,M.
Previous EOF-based techniques then proceeded by performing an equivalent EOF decomposition on the observational dataset, . A given forecast F anomaly, which is to be bias corrected, was projected on to the hindcast-derived EOFs to derive the forecast . Using these forecast PCs, the forecast field was then reconstructed, but replacing model spatial EOF modes with their equivalent observed pattern, relying on temporal correlation of the EOF PCs to identify such ‘paired’ modes. The drawback of this approach was the necessity for the model to reproduce the leading-order observed modes in their correct order.
Here a more generic approach is taken by posing the question whether any of the model modes reflect variability in the observations. This can be seen by projecting (or ‘mapping’) the observational data G on to the decomposed model modes, to derived artificial mapped EOFs, denoted MEOF, as
The mapped projections are deemed artificial since they are not true orthogonal modes. The MEOFs are then used to bias correct the forecasts in a similar approach to previous techniques, by direct substitution of the respective equivalent mode. This is done systematically for all M mapped MEOFs, since by construction any model mode that does not correlate well with observed modes of variability will result in a zero-mapped anomaly. In summary, given a forecast precipitation anomaly, the respective leading-order PCs are derived in the usual way, performing the spatial dot product of the full field and the model MEOFs derived from the hindcast analysis, and the bias-corrected precipitation anomaly field is reconstructed from the combination of the PC and its respective MEOF:
The precipitation field is then constructed adding the mean of the observed precipitation to the summed anomaly modes. The outlined approach directly accounts for model skill since if the model has no skill at all then the MEOFs will be null, and the resulting corrected field will simply reproduce the observational climatology.
One potential drawback of the method is that it relies on past predictive skill over the hindcast period for its operation. While the model structure such as the physics parametrizations and the data assimilation framework is invariant over the 18-year hindcast period, the atmospheric and oceanic observational network is not, with satellite and buoy observations becoming increasingly beneficial, especially for the tropical regions where climate anomalies are more directly impacted by sea surface temperatures and where land-based conventional observational networks are typically sparse (e.g. Tompkins and Feudale, 2010; Fink et al., 2011). Poor-quality initialization reducing skill in the earlier hindcast periods will correspondingly weaken the MEOFs and lead to forecast anomalies being under-predicted for the present-day forecasts.
2.4. Operational implementation at ECMWF
The bias correction operational implementation at ECMWF is schematically illustrated in Figure 2. It has to be stressed that some of the choices for the operational implementation suggested here were imposed by the existing product specifications at ECMWF and therefore should not be considered a strict constraint of the calibration method itself. Thus, for example, the 18 years of hindcast daily precipitation from the EPS monthly forecast system is temporally averaged to create consecutive pentads for identical dates to the GPCP data. As the EPS monthly forecast system integrations are available only once a week each Thursday, this sometimes necessitates the splicing of two consecutive forecast ensembles for the training dataset. With reference to Figure 2, if a pentad period starts on 0000 UTC Tuesday, the pentad P05 will consist of days 6 and 7 of the previous forecast ensemble spliced with the first 3 days of the subsequent forecast ensemble. Therefore pentads will be created out of an amalgamation of daily precipitation at different lead times. This splicing ensures, nevertheless, that all pentads have on average similar lead times. This is at the obvious expense of discarding forecast continuity, which is deemed acceptable since the splicing technique is only used in the training period, for which mean bias statistics are required. For forecasting purposes the bias correction is instead applied to a single time-continuous forecasting ensemble.
The model hindcasts are then decomposed into the leading EOFs on a month-by-month basis, but with the seven pentads that are representative of a given month presented as separate fields (i.e. not averaged), meaning that for the 18-year time series (1991–2008) a total of 126 fields are analysed. Variability on the sub-month time-scale is thus included by retaining separate pentad fields in the analysis. An example decomposition for June for macro region A is shown in Figure 3, which gives the first three EOFs for the model and observations over Sahel, while Figure 4 shows the associated first PC. The EOFs are similar, comparing the model forecasts and the observations, demonstrating the capability of the model to capture the main observed spatial mode of variability. That said, the model first PC instead indicates an excessive level of month-to-month variability in the forecast ensemble. The first three model EOFs explain 26%, 12% and 10% of the total dataset variance, respectively, while the corresponding first three GPCP EOFs explain 15%, 10% and 7% of the total dataset variance. For this example case study, the MEOFs obtained by projecting the observational anomaly time series on to the model PCs for the same 126 time slices (18 × 7 pentads in June) are shown in Figure 5. The MEOFs are weaker relative to the raw forecast EOFs, but this is expected as it should be recalled that the MEOFs do not have the same physical and statistical interpretation. For a perfect model initialized from perfect boundary conditions each MEOF would be identical to its corresponding forecast EOF (which would of course match the observed GPCP EOF if the observations were error free). Any reduction in skill due to an imperfect model or initial conditions necessarily leads to a weaker MEOF pattern.
The MEOFs are subsequently employed to reconstruct the precipitation anomaly for the forecast and for each of the 51-member ensembles of the EPS monthly system using the PCs obtained by projecting the actual forecast on to the hindcast EOFs. Figure 6 shows the final calibrated precipitation field resulting from this process for the control member of the EPS monthly system applied to MacroRegion A (see Figure 6). Even if the MEOFs were weaker than the raw model EOFs, the final reconstructed precipitation anomaly pattern is improved, with the rainfall overestimation reduced and the rain band moved to the south.
To highlight the potential of the new generalized method compared to the previous simpler technique in which only the EOF swapping is performed, Figure 6(C) also shows the calibrated field resulting from the application of the method outlined in Feudale and Tompkins (2011). The original work of Feudale and Tompkins (2011) applied the method to monthly mean data for which the latitudinal monsoon offset was well described by the first EOF in both the model and in the observations. In contrast, the higher temporal variability of the pentad data here confounds this signal and the method performs poorly relative to the MEOF correction presented here.
The MEOFs constructed from the short-range forecast (days 1–5) pentad are then also used to correct the subsequent four model forecast pentads for lead times up to 25 days. This implicitly assumes that model biases that develop within the 5 five days of the forecast system resemble those of longer lead times, which is a common and reasonable assumption (Rodwell and Palmer, 2007).
3.1. Experiment configuration
Validation of the approach is performed using operational forecasts spanning April 2008 to August 2009; the largest interval in which both the EPS monthly forecast (which started in April 2008 in its fully operational configuration) and the GPCP dataset are available simultaneously. A total of 17 individual bias correction cases are formed using seven pentads for the training dataset which sample different months. The pentad range selections for the experiments are shown schematically in Figure 7. This highlights that the choice of a pentad-based system is also pragmatic since an overlap of one pentad allows a year (containing 73 pentads) to be divided into 12 ‘monthly’ segments of seven pentads (35 days) each, facilitating the eventual aim of moving towards a seamless combination of the monthly and seasonal systems.
Each of the seven pentad slices is then taken as representative of the correction in individual calender months. The MEOF calculated for the hindcast dataset for each seven-pentad slice are then used for the following five pentads of the forecast as explained above, thus producing 17 examples of corrected forecast ranging from one to five pentads in lead time. The mapping is performed separately on each region (A–H) and then global fields are obtained by merging the predicted precipitation. This is achieved taking the precipitation average where regions overlap.
The bias correction using the new mapped EOF methodology is referred to as the MEOF method in the following, and is compared to a point-wise correction technique of simply removing the mean bias with respect to the observational dataset. The latter will be referred to as the ‘standard methodology’.
3.2. Calibration assessment
The seasonal anomaly with respect to the GPCP dataset of the uncalibrated forecasts, the classical mean bias correction and the new mapped approach for the DJF 2009 and JJA 2009 periods is shown in Figures 8 and 9 respectively. The raw forecast model anomaly is dominated by the bias in the model and highlights the need to bias correct model output to drive sectoral impact models. The middle two panels show that both the mapped EOF method and the simple point-wise bias correction method correct the main systematic bias, but in terms of the spatial pattern of the forecast anomalies the improvement depends on the region in question. In western Africa, where the standard model not only overestimates rainfall but also the rain band is generally displaced to the south, the MEOF is clearly superior, shifting the forecast anomaly into the correct latitudinal location, performing similarly to the simpler EOF-based monthly mean approach of Feudale and Tompkins (2011). In contrast, the difference between the methods is less clear cut over southern Africa for the season of DJF and, in particular, a simple removal of the point-wise bias seems to better reveal the tri-pole precipitation anomaly observed in this season along the west coast.
To provide a more detailed assessment month by month, Figure 10 gives time series of precipitation anomalies for every region in Africa using simple point-wise bias correction and the new approach. The box-whiskers are constructed using the 51 forecast ensemble members both from the control and the mapped precipitation. Again, the GPCP mean observed precipitation is reported for comparison. The figure confirms that the the new method performs poorly for Africa region D during DJF but improves during the subsequent months of March and April. For area G, which contains the target country Malawi, there is an improvement in the main rainy season months of January and February. While one or two individual exceptions can be found, in general the forecast anomaly is improved with the new method, in particular for western African regions.
As the spatial-based approach seems to function well in western Africa (which contains two of the target countries: Ghana and Senegal) by shifting the monsoon rains to their correct location, Figure 11 examines the temporal evolution of the monsoon onset and cessation. It is recalled that the methodology is unable to explicitly bias correct in time, and this is confirmed, with the monsoon advance and decay very similar using both bias correction methodologies. However, it is seen that the intensity and location of the monsoon events are improved. In particular, one problem of the standard point-wise bias correction is clear when examining the spurious anomalies that occur between 13 and 18°N in April–May of both years (2008 and 2009) in the dry season, when no rain in the climate is expected (as confirmed by the GPCP panel of Figure 11). If the model (hindcast) climate suffers from some spurious events in isolated years, the standard correction method can even produce negative rainfall, which is usually simply corrected by clipping to produce a positive-definitive final field. The mapping method avoids this and retains a correct mean field value by construction.
3.3. Precipitation forecast assessment
The impact of the mapping bias correction is assessed at a range of forecast lead times. One of the greatest challenges in precipitation forecasting is the correct location of a given event. The mapping technique has been demonstrated to improve both the forecast precipitation amount and the location of events. A quantitative method to assess this is the intensity-scale verification method described by Casati etal. (2004). To apply the technique the pentad fields at lead times of one through to five pentads for both the standard and MEOF-corrected precipitation fields, in addition to the GPCP observations, are interpolated to equivalent increasing resolution, starting from the common 1° grid. Following Casati etal. (2004) the precipitation of a selected macro area is first interpolated over a square grid of dimensions 2n with n = 7, then successive grid aggregations are performed by averaging over grids of dimensions 2n with decreasing n. For every aggregation scale, the number of pixels which exceed a particular accumulation threshold (e.g. 1 mm as a pentad average over a 24 h period) are considered in turn. This is accomplished for both the GPCP data and the forecast fields. For every aggregation scale, the corrected forecast field pixel is compared to its equivalent GPCP pixel to construct a contingency table as a function of aggregation scale and precipitation threshold. This is then used to calculate dichotomous scores, namely the fraction of observed and/or forecast events that were correctly predicted. The threat score measures the forecast accuracy and has a range between 0 and 1, with 0 indicating no skill.
The threat skill score contouring for Sahel region A (Figure 12) shows that the skill generally increases with aggregation area for both the standard and MEOF bias, as expected. As the aggregation scale increases (eventually reaching the whole macro region domain when n = 0), the corrected forecast and observations will become more alike because the spatial errors in the forecast such as the misplaced monsoon rain belt in western Africa will have less significance. This apparent improvement is associated with a loss of resolution rather than with a real gain in forecast skill. The skill also reduces for increasing rainfall thresholds that become increasingly rare, also as expected. While the standard method shows positive skill (upper panels), the MEOF is superior, especially at lower rainfall thresholds. Skill levels in macro region E are much lower (not shown) but also show an improvement when the MEOF correction is applied.
3.4. Case study
Figure 8(D) shows that JJA 2009 was an overall dry year when compared to the 1991–2008 period, with the exception of a localized region centred in Ghana (centre in 3°W–6°N). During June–July 2009 Ghana experienced serious localized flooding after several rivers, including the Volta, broke their banks, causing destruction of houses, bridges, roads and crops.
Figure 13 depicts the observed climatological variability in precipitation for the region as the minimum, average (white line in Figure 13) and maximum precipitation recorded in the 18 years specified by the hindcast. Isolated precipitation events in 2009 (black line in Figure 13) exceed the average climatological precipitation, but for most of the JJA period the GPCP-observed precipitation was below average. The flooding in Ghana was localized and there is indications that changes in land surface cover and urban hydrological planning have exacerbated flooding impacts in the region over time (Douglas et al., 2008; Yengoh et al., 2010). The EPS monthly prediction 10 days (pentad 2) ahead is also reported for both the standard (dark grey) and the mapped calibrated forecast (light grey). The mapped calibration improves at this lead time, and also at longer lead times (not shown). In particular, the mapping technique is able to better capture the precipitation peak which occurred in pentad 38.
3.5. Ensemble spread
One characteristic of the mapped correction approach stands out clearly in Figure 13, namely that that the spread of the ensemble is reduced by the method, while the EPS monthly spread spans the climate variability as it is tuned to do. This is a natural result from the method since it is recalled that a reduction in predictive skill results in weaker mapped modes. The quantification of this effect is shown as a function of lead time is given in Figure 14, with spread reduced by over a third across lead times to five pentads. Steps could be taken to correct this, possibly through post-processing dressing methods (Roulston and Smith, 2003) or by retuning the magnitude of initial condition perturbations to compensate. It should also be pointed out that the spread evolution retains it shape of growing spread saturating after approximately 20 days due to the choice of deriving the MEOFs from the initial 7 days lead time range and applying this to the entire monthly lead times. Had the method instead derived MEOFs at each lead time to correct that specific lead time, the spread would collapse at long lead times as predictive skill is lost. On the one hand, this would appear to indicate an over-confident model; on the other hand, ensemble spread around an unskilful forecast essentially contains no useful information. Addressing this issue is left to future work, but it appears that the ratio of the ensemble spread between mapped and raw forecasts in each region could provide useful information for marking forecast skill at that particular range of use to the forecast end user.
Operational forecasting system errors imply that bias correction of forecast precipitation is necessary, especially for use in sectoral applications. Several methods have been previously suggested, which are either applied point-wise or rely on spatially based methods. Spatial-based methods have the advantage that they can shift rainfall patterns to their correct locations, but have previously only been applied to correct monthly anomalies of leading modes of variability. Here a revised spatial-based method is suggested that projects observed rainfall on to forecast leading-order modes. The consequent mapped patterns of precipitation anomaly essentially express the observed precipitation modes for which the model has skill in predicting. The mode projection is accomplished using chains of pentad hindcasts in the day 1–7 range of the operational monthly ensemble prediction system of ECMWF, which routinely produces an 18-year hindcast ensemble for every forecast ensemble. These mapped patterns are then used to correct the forecast ensemble pentad precipitation out to a range of 25 days, in a so-called MEOF bias correction approach.
The MEOF method was tested for a number of climatologically homogeneous regions in Africa using a downscaled GPCP pentad precipitation product. The method is shown to improve maps of precipitation anomaly for the majority of these regions, in particular for western Africa, where the model monsoon is systematically latitudinally displaced. The MEOF improves on both a simple point-wise mean bias correction, and also a simple version of previous EOF-mode substitution methods. The method is able to import and correct sub-monthly time-scale (pentad) information – a critical requirement for dynamical impact models which often require information on daily to weekly time-scales, although an extensive validation of the precipitation on the pentad time-scale is left to future work. Another side benefit of the approach is that it filters for model skill. An unskilful model will result in zero mapped anomalies and the model forecast will simply be the observational climatology.
Despite the apparent advantages of the approach there are a number of caveats and issues that will require further research to address. As with previous spatially based correction techniques, the method is currently unable to correct anomalies in the temporal dimension, such as a systematically late or early monsoon onset, for example. The fact that the method automatically filters signals to reveal model deterministic skill also leads to a caveat on the method, since the mapped patterns have to be derived for short-range forecasts when there is deterministic predictability. Applying these patterns to correct greater lead times implicitly assumes that the nature of model biases grows in a linear sense, and does not permit nonlinear error growth that changes its spatial structure with time.
The above caveat becomes particularly severe if the intention is to use the method to concatenate two diverse forecast systems together. A good example is the ECMWF monthly EPS and seasonal forecasting systems, which rarely use the same version of the atmospheric model, and utilize different ocean initialization techniques and are therefore likely to have distinct and potentially contrasting error characteristics. It may be that for applications use the preferred approach would be to separately derive the MEOF at each forecast lead time, possibly for the ensemble mean to filter out the predictable signal, with the derived mapped patterns then applied to correct each individual ensemble member. Beyond the limit of prediction this method would then simply produce a climatological forecast. Such considerations will be incorporated in the next step of this work, which is to apply a variant of this approach to correct the newly released version 4 of the ECMWF operational seasonal forecast system, with the goal of producing a seamless forecast system for sectoral applications.
We are in debt to Frederic Vitart for assistance in the use of the EPS monthly forecasting system. This work is part of the ECMWF and ICTP activity funded by the FP-7 European project ‘Quantifying the impact of weather and climate on health in developing countries’ (QWeCI).
Appendix A. Rainfall resolution enhancement
The resolution of the 2.5° GPCPv1.0 pentad data is improved to 1.0° with the application of a simple empirical sharpening algorithm which utilizes the ratios of a spatially smoothed and original higher-resolution version of the same dataset. The lack of a higher-resolution contemporaneous product means that an interpolated 1° resolution short-range forecast ERA interim dataset (Dee et al., 2011) is used for this purpose (referred to as ℰ(1.0)). As models suffer from numerical diffusion effects at the near grid-scale the use of ERA interim implies a likely general underestimation of the precipitation variability on small scales resulting from the application of the technique (Lander and Hoskins, 1997). In contrast, the ERA interim forecasts can also produce isolated spurious intense rain events, which are controlled by empirical factors detailed below. The rescaling is applied at each specific time interval (in this application, pentad fields).
The ERA interim product ℰ(1.0) (Figure A1(c)) is aggregated to a 2.5° grid equivalent to that used by the GPCP v1.0 pentad and then bilinearly interpolated back to a 1.0° resolution. This interpolated product (ℰ′(1.0)) is, by construction, smoother than the original forecast field. The offset ‘sharpening’ ratio between the fields is defined at each grid point and each pentad time slice:
The empirical factor is introduced to scale back the impact of isolated extreme rain events and to avoid infinities when ℰ′(1.0) = 0, and is estimated as a function of the long-term mean rainfall:
The GPCP pentad data product is then bilinearly interpolated to a 1.0° (G′(1.0)) (Figure A1(b)) and scaled by a factor r to give the final 1.0° sharpened rain product as shown in Figure A1(d):
which is applied when r < 2. For larger values of r, a further empirical adjustment was required to prevent spurious events:
Figure A1 gives an example of the application of this sharpening algorithm, and shows that the method is able to add smaller-scale information in regions where the ERA predicts precipitation, but without adding spurious signals in regions where the model obviously suffers from biases. Quality control demonstrated no discrete transitions in the spatial mean and variance of precipitation at the bounds between the two GPCP products in the resulting dataset that spans January 1979 to August 2009 (although only the period 1991–1995 is used in this study). Although highly empirical, it is emphasized that the sharpening method is only providing a long-term rainfall dataset for evaluating the bias correction methodology, which should work equally well with future improved rainfall datasets under development.
Appendix B. The clustering algorithm
A clustering algorithm is implemented to define distinct subregions of Africa based on the seasonality of the rainfall. Since subseason variability is neglected, the continuous GPCP-2.2 monthly precipitation product for the period 1979–2010 is used in the algorithm. The regions are defined quantitatively using a k-means clustering algorithm based on the method of Michelangeli et al. (1995) and Straus etal. (2007). Optimal cluster partitions with number of clusters ranging from two to ten are computed independently (the algorithm is non-hierarchical). The clustering then uses an EOF analysis of precipitation to define regions in which the leading EOF PCs are correlated. EOFs are computed for all months together and the time series at each grid box is projected on to the first ten (orthonormal) PCs. As a result, each location is represented by a set of ten coordinates. The k-means method uses the coordinate-defined clusters of grid points with coherent time variability, according to Euclidean distance in the 10-degrees-of-freedom subspace. For each assigned number of clusters, the optimal partition is that which maximizes the variance explained by the cluster means (centroids).
The regions produced by the method (Figure B1) agree with the common qualitative assessment of the African rainfall climatology. Western Africa, for example, is divided between a uniform Sahelian zone, which undergoes a single monsoon between July and September, and the coastal zone, which undergoes two rainy seasons. Likewise, the highland regions of eastern Africa are distinct. Note that there is no constraint in the clustering method about geographical proximity, so the geographical boundaries of the cluster regions are defined only by the ‘closeness’ of the anomaly time series. Since the coordinates are dependent on the anomaly amplitude, very arid regions with negligible rainfall values are grouped into a single cluster.
Using the results of the clustering analysis as a guideline, eight latitude–longitude regular African regions have been drawn (from A to H) with boundaries given in Table B1. This is done in the interest of simplifying the operational implementation of the calibration.
Table B1. Geographical boundaries of the eight African regions depicted in Figure 16.
NS = [20°, 8°]
NS = [10°, 5°]
NS = [10°, −5°]
NS = [5°, −15°]
EW = [50°, −20°]
EW = [30°, −20°]
EW = [50°, 30°]
EW = [30°, 10°]
NS = [−10°, −30°]
NS = [−15°, −35°]
NS = [−10°, −30°]
NS = [−10°, −30°]
EW = [40°, 25°]
EW = [35°, 10°]
EW = [40°, 20°]
EW = [50°, 35°]
From October 2011 the extended monthly integrations have been conducted twice a week.