Hydrostreamer has been designed to work with very low data requirements consisting of, at minimum, a distributed runoff dataset as the source zones and an explicit river network as the target zones, both of which can be obtained from free and open repositories. Additional data in the form of a digital elevation model (DEM) or ancillary information on the target river network can be used to further improve the streamflow estimates.

Hydrostreamer is designed to take runoff in a raster time series format – a multi-layer raster where each layer corresponds to a specific time step – as a plug-and-play solution. Such datasets include, for instance, LORA (Linear Optimal Runoff Aggregate; Hobeichi et al., 2019) and GRUN (Ghiggi et al., 2019), both of which are optimised global runoff datasets at 0.5∘ resolution. Outputs of a large number of global hydrological and land-surface models can also be obtained from the ISIMIP archive (as is done in the case study described in Sect. 4; https://www.isimip.org/, last access: 5 May 2021) at the same 0.5∘ resolution. In order to use this type of data in Hydrostreamer, one needs to read it in the R session, and use the raster_to_HS() function to format the data for Hydrostreamer to use. Non-gridded data may also be used, and one potential source dataset is GRADES (Lin et al., 2019), which is used as a benchmark dataset in this study. While Hydrostreamer supports non-gridded input runoff via the function create_HS(), its use may not be trivial and depending on the source format may require additional preprocessing steps.

The river network required by Hydrostreamer can be obtained from various sources. Any network which is topologically clean can be used as a plug-and-play solution without any preprocessing needed when Step I (Figs. 1 and 2) is applied. A clean network refers to a network where connected lines share a node and lines are split at junctions so that connected line segments start or end at a shared node. With a clean network, the river_network() function is able to extract the topological relationships Hydrostreamer needs. Further, if the used river network product contains topological information on the next river segment, the cleanliness requirement can be relaxed. The case study presented in Sect. 4 uses the HydroSHEDS river network (Lehner and Grill, 2013), available from https://www.hydrosheds.org/ (last access: 5 May 2021), which is an example of a clean river network. The HydroSHEDS website provides additional river network datasets with a large amount of attribute information describing each segment: Global River Classification (Dallaire et al., 2019) and HydroATLAS (Linke et al., 2019). These two datasets are particularly useful in Hydrostreamer, because their attribute information can be (but is not required) used as ancillary variables in dasymetric mapping (DM; see the following Sect. 2.2).

Hydrostreamer further provides an optional auxiliary function create_river() which can be used to extract a river network and catchment areas from a DEM for each river segment. The function requires an external program, SAGA GIS (Conrad et al., 2015), to be installed and requires definition of a threshold for the size of the stream (the Strahler stream order) at which point river line extraction starts. The selection of the threshold should be guided by the resolution of the source zones as well as understanding of the hydrology within the basin. We recommend visual inspection of the extracted river network as well as the corresponding catchment areas. Catchment areas can also be delineated for each individual river segment from a flow direction raster using function delineate_basin(), if the flow direction information from which the river network is derived is available. For HydroSHEDS, HydroATLAS, and Global River Classification, the flow direction information can be obtained from the same repository and can be used with the delineate_basin() function. Further, when an applicable DEM or flow direction is not available, function river_voronoi() approximates catchment areas by building a Thiessen polygon (Voronoi diagram) network from the line segments of the river network (Karimipour et al., 2013). This is particularly useful for river networks derived from surveying or from satellite measurements.

Finally, in order to use the model averaging functionality (optional, Step III in Figs. 1b and 2) of Hydrostreamer, one needs to obtain a time series of discharge measurements at gauges corresponding to the catchment of interest. We recommend obtaining time series from the authorities of administrative area(s) where the catchment is located, but when this is not possible, one can search the Global Streamflow and Metadata archives (Do et al., 2018) or the Global Runoff Data Centre (https://www.bafg.de/GRDC/EN/Home/homepage_node.html, last access: 5 May 2021) for appropriate data. Hydrostreamer requires observation data in a standard table format with columns for date and station observations in units of m3 s-1.

The first step in the core Hydrostreamer workflow is the areal interpolation step, which is also the key focus of the software package. In this section we give a brief background for the areal interpolation methods and how they are implemented in Hydrostreamer. For a more thorough overview and applications of areal interpolation methods, we recommend Comber and Zeng (2019). The current implementation in Hydrostreamer assumes that the interpolation is constant and does not change through time.

The second step in a typical Hydrostreamer workflow involves routing runoff down a river network to estimate discharge. Two simple routing solutions, instantaneous routing and constant-velocity routing, as well as one more advanced routing solution, the Muskingum–Cunge method, are implemented in Hydrostreamer. More sophisticated schemes could be used by exporting interpolated runoff to other tools specialising in routing.

In instantaneous routing, discharge Q is the sum of runoff, in volume per time (e.g. m3 s-1), from all upstream catchments, as shown in Eq. (8): 8Qi,t=ROi,t+∑ROup,t. Here Qi,t is the discharge at river segment i at time step t, ROi,t is the runoff contribution from the catchment area of segment i at the same time step t, and ROup,t is the runoff from a river segment upstream of segment i, at time step t. Instantaneous flow is the simplest form of river routing and has the advantage that it is intuitive. However, it assumes that all runoff generated at a time step t will drain through the entire river network within that same time step. The applicability of this assumption is therefore limited to catchments where the time step length far exceeds the maximum river network length. One can evaluate the applicability of the instantaneous routing (which in fact takes one time step) using Eq. (9): 9M=LupVmax1s, where M is a dimensionless ratio between the time it takes for water to flow through the maximum upstream length of the river system Lup (in metres) at a maximum realistic average flow velocity Vmax (default 1 m s-1) during a time step of length s (in seconds). M can be interpreted so that, for example, when M=0.1, 10 % of the runoff generated at the most distant upstream location does not flow through the outlet within a single time step. The evaluation can be carried out using the function evaluate_instant_routing(). If M is found to be too large for the application, the constant flow velocity or Muskingum–Cunge option may be more appropriate.

The second option, constant velocity routing, assumes that water drains through the river network at a constant pace, which is the solution also adopted in routing tool HydroROUT (Lehner and Grill, 2013) and a number of global hydrological models (GHMs) (Telteu et al., 2021). The default flow velocity of 1 ms-1 is adopted in HydroROUT and LPJmL (Telteu et al., 2021). Assuming that the generation of runoff is uniformly distributed within a time step, we can think of a block of runoff moving downstream. Given a distance S covered in time τ at velocity V, the block of runoff typically finds itself spanning several river segments (depending on the segment size and time step length). The discharge in a particular segment comes from two blocks of runoff (two consecutive time steps) from each upstream segment. We calculate the number of whole time steps Ti,upwhole elapsed for the runoff to cover the distance Di,up and the fractions Ti,up- that will come from that time step and Ti,up+ from the following time step. See Appendix B, Fig. B1. The process is described in Eqs. (10)–(14). 10S=Vτ11Ti,upwhole=floorDi,upS12Ti,up+=fracDi,upS13Ti,up-=1-fracDi,upS14Qi,t=∑upupstreamROup,Ti,upwhole×Ti,up-+ROup,Ti,upwhole+1×Ti,up+ The third routing option implemented in Hydrostreamer is the Muskingum–Cunge routing algorithm (Cunge, 1969; Ponce, 2014). Muskingum–Cunge is a modified version of the original Muskingum routing method (Chow, 1959) where routing parameters k and x are derived from hydraulic data and do not require observation data to calibrate against. Full derivation and explanation of the Muskingum–Cunge routing can be found in Ponce (2014). The algorithm requires extensive user input in the form of river cross sections (i.e. shape, channel width, flow depth), river bed roughness (Manning's roughness coefficient), and river bed slope, which are commonly available only for certain locations. Consistent with the desire to minimise data requirements, the Hydrostreamer implementation of Muskingum–Cunge provides defaults and therefore requires the user only to provide main parameters: (1) Manning's roughness coefficient (for readers unfamiliar with Manning's coefficient, Arcement and Schneider (1989) provide an extensive guide on its estimation), (2) bed slope (precomputed bed slopes are available from the HydroATLAS (Linke et al., 2019) database which can be directly used in Hydrostreamer), and (3) channel width. An estimate of the channel width can be computed using a power-law relationship (Leopold and Maddock, 1953): 15W=aQrefb, where a and b are parameters to be estimated and Qref is the reference discharge, and W is the channel width. Hydrostreamer has a built-in estimates for a and b from Moody and Troutman (2002) and Allen et al. (1994). Qref is estimated from the inflowing discharge time series for each river segment using Eq. (16), 16Qref=min⁡Qin+max⁡Qin-min⁡Qin2, where Qin is the time series of discharge inflowing to the river segment. Alternatively, the user can provide their own parameters for each a, b, and Qref. Vatankhah and Easa (2013) derived a relationship between discharge Q and flow area based on channel width. Their approach is used here to estimate flow depth assuming a rectangular river cross section.

All three routing solutions support setting boundary conditions which modify RO at specified river segments. The boundary conditions, termed control time series, include addition, subtraction, multiplication, and setting RO to a user-specified value. This allows inclusion of, for example, controlled inflows, water extraction, simple fractional environmental flow considerations, and specified dam releases.

For beginners, use of coarse timescales and instantaneous routing is recommended, subject to evaluation of performance. This will avoid the difficulties in estimating the parameters required for more complex algorithms. If additional information or expertise is available, the more complex routing algorithms may be selected to further improve performance or allow discharge estimation at shorter timescales. We provide a comparison of the three routing methods in Appendix B, applied to the case study presented in Sect. 4.

Note that our case study example and Appendix B provide validation for the routing with monthly time series only. We therefore recommend caution and careful review of Hydrostreamer outputs in applications using sub-monthly time series, until proper validation for the method is published.

The third step in the typical Hydrostreamer workflow is model averaging. Model averaging using varying sizes of ensembles is a common approach to data assimilation in hydrological sciences (see e.g. Arsenault et al., 2015; Arsenault and Brissette, 2016; Gosling et al., 2010; Skøien et al., 2016; Velázquez et al., 2011; Zaherpour et al., 2019). In model averaging, an ensemble of time series is combined into a multi-model combination (MMC), commonly using a weighted approach. Provided that streamflow records are available, Hydrostreamer provides facilities for at-location model averaging, as well as regionalised model averaging. Table 1 provides an overview of all the implemented model averaging methods in Hydrostreamer along with some of their properties. The methods are based on minimising error between observations and the weighted ensemble average.

In a regionalisation experiment Arsenault and Brissette (2016) explore MMC weights using a small three-member ensemble, finding that MMCs are nearly always outperformed by the best-performing individual model member at regionalised locations and that a simple ensemble mean (each ensemble member receiving equal weights) performs reasonably well across all locations. They further conclude that regionalising model averaging weights is not a reasonable task; however their conclusions are based on a limited analysis of three methods which all allow negative weights. We argue that their conclusion applies to methods that allow for negative weights due to the fact that relationships between hydrological time series at different locations likely differ considerably. The lack of negative weights, we assume, is one reason that the ensemble mean is able to outperform their selected model averaging methods.

To help extend model averaging to ungauged basins (that is, any river segment which does not contain a monitoring station), we must find a way to regionalise model averaging weights. In Hydrostreamer, we consider those methods which do not allow negative weights as fit for regionalisation. This ensures that, while the performance of regionalised MMC weights may be worse than the best individual ensemble member (which we cannot know in an ungauged basin), the output hydrological time series is ensured to be positive.

In practise, in Hydrostreamer regionalisation of MMC weights is done so that each river segment receives model averaging weights from the nearest downstream gauging station, and if there are no downstream gauging stations, the ensemble mean is used instead.

The functions used in a typical workflow and data requirements in Hydrostreamer are shown in the flow diagram in Fig. 2. Due to the large number of optional arguments and functionality, the figure only shows minimum required inputs. Complete up-to-date documentation and default values can be found on the model documentation website at https://mkkallio.github.io/hydrostreamer/, last access: 5 May 2021, or using the internal R help() function. The documentation site also provides a vignette which gives further information on the workflow of Hydrostreamer in the form of a practical example.

The workflow starts from pre-processing data to a format with which Hydrostreamer is able to work. Hydrostreamer supports providing input hydrological variables either as raster or vector formats, each with dedicated functions, raster_to_HS() and create_HS(). Each raster cell or each polygon in the input data is considered a source zone. For the interpolation step, a network of target river lines and/or target zones need to be provided in addition to the HS object output from the pre-processing steps. The routing step requires the output from the interpolation step, the routing method (instantaneous, or constant velocity), and parameters for that routing method. Finally, model averaging can be performed (1) at the point location of the monitoring station, giving optimal time series for the monitoring station locations only; (2) regionally, where MMCs are provided for each river segment, optimised at the nearest downstream monitoring station, and (3) with user-provided combination weights.

Once runoff data and the river network have been read into R, the full workflow can be achieved with only three to five chained commands, depending on whether the user wishes to apply model averaging. Hydrostreamer provides some additional functionality which supports the optional components in Fig. 2. These and further supporting functions are given in Table 2.

The data structures used in Hydrostreamer are compatible with the packages from the tidyverse (Wickham et al., 2019) suite of packages, including chaining of commands with the pipe operator commonly associated with the tidyverse workflow. For spatial representation, R makes use of the simple features implementation for R (sf; Pebesma, 2018).

Hydrostreamer objects have class HS, which are essentially standard R data frames and which can be modified with any function that works on standard data frames. In a HS data.frame object, each row is either a source or target zone which is described by variables and time series in list columns. Each HS object contains at least a unique ID (riverID for target zones, or zoneID for source zones) and a time series column. Depending on the usage, the object can also have a number of other columns with variables such as topological network information (previous and next river segments) for the routing algorithms, names of monitoring stations, and various time series (e.g. runoff, discharge, control, observation time series) and variables which are used in the interpolation step. As the HS object is a data frame, additional columns with information the user wants to include can be added. For all the functions which add or modify HS specific columns, see Appendix A, Fig. A1.

Each time series is stored in a list column, where each element of the list is a data frame giving the time series for the target or source zone in question. Each of these tables is structured so that each row is a time step, for which the date is given by a column named “Date”. In runoff or discharge time series, each additional column is the estimate of an ensemble member. For control and observation time series, the table may contain only one column in addition to Date.

The river network structure follows a hierarchical node–link network, where each river segment is represented by a node which has links to previous and next river segments. This data model is simple and intuitive. Demir and Szczepanek (2017) find that this type of river network representation is generally more performant in different types of queries than alternative network representations. The adjacency information is stored in HS as list columns NEXT and PREVIOUS, which store all the river segment IDs which flow into the segment in question and which segments are topologically immediately downstream from it.

The experiments build on three main data sources and two distributed global discharge estimates as benchmarks. First, we used the aforementioned ISIMIP 2a data archive (accessed in August 2018). We obtained all total runoff (variable “mrro” in the ISIMIP archive) and discharge (“dis”) time series, modelled with the variable social forcing (“varsoc”) scenario, available in the archive in monthly time steps. When monthly data products were not available, we used the daily product and aggregated it to monthly averages. The obtained datasets are summarised in Table 3. We did not use the products forced with the WATCH dataset, as it only extends until the end of 2001 and would have meant discarding some of our observation stations with records only after 2001. In total, we obtained products from 10 global hydrological models and land-surface models from the archive (from now on, both referred to as GHM). From the total of 24 runoff products, we only use those which also provided discharge output (n=15). The ISIMIP outputs are delivered with a spatial resolution of 0.5∘ (approximately 55 km at the Equator).

The runoff time series were downscaled to the HydroSHEDS 30 arcsec resolution river network for Asia (Lehner and Grill, 2013). The total size of the river network within the 3S was 2115 river segments with a median length of 5055 m. To accommodate the evaluation of areal interpolation techniques, we also obtained the HydroSHEDS 30 arcsec resolution flow direction raster from which the river network was derived. We likewise obtained the HydroSHEDS DEM in order to derive an ancillary variable (see Sect. 4.3).

Similarly to the runoff and discharge GHM time series, the daily observed streamflow for the 10 MRC hydrological stations was aggregated to monthly by taking the mean monthly streamflow. For comparison, we additionally use two recent global streamflow products: GRADES (Global Reach-Level A Priori Discharge Estimates for SWOT; Lin et al., 2019) and GLOFAS reanalysis streamflow dataset (GLObal Flood Awareness System; Alfieri et al., 2020). GRADES is provided in zones similar to the HydroSHEDS river network product, albeit derived from a higher-resolution DEM. GLOFAS comes as a global grid with 0.1∘ resolution (∼11 km at the Equator). Both datasets are provided with a daily time step and were also aggregated to monthly means.

Figure 1a shows the 3S basin and the used HydroSHEDS dataset overlaid on the 0.5∘ model grid used in the ISIMIP data. The figure additionally shows the locations of the monitoring stations and the basins they drain from.

We assessed the performance of the Hydrostreamer streamflow predictions and all benchmark datasets to the observation time series using commonly used metrics:

root-mean-square error (RMSE), a commonly used model performance metric (in principle, all model averaging techniques in Hydrostreamer minimise RMSE);

We conducted three experiments building upon one another. In the first experiment, we performed downscaling of the total runoff inputs using AWI (for DEM-delineated catchment areas, as well as Voronoi diagram-based delineation), DM (DEM-derived catchments with an ancillary dasymetric variable), and area-to-line interpolation (without dasymetric variable). As a dasymetric variable, we used a recently developed topographic index DUNE (Dissipation along unit length; Loritz et al., 2019) that is capable of distinguishing different runoff formation regimes and is computed from the HydroSHEDS DEM. We used instantaneous routing for this experiment, because the flow timing error M (Eq. 9) was found to be insignificant considering other potential sources of error (0.028 for the most upstream location and 0.004 for the 3S basin on average). The most representative river segment was selected from the HydroSHEDS network for each monitoring station location based on comparison to the location on an actual river network. For assessment of global model performance, we likewise selected the grid cell which best represents the monitoring station in the low-resolution DDM30 river network (Döll and Lehner, 2002) used in the ISIMIP framework. We expected that the downscaled Hydrostreamer time series should perform at least as well as, or better than, the discharge time series from the GHMs due to better representation of the drainage basins associated with each monitoring station.

In the second experiment, we performed model averaging at the monitoring stations and assessed how the uncertainty related to the model averaging weights affects performance of the optimised MMC. In particular, we use the constrained least squares (CLS) technique (Diks and Vrugt, 2010). CLS is constrained to positive weights only, and the sum of weights must equal 1 – this means that the MMC time series will never have higher or lower discharge than any individual ensemble member. The combinations are performed multiple times with different training periods to assess the uncertainty in the model averaging weights. We used three sampling strategies (each available in Hydrostreamer) for the selection of the training period: (1) random selection of 50 % of all the time steps in the observation record (performed 100 times for all stations), (2) random selection of 50 % of calendar years in the observation record (performed 50 times for all stations), and (3) training combinations for each calendar month separately, with random 50 % of available time steps for each month (performed 50 times for all stations). The time steps in the observation record not included in the training period were used for model evaluation. We also created 10 000 random combinations with a multi-stage sampling technique, first randomising the n number of models to include, second picking n random models, and third randomising positive weights among the randomised model selection using uniform distribution, with weights summing to unity in order to have comparable constraint in the randomisation as we have in CLS.

Finally, in the third experiment, we regionalise the optimised MMC weights at each monitoring station and evaluate their performance at the other monitoring stations on the same river.

The four tested downscaling methods (see Sect. 4.3) show a negligible difference across all used performance metrics, when averaged over monitoring stations (see Appendix A, Table A1). Station-wise, there are very small differences in all stations except Sesan Upstream-East, where the largest differences in performance between downscaling methods are up to 0.40 for NSE and 0.10 for KGE, both when downscaling H08 forced with PGFv2. Across the entire 3S Basin, the difference between downscaling methods becomes smaller as the basin size increases. This is expected: with the chosen instantaneous routing method being the only difference in discharge that comes from the basin boundary, since all runoff from the middle of the basin instantly flows through the station. As the basin size increases, the proportion of runoff contribution from the catchments at the basin boundary becomes increasingly small and thus is shown in decreasing difference in performance metrics. This is in line with Cunha et al. (2012), who find that in ensemble modelling uncertainty gets smaller with increasing basin size. Because the differences between the downscaling are so small apart from the Sesan Upstream-East station, we opt to continue the analysis on the simplest downscaling method: area-to-line interpolation. It should be noted, however, that Virkki (2019) showed that area-to-line interpolation causes larger uncertainties in the reach level than approaches using reach-specific catchment areas. Furthermore, Kallio et al. (2019) found in a study that included 126 catchments with natural flow regime that DM using DUNE as an ancillary variable does improve the performance of downscaling in topographically varying terrain.

The performance of individual downscaled GHMs varies much more than the performance between tested downscaling methods (see Appendix A, Tables A2–A4). Compared to the discharge output from GHMs, the downscaled ones are similar or better in their performance, as seen in Fig. 3. Moriasi et al. (2015) recommend that for watershed scale models at monthly temporal resolution acceptable model performance is R2>0.60, NSE >0.50, and PBIAS ≤±15 %. Using these criteria, we find that they are fulfilled in 47 % and 55 % (R2), 22 % and 43 % (NSE), and 27 % and 53 % (PBIAS) of cases in downscaled GHMs and GHMs, respectively (see Table 4). Volume-wise the downscaled estimates fare better with mean (across all 15 GHM–climate forcing pairs, and all monitoring stations) PBIAS of -0.3 % against GHM's 11.2 %. The difference and direction in performance are visualised in Fig. 4, confirming our expectation that downscaling does improve the performance of GHM outputs.

Comparing to the openly available benchmark products GRADES and GLOFAS, the downscaled GHMs fare reasonably well. While the individual ensemble members have large spread in their performance, often being worse than either of the benchmarks, the ensemble mean provides consistent good performance with KGE >0.5 at all stations except Srepok Downstream (Fig. 3). The ensemble mean is considerably better performing than GRADES, at all stations but Srepok Downstream and Midstream. GLOFAS performs better than the ensemble mean at 6 of the 10 stations.

In the second experiment we performed model averaging on the 15-member ensemble with three combination strategies. For all stations, the time series and annual combination strategies produce very similar distribution in performance (Fig. 3, distributions 2T and 2A). Monthly combination (Fig. 3, distribution 2M) strategy can, however, produce better performance at point locations for PBIAS, which is lower than the threshold (PBIAS <15 %) in 91 % of all combinations (Table 4). However, when taking an ensemble mean from the 50 monthly combinations for each station, PBIAS threshold is satisfied in only half of the stations – and on the other hand the ensemble mean from time series or annual combinations performs considerably better. Comparing against the benchmarks, GRADES and GLOFAS, we see that all of the combination strategies can produce better performance (Fig. 3), with only a small minority of optimised MMC combinations showing worse performance for KGE.

When looking at individual ensemble members in Fig. 5, and their performance with random MMC weights, we see that the models have highly variable performance at different stations. The performance of individual ensemble members varies between stations (Fig. 5) and between indicators (Appendix A, Table A2). MATSIRO in particular shows high sensitivity to climate forcing; the performance when forced with PGFv2 has the largest mean RMSE, but forced with GSWP3 results in the smallest RMSE. In general, models forced with PGFv2 perform considerably worse at the 3S than when forced with WFDEI or GSWP3 (see Appendix A, Table A3).

The ensemble mean is robust throughout the basin; it is among the best performing individual members for most of the stations, and for Srepok Upstream it performs better than any single ensemble member. We can further infer from Fig. 5 that the skill of the ensemble mean stabilises at around 10–12 random ensemble members.

In the third experiment we tested the applicability of regionalising weights derived at one station to the other stations in the basin. We used weights derived from the stations with direct upstream or downstream linkage – Sekong, Sesan, and Srepok stations separately (refer to Fig. 1a). Our results suggest that the performance of regionalised model averaging weights is variable, as seen in Fig. 3. Regionalised time series and annual model averaging strategies produce commonly higher performance than the ensemble mean or the distribution of the random ensemble combinations, and in some stations can produce similar performance to the optimum for that station. This is desirable, as regionalisation of MMC weights would make no sense if the simple ensemble mean would perform better. We explored the distribution of weights at different stations and found that Sekong and Srepok stations produce an entirely different weighting of ensemble members. Sesan stations are somewhere between, with Sesan Downstream showing similar MMC weights to Sekong stations, and Sesan Upstream-East similar to Srepok River (the distribution of weights not shown). Sesan Midstream and Upstream-North appear unique in their sets of MMC weights. This is clearly seen in the distribution of performance of the Sesan stations in Fig. 3; there are clear clusters of performance from weights from the other stations, whereas Sekong and Srepok stations give a more uniform distribution in regionalised MMC performance.

The distribution of performance of the regionalised weights from monthly combinations is a clear case of overfitting – the distribution of performance (3M – the rightmost distribution in each facet in Fig. 3) is very large and is similar to or worse than the ensemble mean (Table 4). This is natural, since in the monthly weighting strategy we develop a set of 12 weights (one set for each month of the year) instead of a single set with time series and annual combination strategies. This suggests that monthly combinations are more useful for point optimisations but are not advantageous for use cases requiring regionalisation.