Integrated assessment models (IAMs) are scientific modelling tools that integrate knowledge from different academic disciplines with the aim of exploring and informing policy decisions (Clarke et al., 2014; Rogelj et al., 2018a; Weyant, 2017). They are widely used in climate change research to combine insights from energy, economy, agricultural, and natural sciences, with the aim of creating scenarios that explore how societal decisions can affect projected greenhouse gases and other emissions, as well as their related climate outcomes (Clarke et al., 2014; Huppmann et al., 2018; Riahi et al., 2017; Rogelj et al., 2018b).

However, IAMs do not always exhaustively represent all possible processes or sources of climate-relevant emissions. Thus, many IAM scenarios lack projections for some climate forcers, be it specific greenhouse gas emissions or aerosol precursors. A complete set of these climate forcers is required to accurately estimate the overall climatic effects of a given scenario (Meinshausen et al., 2011; Smith et al., 2018), as a large number of supposedly minor emissions may collectively exert a significant radiative forcing (Meinshausen et al., 2017; O'Neill et al., 2016).

Scenarios that only report a limited set of greenhouse gases or climate forcers must thus be complemented by estimated evolutions of missing emissions derived without further economic analysis. We term this estimation “infilling”. If no infilling is attempted, the unevaluated emissions would effectively be considered zero, which would clearly create systematic biases and potential artefacts in the projected temperatures. Depending on the radiative forcing of the species in question, this bias may be positive or negative, so infilling with zeros would not necessarily be a conservative choice. Most earlier studies overcame this problem in one of two ways: with expert-based ad hoc decisions on how to adequately fill in missing species (Schaeffer et al., 2015) or by assuming that a pathway will occur at the same quantile for each set of emissions in a particular year, although the quantile can vary over time (Gütschow et al., 2018; Meinshausen et al., 2006; Nabel et al., 2011). However, the former clearly does not scale easily to larger databases (because making ad hoc decisions for a thousand scenarios requires a significant time input), and the latter approach, termed the “equal quantile walk” (EQW), ignores trade-offs and specific relationships between emission species resulting from how competing technologies, behaviours, and industrial practices result in different emissions. A few alternative approaches have been used recently: for instance, using the pathway with the smallest mean squared distance over time was used in Robiou du Pont and Meinshausen (2018). This works well for large databases containing similar paths but is less reliable for smaller databases or for paths with an unusual behaviour over time. A more sophisticated “generalised quantile walk” technique can capture the effect of trade-offs and was recently introduced in Sect. 3.8.1 in Teske et al. (2019), involving quantile regression between a lead variable (fossil CO2 emissions) and other gases for every individual year. Unfortunately, the implementation there did not consistently guarantee that higher quantiles resulted in higher emissions, and has not been followed up with any peer-reviewed work that does so. A tool for infilling was provided by Rogelj et al. (2014) using a cubic spline between specific points in a small database; however, this type of infiller behaves chaotically when applied to large databases incorporating many different models. It was also coded in Excel, limiting the ease of open-source development.

Here we present a new toolbox of methods to address these recurring infilling challenges in the climatic assessment of socioeconomic emissions scenarios. The toolbox introduces new approaches as well as building on and combining previous approaches. The code base is a significant improvement compared to existing options in terms of flexibility, applicability, reproducibility, and versatility.

Silicone is a Python package designed to enable users to expand scenario projections of a limited set of climate forcers to a broader set required for a sensible climate assessment. In essence, its methods are grounded in a comparison of the co-evolution of anthropogenic emissions in scenarios that are readily available in the literature (Huppmann et al., 2018; Riahi et al., 2017; Rogelj et al., 2018b). Silicone aims to provide IAM teams that do not represent all individual climate forcers with robust methods to complement their model output and facilitate a climatic assessment of their work. Furthermore, Silicone also aims to provide geoscience researchers with a tool to easily develop stylised, yet internally consistent future emission pathways of the most important climate forcers. It can also estimate or calculate missing emissions from particular sectors. Notebooks describing how to use these tools are available on the accompanying GitHub repository (Lamboll et al., 2020b) and the formal documentation is available in Lamboll et al. (2020a). Additional examples of using Silicone for the specific situations outlined below are included in a separate GitHub repository (Lamboll, 2020). The package is open-source and intended to allow groups to write their own infilling methods if desired. Users and collaborators are encouraged to add any such developments to the code base via GitHub.

Silicone is compatible with a suite of Python tools that make up the IAM climate assessment pipeline developed under the umbrella of the Integrated Assessment Modelling Consortium (IAMC). Compatibility with these tools allows us to load, manipulate, and save files using a common file format. The pipeline is based around the pyam package (Gidden and Huppmann, 2019), specifically its IamDataFrame class, which Silicone makes extensive use of; pyam data frames easily convert to and from widely used pandas data frames, which pyam and Silicone also use internally (McKinney, 2011). The pipeline also includes tools to harmonise (i.e. correct projection made in the past to match now-known emissions) (called aneris; Gidden et al., 2018) before infilling and to pass the complete projections to climate simulators. The estimation of climatic impact is performed by OpenSCM (Nicholls et al., 2020), which is compatible with the data structure of the pipeline. This pipeline is being developed in support of the IAM community and the IAM scenario assessment for the forthcoming Sixth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC AR6) in particular.

This paper is structured as follows: Sect. 2 presents an overview of the different infiller methods, then goes through the infiller techniques in precise and mathematical detail. In Sect. 3, we present our analysis of emissions projections in the SR1.5 database. This includes correlation statistics for the database and how well Silicone reproduces one entry in the database given the other entries. We use this to draw conclusions on the implications for using Silicone on unknown data. In Sect. 4, we present three examples of using Silicone for infilling a pathway with limited information, splitting up an aggregate basket of emissions, and infilling stylised emissions trajectories. We end with a summary of our paper.

Notebooks presenting the benefits and risks of each cruncher type can be found in the Silicone GitHub (Lamboll et al., 2020b) and may be useful to have as examples when analysing the work below, as well as demonstrating how to use them.

There are two main classes of infillers: those based on the ratios between two emission pathways and those based on the absolute emission values in the infiller database. If the results are to be harmonised, then harmonising both the infiller and target data before infilling is required for improved consistency (otherwise, infilling depends on outdated data). Absolute value infilling techniques preserve harmonisation; however, ratio-based approaches do not necessarily do this and may need harmonisation again afterwards.

The ratio-based approaches are better for cases in which the lead values to be infilled are outside the range in the infiller database and we expect the emissions to scale with each other, for instance if we are infilling one incomplete combustion product based on another or splitting up aggregated emissions into their components. However, care needs to be taken when infilling emissions that are non-negative using a lead value that may be of any sign, for example CO2 emissions. In that case, the ratio method might produce values for the target emissions that are unsupported by any available evidence. Singular behaviour may also be encountered when the lead data are close to zero in the infiller database. The different crunchers present different ways to estimate the ratio to use.

The absolute-value-based techniques infill with values derived from the absolute data found in the infiller database or linear combinations of them. This means that they will always return values within the range spanned by the infiller database. This is most appropriate for processes whereby we have a greater number of cases, preferably with both larger and smaller lead emissions in the infiller database or when we expect the follower emissions to be strongly bounded rather than increasing in line with other variables. They may be considered more stable and more conservative. The quantile-rolling-windows (QRW) cruncher can be used in either ratio or absolute (non-ratio) mode, the absolute mode being the default.

As one final detail, we discuss the data model which is assumed by Silicone. Silicone is built around the pyam package (Gidden and Huppmann, 2019). As a result, it assumes that all input data are provided in a particular structure. The structure includes the model which created the time series, the scenario with which the time series is associated (e.g. a high BECS 1.5∘ scenario), the region the emissions occurs in, and the unit of the data (full details available at https://pyam-iamc.readthedocs.io/en/stable/data.html, last access: 2 November 2020). Accordingly, Silicone is able to work on specific subsets of models (e.g. only the MESSAGE model) or subsets of scenarios (e.g. all SSP1-like scenarios). We therefore follow the pyam convention and refer to a “model–scenario combination” to mean a single projected world that in some contexts might be called a “scenario”.

Pyam data frames assign values to variables as a function of different models, scenarios, regions, and times. All methods work on databases with only a single region at a time, although the region can be different between the infiller and target databases.

Note that all of the methods listed above are purely statistical in nature: if the scenarios in the infiller database are fundamentally different from those in the target database, different relationships are likely and the validity of the results is poor. The adequate use of Silicone requires users to select an infiller database most appropriate for each respective application. Using Silicone with an infiller database that has itself been infilled may distort the model democracy of the results. Note also that in version 1.0.0 of Silicone, all methods take only a single lead value, although forthcoming work will add the capacity to use multiple lead values to some crunchers. This will improve the ability to resolve more complex relationships, since it is possible for very different worlds to have similar emission trends in one emission without being similar in other emissions.

The infilling method is important. However, equally important is the choice of lead variable. The best choice is where there is a causal link between the lead and follower variable, particularly if there is a clear understanding of the implications of this link for the relative behaviour of the two variables; for instance, black carbon and carbon monoxide are both produced by incomplete combustion. In most cases, there is no such certainty, and the best choice is then to find the lead variable with the best predictive power. We estimate this by the Spearman's rank correlation coefficient, a measurement of the monotonicity of the relationship between the two variables. In cases in which this value is low, we anticipate the need for higher effort to select relevant cases from the infilling database. We use the data from the IPCC Special Report on Global Warming of 1.5 ∘C (Huppmann et al., 2018) as our database of scenarios and compare the correlations between the different variables. The Silicone package has a function in the statistics section called calc_all_emissions_correlations, which will produce tables of both the Spearman (rank) and Pearson correlation coefficients, calculated separately for each year requested, and the time-averaged magnitude of the correlations. Since there is no reason to expect the relationships between variables to be linear, we will focus on the rank correlation in this analysis. We also plotted the relationships between CO2 and all other variables (using the plotting function in the Silicone examples on GitHub) to check that there were no obvious cases of a non-monotonic relationship. All the crunchers work just as well with negative trends as with positive, so the sign of the correlations is not relevant for considering goodness of fit. Using this tool, we can calculate the decadal-averaged magnitude of the rank correlation coefficient, found in Table 3. We also calculate the variation of this value with time, and in cases in which this exceeds 0.03 (chosen to highlight only extreme cases) we write the values with asterisks. This is to indicate cases in which more care needs to be taken to ensure that values are representative for the times of interest.

The immediate observation from the study of absolute rank correlations is that there is no clear, overall best infiller gas. CH4 has a slightly higher average than other emissions and is reported by most models. CO2 is reported by all models and has the second-highest correlation; however, this is somewhat inflated by having two of its constituents listed separately (agriculture, forestry, and other land use – AFOLU – and energy and industrial processes; a similar concern can be raised about F gases). Generally, CO2 and CH4 are therefore the best choices for a default lead variable. However, there are some specific cases in which the correlations are low, and much better choices could be made.

There is a cluster of emissions species, specifically black carbon, organic carbon, and carbon monoxide, that correlate well with each other but less well with other emission pathways. Physically, these relate to incomplete burning and are best infilled using each other. The F gases, SF6, hydrofluorocarbons (HFCs), and perfluorinated compounds (PFCs) also primarily relate to each other. Many models report F-gas emissions as a basket. Infilling these should best be done by splitting the F-gas basket into its constituents. Otherwise, the default infillers, CO2 and CH4, should do reasonably.

The choice of cruncher to use in different situations will depend on the expectations about the specific emissions in question. However, in cases in which there are no clear expectations, it is good to have a default. In this section we assess to what degree the cruncher reproduces the follower data from one model and scenario given the lead data from that case and all data from all the other model–scenario combinations in the SR1.5 database. We try this with both CH4 and CO2 as our lead variables. We use the crunchers that are designed for use on complete datasets with only default settings: QRW (default settings mean in absolute mode and for the 0.5 quantile), RMS closest, EQW, time-dependent ratio, and linear interpolation. “Interpolate selected model” behaves identically to linear interpolation with default settings and is not treated separately here. We perform the infilling for each model–scenario combination for each decade from 2020 to 2100 and report the root mean squared difference between the original value and the infilled value, normalised by the standard deviation in the follower value in the database at that time (σ), i.e. Ef,inf-Ef,actσ2idecade, with the subscript text “inf” indicating that the value is infilled, “act” indicating actual, and i/decade indicating averaging over model–scenario cases or decades. These results are found in Tables 4 and 5. Given the definition of standard deviations, values larger than 1 would indicate that this infiller is worse than simply using the mean value in the database.

We see with this fairly large infiller database that for both CO2 and CH4 the approach that generates follower pathways most similar to those removed from the initial scenarios (i.e. the smallest errors) is the RMS technique, with the QRW technique being the next smallest. Linear interpolation without smoothing is expected to produce a noisy fit when given a large infiller dataset, so its performance is unsurprisingly worse. The equal quantile walk (EQW) performs similarly poorly due to effectively ignoring the relationship between the lead and follower data. The time-dependent ratio method is worst of all – its errors are potentially unbounded and for CO2 the average error far exceeds 1. To determine the appropriate statistics to apply on the errors, we first perform a Shapiro–Wilk test to detect any non-Gaussian aspect for the error distribution (details can be found in the statistics_for_ paper notebook of the examples on the GitHub repository). This indicated that the distributions are statistically significantly non-Gaussian for several crunchers when analysed separately and most clearly as an aggregate. We will therefore use non-parametric tests where possible. The small differences in rank between CH4 and CO2 manifest in slightly lower values for CH4. Performing a Wilcoxon's t test on the results indicates that this result is statistically significant for the data as a whole (relative t-test statistic 376, p=0.00007), although when considering each of the crunchers individually, only the RMS-closest and time-dependent ratio crunchers are significantly better with CH4 than CO2 (p values for time-dependent ratio = 0.012, QRW = 0.48, RMS closest = 0.041, linear interpolation = 0.060, EQW = 0.39). We therefore conclude that using either CO2 or CH4 as the default will produce the most reasonable results when using one infiller species, with CH4 performing slightly better while also generally having a slightly lower availability of data.

We perform similar pairwise Wilcoxon t tests on the results of different crunchers and find that the ordering of mean errors (RMS closest < QRW < linear interpolation ≈ EQW < time-dependent ratio) is statistically robust. The p values are < 0.01 for almost all pairs except linear interpolation and EQW, which are much greater than 0.1 whether the comparison uses CO2 lead data, CH4 lead data, or all data combined. The one pairwise exception to this is time-dependent ratio and EQW for CH4, which has only p=0.028, though the values for other combinations still have p<0.01.

We stress that this does not always mean that the RMS-closest technique is the best default, as it makes the assumption that the pathway being infilled is similar to a whole pathway found in the database. The advantage of the quantile-rolling-windows technique is its choice of conservativity – for example, it tends to produce values more towards the median value if the default 0.5 quantile is used – and time independence, whereas RMS closest is better at reconstructing the data and has better consistency over time. Linear interpolation, EQW, and the time-dependent ratio are best used in cases in which there is a large degree of knowledge about the expected relationship between variables.

To demonstrate the uses of this package alone, we will apply the methods directly using unharmonised data in the SR1.5 repository (Huppmann et al., 2019) to infill the emission pathways of the POEM scenario B from the AR5 database (Clarke et al., 2014). The POEM scenarios only report CO2 from certain sources and are thus an excellent use case. The crunchers are all used via the multiple infiller, infill_all_required_emissions_for_openscm. No active decisions are taken except to use the SSP2 scenarios from the MESSAGE model for the specified model interpolation. The choice of SSP2 in this case is ultimately arbitrary but supported by POEM scenario B in being fairly middle of the road and usually fitting in the SSP2 range. The choice of MESSAGE model is because this is the marker model for SSP2 (Riahi et al., 2017). Other POEM scenarios would need different ranges of scenarios for infilling.

We see from Fig. 3 that the linear interpolation model (without filtering the database) provides a chaotic pathway due to its value being determined only by the two points either side of it in the database, which changes semi-randomly with time and should not be used here. Although the “interpolate specified model” approach is also determined by only a few model–scenario pairs because there are only data from a small number of related scenarios, the pathway is smoother and more consistent. The EQW pathway assumes a strong, direct relationship between CO2 and CH4 emissions, which the other crunchers do not uphold at early times, although this would disappear if the data were harmonised. The other cruncher results are all fairly similar and look consistent. The RMS-closest pathway is consistent by construction (and precisely overlines a point in the original database). The quantile-rolling-windows result also looks consistent and tends to move closer to dense clouds of values in the infiller database. In deciding which is the best infiller to use, the RMS-closest result is more consistent over time but more arbitrary in its selection of the pathway, while quantile rolling windows is more conservative in the sense of giving results closer to the median behaviour of the whole dataset.

The Silicone package has features that can split a basket of gases into its constituents. In this example we take data from the Climate Action Tracker (CAT) website (https://climateactiontracker.org/, last access: 9 July 2020; Climate Action Tracker, 2020), which reports projected global emissions in terms of Kyoto gas totals, shown in Fig. 4. While it is possible to use this to infill all other values directly as above, the subcategories of Kyoto gas will not necessarily add up to the Kyoto gas total. Therefore, one of the multiple infillers designed for this use is preferable. The symmetric way to divide the basket into its constituent parts (CO2, CH4, N2O, and F gases) is using the “decompose collection with time-dependent ratio” multiple infiller, which uses a ratio-based technique to ensure conservation of the total amounts. Alternatively, the “split collection with remainder” multiple infiller can estimate the fractions of CH4, N2O, and F gases, then assign the remainder to CO2. F gases could be further subdivided using similar methods.

As can be seen in Fig. 5, the curves that result from “decompose collection” are generally smooth, in spite of being separately calculated at each time point. It is important to ensure that the number of scenarios reported at each time are consistent. In the SR1.5 database, some scenarios only report values at decadal intervals, whereas others use 5-year intervals. We interpolated all models to 5-year intervals to give consistent representation. In the CH4 and F gases, the lowest orange line is clearly seen to rise discontinuously after 2060. This is the last point before the Kyoto total goes negative. To ensure that the sign of the constituents is correct, the formula only considers data from SR1.5 paths for which the Kyoto total has the same sign as in the data being infilled. In this way, emissions that are unlikely to go negative like CH4 are ensured positive; however, their magnitude increases the more negative the aggregate is.

For this reason, the “split collection with remainder” method produces more robust results with sign changes in the lead variable. This technique can use any cruncher, usually RMS closest or (probably non-ratio) quantile rolling windows, to infill the positive values and then allow the value that may be negative (CO2) to make up the rest. This produces the results seen in Fig. 6. Here the behaviour of all curves is fairly smooth, with no obvious features around zero-crossing points and no negative values except in CO2, as expected.

Another use of this software is to infill simple, stylised trajectories generated to explore a wide range of possibilities without detailed economic modelling. For example, Sanderson et al. (2016) suggest simple formulae whereby one may construct emissions trajectories characterised by a few free variables – in this case, based on rates of transition between the RCPs and a long-term emissions value. They present general formulae for generating plausible total CO2 pathways with several free variables. Silicone provides an alternative means of complementing such results – instead of specifying the functional forms of all emissions, you can have a few key emissions prescribed and infill the remainder using scenarios with similarities to the desired narrative. A notebook can be found in the Silicone examples on GitHub detailing the calculations and demonstrating this usage, titled Infill_stylised_path.ipynb (Lamboll, 2020), using data from Riahi et al. (2011) and van Vuuren et al. (2011). It shows that curves with different values in some of the parameters, termed E∞ and τ, can be complemented using a number of techniques. Here we highlight the method of interpolating results from any of the SSP scenarios as implemented by variants of the MESSAGE model. As the different SSPs have different narratives, this allows the user to decide what narrative is relevant to the infilling, rather than adding more arbitrary values (Gidden et al., 2019). An example of this output can be found in Fig. 7.