Reconstructing past climates, while being an important element in the global effort to understand climate system dynamics and their potential future structure and characteristics, is often limited to qualitative assessments of past conditions. This limits the potential for comparisons with the general circulation model (GCM) simulations, and the integration of palaeoenvironmental information in modelling initiatives . As a result, while inconsistencies exist both between GCM simulations and between GCM simulations and fossil records, it is difficult to use the bulk of the palaeodata available to evaluate GCM simulations in an efficient and effective way.

Many techniques have been developed to quantitatively reconstruct past climates from palaeobotanical data . They rely on the fundamental hypothesis that a causal relationship exists between the modern distributions of plants and the associated climates (Jackson and Williams, 2004, and references therein). These techniques can be divided into two types: (1) those based on plant assemblages – modern analogue technique (MAT) , weighted averaging (WA) , weighted averaging–partial least-squares regressions (WA-PLS) , artificial neural networks (ANNs) or regression trees – and (2) those based on plant distributions – mutual climatic range (MCR) , the coexistence approach (CA) or probability density functions (PDFs) . These methods are fully detailed in . Amongst these methods, MAT, WA and WA-PLS are the most commonly used. To date, no consensus has been reached as to which performs the best, and since the paper of the debate has been focussed on the sensitivity of different methods to spatial autocorrelation in the training data set, which can cause an overestimation of the performance of a method .

Beyond these conceptual issues, the calibration data set is another key concern. Methods based on plant assemblages need to be calibrated on robust modern data sets covering various environmental and biotic conditions, data sets that are not available from many parts of the world (e.g. drylands where pollen is often poorly preserved in surface samples). In such situations, the flexibility of methods based on plant distributions (MCR, CA, PDFs) becomes more evident, expanding the range and scope of “reconstructible” environments. They also offer the possibility to reconstruct climate from non-analogue assemblages, provided that most species from that palaeoassemblage still exist. Conceptually, PDF-based methods evolved from MCR techniques as a way to model the strength of the relationship between plants and climate. Indeed, MCR (which considers a rectangular envelope defined by minimum and maximum values for a given climate variable) can be seen as the simplest PDF-based method. These methods are based on the correlation between plants' modern geographical distribution and climate gradients, with the climate value that is the most common in the plant distribution being its “optimum”. Among the approaches that have already been proposed within the last decade , a recurrent issue concerns the assumptions made about the morphological characteristics of the envelope (width, skewness, central tendencies). fitted a multidimensional Gaussian surface that excluded both multimodality and asymmetry, which are however common features when dealing with botanical assemblages. Later, proposed to fit mixture models (combination of several Gaussian surfaces) to relax the constraints of a unimodal Gaussian shape, and more recently proposed the application of non-parametric PDFs to improve the fit between PDF and data.

In addition to the issue of the shape, the accuracy of such models is also a function of the taxonomic resolution at which pollen can be identified (usually family to generic level) and the number of species making up a given pollen type. Pollen types often become climatically non-informative due to a saturation effect wherein too many species result in the climatic information conveyed by each species being averaged and lost. Contrary to the problem of the shape of the climate envelope, the problem of low taxonomic resolution has rarely been discussed as its effects are usually not significant when plant diversity is relatively low. However, in areas where pollen types can comprise a high number of plant species (>30), it becomes increasingly significant and can result in saturated PDFs. proposed a species selection method (SSM) that recursively alters the taxonomic composition of a pollen type by taking into account the coexistence with other pollen types. In order to minimize PDF saturation, the SSM removes species that have climate requirements that are different from that of the assemblage.

However, the SSM only removes species with optima at the extremes of climatic gradients, leaving a certain number of climatically undifferentiated species around the median climate. We believe that the problems of PDF shape and plant diversity are in fact intimately related to the strategy used for fitting PDFs. A pollen type is not a homogeneous ecological unit in the sense that many species with different climate requirements can be classified in the same pollen type. From this point of view, fitting a density function directly to a pollen type is questionable. On the basis that species are the ecological units that respond to climate gradients, we propose a two-step procedure to define the PDF of the pollen types: (1) univariate and unimodal parametric PDFs are fitted for the species (PDFsp's), and (2) those parametric PDFsp are combined to produce the PDF of the pollen-type (PDFpol). The PDFpol reflects the diversity that exists among its species by considering independently each species. To reconstruct a climate value, we propose to combine the PDFpol with a weighted geometric mean. The multiplication of PDFpol ensures the conservation of the mutual climatic range.

To quantify the method's capability to reconstruct different variables in different environments, we have reconstructed a set of modern climatic conditions (20 variables) over a large area (3389 quarter-degree grid cells representing southern Africa). Southern Africa – composed of South Africa, Botswana, Lesotho, Swaziland and Namibia – is well suited for such a test as it is characterized by a strong topographic, geological and climatic heterogeneity , leading ultimately to a great diversity of plant species . Statistical tests were performed on climate anomalies (1) to analyse where and why the model was reliable, and (2) to measure the effects of parameters, such as the type of variable, the number of taxa used and/or the vegetation type.

The method presented here has been implemented in a software package entitled CREST (Climate REconstruction SofTware). CREST – presented in Appendix  – is intended to make quantitative climate reconstructions more accessible to the wider community. Our hope is that a proliferation of quantitative reconstructions of past climate conditions will facilitate the consideration of palaeoenvironmental data in the assessment of GCM performance and ultimately allow for an improved understanding of both past and potential future climate change.

The climate reconstruction method we propose is based on univariate PDFs. Here, a PDF represents the probability of a species existing along a climate gradient and is a surrogate for the species' realized niche see for example. The process follows three general steps: (1) the plant–climate relationship is quantified (i.e. PDFs are fitted), (2) information conveyed by each taxon is combined and finally (3) a climate value from the resulting climate likelihood function is extracted. This method relies on the assumption that the plant–climate relationship has remained relatively constant since the deposition of the fossil assemblage.

As a case study, we have used a modern botanical database to reconstruct a set of contemporary climate values to highlight and explore the strengths and weaknesses of the approach, and to quantify its accuracy and robustness. Using southern Africa as a study area, we consider five countries: South Africa, Namibia, Lesotho, Swaziland and Botswana (from 17 to 34.5∘ S and from 12 to 32.5∘ E, Fig. ). This area is composed of 3913 quarter-degree grid cells. The region provides an excellent case study as it is characterized by strong topographic, geologic and climatic heterogeneity , which has resulted in the existence of many vegetation types, which often change rapidly over short distances.

Our results indicate that the PDF-based method of CREST performs well (Table , Figs. and ), even if some differences in terms of reconstruction quality exist between variables. The variables that were best reconstructed were those that have a direct impact on the physiology of plants, and thus strongly constrain their distribution (e.g. Tmean wet Q, Frost days, Prec dry Q or Prec wet Q) (referred to as direct gradients by ). The impact of other variables such as mean diurnal range or temp seasonality on the plant life cycle is indirect. Thus, they are less likely to be accurately reconstructed from pollen data. Variables that are surrogates for direct gradient may show strong ability to describe modern data but poor predictive power to describe past conditions (see, e.g., the palaeolimnological example of ). Selection of variable(s) of interest should always be conditioned to an appropriate analysis of the data. Statistics could help in that process , but the final decision about the variables to reconstruct should always derive from an enlightened choice based on both statistical and ecological/environmental considerations. In the semi-arid to arid environments of southern Africa, precipitation and/or water availability strongly constrain plants distributions, which probably explains why we get lower NRMSDs for moisture-related variables in our case study.

The method performed well regardless of vegetation type. We were not able to show any differences in accuracy between the different biomes and/or ecoregions, provided that the distribution of the biome and/or ecoregion was sufficiently spatially extensive. Based on these results, we have found that the method works best for vegetation types represented by at least ∼25 to 50 quarter-degree grid cells (estimation based on Fig. ) in order to adequately determine the plant–climate relationship.

While our expectation was that a high number of species would result in more precise reconstructions, we were not able to observe any relationship between anomalies and the number of species. Anomalies do decrease when the number of species begins to increase (from 1 to ∼20–30), but then the tendency is reversed, and the large anomalies were observed in samples with the largest number of species. This may be related to a saturation problem, wherein more is not necessarily better. As we used a presence/absence weighting strategy, species far from their climate optimum have the same importance as those living in their optimal climate. The increase in the number of species could increase these marginal elements, biasing the reconstructions. The role of the number of taxa on the accuracy is not yet fully understood and is the subject of ongoing studies.

Other studies have shown that selecting a subset of the recorded taxa was sometimes more appropriate when attempting to capture a given climate signal. In order to improve the quality of the reconstructed variables, consideration should be given to reconstructing each variable with a different subset of the total of the available species list. Reducing this list to a shorter list of responsive species reduces the noise and consequently leads to better reconstructions. These choices, however, are not wholly objective. A first approach consists in observing directly the PDFpol's since flat and multimodal PDFs may indicate insensitivity to a given climate variable. made choices based on considerations of the ecology of the given species, while opted for a more statistical approach. To avoid using redundant information, only species that were statistically different (based on the Mahalanobis distance between the PDFs) were conserved. combined those two approaches and based their choices both on niche-based modelling and ecological considerations. The process of selection is thus not straightforward, and, while it may improve reconstructions, care needs to be taken to avoid undue bias in the results. The selection of the different sets of pollen types probably lowers the impact of spatial autocorrelation . However, it should be kept in mind that the “supposed” reconstructed variable may in fact be a composite of primary and secondary variables. Secondary variables have the power to strongly bias reconstructions when spatially correlated with the variable of interest , so that only part of the reconstructed palaeovariability can be related directly to it. CREST (Appendix 1) provides a range of outputs that indicate the sensitivity of different taxa to given climatic parameters and allow the user to assess the data being considered and make informed choices in the selection of such subsets.

When plotted on a map (Figs. and ), the reconstruction anomalies appear to be spatially clustered. Those patches of large anomalies can be explained by the position of the local climate along the climate gradients (Fig. ) and are a direct consequence of the hypotheses underlying the model. The method is correlative, and consequently it is biased towards the best-represented climate values. This uneven sampling of the environmental gradients biases the estimation of plants' optima by shifting the “real” optimum towards the portion of the gradient with the most observations . In most cases, lowest/highest values along the studied climate gradient are rare, but there are exceptions. For example, low rainfall amounts are common in southern Africa, and as a result they are well represented and the signal is easily captured by the model.

To offset the impact of the climate distribution's heterogeneity, we upweighted rare climate values as proposed by and . This method shifts PDF optima towards the rarest climate values. The climate abundance weighting did decrease the errors for the extreme climates but also increased them for the most common ones (data not shown). The overall impact is nevertheless positive since it decreased the RMSDs for all the variables. It also reduced the clustering of errors. Despite its advantages, the strategy has the drawback that artificial geographical limits must be selected e.g. mountain ranges or country borders; to compute the weights. A finite number of grid cells must be selected and sorted into bins. Any change in the boundaries would affect – potentially significantly – the weights, and thus the reconstructions. It is also possible that the climate abundance weighting may be the cause of the small but significant bias observed between temperature and moisture variables, which are, respectively, under- and overestimated for rare climates in the region.

Even with the climate abundance weighting, it is apparent that reconstructing the rarest climates is extremely complicated with models such as those described here. This is why, for example, the Cape region is poorly reconstructed for the precipitation-like variables but not for the temperature-like variables. The temperature of the area is common in southern Africa – so its signal is well captured – but it is an outlier in terms of quantity and seasonality of rainfall. Other areas of notable climatic rarity in southern Africa include (1) the eastern portion of the Great Escarpment (high precipitation); (2) the high mountains of Lesotho, where temperatures are very low and precipitation is high; (3) the thin coastal band along the southern coast of South Africa, where moist forests can develop as a result of significant aseasonal rainfall; and (4) along the Namibian coast (stable temperature and extremely low precipitation). All these areas lie at an extreme (relative to the study area) of one or several climatic gradients, giving rise to clusters of high anomalies.

In terms of reconstructing quantitatively long-term climate variations it should be kept in mind that PDFsp is defined by the modern climatic space. Climatic space varies over time, and certain elements of some past climate regimes may be more or less abundant and/or more or less accessible to some species than in the modern climatic space . Depending on the location of the site vis-à-vis the climatic space, the potential to estimate the amplitude of climate change varies. As shown schematically in Fig. , samples located in the mean climate space have greater potential to “move” in several directions and with greater amplitude than samples that are already at the margin of the climatic space. In the latter case, the exact amplitude of change may be underestimated, but the overall trends and direction of change may still be accurate. It is expected that, even under a different climate, the relative position of the different taxa along a climatic gradient would stay the same, so that the replacement in the past of a taxon by another that currently lives in colder environments will effectively indicate colder conditions with – possibly large – uncertainties regarding the amplitude of change .

The PDF-based method we have presented in this paper provides robust results across a range of climates and vegetation types. We have demonstrated that the accuracy does not vary significantly as a function of vegetation type or the number of species considered, and it is thus a useful tool for reconstructing climates in many regions and biomes. The accuracy of the reconstructions is, however, strongly impacted by the climate variable being reconstructed (direct or indirect gradients) and primarily by the position of the targeted climate on the climate gradient of the study area. To ensure a robust reconstruction, one should

select climate variables that directly impact the distribution of the species and, inversely, use only species whose distributions are significantly defined by the climatic variable;

CREST is freely available from the authors as well as at www.hyrax.univ-montp2.fr.