Improved estimates of future precipitation change in climate models

Abstract

Future change in precipitation driven by anthropogenic influences on the Earth’s radiative balance will further affect ecosystems, water resources, agriculture, economies, lives and livelihoods. Increased clarity on anthropogenically forced precipitation change can assist adaptation in some contexts. Climate scientists typically quantify precipitation change in models using the average value of the percentage change evident in many different models, i.e., $\% \Delta {P}^{j}=100\left(\,\frac{{{P}_{2}}^{j}-{{P}_{1}}^{j}}{{{P}_{1}}^{j}}\right)$, where ${P}_{i}^{j}$ is the average value of precipitation over Period $i$ in model $j$. Here we use theory and results from CMIP6 climate models under preindustrial, historical and future forcing to assess the accuracy of this approach. We show that this standard approach inaccurately estimates precipitation change evident in models, even in infinitely large ensembles. Under a wide variety of circumstances, the discrepancy is approximated by $100({\mu }_{2}/{\mu }_{1})/(m/{{{\rm {CoV}}}}^{2}-1)$, where ${\mu }_{i}$ is the population mean for Period $i$, $m$ is the number of years in the reference period, and CoV is the Coefficient of Variation (i.e., the standard deviation of precipitation variability divided by the mean). The discrepancy is therefore greater for shorter reference periods and is greatest where the ${CoV}$ is large (which tends to occur in dry regions) and anthropogenic forcing increases precipitation. The discrepancy using climate model output under SSP370 forcing has an average value of 5.7% over the tropics in December–January–February, with far greater values in many subregions. Alternative approaches to quantifying precipitation change are described.

Introduction

Future externally forced change (EFC) in precipitation driven by anthropogenic changes to the Earth’s radiative balance will affect ecosystems, agricultural production, water resources, economies and people^1,2,3,4. In recognition of this importance, the last two sets of the highly influential Intergovernmental Panel for Climate Change (IPCC) reports on physical climate science provided estimates of the percentage change in seasonal and annual precipitation across the globe relative to the recent past, using climate model output obtained using a range of different future scenarios^5,6,7,8,9,10. Percentage change in precipitation is used in many peer-reviewed journal articles and important international, national and sub-national reports^{8,11,12,13,14,15,16,17,18}, as well as popular online tools, including the IPCC AR6 WGI Interactive Atlas and the World Bank Group’s Climate Change Knowledge Portal). See, for example, Fig. SPM.5 in the Summary for Policymakers, Figs. 4.13 (Chapter 4) and 8.1 (Chapter 8), and Cross-Chapter Box Atlas.1, Fig. 1, all in the latest IPCC AR6 WGI report^8,9,10,13. Percentage change is a useful statistic because it allows us to map precipitation change across the globe, including very wet regions, as well as much drier regions.

Climate scientists seek estimates of the percentage change in the population mean of precipitation from a past reference period (Period 1, say) to a future period of interest (Period 2 say), %$\Delta P=$100*$({\mu }_{2}-$ ${\mu }_{1})/{\mu }_{1}$, where ${\mu }_{i}$ is the population mean¹⁹ of precipitation for Period $i$.

To understand the primary focus of this study, it is essential to understand the method that climate scientists typically use to estimate the model-based value of $\% \Delta P$. In the last IPCC report, for example, model-based estimates were calculated using the percentage change evident in single runs from many different models, i.e., $\% \Delta P^{j}=100\left(\frac{{{P}_{2}}^{j}-{{P}_{1}}^{j}}{{{P}_{1}}^{j}}\right)$, where ${P}_{i}^{j}$ is the average value of precipitation over Period $i$ in model $j$. The model-based estimate of $\% \Delta P$ is then created by averaging $\% \Delta {P}^{j}$ over all the available models, i.e., $\% \overline{\Delta {P}^{j}}=\frac{1}{{N}_{m}}{\sum }_{j=1}^{{N}_{m}}\{ \% \Delta {P}^{j}\}$, where ${N}_{m}$ is the number of models. This approach is taken in, e.g., Fig. SPM.5 in the Summary for Policymakers, Figs. 4.13 (Chapter 4) and 8.1 (Chapter 8), and Cross-Chapter Box Atlas.1, Fig.1, all in the latest IPCC AR6 WGI report^8,9,10,13. We will refer to this approach (i.e., the calculation of $\% \Delta {P}\,^{j}$ for each model and the subsequent averaging over all models, giving $\% \overline{\Delta {P}^{j}}$, as the standard method.

While the standard method used to estimate $\% \Delta P$ has proven very useful, issues can arise. For example, a problem occurs if ${{P}_{1}}^{j}=0$ in any of the models because then an estimate of $\% \Delta P$ cannot be created using the standard method. It is also recognised that very large % increases can occur in arid regions⁸. While these issues are anecdotally widely recognised, there does not seem to have been a systematic assessment in the literature as to how extensive the problem is and how large the % changes can actually become in practice. So a brief assessment of these issues will be conducted here. However, they are not the issues that concern us most. Nor are we concerned here with the fact that models and the forcing applied to them have deficiencies that can lead to unrealistic model-based estimates of what $\% \Delta P$ is in reality, or the fact that sampling error will typically arise from estimations based on finite samples. Instead, we are primarily concerned about the possibility that $\% \overline{\Delta {P}\,^{j}}$ provides an inaccurate estimate of the model-based value of $\% \Delta P$ even if we have infinitely many models, even if those models are perfectly realistic.

To understand how such a problem might occur, we first note that the standard method can also be used to estimate percentage change in a single model with multiple simulations (in which case $j$ is an index for the model run and ${N}_{m}$ is the total number of runs available for the model).

Consider now the following simple, highly idealised situation. Suppose we have one model with infinitely many runs, and we wish to estimate the EFC in precipitation in that model between two years of interest: one in the past Period 1 and the other in the future Period 2. We consider the change between these two years rather than two multidecadal periods because this greatly simplifies the analysis while still illustrating the potential problem we are interested in.

Suppose that in this model, precipitation during the first year of interest can only take one of two values, 100 or 200 mm, and both are equally likely (i.e., probability = 0.5). Similarly, in our year of interest in Period 2, precipitation can only take one of two values, 150 or 250 mm, and both values are again equally likely (i.e., probability = 0.5). In this model, external forcing has increased the population mean from ${\mu }_{1}=\,$(100 mm + 200 mm)/2 = 150 mm to ${\mu }_{2}=$(150 m + 250 mm)/2 = 200 mm. Therefore, the percentage increase in precipitation in this model, $\% \Delta P$, is 100*(200 mm−150 mm)/150 mm = +33.3%.

Now let’s use the standard method to estimate $\% \Delta P$ in this case. If ${p}_{i}^{j}$ is the precipitation in run $j$ and in the year of interest during Period $i$, then each run can only have $\left(p_{1}^{j},\right.$ $\left.{p}_{2}^{j}\right)$= (100, 150), (100, 250), (200, 150), (200, 250). There are no other possibilities. And as each of these possibilities is equally likely, then one quarter of the simulations will be (100, 150), one quarter will be (100, 250), one quarter will be (200, 150), and the remaining quarter will be (200, 250). The corresponding $\% \Delta {P}^{j}$ for these four groups of runs is therefore 100*(150−100)/100 = +50%, 100*(250−100)/100 = +150%, 100*(150−200)/200 = −25% and 100*(250−200)/200 = +25% respectively, and so $\% \overline{\Delta {P}^{j}}$ is (+50 + 150−25 + 25)%/4 = +50%. So in this simple case, the method used to estimate $\%\Delta P$ is inaccurate, even though there are infinitely many simulations, so that sampling error is not an issue, and precipitation in all years is neither zero nor small.

Of course, the example above is highly idealised and we only looked at the difference between two specific years, not two multidecadal periods. So we should not necessarily conclude from this that the same type of inaccuracy arises when we estimate EFC between two multidecadal periods using real climate models with more realistic precipitation probability density functions. However, the results are sufficient for us to hypothesize that (i) the standard method produces inaccurate estimates of the modelled EFC even in the absence of sampling error, and (ii) the inaccuracy is large enough to be consequential in some circumstances. Whether (i) and/or (ii) are true remains to be seen.

The primary purpose of this paper is to test hypotheses (i) and (ii). We do this in several ways. In the first test, we derive analytic expressions for the Expectation Value of the difference, $D= \% \overline{\Delta {P}^{j}}- \% \Delta P$. In the derivation in the following section, precipitation is assumed to have much more complicated and realistic probability density functions than assumed in the simple example above. The derivations will show that the standard estimation of $\% \Delta P$ does indeed inaccurately represent the EFC under more realistic conditions.

The derivations also show that the standard method gives an inaccurate estimate of $\% \Delta P$ even if the modelled EFC is actually zero. More specifically, it creates an apparent positive EFC where none is actually present. We explicitly examine this issue in the second test, in which we use the standard method to estimate $\% \Delta P$ in preindustrial control runs. In these runs, there is no EFC, and so the estimate of $\% \Delta P$ should be close to zero. We will see that in many places this is not the case. The standard method suggests that there is a robust increase in precipitation in the preindustrial runs when one does not actually exist.

In the third and final test, we estimate the Expectation Value of $D$, i.e., $E(D)$, using a large (40-member) ensemble of a single CMIP6 model (ACCESS-ESM1.5) and again find that $E(D)$ is nonzero (and positive) in many locations.

If $E(D)\,\ne\, 0$, i.e., if $E( \% \overline{\Delta {P}^{\,j}})$ $\,\ne\, \% \Delta P=100({\mu }_{2}-{\mu }_{1})/{\mu }_{1}$, then the standard method provides what statisticians call a biased estimate¹⁹ of the modelled percentage change. This is a different form of bias from the “bias” frequently discussed by climate scientists²⁰. In climate science bias typically refers to systematic differences between modelled and real-world values of simulated quantities caused by climate model deficiencies and/or errors in the forcing applied to the models. Here we are instead interested in knowing if the standard method used to estimate $\%\Delta P$ provides an unbiased (using the statistician’s definition, not the definition typically used by climate scientists) estimate of the actual value (i.e., the population value) of percentage change in models, not how well models replicate reality.

We also identify key relationships between $E(D)$ and key statistical properties of precipitation; examine $E(D)$ in regions where precipitation in individual years is zero; specify the circumstances in which the inaccuracy is most problematic; and provide recommendations on how to produce better estimates of EFC in precipitation that do not employ the standard approach.

Methods, including details on the climate models and observational data used, are given near the end of the manuscript.

Results

Analytic solutions for E(D)

In this section, we present and discuss mathematical formulae for the Expectation Value of the difference, i.e., ${E}(D)$, where $D= \% \overline{\Delta {P}\,^{j}}- \% \Delta P$ (see Table 1 for a list of symbols and acronyms. The quantity $\% \overline{\Delta {P}\,^{j}}$ is again the percentage change in mean precipitation estimated using the standard method as outlined in the Introduction, and $\%\Delta P$ is again $100({\mu }_{2}-{\mu }_{1})/{\mu }_{1}$, where ${\mu }_{i}$ is the population mean of precipitation in Period $i$. $\%\Delta P$ is the actual % change in the mean, whereas $\% \overline{\Delta {P}\,^{j}}$ is an estimate. For finite samples $D=$ $E(D)+$ sampling error. If $E(D)=0$ then statisticians would say that the standard method provides an unbiased estimate of precipitation change in the model/s of interest.

Table 1 List of acronyms and symbols used in the main text

The formulae derived below apply in a very wide range of cases where precipitation in individual years, seasons, months or some other period of interest, $p(t)$, the precipitation in year $t$, has a Gamma distribution. Gamma distributions have the property that $p(t) > 0$ for all time $t$, and—depending on the parameter values—the distributions are Gaussian-like (though with one tail, not two), highly skewed, or something in between. We will see below that $p(t) > 0$ for all time in the vast majority of the world, for the annual and seasonal data in climate models (and observations).

As shown in the Supplementary Material, the Expectation Value of the difference, $E(D)$, is given by

$$E(D)=100({\mu }_{2}/{\mu }_{1})/(m/{{{ {CoV}}}}_{1}^{2}-1)$$

(1)

where $m$ is the number of years in the reference period (i.e., Period 1), and ${{{\rm {CoV}}}}_{1}$ is the coefficient of variation (=standard deviation of precipitation variability divided by the population mean) in Period 1. The coefficient of variation is a commonly used statistic to help quantify the importance of variability relative to the mean.

Note also that if the EFC is zero, then ${\mu }_{2}={\mu }_{1}$ and so Eq. (1) simplifies to

$$E\{D\left({{ {EFC}}}=0\right)\}=E(D_{0})=\frac{100}{\frac{m}{{{{{ {CoV}}}}_{1}}^{2}}-1}$$

(2)

where ${D}_{0}$ is the difference in the case where the EFC is zero. If the EFC in the model is zero, then $\% \triangle P$ in this model, is also zero. However, ${E(D}_{0})$ is nonzero, indicating that (i) the standard method tends to give an inaccurate estimate of $\% \triangle P$ in cases where the EFC is zero; (ii) ${E(D}_{0})$ is positive if ${{{\rm {CoV}}}}_{1} < \surd m$ (which is always the case—see Supplementary material); (iii) that it is only a function of $m$ and ${{{\rm {CoV}}}}_{1}$; and (iv) it increases as either $m$ declines or ${{{\rm {CoV}}}}_{1}$ increases. This indicates that the standard method will tend to suggest that precipitation increases in regions are occurring where there is, in fact, no EFC in mean precipitation.

Thedependence of $E({D}_{0})\,$(using Eq. (2)) on ${{{\rm {CoV}}}}_{1}$ for $m=$ 10, 20, 30, 40 is illustrated in Fig. 1a. It shows that ${E(D}_{0})$ is positive for all values of ${{{\rm {CoV}}}}_{1} > 0$, and it increases monotonically as ${{{\rm {CoV}}}}_{1}$ increases for all values of $m$. ${E(D}_{0})$ is greatest for m = 10 and smallest for m = 40 for all values of CoV₁. For m = 10, ${E(D}_{0})$ exceeds 40% if CoV₁ > 1.7, while E(D₀) is ~25% at CoV₁ = 2 if m = 20. If m = 20 then ${D}_{0}$ > 5% if ${{CoV}}_{1} > 1.0$.

Fig. 1: Analytic solutions showing ${E}({D})$ vs. the Coefficient of Variation (CoV).

For a EFC = 0 (so ${D}={D_{0}}$) and the number of years in the reference period = 10, 20, 30, 40 years, and for b EFC = 0, and EFC = ±50% of ${\mu }_{1}$ with the number of years in reference period = 20.

Equation (1) shows, for precipitation with a Gamma distribution, that:

• if EFC$\ne 0$ then $E(D)$ is different to $E({D}_{0})$, but only if the external forcing causes a change in the population mean, so that ${\mu }_{2}\ne {\mu }_{1}$. An externally forced change in variance alone does not have any effect on $E(D)$.

• $E(D)$ with EFC$\,\ne\, 0$ is given by ${\frac{{\mu }_{2}}{{\mu }_{1}}E(D}_{0})$. So, the magnitude of $E(D)$ is greater than or smaller than $E({D}_{0})$ depending on whether the external forcing increases or decreases the mean. This is illustrated in Fig. 1b, which shows $E(D)$ as a function of the CoV for EFC = 0 (solid curve; i.e., ${E(D}_{0})$), and when EFC = −50% and +50% of the reference value mean (red and blue curves, respectively).

The latter means that the inaccuracy will tend to be greater in regions where external forcing causes an increase in precipitation, and that it will be greatest in regions where the EFC in precipitation is greatest, for any given value of CoV.

Before we consider output from CMIP6 climate models, let’s consider an idealized case similar to that given in the Introduction to help illustrate why the standard method might give rise to an inaccurate estimate of the modeled EFC in the case where the EFC is zero. Suppose we have single runs from infinitely many perfect models, but this time suppose that there is no EFC. As noted in the Introduction, precipitation during the first year of interest in Period 1 can only take one of two values, 100 or 200 mm, and both are equally likely (i.e., probability = 0.5). But as the EFC = 0 in this second case, then in our year of interest in Period 2, precipitation can only take one of the same two values, 100 and 200 mm, and both values are again equally likely (i.e., probability = 0.5). In this case, external forcing has not changed the population mean, i.e., ${\mu }_{2}={\mu }_{1}=\,$(100 mm + 200 mm)/2 = 150 mm, and so the percentage increase in precipitation in this case, $\%\Delta P$, is zero.

Now let’s use the standard method to estimate $\%\Delta P$ in this case. Each model run can only have (${p}_{1}^{j},$ ${p}_{2}^{j}$)= (100, 100), (100, 200), (200, 100), (200, 200). There are no other possibilities. And as each of these possibilities is equally likely, then one quarter of the simulations will be (100, 100), one quarter will be (100, 200), one quarter will be (200, 100), and the remaining quarter will be (200, 200). The corresponding $\%\Delta {P}^{j}$ for these four groups of runs is therefore 100*(100-100)/100 = 0%, 100*(200−100)/100 = +%100%, 100*(100−200)/200 = −50% and 100*(200−200)/200 = 0%, respectively, and so the average value of $\%\Delta {P}^{j}$ (i.e., $\% \overline{\Delta {P}^{j}}$) in this case is (0 + 100−50 + 0)%/4 = +12.5%. So in this simple case, in which we know that the EFC is zero, the standard method erroneously suggests that the EFC is non-zero (and positive).

In this simple case we (again) examined the change between two specific years, not two multidecadal periods, so it (again) offers only a hint as to what might occur in a more realistic situation. Much more realistic cases are considered in following sections.

Inaccuracy arising from the standard method using the climate model output

Let us now use climate models under preindustrial control runs to estimate $E(D)$ in the absence of external forcing (i.e., ${E(D}_{0})$). To do this, we use the value of the percentage differences between successive, non-overlapping 20-year averages of the pre-industrial control runs, averaged over the entire pre-industrial control run (=$\% \overline{\Delta {P}^{j}}$ say). If the individual model output had no long-term trend, then $\%\Delta P$ in each model would be zero, and we’d expect $\% \overline{\Delta {P}^{j}}$ to be small in magnitude, but vary randomly about zero due to sampling error. We would not expect there to be a high degree of agreement on the sign of $\% \overline{\Delta {P}^{j}}$ among the models.

Maps showing the multi-model mean (MMM) values of ${E(D}_{0})$ across the globe are presented in Fig. 2 for annual, December–January–February (DJF) and June–July–August (JJA) data. Stippling indicates that there is agreement in over 66% of the models on the sign of ${E(D}_{0})$. The % area of the globe showing model agreement is 23.7%, 55.5%, and 52.3% for annual, DJF, JJA data, respectively, and MMM(D₀ > 0) in the vast majority of the grid points where there is model agreement.

Fig. 2: Multi-model mean (MMM) values of the $E({D}_{0})$ for precipitation, from pre-industrial control runs.

a Annual, b DJF, and c JJA. Stippling indicates that there is agreement in over 66% of the models on the sign of $({D}_{0})$. Models with zero precipitation averaged over Period 1 at particular grid points are excluded from the MMM at those grid points. ${E(D}_{0})$ is the value of $E(D$) with no external forcing, i.e., $E(D({EFC}=0))$.

This suggests that there are robust positive trends in the pre-industrial control runs. However, this is not the case: there are only a few places where the models agree on the sign of the linear trends (Supplementary Fig. 1).

The relationship between the $E({D}_{0})$ estimated using the pre-industrial control runs and the CoV is illustrated in Fig. 3. It is a scatter plot created using ${({CoV},E(D}_{0}))$ at every grid point in every model. The curve-of-best-fit (blue) agrees fairly well with the curve representing theoretical values (black), though the climate model estimates tend to exceed the theoretical estimates somewhat. Both curves indicate that ${E(D}_{0})$ increase as the CoV increases. Some of the values obtained are very large, with many values > 20% using seasonal data. Some values >50% (annual) and >100% (seasonal) are evident, indicating that the change estimated using the standard method can be extremely inaccurate in some cases.

Fig. 3: Scatter plots of $E({D}_{0})$ versus the Coefficient of Variation (CoV) based on the MMM of preindustrial control runs.

a Annual, b DJF, and c JJA data. Each blue dot shows the value of (x, y) = (MMM(${D}_{0})$, MMM(CoV)) at a particular grid point. The blue curve shows the cubic curve of best fit. The black curve shows the analytic solution for a 20-year reference period. Only grid points for which ${p}_{1}^{j}(t) > 0$ for all models and all t are included.

While we now have an estimate of ${E(D}_{0})$, i.e., $E(D)$ in the case where the EFC = 0, we previously saw that the magnitude of $E(D)$ is affected by the presence of ${{\rm {EFC}}}\,\ne\, 0$. We will therefore now examine $E(D)$ in a large ensemble (with 40 members, see the “Methods” section for further details) of a single climate model in the presence of non-zero EFC. We use the ACCESS-ESM1.5 model, forced using historical and SSP370 forcing, and we estimate the $E(D)$ between 1995–2014 and 2080–2099.

The value of $E(D)$ in this case is estimated using $100\{{{ {EM}}}(\frac{\varDelta {P}^{\,j}}{{P}_{1}^{\,j}})-\frac{{{ {EM}}}(\varDelta {P}^{\,j})}{{{ {EM}}}({P}_{1}^{\,j})}\}$, where EM is the ensemble mean. The results are presented in Fig. 4. Here ${{\rm {EM}}}(\frac{\varDelta {P}^{j}}{{P}_{1}^{j}})$ is the estimated fractional change using the standard method, while $\frac{{{ {EM}}}(\varDelta {P}^{j})}{{{ {EM}}}({P}_{1}^{j})}$ is an estimate of the corresponding population value. The maps presented in the last row collectively indicate that $E(D)$ in the presence of external forcing is positive almost everywhere.

Fig. 4: Maps showing results from the ACCESS ESM1.5 ensemble.

Raw precipitation change (EM $(\Delta{P}^{j})$; first row); (a) annual, (b) DJF, (c) JJA), the estimated population mean of % change $\left(\frac{{\rm{EM}}(\Delta{p}^{j})}{{\rm{EM}}(\Delta{p}^{j}_{1})}\right)$;second row; (d) annual, (e) DJF, (f) JJA), the % change estimated using the standard method (EM$\left(\frac{\Delta{P}^{j}}{{{P}_{1}}^{j}}\right)$; third row; (g) annual, (h) DJF, (i) JJA), and ${\rm{EM}}(D)={\rm{EM}}\left(\frac{\varDelta{P}^{j}}{{{P}_{1}}^{j}}\right)-\frac{{\rm{EM}}(\varDelta{p}^{j})}{{\rm{EM}}({p}^{j}_{1})}$ (last row; j Annual, k DJF, l JJA). EM is the Ensemble Mean. Each row contains results using annual (first column), DJF (second column), and JJA (third column) means. Stippling in panels (a–g) indicates where more than 66% of the ensemble members agree on the sign of $\varDelta {P}^{j}$. For consistency with IPCC (2021), we compare 1995–2014 from the Historical scenario with 2080–2099 from SSP370.

The largest values of $E(D)$ occur in North Africa, the Horn of Africa, the Middle East, parts of India, Pakistan, Afghanistan, Uzbekistan and Tajikistan, northwest Australia, the equatorial Atlantic and near-zonal bands in the equatorial Pacific off the equator.

The values of $E(D)$ at each grid point are given in Fig. 5 for DJF (top row) and JJA (bottom).

Fig. 5: Estimated $E( \% \varDelta P)$ at each grid point using the standard method versus $100\frac{{{\rm {EM}}}\left(\Delta {P}^{j}\right)}{{{\rm {EM}}}({P}_{1}^{j})}$ (left column), and $E(D)$ versus $100\frac{{{\rm {EM}}}\left(\Delta {P}^{j}\right)}{{{\rm {EM}}}({P}_{1}^{j})}$ (right column).

Top row (DJF), bottom row (JJA). The red lines in the left column correspond to $y=x$. The data used here is the same as the data used in Fig.4.

$E(D)$ is very small in the overwhelming majority of grid points. However, $E(D)$ is positive almost everywhere, and it is very large in some regions. In DJF, for example, 11%, 4.5%, 2.5% and 1.4% of the world have values of $E(D)$ greater than 2%, 5%, 10% and 20%, while the equivalent figures for JJA are 8.7%, 4%, 1.8%, and 0.76%. In some regions $E(D)$ is much greater than this, with values exceeding 100% in both seasons.

$E(D)$ tends to be much larger in individual months: in February 33%, 9.6%, 5.1% and 2.9% of the world have a value of $E(D)$ exceeding 2%, 5%, 10% and 20%, while during August 22%, 7.2%, 3.6% and 2.0% of the Earth exceeds the same thresholds, respectively. Values far in excess of 10% are evident at many locations. The largest values of $E(D)$ (and the largest CoVs) occur in some of the driest regions (Fig. S2).

This tendency is also reflected in the tropically averaged value of $E(D)$, which is larger in February (14.3%) than in DJF (5.7%), and it is larger in August (1.6%) than it is in JJA (3.7%).

It is interesting to note that there is no stippling in the last row of Fig. 4. This indicates that there is not strong agreement on the sign of the individual contributions to $E(D)$ (i.e., $100[{{ {EM}}}\left(\frac{\Delta {P}^{j}}{{P}_{1}^{j}}\right)-\frac{{{ {EM}}}\left(\Delta {P}^{j}\right)}{{{ {EM}}}({P}_{1}^{j})}]$). It seems counterintuitive to have positive ensemble mean values of $D$ at the overwhelming majority of grid points, consistent with theory, without having agreement on the sign of $D$ among the contributing ensemble members at any grid point. This situation arises because the values of $D$ at each grid point form an ensemble that tends to have a median that is close to zero or at least closer to zero than the generally positive mean (Fig. S3 (top row); Figs. 4j–l and 5 (right column)). This can only arise if the data is positively skewed, which indeed it tends to be (Fig. S3, bottom row).

To further examine this issue, we conducted Monte Carlo experiments examining $D$ in thousands of simulations of random precipitation time series drawn from Gamma distributions, with and without external forcing (see the “Methods” section for further details). In every case we found that the behavior was the same as the behavior tended to be in the climate models: the median of the contributions to $D$ from each member was close to zero (<~1% in all cases), and so there is not strong agreement on the sign of the contributions, whereas the corresponding ensemble mean is positive in every case (ranging from ~8% in the case with the CoV = 0.5 and with the EFC halving the mean, through to ~88% in the case with the CoV = 1.5 and the EFC increasing the mean by 50%).

It can also be shown theoretically (see Supplementary Material) that the median of the distribution of $D$ is close to zero in a wide range of cases, despite the fact that the average value of $D$ is always positive. This arises because $D$ distributions are skewed to the right (Fig. S4).

These results highlight the fact that the absence of agreement among members on the sign of a change does not necessarily indicate a lack of robustness if the underlying change data is skewed, as it is here.

We will now examine links between ${E}({D})$ and CoV. Scatter plots of $E(D)$ versus CoV are presented in Fig. 6. The scatter plots in the top row were created by restricting attention to grid points for which $p(t) > 0$ for all $t$ in all ensemble members during Period 1, as it was in the derivation of the mathematical formula in Eq. (1). As hypothesized, for a given CoV the magnitude of $E(D)$ tends to increase as the CoV increases, and $E(D)$ tends to be greater than ${E(D}_{0})$ (black curves) in regions where precipitation is projected to increase (blue dots and curves). In addition, for a given CoV, $E(D)$ also tends to be larger in regions where external forcing causes an increase in precipitation than in regions where external forcing causes a decrease in precipitation (red dots and curves). All of these results are consistent with theory (see e.g., Fig. 1b).

Fig. 6: Scatter plots showing $E(D)$ versus the Coefficient of Variation (CoV) with ${{\rm {EFC}}}\,\ne\, 0$. $E(D)=100[{{ {EM}}}(\frac{\varDelta {P}^{\,j}}{{P}_{1}^{\,j}})-$ $\frac{{{ {EM}}}\left(\Delta {P}^{\,j}\right)}{{{ {EM}}}({P}_{1}^{\,j})}]$, with 1995–2014 from the Historical scenario as Period 1 and 2080–2099 from SSP370 as Period 2.

Each dot represents the value of $E(D)$ estimated using a 40-member ensemble of ACCESS-ESM1.5 simulations at a particular grid point. Red dots indicate drying regions with ${\rm {{EM}}}({P}_{2}^{\,j})/{\rm {{EM}}}({P}_{1}^{\,j}) < 0.9$ and blue dots indicate regions that get wetter ${{\rm {EM}}}({P}_{2}^{j})/{{\rm {EM}}}({P}_{1}^{j}) > 1.1$. The coloured curves show the quartic curves-of-best-fit, and the black curve shows the analytic solution for a reference period of 20 years. Only grid points with ${p}_{1}(t) > 0$ for all $t$ during Period 1 in all ensemble members were considered for the top row (a-c). Additional grid points at which ${p}_{1}(t)=0$ for some $t$ during Period 1 in some ensemble members, are included in the bottom row (d-f). Note that ${P}_{1}^{j} > 0$ in all ensemble members.

The results in regions where precipitation is projected to decrease (red dots and curves) are only below the value of ${E(D}_{0})$ for some values of the CoV (e.g., in DJF for values of CoV < 1.2). For annual and JJA data, the red curves are only very marginally below the black curve representing ${E(D}_{0})$ for lower values of the CoV, and the red curves rise above the black curves for larger values of CoV.

At the grid points considered in the top row in Fig. 6 (i.e., where $p(t) > 0$ for all $t$ during Period 1, in all ensemble members), the CoVs are all <1.0 for the annual data and 1.75 for the seasonal data. This restricts the value of $E(D)$. The largest values evident are ~17.5% (annual), 29% (DJF), and 42% (JJA).

The scatter plots in the bottom row also include data from grid points at which ${p}_{1}(t)=0$ for some $t$ during Period 1. This is a situation not considered in the derivation of the equations, and so the theory does not necessarily apply. Nevertheless, $E(D)$ again tends to increase as the CoV increases. However, much larger CoVs now occur (up to approximately 1.0, 2.6, and 2.2 for annual, DJF and JJA data, respectively), and very much larger values of $E(D)$ occur with many values over 10%, 1000% and 100% for annual, DJF, and JJA data, respectively. $E(D)$ again tends to be greater than ${E(D}_{0})$ in regions where precipitation is projected to increase (blue dots and curves), and the results in regions where external forcing decreases precipitation are not consistent with theory.

In summary, the theory is consistent with several, but not all, of the climate model results in regions where ${p}_{1}(t) > 0$ for all $t$ during Period 1. The theory underestimates $E(D)$ in some regions where the CoV is largest. Similar comments can be made for regions where ${p}_{1}^{\,j}(t)=0$ for some $t$, in at least some ensemble members $j$, though the underestimation is much greater in some regions where the CoV is largest.

As the theoretical and the model results indicate that the CoV plays a major role in setting the value of $E(D)$, we will now examine the CoV in more detail, in models and observations.

Maps showing the spatial distribution of the CoV are given in Fig. 7 for observations (a–c) and models (d–f; MMEMs) for annual, DJF, and JJA, respectively.

Fig. 7: The Coefficient of Variation (CoV) in observations and models.

Observations (GPCP; a–c) and models (d–f), for annual (a, d), JJA (b, e), and DJF (c, f). All based on 1979–2014. Model values are multi-model ensemble means (MMEMs).

The CoV generally lies below 0.5 in the annual data, and below 1.0 in the seasonal data. However there are exceptions in several regions. For example, the CoV exceeds 1.5 in the eastern equatorial Pacific, and it exceeds 2.0 in the equatorial Pacific, North Africa and the Middle East during JJA. Larger values occur more extensively in the monthly data (Fig. S5).

The global average CoVs in the observations are 0.216, 0.425, 0.349, 0.630, 0.511 for annual, DJF, JJA, February and August data. Approximately 0.27%, 4.76%, 2.33%, 13.70% and 7.06% of Earth have CoVs > 1.0. The corresponding MMEM global averages are 0.183, 0.380, 0.342, 0.595, 0.528.

Approximately 0.01%, 1.0%, 0.72%, 3.07%, and 1.44% of Earth have observed CoVs > 1.5, while 0.13%, 1.70%, 1.31%, 3.53% and 3.06% of Earth have MMEM CoVs > 1.5. If we restrict attention to regions where $p(t) > 0$ for all available data, these last figures remain the same or very similar (0.14%, 1.70%, 1.31%, 3.53%, and 3.06%).

Dry years in observations and models

As noted in the Introduction, $\% \overline{\Delta {P}^{j}}=\frac{1}{{N}_{m}}{\sum }_{j=1}^{{N}_{m}}\{ \% \Delta {P}^{j}\}$ cannot be used if any model has ${P}_{1}^{j}=0$, i.e., if the average value of precipitation over Period 1 in model $j$ is zero. This can occur if (i) the precipitation is permanently zero so that the mean of precipitation in Period 1 in model j, ${\mu }_{1}^{j}$, is zero, or (ii) if ${\mu }_{1}^{j} > 0$, but if the sample ${P}_{1}^{j}$ just happens to be zero by chance for some $j$.

Maps showing the percentage of years that precipitation fell to zero (i.e., percentage of years t, in which p(t) = 0) in three different observational data sets, and their MMEM counterparts, are given in Fig. S6. The % area of the Earth in which p(t) falls to zero in at least one year, varies between 0–0.05%, 0.81–2.07%, 0.53–2.24%, for annual, DJF, and JJA, in the observational data sets. The equivalent figures for February and August are 4.13–13.72% and 3.88–6.73%, respectively. The corresponding MMEM % area is greater than its observational counterparts except in February (MMEM values: 4.69, 3.62, 5.41, 7.85, and 7.51, respectively).

Only the IPSL-CM5A2-INCA and IPSL-CM6A-LRa models simulate ${\mu }_{1}=0$ (in DJF and February). Though the extent of such regions is limited to North Africa. Note that we cannot be 100% certain that ${\mu }_{1}=0$ because it may be that if the ensembles were larger or the internal variability in the real world had been different, precipitation in at least one year might have occurred. We can only say that no precipitation at all fell during 1979–2014, and that ${\mu }_{1}=0$ is a possibility.

The only models that have grid points with $p(t)=0$ in some years, but not all years, are the IPSL models (annual data), the IPSL models, ACCESS ESM1.5 and UKESM1-0-LL (DJF), and the IPSL models, ACCESS ESM1.5 and Can ESM5 (JJA). Regions with ${\mu }_{1}=0$ or $p(t)=0$ in some, but not all years during Period 1, using February and August data, are more extensive and occur in more models (Fig. S7).

The GPCP and ERA5 observations (see the “Methods” section for further details) have ${\mu }_{1} > 0$ everywhere, for annual, DJF, JJA, February and August, while in the GPCC data set 0%, 0.04%, 0.01%, 0.08% and 0.06% of Earth respectively has ${\mu }_{1}=0$. The locations where ${\mu }_{1}=0$ in the GPCC observations (indicated using magenta crosses in Fig. S6e, f) only occur in very limited regions in North Africa, during JJA and DJF (Fig. S6e, f), as well as individual months August and February (Fig. S7). Precipitation in ERA5 also occasionally drops to zero in this region during JJA and DJF (as well as parts of South Africa and the Middle East during JJA), but only during half or less of the historical period (Fig. S6). This occurs more extensively in the monthly data (Fig. S7).

Alternative presentation of seasonal precipitation change

Given the widespread presence of the inaccuracy when the standard method is used to estimate % change, and the fact that some models only have single runs available, it is of interest to know what the MMEM changes in precipitation actually are. As maps showing estimates of $\varDelta P$ are not available in the latest IPCC report or in the IPCC Atlas, they are presented here in Fig. 8a DJF and 8b JJA.

Fig. 8: Projected changes in MMEM precipitation from 1995–2014 to 2080–2100.

a DJF and b JJA. Results are based on simulations from the CMIP6 multi-model ensemble under the SSP3-7.0 scenarios. Stippling indicates regions where at least 70% of models agree on the sign of change.

Figure 8 indicates that increases in precipitation are more widespread than decreases during both DJF and JJA. The largest increases tend to occur near the equator, while the greatest drying occurs in the tropics and subtropics, generally poleward of large increases. The magnitude of some changes in JJA exceeds the magnitude of changes evident anywhere in the world during DJF.

During JJA, the greatest drying occurs in the far eastern equatorial Pacific in the Northern Hemisphere and over nearby Central America, with a large secondary maximum in the northeast Indian Ocean, southwest of Sumatra (Indonesia), Singapore and Malaysia. Large increases are also evident over Burma, Bangladesh and western India.

Both the Arctic and Antarctic display increases almost everywhere, and in JJA there is a band of drying between 15°S and 40°S that extends right around the globe, incorporating the Indian, Pacific and Atlantic Oceans, much of Australia and southern Africa, and parts of South America.

In the southern hemisphere, the greatest drying over land occurs over Chile in both DJF and JJA, whereas the largest increase over land during winter occurs over the South Island of New Zealand.

The sign of change is the same in JJA and DJF in many locations. For example, precipitation increases are evident in both seasons in the Arctic, Antarctic, equatorial Pacific, northern Norway, northern Russia, eastern China, Alaska, northern Canada, Greenland, and northern Japan, while drying is evident in both seasons in the Mediterranean, southern Spain, far northern Africa, much of the eastern South Pacific and southeastern Indian Ocean, the Great Australian Bight, central Chile, parts of northern South America, southwestern Australia, and parts of southwestern Africa. In some areas, the signs of change are different, with wetter DJFs and drier JJAs projected in much of Europe (including the United Kingdom), the western half of Russia, the central USA, southern Florida, southeastern Brazil, eastern and northern Australia, eastern South Africa, and southern Indonesia. The last combination, i.e., a drier DJF and a wetter JJA, is only projected in a smaller area of land, including a patch covering parts of Bangladesh, Nepal, parts of southeast Asia and southwest China, and another patch covering Taiwan and another over the southern tip of Chile.

Discussion

We showed that a widely used, standard approach to estimating percentage change is problematic because the Expectation Value of the standard estimate (i.e., $E( \% \overline{\Delta {P}^{j}})$; see Table 1 and the Introduction), differs from the population mean of percentage change (i.e., $\% \varDelta P$; see Table 1 and the Introduction). In other words, even with an infinite amount of data so that sampling error is not an issue, the standard method would still inaccurately estimate the modelled % change in precipitation.

As noted in the Introduction, statisticians refer to statistics with Expectation Values that differ from the corresponding population means as biased statistics. So, in this sense, $\% \overline{\Delta {P}^{j}}$ is a biased statistic. However, we largely avoided the use of the term “bias” to this point because in climate science bias typically refers to systematic model errors arising from deficiencies in either the climate model and/or the forcing applied to the model. So for climate scientists bias is a difference between a model solution and reality, whereas statistical bias is a nonzero difference, $D$, between the true (or population mean) value of the modelled change (i.e., $\% \varDelta P$) and the Expectation Value of the estimation of that change (i.e., $E( \% \overline{\Delta {P}^{j}})$).

In practice, we have finite samples, and so $\% \overline{\Delta {P}^{j}}$ will differ from $\% \varDelta P$ for that model for two reasons: sampling error and the fact that the statistic is biased. This means that the standard method will misrepresent the percentage change in precipitation in the model ensemble, even if an infinitely large ensemble is available.

We derived an equation for the discrepancy (i.e., for the Expectation Value of $D= \% \overline{\Delta {P}^{j}}- \% \varDelta P$; i.e., Eq. (1)) in a wide range of cases in which precipitation has a Gamma distribution. Gamma distributions have the property that precipitation is always nonzero (i.e., $p(t) > 0$ for all time $t$), and—depending on the parameter values—the distributions are Gaussian-like (though with one tail, not two), highly skewed, or something in between. We showed that precipitation is always greater than zero in the vast majority of the world for the annual, seasonal and monthly data in climate models and observations (Figs. S6 and S7).

Equation (1) shows that $E(D)$ increases as the coefficient of variation (CoV) increases, and as the number of years in the reference period decreases. Equation (1) also shows that $E(D)$ is positive for practical parameter values and it increases further if external forcing causes an increase in precipitation.

The theory is consistent with several, though not all, of the CMIP6 climate model results in regions where ${p}_{1}(t) > 0$ for all $t$ during Period 1. For example, $E(D)$ is positive even if there is no externally-forced change, and it tends to increase as the CoV increases (Figs. 3 and 6) or if EFC increases precipitation (Fig. 6), all of which is consistent with theory.

The largest values of $E(D)$ evident in the ACCESS ESM1.5 climate model in regions where precipitation in Period 1 is always positive (Fig. 6, top row) are approximately 17.5% (annual) 29% (DJF), and 42% (JJA). The equivalent figures for February and August (not shown) are 105% and 125%, respectively. The CoV and $E(D)$ can be much larger in regions where ${p}_{1}(t)=0$ for some years, with many values over 10%, 1000%, and 100%, for annual, DJF, and JJA (Fig. 6, bottom row).

The theory underestimates $E(D)$ in some regions where the CoV is largest. Similar comments can be made for regions where precipitation is zero for some $t$, in some ensemble members, and the underestimation can be much greater in some regions where the CoV is largest.

Discrepancies between theory and climate model output can arise because the theory assumes that (a) precipitation is always positive, (b) the climate during Period 1 (and Period 2) are both stable, (c) precipitation has a Gamma distribution, and (d) precipitation has zero autocorrelation between successive years, successive winters, or successive summers, and from one February to the next, and from one August to the next. We also assumed that (e) $100{{\rm {EM}}}(\Delta {P}_{\mathrm{1,2}}^{k})/{{\rm {EM}}}\left({P}_{1}^{k}\right)$) provides a good estimate of $\% \Delta P$ in our large ensemble of the ACCESS ESM1.5 climate model (which has 40 members). Assumption (a) is not true everywhere (see lower panels in Figs. 6 and 7 in which ${p}_{1}(t)=0$ for some $t$), (b) the model climate is actually changing during both Periods 1 and 2 in response to external forcing, (c) precipitation does not have a Gamma distribution at every grid point (we know this because precipitation is zero in some years and this is not a feature of a Gamma distribution), (d) model output (and observations) does exhibit autocorrelation in some locations²¹, and (e), $100{{\rm {EM}}}(\Delta {P}_{\mathrm{1,2}}^{k})/{{\rm {EM}}}\left({P}_{1}^{k}\right)$ will typically differ from $\% \Delta P$ to some extent because of sampling error even in this large ensemble. Equation (1) is nevertheless helpful in understanding many aspects of the inaccuracy.

Clearly, alternatives to using $\% \overline{\Delta {P}^{j}}$ are needed. If one has a large ensemble for a given model, one can instead use 100EM($\Delta {P}^{j}$)/EM(${P}_{1}^{j}$) rather than the standard approach, although one must ensure that there are at least some years in Period 1 with nonzero precipitation, and that there are sufficient nonzero values to be reasonably sure of obtaining a reliable estimate of the population mean for Period 1, ${\mu }_{1}$.

A commonly used standard approach^{5,6,7,8,9,10,11,12,13,14,15,16} is used to provide estimates based on all available CMIP models, even though some models might only have a very limited number of runs available. So, in this case it is not possible to reliably estimate 100EM($\Delta {P}^{j}$)/EM(${P}_{1}^{j}$), and so the resulting estimate of % change using the MMM value of 100 EM($\Delta {P}^{j}$)/EM(${P}_{1}^{\,j}$) will be (statistically) biased.

A simple way to avoid this inaccuracy is to calculate and present changes, rather than fractional or percentage changes, i.e., MMEM($\Delta{P}_{\mathrm{1,2}}^{j}$), as shown in Fig. 8. This figure shows that during JJA the largest drying occurs in the far eastern equatorial Pacific in the Northern Hemisphere and over nearby central America, with a large secondary maximum in the northeast Indian Ocean. Large increases are also evident over Burma, Bangladesh and western India. Some changes evident in JJA exceed the magnitude of changes evident anywhere in the world during DJF. In the Southern Hemisphere, the largest drying over land occurs in Chile during both DJF and JJA, while the largest increase over land during southern winter (i.e., JJA) occurs over the South Island of New Zealand. Further details are provided in the previous section.

While this measure of change is free of the inaccuracy we have focused on in this study, it, too, has limitations. For example, it will, in general, still exhibit biases (in the sense it is typically used by climate scientists) arising from imperfections in either the climate models or the forcing applied to them²⁰, not statistical bias. In addition, a change of a given magnitude (e.g., 0.5 mm/day) will tend to be much more impactful in dry climates than it will tend to be in wet climates, and so some users may need to have estimates of percentage changes, e.g., for impact assessment.

Also note that using change rather than fractional or percentage change is not new. While the reasons were not stated, it was actually used in the First, Second, and Fourth Assessment Reports of the IPCC^22,23,24,25. This paper provides a detailed description of why this is a worthwhile approach.

Additional metrics for assessing change in multi-model ensembles with one run per model include (i) the multi-model median value of the mean precipitation changes evident in the models, i.e., ${{\rm {MMMedian}}}({\Delta P}_{\mathrm{1,2}}^{k})$, (ii) the MM median value of the modelled changes in median precipitation, i.e., ${{ {MMMedian}}}\left({\Delta M}_{\mathrm{1,2}}^{k}\right)={{ {MMMedian}}}({M}_{2}^{k}-{M}_{1}^{k})$, where ${M}_{j}^{k}$ is the median value of the precipitation in individual years in Period $j$, and (iii) the percentage change in the MM median value, i.e., $100* \frac{{{ {MMMedian}}}\left({\Delta M}_{\mathrm{1,2}}^{k}\right)}{{{ {MMMedian}}}({M}_{1}^{k})}$.

Metrics for assessing change in a large ensemble of a single model include the percentage change in the ensemble mean (which we used in our analysis of the ACCESS large ensemble), and the percentage change in the ensemble median.

In the more general case in which we have a multi-model ensemble with multiple runs for some, but not all, models, one could try using the MM ensemble median of the modelled changes in the median, i.e., ${{\rm {MMEMedian}}}({\Delta {{\rm {EM}}}}_{\mathrm{1,2}}^{k})$. If model $k$ has more than one run then ${{{\rm {EM}}}}_{\,j}^{k}$ is the median value of all precipitation values present in the set created by pooling all years in all ensemble members available for model $k$ in Period $j$. If, on the other hand, model $k$ only has one run, then ${{{ {EM}}}}_{\,j}^{k}={M}_{\,j}^{k}$.

Other metrics for a multi-model ensemble with multiple runs for at least some models include using the percentage change in (i) the MM ensemble mean, ($100* \frac{{{\rm {MMEM}}}\left({\Delta P}_{\mathrm{1,2}}^{k}\right)}{{{\rm {MMEM}}}({P}_{1}^{k})}$), and in (ii) the MM ensemble median ($100* \frac{{{\rm {MMEMedian}}}\left({\Delta {{\rm {EM}}}}_{\mathrm{1,2}}^{k}\right)}{{{\rm {MMEMedian}}}({{{\rm {EM}}}}_{1}^{k})}$).

We hope to examine at least some of these metrics in a future study.

It would be useful if future papers and reports provide full details on how the change is estimated. For example, if percentage (or fractional) change is used in future studies, it would be useful to know how grid points with $p(t)=0$ for all years in the reference period, in at least some ensemble members, were treated. Strictly speaking, the statistic MMEM($\Delta {P}_{\mathrm{1,2}}^{\,j}$) is not defined if ${P}_{1}^{\,j}=0$ for any member j. We saw that this occurred in the two IPSL models in some locations. Note also that if $p(t)=0$ for some, but not all years, at least one member with ${P}_{1}^{\,j}=0$ might arise if a larger ensemble was available. In both these cases, the standard approach produces ill-defined statistics.

Methods

We examine annual, seasonal (December–February (DJF), June–August (JJA)) and monthly (February and August) data.

Climate models and climate model output

The CMIP6 climate models used in this investigation are given in Table S1. We use precipitation from preindustrial control simulations and results under both historical and future forcing²⁵. For the future forcing, we use the middle-to-high emissions scenario corresponding to Shared Socioeconomic Pathway (SSP) 370.

As each CMIP6 model has native grids of different resolutions, all climate model output is bilinearly interpolated to a common 1° by 1° grid.

For the preindustrial control runs, we use quadratically detrended output from the last 200 years of available output to reduce any artificial climate drift. One run per model is used. Key results are presented using multi-model means (MMMs).

For the historical and future periods, all available runs are used, and results are given using multi-model ensemble means (MMEMs) for those models that have more than one ensemble member. We examine the large ensemble with 40 members available for the ACCESS-ESM1.5 model. This model has been described in detail previously²⁶.

In calculations of the Coefficient of Variation only, Historical output from 1979 to 2014 is quadratically detrended to ensure that we are capturing only internal variability. However, in calculations of the metrics for forced changes, raw precipitation values are used.

In calculations of precipitation change, for consistency with a recent IPCC report⁸, we compare the periods 1995–2014 from the Historical scenario with 2080–2099 from SSP370. Finally, the figure given in the “Abstract” (i.e., 5.7%) is based on an average over 23°S–23°N.

Observational data

Observational data is only used to estimate the Coefficient of Variation (CoV) and to identify where precipitation falls to zero in individual years, seasons or months.

We use observations from three different sources over the 1979–2014 period, common with models’ historical runs. We use the Global Precipitation Climatology Project (GPCP) dataset, which provides monthly mean satellite-based observations spanning 1979 to present at 2.5° spatial resolution²⁷. This is supplemented by reanalysis output from the ECMWF Reanalysis v5 (ERA5) dataset available since 1940 on a 30-km global grid²⁸, as well as monthly precipitation from the Global Precipitation Climatology Centre dataset version 2.3 (GPCC²⁹), which is calculated from global station data and available since 1971.

Derivations

Mathematical formulae for the expectation value of $D$ are derived for the case where precipitation, $p(t),$ in individual years, seasons (e.g., December–February) or months (e.g. August) of interest, is randomly sampled from a Gamma distribution. The Gamma distribution, depending on parameter choices, incorporates a wide range of distributions with skewing to the right that is very weak (when the distribution resembles a bell-shaped curve with no negative tail) to very strong²⁹. In some of the cases examined, it is assumed that external forcing alters at least one of the parameters used to define the Gamma distribution.

The gamma distribution also has the property that $p(t) > 0$ for all $t$, where $t$ = time. The formulae derived are average values of random variables defined below. Consequently, the theoretical estimates strictly only apply where this condition is met.

Monte Carlo simulations

To help interpret some of the results, we conducted Monte Carlo experiments. In these experiments, time series of precipitation are created by randomly choosing values for individual years from gamma distributions for two 20-year periods, one with parameters ${k}_{1}$ and ${\theta }_{1}$, another with parameters ${k}_{2}=\lambda {k}_{1}$ and ${\theta }_{2}={\lambda \theta }_{1}$. Three sets of parameters are considered for the first/reference 20-year period: (i) ${k}_{1}=0.4444$ and ${\theta }_{1}=80$ (for which CoV = 1.5); (ii) ${k}_{1}=\,1.0$ and ${\theta }_{1}=80$ (CoV = 1.0), and (iii) ${k}_{1}=4.0$ and ${\theta }_{1}=80$ (CoV = 0.5), and in each case three values of $\lambda$ are considered for the second 20-year period: 0.5, 1.0, and 1.5. Twenty thousand simulations are produced for both periods, and the median and average values of the contributions to $D$ from each set of 20-year simulations is recorded. This whole process is repeated three times, giving a total of 1,440,000 years of simulation and a total of four estimates of the mean and median for each set of parameters.