We use the example of UTCI and WBT to explain our method of comparison between HSIs but expand to comparison of more HSIs in a later section.
Figure 1 shows isopleths (lines of constant value) for two HSIs plotted on a chart of air temperature and relative humidity (RH). The same is shown using specific humidity in Supplementary Fig. 1. Although both UTCI and WBT are in units of °C, their values are not directly comparable. Firstly, they intersect at multiple points: for example, a WBT value of 20°C could correspond to UTCI values ranging from just over 25°C to over 40°C for different combinations of temperature and RH (Fig. 1). Secondly, they are clearly calibrated to different scales: the 35°C isopleth in WBT approximately corresponds to the 55°C isopleth in UTCI.
Visually comparing the gradients of the isopleths in Fig. 1 gives an intuitive way of comparing these indices. Where the line is steeper, the HSI is more sensitive to temperature; where the line is shallower, the HSI is more sensitive to humidity. At high values (WBT = 35°C and UTCI = 55°C) the WBT and UTCI lines are nearly parallel: this means that the two indices agree on the relative importance of temperature and humidity, but only for these high values. For most of the temperaturehumidity space the gradients of the lines are very different between the indices, showing that they disagree about the relative importance of marginal changes to temperature and humidity.
For this example, we use relative humidity because it will be most familiar for most readers, but the same process could be applied to a different measure of humidity. Calculations are made with all else being equal, so there is implicitly a small change in the specific humidity of the air parcel if relative humidity is held constant with changing temperature and vice versa.
We now describe this approach using calculus of variations to generalise the argument. Firstly, consider a point in the temperaturehumidity space \((T,h)\) where T is temperature and h is humidity; the HSI calculated at that point is \(U(T,h)\). We can imagine increasing the temperature by a small amount \(\delta T\), which will change the heat stress to \(U(T+\delta T,h)\) assuming humidity remains the same. Then, we can imagine increasing the humidity by a small amount \(\delta h\), which will change the heat stress to \(U(T,h+\delta h)\).
If \(\delta T\) were fixed and \(\delta h\) were solved to give the same change in the HSI \(U\), i.e., solving
$$U(T+\delta T,h)U(T,h)=U(T,h+\delta h)U(T,h)$$
for \(\delta h\) at fixed \(\delta T\), then the quantity \(\frac{\delta h}{\delta T}\) would tell us what change in humidity would produce a change in \(U\) equivalent to a \({1}^{o}C\) change in temperature. This is illustrated in Fig. 2. If \(\delta T\) and \(\delta h\) are small, then this is equivalent to
$$\frac{\delta h}{\delta T}=\frac{dU}{dT} / \frac{dU}{dh}$$
It is more practical numerically to estimate \(\frac{dU}{dT}\) and \(\frac{dU}{dh}\) directly using a finite differences method, rather than solving the equation. If \(\delta T\) and \(\delta h\) are small, then we can estimate the onedimensional gradients of the function \(U(T,h)\) using the forward difference approximation:
$$\left(U\right(T,h)U(T+\delta T,h\left)\right)/\delta T\approx \frac{dU}{dT}(T,h)$$
and
$$\left(U\right(T,h)U(T,h+\delta h\left)\right)/\delta h\approx \frac{dU}{dh}(T,h).$$
We choose to do this numerically because not all the HSIs are analytically differentiable, and we choose this approximation because it is easiest to explain.
We will call the gradient of the isopleth of a heat stress function \(U\) in temperaturehumidity space the point \((T,h)\) the marginal temperatureequivalent change (MTEC)
$$MTEC=\frac{dU}{dT}/\frac{dU}{dh}.$$
MTEC can be interpreted as how much of a change in humidity produces a change in the HSI that is equivalent to a unit change in temperature. The smaller the value of MTEC the more sensitive the HSI is to changes in humidity compared to changes in temperature.
MTEC can be compared freely between different HSIs under the same conditions \((T,h)\) as the HSI is not one of the dimensions of MTEC. This procedure is valid whether humidity is represented as relative humidity, specific humidity, vapour pressure or even WBT; we will use relative humidity in this paper as it is most intuitive. This procedure is not restricted to the temperaturehumidity space, and could be equivalently performed for other variables, for example, in the temperature wind speed space.
To make the comparison between different HSIs more explicit, we can take the difference in MTEC between them. Figure 3c shows the difference in MTEC (\(\varDelta\)MTEC) between UTCI and WBT, subtracting Fig. 3b from Fig. 3a. In this case a positive value of \(\varDelta\)MTEC means that WBT is more sensitive to a change in relative humidity, while a negative value means that UTCI is more sensitive to a change in relative humidity. A value close to zero indicates that the HSIs agree. Generally, humidity is much more important for WBT than UTCI for lowtemperature lowhumidity conditions, and the two HSIs agree under hightemperature highhumidity conditions. UTCI and WBT are closest in MTEC at around 35°C WBT or 55°C UTCI: \(\varDelta\)MTEC is generally below 7.5% for a WBT of > 30°C, below 5.0% for a WBT of 32°C or above, and closest to zero around a WBT of 35°C, that is, the difference in the change in relative humidity required to produce a change in the two HSIs equivalent to a 1°C change in temperature is least in this range. For context, in the present climate WBT rarely exceeds 31°C, but WBT > 35°C occurs in some locations for a few hours at a time and could begin to occur at a large scale with global warming of around 7°C20,21.
Examples of heat stress index selection reversing the conclusion
In this section, we examine two illustrative examples of recent studies where the conclusions were strongly determined by the choice of HSI.
In Wouters et al.11, a combination of observations and atmospheric modelling is used to argue that soil droughts lead to a reduction in heatwave lethality. For selected heatwave events, they use atmospheric modelling to estimate what the effect would be if at the start of the heatwave the soil moisture had been at the mean climatological level for that location rather than its actual value. They found that higher soil moisture leads to lower temperature and higher specific humidity. Based on Mora et al.22 they introduce a new heat stress metric
$${T}_{s}={T}_{w}+4.5(1{\frac{RH}{100}}^{2}).$$
The counterfactual used meant that some cases had soil moisture increased while others had soil moisture reduced; those which had soil moisture increased had increased air humidity and vice versa. They found that lower soil moisture leads to lower values of \({T}_{s}\), which they argued means soil drought reduces heatwave lethality. However, using their Table 1 we calculated changes in a variety of HSIs (our Table 1) which highlights that some other HSIs have the opposite correlation with specific humidity, indicating that increased soil moisture would decrease heat stress.
The reason for this can be understood by comparing MTEC for the various HSIs in temperaturehumidity space for the temperature and relative humidity range of interest for their study. Figure 4 shows the difference in MTEC between \({T}_{s}\) and UTCI (shading), as well as the changes in temperature and humidity given by Wouters et al11 Table 1. In the heatwaves conditions from Wouters et al, UTCI is much more sensitive to temperature (as opposed to relative humidity) compared to \({T}_{s}\) in the same range of conditions. This does not necessarily mean that the choice of HSI in Wouters et al is wrong, but it is clearly problematic that different HSIs give such opposing conclusions.
Table 1
Rank correlation (Kendall’s tau and Spearman’s R) between the changes in specific humidity and the changes in heat stress indices (HSIs) calculated from the temperature and humidity changes induced by soil moisture changes modelled by Wouters et al. A positive number means that increasing humidity (and decreasing temperature as a result) led to an increase in that HSI, and vice versa. While the changes in the HSI used in Wouters et al are positively correlated with the corresponding changes in humidity, other commonly used heat stress indices have weaker or the opposite correlations.

Kendall

Spearman

\({T}_{s}\)

0.66

0.83

WBT

0.85

0.94

UTCI

0.85

0.95

WBGTindoor

0.18

0.24

Humidex

0.33

0.50

Heat index

0.30

0.37

sWBGT

0.64

0.75

Apparent temperature

0.50

0.61

As another example, we examined Mishra et al.12, in which the effect of irrigation was modelled using the Weather Research and Forecasting regional climate model at \({0.25}^{o}\) horizontal resolution driven by ERA5 reanalysis and covering the 2000–2018 period. Two scenarios were modelled: with and without irrigation. Mishra et al. find that irrigation decreases drybulb temperature but increases WBT, and therefore conclude that increased irrigation area can be detrimental to human heat stress. The accuracy of the assumptions Mishra et al. made about when and how much irrigation occurs have been discussed elsewhere23. The conclusion of Mishra et al. is in contrast with the interpretation provided by other studies e.g., Thiery et al10 which found that expansion of irrigation had a large enough effect to cancel out the effect of global warming on temperature extremes in some regions. However, the results in Mishra et al. show that while 95th percentile WBT increased, 95th percentile temperature and HI decreased (their Fig. 4). The increase in WBT was emphasised rather than the decrease in HI, which is not mentioned in the text.
When do commonly used indices agree?
Figure 5 shows MTEC for several commonly used HSIs. MTEC differences between each of the variables are shown in the Supplementary Fig. 4. We have chosen these HSIs to show a range of different outcomes and this is not intended to be exhaustive. MTEC enables comparisons between different conditions (e.g., between hothumid and hot dry conditions) as well as between different heat stress indices even if calibrated to completely different scales. Patterns of agreement and disagreement between these HSIs can be summarised by examining MTEC in three regimes: lower temperature regime (LTR), hothumid regime (HHR), and the hotdry regime (HDR). We define representative values of temperature and humidity for these regimes, but these are only intended to be illustrative. The LTR is the lefthand side of the plot, which we characterise as 20°C temperature and under. The HHR is the top righthand side of the plot, which we characterise at 35°C temperature and 80% relative humidity (this is extremely hot and humid but occurs in the presentday climate). The HDR is the bottom righthand side of the plot, which we characterise as 40°C with 20% relative humidity.
Figure 5 shows that AT, HI, and UTCI are mainly sensitive to temperature in the LTR. In fact, HI is defined as equal to temperature when below 26.7°C (80°F). In comparison, WBT has very low MTEC even in the LTR. Humidex, WBGTindoor, and sWBGT are slightly higher MTEC in the LTR compared to WBT, but clearly have lower MTEC compared to AT, HI, UTCI. Physiologically, we would not expect the effect of humidity to be important in the LTR: changes in the air temperature will dominate changes in heat balance as the sweat evaporation rate required to maintain heat balance is low24.
In the HDR of Fig. 5, we see a different pattern of disagreement between the HSIs. MTEC is relatively low in AT, WBT, humidex, WBGTindoor, and sWBGT. In contrast, HI and UTCI have an area of relatively high MTEC extending into the HDR, more extensive in UTCI. In the HDR, the sweat evaporation rate required to maintain a stable body temperature is high but sweat evaporation efficiency is also high as the vapour pressure is low25. Therefore, it makes sense that MTEC is low. However, if one assumes that there is a limit to the sweat secretion rate, the HDR is the region where that limit is most likely to be reached. If the sweat secretion rate is reached, then increasing the relative humidity (and therefore the vapour pressure) will not necessarily decrease the sweat evaporation rate as there is no more sweat to evaporate25,26. Sweat secretion rate limits in the underlying model may explain the higher MTEC values seen in the HDR for UTCI; this can be reproduced in a heat balance model (e.g. ISO standard 7933:2004) by reducing the sweating rate limit.
In the HHR, we would expect changes in relative humidity to be at their most important, as it is the regime where vapour pressure is likely to be the limiting factor in sweat evaporation rate25. AT, HI, UTCI, WBT, humidex, WBGTindoor, and sWBGT all have low MTEC in the HHR; so are roughly in agreement.
UTCI displays some unexpected properties at very high temperature and humidity and at very low humidity. This is an artefact of the UTCI being calculated as a polynomial beyond the range for which is it well conditioned and is related to Runge’s phenomenon wherein highdegree polynomial interpolations show large oscillations at the edges of the domain. We prevented this behaviour in this study by limiting the UTCI calculating to vapour pressure values below 7 kPa, which is approximately the saturation vapour pressure at 40°C. The values at which this occurs are so extreme that it is not a problem for general use of the UTCI but could limit its applicability in future extreme conditions. We have made assumptions about the mean radiant temperature and wind to calculate the UTCI, but our general qualitative description is still valid under different assumptions, see the Supplementary Fig. 3.