Equivalent Temperature: HadISDH.land and ERA5

The following was contributed by Dr. Tom Matthews, August, 2025:

What are the typical research applications of this data set?

Radiative forcing leads to heat accumulation in the Earth system (von Schuckmann et al., 2020). In the atmosphere, $T$ is an incomplete metric of this accumulation because some of that heat is used to increase the moisture content of the air. Indeed, across much of the tropics and sub-tropics, the majority of near-surface atmospheric heat accumulation in recent decades has been in the latent rather than the sensible term (i.e., the change in $q L_v$ has been larger than the change in $T C_p$) (Matthews et al., 2022). This geographical pattern is expected because the share of latent heat accumulation should, under the assumption of near-constant relative humidity, be higher in warmer climates. Because $T_e$ captures changes in both sensible and latent heat content, it is more suitable than $T$ for tracking atmospheric heat accumulation (Pielke et al., 2004) – especially when comparing rates of change between regions (Matthews et al., 2022). For example, tropical and subtropical regions – including densely populated South Asia and parts of the Amazon basin – generally emerge more as hotspots of warming when $T_e$ is used in place of $T$ (Fig. 1).

Whilst mean $T_e$ is a desirable quantity for tracking heat accumulation, extremes in $T_e$ are also of interest from an impacts’ perspective because this metric’s inclusion of humidity makes it a good proxy for potential heat stress in humans, and for convective precipitation intensity (Matthews et al., 2022; Song et al., 2022). The monthly temporal resolution in HadISDH.land is, however, too low, to assess such impacts directly; convective precipitation extremes and episodes of human heat stress occur at sub-daily timescales. Moreover, some stations contributing to HadISDH.land report at only three- or eight-hourly frequency so the true minimum and maximum values will not contribute to the monthly statistics. In ERA5, $T_e$ can be computed at hourly resolution, but the spatial resolution (0.25°) means that extremes will be underestimated. 

How is uncertainty characterized?

In HadISDH.land, 2 sigma $T_e$ uncertainty (~95% confidence interval) is shown for each grid box month. The uncertainty analysis accounts for errors at the individual station level—including those from measurements, bias corrections, and climatological estimates. It also incorporates sampling uncertainty from representing a 5°×5° grid using point data. For regional average time series, an additional term is included to reflect uncertainty due to missing data within some grid cells (Willett et al., 2014).

In ERA5, uncertainty can be computed using the 10 ensemble members, which accounts for random errors in the observations and parameterizations in the forecast model (ECMWF, 2024).

What are some comparable data sets, if any?

In terms of metrics, the wet-bulb temperature—also available in HadISDH.land (and easily calculated from ERA5 air temperature, dewpoint, and surface pressure)—is another way of expressing the moist enthalpy and is therefore similar to the equivalent temperature. However, the relationship is non-linear, with ever greater increases in moist enthalpy required to generate a 1°C rise as wet-bulb temperatures climb (Fig. 2). Wet-bulb temperature increments can, therefore, downplay rates of heat accumulation in hotter and more humid climates (i.e., where $T_e$ is already high).

HadISDH.land is the only near-surface, in-situ humidity dataset, and hence currently the only the only source of station-based equivalent temperature. Reanalyses products other than ERA5 (e.g., JRA-3Q, and NCEP/NCAR) can be used to compute the equivalent temperature, but users should be aware of any known biases (e.g., the spurious drying in NCEP/NCAR Reanalysis 1: (Dessler and Davis, 2010).

What are the key limitations of the available datasets?

Equivalent temperature in HadISDH.land and ERA5 inherit any issues from the underlying temperature, humidity, and pressure data. For example, some spurious ERA5 changes in tropospheric humidity over oceans in the mid-1990’s are present due to the introduction of microwave imager data (Hersbach, 2024). HadISDH.land $T_e$ is spatially coarse (5°) and spatially incomplete, with tropical Africa – a region expected to have particularly steep Te trends as the climate warms – a notable data gap (Matthews et al., 2022; Willett, 2025). An additional minor limitation of HadISDH.land is the constant, estimated surface air pressure used in HadISDH.land to compute $T_e$ (Willett et al., 2014).

Users should also be aware that the treatment of $L_v$ and $C_p$ in the calculation of temporal averages can introduce error into $T_e$. $L_v$ and $C_p$ are temperature ($L_v$) and humidity ($C_p$) dependent (Raymond et al., 2021), and $T_e$ should ideally be calculated from $L_v$ and $C_p$ at the respective, concurrent $T$ and $q $before averaging, rather than being evaluated from their already time-averaged quantities:$$\bar{T_e} = \overline{T_t + \frac{L_v(T_t)}{C_p(q_t)} q_t} \tag{2}$$in which the overbars denote temporal means, and the subscript $t$ is time. The below is an approximation: $$\bar{T_e} \approx \bar{T} + \frac{L_v(\bar{T})}{C_p(\bar{q})} (\bar{q}) \tag{3}$$ which will contain errors (e.g., a general underestimate reaching approximately -0.6 K in July, 2024: Fig. 3). Users should consider the impact of this for their application and, where possible, use Eq. (2) (e.g., with hourly rather than with monthly-mean ERA5 data) to calculate $T_e$ if the difference is likely to be important. Note that in HadISDH.land, Eq. (2) is used to compute monthly $T_e$.

 How does one best compare these data with model output?

Equivalent temperature is often not directly available in climate model output (e.g., from CMIP6 MIP), but can be readily calculated from near-surface air temperature (‘tas’), specific humidity (‘huss’), and surface pressure (‘ps’).