Analysis: Location and time components of the emissivity boost factor

The surface temperature, $\tsurf$ , varies both spatially, between different locations on Earth, and temporally, between different times. So, it may be of interest to examine what how much of the temperature-variation emissivity boost factor, $\tveb$ , depends on each type of variation. How we assess these two contributions will depend on whether we average over time or location first.

Case: First average over time window

First, we’ll address the case where time averaging occurs first.

(1) $\begin{equation*} \ex{\tsurf^4} = \ex{\ex{\tsurf^4}_t}_\loc = \ex{\frac{\ex{\tsurf^4}_t}{\ex{\tsurf}_t^4} \ex{\tsurf}_t^4}_\loc = \ex{\gammasub{T:t} \cdot \ex{\tsurf}_t^4}_\loc \end{equation*}$

where

(2) $\begin{equation*} \gammasub{T:t} = \frac{\ex{\tsurf^4}_t}{\ex{\tsurf}_t^4} \end{equation*}$

We can further rewrite $\ex{\tsurf^4}$ as:

(3) $\begin{equation*} \ex{\tsurf^4} = \frac{\ex{\gammasub{T:t} \cdot \ex{\tsurf}_t^4}_\loc }{\ex{\ex{\tsurf}_t^4}_\loc }\frac{\ex{\ex{\tsurf}_t^4}_\loc}{\ex{\tsurf}^4} \ex{\tsurf}^4 \end{equation*}$

Based on this, we can write:

(4) $\begin{equation*} \tveb = \avg{\gammasub{T:t}} \cdot \gammasub{T:t:\loc} \end{equation*}$

where

(5) $\begin{equation*} \avg{\gammasub{T:t}} = \exw{\gammasub{T:t}}{\ex{\tsurf}_t^4}_\loc = \frac{\ex{\gammasub{T:t} \cdot \ex{\tsurf}_t^4}_\loc }{\ex{\ex{\tsurf}_t^4}_\loc} = \frac{\ex{\tsurf^4}}{\ex{\ex{\tsurf}_t^4}_\loc} \end{equation*}$

(6) $\begin{equation*} \gammasub{T:t:\loc} = \frac{\ex{\ex{\tsurf}_t^4}_\loc}{\ex{\tsurf}^4} \end{equation*}$

In summary, the analysis has identified a quantity that depends on latitude and longitude:

$\gammasub{T:t}$ ― the location-dependent temperature-time-variation emissivity boost factor

It has also identified two fully-averaged quantities whose product equals $\tveb$ :

$\avg{\gammasub{T:t}}$ ― the standardly-averaged or global temperature-time-variation emissivity boost factor; this is the time-variation component of $\tveb$
$\gammasub{T:t:\loc}$ ― the post-time-averaging temperature-location-variation emissivity boost factor; this is the location-variation component of $\tveb$

All of these quantities are greater than or equal to 1.

Case: First average over latitude and longitude

If we instead average by longitude and latitude and then by time, we obtain:

(7) $\begin{equation*} \tveb = \avg{\gammasub{T:\loc}} \cdot \gammasub{T:\loc:t} \end{equation*}$

where

(8) $\begin{equation*} \gammasub{T:\loc} = \frac{\ex{\tsurf^4}_\loc}{\ex{\tsurf}_\loc^4} \end{equation*}$

(9) $\begin{equation*} \avg{\gammasub{T:\loc}} = \exw{\gammasub{T:\loc}}{\ex{\tsurf}_\loc^4}_t= \frac{\ex{\gammasub{T:\loc} \cdot \ex{\tsurf}_\loc^4}_t }{\ex{\ex{\tsurf}_\loc^4}_t} = \frac{\ex{\tsurf^4}}{\ex{\ex{\tsurf}_\loc^4}_t} \end{equation*}$

(10) $\begin{equation*} \gammasub{T:\loc:t} = \frac{\ex{\ex{\tsurf}_\loc^4}_t}{\ex{\tsurf}^4} \end{equation*}$

In summary, the analysis has identified a quantity that depends on unaveraged time:

$\gammasub{T:\loc}$ ― the unaveraged-time temperature-location-variation emissivity boost factor

It has also identified two fully-averaged quantities whose product equals $\tveb$ :

$\avg{\gammasub{T:\loc}}$ ― the standardly-time-averaged temperature-location-variation emissivity boost factor; this is the location-variation component of $\tveb$
$\gammasub{T:\loc:t}$ ― the post-location-averaging temperature-time-variation emissivity boost factor; this is the time-variation component of $\tveb$

All of these quantities are greater than or equal to 1.

Conclusion

The formulas provided by this analysis make it possible to factor the temperature-variation emissivity boost factor into contributions from variations in time and variations between locations.