*Note: This page is highly mathematical.*

## Generic temperature variation

Suppose that the temperature is given by:

(1)

where is a temperature-variation with zero mean, so that . Then, the temperature-variation emissions boost factor, or , will be given by:

(2)

(3)

If is small, i.e., , then this may be approximated by:

(4)

From this, it also follows that, if is small, then:

(5)

## Contributions to emissions boost factor

Suppose we re-arrange one of the equations above to write:

(6)

where

(7)

(8)

This allows us to think of or as being made up of contributions from different times and places, contributions that are likely all positive (though I haven’t rigorously proved that must be positive). We might think of as the *local TVEB Contribution*, .

can alternatively be computed as:

(9)

Filling in the definition of , this becomes:

(10)

If is small, this may be approximated as:

(11)

The quantity or is interesting insofar as it offers a way of attributing contributions to or to different places and times (though is ultimately a non-local quantity).

This local version of could have been defined differently: adding or subtracting any multiple of to the definition of would yield a quantity that averages to . So, it’s still unclear, as yet, if this way of defining is meaningful.

## Sinusoidal time variation

Suppose that the temperature varies as:

(12)

where and are constants.

If is small compared to 1, this leads to the approximate result:

(13)

(14)

As an example, suppose 288 K and 5℃. Then this diurnal temperature variation would lead to 0.017, 9e-4 and 1 – 2.3e-4, so that there the emissions boost would lead to a net temperature reduction (for the same average emissions) of about 0.065℃.

Studies show that global warming has been leading to night-time low temperatures increasing more than in day-time high temperatures. In other words, has been decreasing. The above result indicates that the reduction in the amplitude of diurnal temperature variation, , is not likely to contribute significantly to changes in the GMST (global mean surface temperature).