Analysis: Sequential-averaging components of the emissions boost factor

Note: This page is highly mathematical.

The temperature-variation emissions boost factor, $\tveb = \ex{T^4}/\ex{T}^4 - 1$ , is a quantity that emerges as a result of averaging.

In practice, averaging is likely to be done in a variety of ways:

over various time windows, such as day, month, season, or year
over longitude (a “zonal” average), or globally (over both longitude and latitude)

Various types of averaging may be done in sequence.

So, it will potentially be useful to be able to talk about the contributions to the emissions boost factor that are associated with each averaging operation.

Setup

Suppose quantities of interest are subject to a sequence of $n$ averaging processes. Denote averaging process $k$ of quantity $X$ by $\ex{X}_i$ . Also, denote the composite process of averages $i$ through $k$ :

(1) $\begin{equation*} \ex{X}_{i:k} = \ex{\ldots\ex{\ex{X}_i}_j\ldots}_k \end{equation*}$

It will be convenient to us use the notation $\ex{X}_0$ to refer to the unaveraged value of $X$ .

Decomposition A

Define:

(2) $\begin{equation*} T_{k} = \ex{T}_{0:k} \end{equation*}$

$T_{k}$ is the temperature as averaged by the first $k$ averaging processes. Also, define:

(3) $\begin{equation*} E_{k} = \sigma \ex{E}_{0:k} \end{equation*}$

where $E = \sigma T^4$ and $\ex{E}_{0:k}$ is the value of this as averaged by the first $k$ averaging processes. Then, define:

(4) $\begin{equation*} U_{k} = \ex{ \frac{E_{k}}{\sigma T_{k}^4}}_{k+1:n} = \ex{ \frac{\ex{E}_{0:k}}{ \left( \ex{T}_{0:k} \right)^4}}_{k+1:n} \end{equation*}$

and take $U_0 = 1$ . Basically, $U_k$ is $\tveb$ as calculated after the first $k$ averaging operations, and then with the value of $V$ averaged. Note that $\tveb = U_n$ .

We can then define:

(5) $\begin{equation*} 1+V_{k} = \frac{U_k}{U_{k=1}} \end{equation*}$

It follows that:

(6) $\begin{equation*} 1+\tveb= (1+V_1) \, (1+V_2) \cdots (1+V_{n}) \end{equation*}$

so we can interpret $V_k \ge 1$ as the contribution of averaging number $k$ to the total value of $\tveb$ .

Decomposition B

Define:

(7) $\begin{equation*} B_{k} = \left( \ex{T^4}_{0:k} \right)^\frac{1}{4} \end{equation*}$

If we take emissivity $\emis$ to be 1, or assume that $T$ is actually a surface “effective temperature” (i.e., “effective radiative emission temperature”) then $B_{k}$ can be interpreted as the “effective temperature” deduced from averaging the emitted radiative flux through the first $k$ averaging processes. Note that $B_0 = T_0$ and $B_n^4 = \ex{T^4}_{1:n}$ .

Define:

(8) $\begin{equation*} F_{k} = \sigma \, \left( \ex{ B_k}_{k+1:n} \right)^4 = \sigma \, \left( \ex{ \left( \ex{T^4}_{0:k} \right)^\frac{1}{4}}_{k+1:n} \right)^4 \end{equation*}$

Note that $F_0 = \sigma \,\ex{T}^4$ and $F_n = \sigma \,\ex{T^4}$ .

We can express $\tveb$ in terms of $B_{k}$ as follows:

(9) $\begin{equation*} 1+\tveb = \frac{\ex{T_0^4}_{1:n}}{\left(\ex{T_0}_{1:n}\right)^4} = \frac{F_n}{F_0} \end{equation*}$

This can be written as:

(10) $\begin{equation*} 1+\tveb = \frac{F_1}{F_0} \cdot \frac{F_2}{F_1} \cdots \frac{F_n}{F_{n-1}} \end{equation*}$

(11) $\begin{equation*} 1+\tveb = (1+W_1) \, (1+W_2) \cdots (1+W_{n}) \end{equation*}$

where

(12) $\begin{equation*} 1+W_{k} = \frac{F_{k}}{F_{k-1}} = \frac{ \left( \ex{ B_k}_{k+1:n} \right)^4 }{ \left( \ex{ B_{k-1}}_{k:n} \right)^4 } \end{equation*}$

$W_{k}$ is always greater than or equal to 0.

Decomposition C

Define:

(13) $\begin{equation*} T_{k} = \ex{T}_{0:k} \end{equation*}$

$T_{k}$ is the temperature as averaged by the first $k$ averaging processes. Also, define:

(14) $\begin{equation*} F^{\prime}_{k} = \sigma \, \ex{ T_k^4 }_{k+1:n} = \sigma \, \ex{ \right( \ex{T}_{0:k} \left)^4 }_{k+1:n} \end{equation*}$

Note that $F^{\prime}_0 = \sigma \,\ex{T^4}$ and $F^{\prime}_n = \sigma \,\ex{T}^4$ .

The temperature-variation emissions boost factor, $\tveb$ , is given by:

(15) $\begin{equation*} 1+\tveb = \frac{\ex{T_0^4}_{1:n}}{\left(\ex{T_0}_{1:n}\right)^4} = \frac{F^{\prime}_0}{F^{\prime}_n} \end{equation*}$

This can be written as:

(16) $\begin{equation*} 1+\tveb = \frac{F^{\prime}_0}{F^{\prime}_1} \cdot \frac{F^{\prime}_1}{F^{\prime}_2} \cdots \frac{F^{\prime}_{n-1}}{F^{\prime}_n} \end{equation*}$

(17) $\begin{equation*} 1+\tveb = (1+W^{\prime}_{1}) \, (1+W^{\prime}_{2}) \cdots (1+W^{\prime}_{n}) \end{equation*}$

where

(18) $\begin{equation*} 1+W^{\prime}_{k} = \frac{F^{\prime}_{k-1}}{F^{\prime}_k} = \frac{ \ex{ \right( \ex{T}_{0:k-1} \left)^4 }_{k:n} }{ \ex{ \right( \ex{T}_{0:k} \left)^4 }_{k+1:n} } \end{equation*}$

$W^{\prime}_{k}$ is always greater than or equal to 1.

Discussion

We’ve developed three distinct ways of decomposing a series of successive averages.

Decomposition A assumes we have data for both $T$ and $T^4$ which is well-resolved both geographically and temporally. Given that well-resolved information, it offers $V_k$ as the contribution associated with that level of temperature non-uniformity.

Decompositions B and C assume that up until some level of aggregation, we only have accurate information about either $\ex{T^4}$ (in the case of decomposition B) or $\ex{T}$ (in the case of decomposition C), but not both. It’s assumed that at that point one tries to infer the value of the unknown quantity from the known quantity. The decompositions into $W_k$ or $W^{\prime}_k$ indicate the loss in accurate estimation of $\tveb$ that occurs if one aggregates through that level before beginning to work with separate averaging of $T^4$ and $T$ .

These generic decomposition results can be applied to averaging over time and location, or over different time windows, or over different granularities of geographic location (e.g., regional, zonal, or global).