Lucia Liljegren graciously agreed to permit Climate Science to post her weblog as a guest contribution on Climate Science. Her weblog was motivated by the Climate Science weblog on February 8 2008 titled

An Error In The Construction Of A Single Global Average Surface Temperature

**Lucia Liljegren’s Guest weblog**

In my post at ‘The Blackboard’ on February 13, 2008, I discussed a Global Climate Change Blog kerfuffle over the IPCC equation describing the radiative balance for the Earth’s climate system. The kerfuffle involves one of the conclusions Dr. Roger Pielke Sr. reported in a recent peer-reviewed article and two blog posts (see January 24, 2008 and February 8, 2008 at climatesci.org).

Outside of the blog kerfuffle, Dr. Pielke’s fuller point is rather arcane and relates to the ability of scientists to accurately estimate the magnitude of the climate sensitivity, λ, based on Global Mean Surface Temperature anomalies, T’, as measured and reported using a very specific equation in an IPCC document. Dr. Pielke Sr. made several points on his weblog of February 6 regarding the uncertainties associated with determining the λ, all related to issues associated with the anomaly, T’.

One point in particular, discussed in the following paragraph by Pielke Sr., bothered Eli Rabett:

“Indeed, it is easy to show that weighting by (T+T’)**4 significantly emphasizes the lower latitudes, since the relationship is to the 4th power of temperature. I look forward to your analysis as we have recommended.”

“[This] is just wrong, as anyone who learned about series expansions of functions in Cal I would know.”

Turns out *Eli *is wrong; Dr. Roger Pielke Sr. is exactly right.

The correctness of Dr. Roger Sr.’s claim can be shown *either* using series expansions of functions *or* the simpler arithmetical method Dr. Roger Sr. used. Both give the same results for the magnitude of the effect Dr. Roger Sr.’s describes (as they should.)

Now, it is my opinion, that when possible, it is much wiser to rely on simple arithmetic to illustrate the importance of phenomena: One is less likely to go horribly wrong. However, as Eli claims to have dis-proven Dr. Roger Sr.’s arithmetic using series expansions of nonlinear functions, I think it’s become necessary to illustrate how to do this correctly.

Doing so provides an added bonus: The terms describing the phenomenon Dr. Roger Sr. described will be identified and shown to be leading order mathematically. We’ll also see the magnitude of the effect is important enough to require including the terms in any empirical analysis to estimate the magnitude of climate sensitivity, λ.

**Conclusions**

For those who don’t want to read the math, the main conclusions, in term of phenomenology and the blog kerfuffle follow.

**Phenomenology**

With regard to the question “Do neglecting spatial variations in surface temperature introduce uncertainty when estimating climate sensitivity using the IPCC equation in question?”

1. Spatial variations in temperature do matter. If I’m not mistaken in my math, we could fix up up the IPCC equation to include their effect, resulting in:

(1) dH/dt = f -( T’/λ + **(3/****λ****) <****δ****T _{o}**

**δ**

**T’> /<T**)

_{o}>where H is the heat content of the land-ocean-atmosphere system, f is the radiative forcing (i.e., the radiative imbalance), λ is called the “climate feedback” parameter, <T_{o}> is the absolute value of the global means surface temperature in the reference case, δT_{o} is the difference between a local surface temperature and the GMST in the reference case (denoted with ‘o’ subscripts) , and δT’ is the difference between the local surface temperature and the GMST at the current time (t).

The final term on the righthand side of (1), shown in bold, describes the effect of variations in the spatial distribution of temperature that exist, and may change as the planet warms (or cools.) Mathematically, the term is leading order in temperature differences, noted with primes ′.

2. A back of the envelope estimate indicates the magnitude of the effect of spatial variation is sufficiently large to retain in (1). Using current measured values of the anomaly T’ and the spatial variations in temperature suggest the new term is roughly 15%-25% the size of the original linear term. Neglecting this physical effect could account for a roughly 1/3 to 1/2 the uncertainty in the estimate for the climate sensitivity to doubled CO_{2} in the IPCC estimate of climate range. (Note to those who read my February 13 post on ‘The Blackboard’: This uncertainty in the radiative balance equation (1) is comparable to that obtained using Atmoz’s estimate of the error in using measurements of T’ based on linear averaging and those based on T’ due to power 4 averaging. )

3. If someone wishes to obtain an empirical estimate of λ_{2xCO2} with uncertainties less than ±0.7K, it is important to account for the effect of spatial surface temperature variations in some way. Since the current uncertainty range is thought to be ±1.5K, this means any sensible researcher should consider these variations.

**The Blog Kerfuffle:**

The main results, with regard to the blog kerfuffle, are:

1. both Roger Sr.’s arithmetic and identification of a physical phenomenon correct.

2. Eli’s series expansion was inadequate.

**The Boring Proof**

The blog kerfuffle is related to this IPCC equation, which is supposed to be an approximation for the energy balance of the earth, expressed in terms of the “Global Mean Surface Temperature Anomaly”, T’, which I will define more precisely later. The equation in question is:

(2) dH/dt = f -T’/λ

where H is the heat content of the land-ocean-atmosphere system, f is the radiative forcing (i.e., the radiative imbalance), and λ is called the “climate feedback” parameter.

The “Series Expansion Kerfuffle” relates only to one term, which I will call Q_{rad}. It’s supposed to describe the excess radiation losses from the earth that occur as its surface temperature responds to forcing due to greenhouse gases (or anything for that matter.) In (2) this term is approximated as: Q_{rad} ~ T’/λ.

That approximation for Q_{rad}accounts for extra heat lost by radiation when the average surface temperature of the planet warms. In so far as it describes that effect, the term is linearized. As far as I can tell, no one is worried about that linearization.

So, what’s the problem? Roger Sr. is concerned that we can’t use historical records for T’ to estimate λ for a number of reasons. The one important to this kerfuffle is this: Equation (2) is missing terms that arise as a result of the spatial variations in the Earth’s surface, and Dr. Roger Pielke Sr. believes these terms matter.

**Fuller Representation of Radiative Losses**

How does Dr. Roger Pielke Sr. describe the issue?

As an example, assume a region of the Earth with a base temperature of 270K and another region with a base temperature of 300K. The difference in the outgoing longwave radiation (assuming blackbody behavior where the emission is proportional to T**4) results in a 34% greater emission from the warmer location. Adding a temperature increase of 1K to each location results in a 38% greater change when this increase is applied to the warmer temperature (i.e., comparing the difference between the incremental change in outgoing longwave radiation at the cold and warm locations).

Dr. Roger Sr. is correct. Full expression for the radiative flux should include a T^{4} dependence where T is an absolute temperature, not the temperature anomaly and T should also account for the spatial variations.

So, let’s fix up equation (2) to account for both features.

Because temperature varies over the surface of the Earth, a more detailed representation of the total radiative heat losses should be replaced by a surface integral of the form;

~ σ ∫∫ εT^{4} d*A*

where σ is the Stefan-Boltzmann constant constant, ε and T are respectively the spatially varying emissivity and integration is performed over the surface area of the Earth, *A*.

However, equation (2) was obtained by taking a difference with a reference case which means that f is a forcing relative to some absolute value of forcing, F, that causes the temperature on the surface of the Earth to achieve some reference value To. So, any instantaneous local anomaly T’ is defined relative to this temperature. That is T’= T-T_{o}.

Recognizing this, the more finicky blogger might wish to replace the simple linear T’/λ term with the more complicated integral:

(3) dH/dt = f – εσ∫∫T^{4}– T_{o}^{4}d*A*

where *A* is surface area.

This results in an equation sufficiently to repel the average blog reader. Luckily, we aren’t interested in the full equation, but only the portion that describes what might be called the “radiative anomaly”, Q_{rad}, which I’ll define as:

(4) Q_{rad}~ εσA < T^{4}– T_{o}^{4}>

Here, the angle brackets are used <> to indicate the surface area average of any quantity “Y” that may vary over the Earth’s surface; which can be written more formally as <Y> =*A* ^{-1}∫∫ Y d*A*

We now have an equation that not only accounts for the full T^{4} dependence for radiation but also permits us to consider the effect of spatial variations in surface temperature.

However, everyone would prefer a simpler, approximate equation, that does not contain a surface integral. We seek something more like (2) but we also wish to do the analysis in sufficient detail to see if the extra terms in (1) appear.

**Expand local temperature, T, into an average (GMST) and spatially varying portions.**

Let’s begin by defining the temperature anomaly in (1) T’ in (1) in terms of global mean surface temperature. Recall that T’ in (1) & (2) itself is a global average anomaly; it happens to be the quantity

(5) T’ = <T>-<T_{o}>

where the angle brackets describe surface area averaging as above and T_{o} describes the temperature that existed at a point of the surface of the Earth under reference conditions.

We know the poles and equator differ in temperature (even under reference conditions). To explore the effect described by Dr. Roger Sr., we’ll define a local temperature deviation, δT, as the difference between the instantaneous local absolute temperature at any point T, and the instantaneous global average temperature <T> as δT = T – <T>. Applying the definition to the reference case, we get δT_{o} = T_{o} – <T_{o}>)

Let us further define a dimensionless temperature distribution function Θ = δT / ΔT_{o} where ΔT_{o} is the standard deviation in the Earth’s surface temperature in the reference case. This definition is convenient because Θ is sort of a “shape” function, its magnitude is order 1. Later on, we’ll also be able to magnitude of terms containing the powers in ΔT_{o} relative to those containing <T_{o}> noting that ΔT_{o} is much smaller than <T_{o}>.

At any time, the Global Mean Surface Temperature is defined as:

(6) <T>= *A*^{-1} ∫∫(<T> +ΔT_{o}Θ)d*A*

But since the definition of any surface average temperature <T> requires <T>= *A*^{-1}∫∫ <T> d*A* we know that:

(7) 0 =<Θ>=*A*^{-1}∫∫Θd*A*

Equation (7) is unremarkable, but it gets used later, so I’m numbering it should anyone later have questions.

**Insert expansion into the integral (4)**

Our next step is to insert the expansion into the integral describing the anomaly in radiative forcings (4), retain only just enough of the leading order terms to ultimately capture the leading order effect desired for (2) and simplify.

Using the definition of Θ (the dimensionless local instantaneous temperature deviation), we can either use the binomial theorem or MacLaurin Series to further expand Qrad. At this point in the analysis, we’ll immediately neglect terms that are in higher order than ΔTo2, or contain temperature difference to similar order as small compared to contributions that scale as the reference temperature, <To>3. We will also assume variations in emissivity are unimportant to this particular calculations. The result is:

(8) Q_{rad}~ εσ< (T^{4}– T_{o}^{4})> ~

εσ< (<T>^{4} + 4<T>^{3}ΔT_{o}Θ +6<T>^{2}(ΘΔT_{o})^{2}..) – (<T_{o}>^{4} + 4<T_{o}>^{3}ΔT_{o}Θ_{o} +6<T_{ o}>^{2}(Θ_{o}ΔT_{o})^{2} +..) >

Next recall that, for any Y, taking the average of the average returns the average, so <<Y>> = <Y>, and <Y – (Y- <Y>)> = 0. With a small amount of manipulation, the leading order terms for Qrad are found:

(9) Q_{rad}~ εσ <( T^{4}– T_{o}^{4})>

~ 1/ λ_{o} { T’ + ( ΔT_{o}/<T_{o}> )^{2} [ 3T’ <Θ_{o}^{2}> + 3 <T_{o}><ΘΘ’> ] }

where the constant is defined as λ_{o} = <T_{o}>^{3} / (4εσ )

**Now for the final equation!**

Equation (9) contains the leading order and first correction terms in “T’/T” required to determine whether the IPCC approximation can be applied to estimate the radiative heat loss. It’s useful substitute dimensional variables (δT), rather than Θ:

(10a) Q_{rad}~ (T’_{o} /λ_{o}) ( 1 + 3 <δT_{o}^{2}>/<T_{o}>^{2}) + (3 /λ_{o}) <δT_{o}δT’>/<T_{o}>

Given the range of temperatures on earth, it turns out we can simplify this equation further, and for all practical purposes, the appropriate approximation could be written as:

(10b) Q_{rad}~ T’_{o} /λ+ **3 (<T _{o}>/**

**λ**

**)(<**

**δ**

**T**

_{o}**δ**

**T’>/<T**

_{o}>^{2})Leading order terms describing the effect of spatial variations on the radiant heat loss from the earth’s surface are highlighted in bold. These terms describe the effect Dr. Roger Sr. was discussing, and which are missing from the IPCC equation (2).

Other forms of (10a and 10b) are possible, and some are more suited to different analyses, but (10b) is a convenient formulation when estimating the order of magnitude of the effect based on published data.

**Why didn’t these appear in Eli’s result?**

A curious reader might wonder why the new terms did not pop-out when Eli did his two line derivation. The new terms appear when one

- Writes down the functional form for radiation losses from the Earth’s surface, (this is the horrible integral)
*including the effect of spatial variations*, - Expands temperature into mean and deviations in a series, and
- Simplifies to leading order in temperature anomalies.

The terms describing the effect of spatial variations did not appear in Eli’s expansion because he skipped the first step: He did not write down the integral representing the anomalous radiative losses; (equation 4 here). Instead, he wrote down the functional form for radiation losses for an *isothermal* body, with no spatial variations in temperature. He included the T^{4} dependence for that simplified case, expanded that already simplified equation, and then wrote done the leading order terms describing radiative losses for an isothermal planet.

Naturally, the terms associated with spatial variations did not magically appear, as Eli neglected these spatial variations in his step 1.

**So, how much do spatial variations matter?**

Whether or not these spatial variations “matter”, depends, of course, on what phenomena one is investigating.

In Dr. Roger Sr.’s case, he discussing the use of GMST data to estimate climate sensitivity to doubling of CO_{2}, λ_{2xCO2}. Current estimates suggest λ_{2xCO2} falls between 1.5K and 4.5K. The relevant question then, is, “Relative to a range of 3K, how much difference does neglecting spatial variations make?”

To answer that question precisely requires either downloading data from a whole bunch of GCM’s and or obtaining possibly non-existent data describing the Earth’s surface temperature and calculating things like the standard deviation in the Earth’s instantaneous surface temperature, <δT_{o}^{2}> or more detailed statistics.

I’m not going to do that on a blog. (I’ll probably never do it. But if someone else would like to, I’d find the results interesting.)

Still it is worth doing a back of the envelope estimate using accessible data; my goal is to determine if the terms contribute less than 1%, 10%, 100% or so on. So, let’s do that.

Suppose:

1. The baseline temperature for the earth’s surface <T_{o}>~ 280K .

2. The global temperature anomaly is T’ ~ + 0.6K which is approximately equal to the current anomaly.

3. To estimate the value of <δT_{o}δT’>, let’s first assume that we can divide the Earth’s surface into three regions: 1/3 of the area (that near the poles) is ‘cold’ , with a base temperature deviation of δT_{o}=-20K , 1/3 or the area (near the equator) is ‘hot’ with a baseline deviation of δT_{o}=+20K and 1/3 is average δT_{o}=0K

(Note: This temperature distribution returns a standard deviation of 16K, which appears roughly consistent compared to the temperature ranges of 90K in the adjacent figure showing annual average temperature from NASA GSFC. Also, recall that estimates of the temperature variations, δT, based on annual averaged values will tend to *underestimate* the instantaneous variation in temperatures about the instantaneous global mean, which is what is actually required to evaluate δT_{o}. Using a low value will tend to underestimate the magnitude of effect of the variance in actual temperatures. So, we will be obtaining a lower bound. Those wishing to estimate an upper bound might use δT_{o}=±30K as might be more appropriate for instantaneous variations.)

4. Assume the temperature in the “cold” part of the Earth 1K compared to the ‘average’ anomaly and that at the equator dropped 1K compared to average anomaly. This is somewhat consistent with this map of the Surface Temperature Anomaly for 2005. (Source: NASA.)

This results in

<δT_{o}δT’> ~ {(20K)(-1K)+ (0K)(0)K + (-20K)(+1K)}/3

Now, comparing the relative magnitude of the neglected terms in 10b, we find the error associated with neglecting this phenomenon is of the order:

error ~ -3(2/3) (-20K^{2} ) /(300K 0.6K ) ~ 17%

If we use a spatial variation for temperatures of ~30K, as might be a more typical value on an individual day, we estimate the error due to neglecting the spatial terms is roughly 25% compared to the term actually retained in the IPCC equation.

Of course, this is only a back of the envelope estimate of the importance of the term. Nevertheless errors of 15%-25% arising from missing terms in an equation are generally thought to be important. These sorts of uncertainties, introduced by imprecise formulation rather than random measurement uncertainty, tend to introduce bias, and cannot be eliminated by taking lots of data.

The only fix is to correct the functional relationship to include the physical process.

**But does an error of 15%-25% really matter in the specific case Dr. Roger Sr. was considering?**

The error of 15%-25% is *quite* important if one is to use the IPCC equation (1) to estimate the climate sensitivity, λ. Typically, those doing this computation try to obtain climate data at near steady state. In this case, neglecting the additional terms, the error will propagate into the computation of climate sensitivity, λ.

The range for the λ_{2xCO2} is thought to fall in the range of 1.5K to 4.5K. So, the current uncertainty range is ±1.5K. In contrast, if we were to use the IPCC equation, and empirical data for T’ to estimate the λ_{2xCO2}, our uncertainty due to neglecting this physical process alone could be as large as ±0.75 K: That’s 1/2 the current uncertainty interval. Worse, measurement uncertainty, or other errors would widen the uncertainty interval in any estimate.

Clearly, if we the goal of further analysis is to reduce the uncertainty in λ_{2xCO2}, below the current value of ±1.5K, a careful scientist or engineer would wish to account for the existence of spatial temperature gradients, particularly the large ones between the poles and the equator!

Dr. Roger Pielke Sr. was correct when he identified this issue as important; he proved himself to be correct using simple arithmetic. It’s a bit sad to see series expansions dragged into the whole mess. Naturally, when done correctly, the results of series expansions agree with arithmetic.

Advice to bunnies young and old: Always remember, when a result can be proven with simple arithmetic, series expansions won’t make the result go away.