Does a Global Temperature Exist?
Christopher Essex
Department of Applied Mathematics
University of Western Ontario
Ross McKitrick
Department of Economics
University of Guelph
Bjarne Andresen
Niels Bohr Institute
University of Copenhagen
J. Non-Equilibrium Thermodynamics
(in press, June 2006)
Abstract
Physical, mathematical and observational grounds are employed to show that there
is no physically meaningful global temperature for the Earth in the context of the issue
of global warming. While it is always possible to construct statistics for any given set of
local temperature data, an infinite range of such statistics is mathematically permissible
if physical principles provide no explicit basis for choosing among them. Distinct and
equally valid statistical rules can and do show opposite trends when applied to the
results of computations from physical models and real data in the atmosphere. A given
temperature field can be interpreted as both "warming" and "cooling" simultaneously,
making the concept of warming in the context of the issue of global warming physically
ill-posed.
Short title: Global Temperature?
1
1 Introduction
It is widely held that the atmosphere and oceans have been warming over the past half
century. The basis of this view is that there is an upward trend in the graph of a statistic
called the "global temperature" ([1], [2]). It arises from projecting a sampling of the fluctu-
ating temperature field of the Earth onto a single number (e.g. [3], [4]) at discrete monthly
or annual intervals. Proponents claim that this statistic represents a measurement of the
annual global temperature to an accuracy of 0.05 
C (see [5]). Moreover, they presume that
small changes in it, up or down, have direct and unequivocal physical meaning.
While that statistic is nothing more than an average over temperatures, it is regarded
as the temperature, as if an average over temperatures is actually a temperature itself, and
as if the out-of-equilibrium climate system has only one temperature. But an average of
temperature data sampled from a non-equilibrium field is not a temperature. Moreover, it
hardly needs stating that the Earth does not have just one temperature. It is not in global
thermodynamic equilibrium -- neither within itself nor with its surroundings.
It is not even approximately so for the climatological questions asked of the temperature
field. Even when viewed from space at such a distance that the Earth appears as a point
source, the radiation from it deviates from a black body distribution and so has no one
temperature [6]. There is also no unique "temperature at the top of the atmosphere". The
temperature field of the Earth as a whole is not thermodynamically representable by a single
temperature.
The global temperature statistic is also described as the average, as if there is only one
kind of average. Of course there is an infinity of mathematically legitimate options. Indeed
over one hundred different averages over temperatures have been used in meteorology and
climate studies [7] with more appearing regularly. For the case of temperature, or any other
thermodynamic intensity, there is no physical basis for choosing any one of these from the
infinite domain of distinct mathematical options.
The international standards organization ISO tried to choose one but failed [8]. The prob-
lem is not a mere absence of a convention for selecting one from among many mathematically
different but physically-equivalent measures for a single underlying property. The problem
is that there is no single underlying property, because there is no global temperature. But
this does not stop averages from being made.
There is no experimental or theoretical way to falsify any particular choice of averag-
ing rule, if averages are (falsely) proclaimed to be temperatures. Proclaiming them to be
temperatures leads to a paradox, as any two ad hoc choices applied to a particular out-
of-equilibrium field can have mutually contradictory behaviors: the system can seem to be
both warming and cooling simultaneously. Paradoxically, whether the system is "warming"
or "cooling" becomes a property of the choice of average -- a choice which is independent
of the system.
The resolution of this paradox is not through adoption of a convention. It is resolved by
recognizing that it is an abuse of terminology to use the terms "warming" and "cooling" to
denote upward or downward trends in averages of temperature data in such circumstances.
Statistics might go up or down, but the system itself cannot be said to be warming or cooling
2
based on what they do, outside of special circumstances.
By the same token, different statistics produced by different averaging rules should not
be expected to behave the same, since they are not measuring a single, well-defined physical
quantity. Unfortunately, it has long been considered a problem in climatological debates that
the well-known global temperature averages behave differently from each other. Debates as
to which is the correct one are fundamentally false, with no correct resolution. They have an
interminable aspect, as has been illustrated in comparisons of satellite and ground-measured
"global temperatures" [9], or of different versions of the satellite-measured series [10] [11],
or in debates about whether the medieval era was "warmer" than the present, e.g. [12]. We
argue herein that these disputes have their root in attempting to estimate a physical quantity
that does not actually exist, and hence there is no prospect of resolution on scientific grounds.
But if the statistics are not actually temperatures, what then are they? Can they be
used as indices to warn us of subtle dangers, like the proverbial canary in the coal mine? Or
are they just examples of an infinity of ad hoc numbers extractable from measurements that
might be followed to no end?
So far the proponents of global temperatures have been able to avoid this question. They
have not substantiated exactly how changes upward or downward in their statistic might
affect dynamics and local states within the atmosphere and oceans. However that has not
stopped many remarkable attributions of cause: small increases in their global statistic have
been cited to explain hurricane formation [13], viral infections in frogs [14], encephalitis in
horses and even pulmonary disease, delirium and suicide in humans [15].
The awkwardness of such claims is obscured because the concept of averaging is so routine.
In fact it is so pervasive that it may even seem implausible to mount a critique. But it
should only be routine where it makes sense. Personal income and height are meaningful at
the individual level, and no conceptual problems emerge when adding or averaging over a
population. But there are certainly examples where averaging or adding destroys the meaning
of a variable. In economics, for example, an exchange rate is meaningful when comparing two
currencies, but the ideas of a "global exchange rate" or a sum over exchange rates are both
nonsensical. Regardless of the fact that enough data exist to compute something analogous
to a "global temperature" for the money markets, neither the level nor the trend in such
a statistic would provide any meaningful information about the global economy. Another
example: Individual telephone numbers are both meaningful and useful, while the sum or
average over telephone numbers in a directory have no meaning.
The notion of being globally "hotter" or "colder" for out-of-equilibrium systems is not
altogether without merit. Miami in January, with temperatures ranging between 20 to 30 
C,
say, is certainly warmer than Toronto at, say, -15 to -5 
C. However this ranking of relative
warmth is not based on averages, but on the ranges in respective temperature fields. Since
the ranges do not overlap, all averages will agree which field seems to be the warmer. It is
independent of the choice of average.
Not so for the case of comparisons of Earth's temperature field at times a few years apart.
The range over the globe (about -80 
C to +40 
C) is essentially the same for both when
compared to statistical trends in averages (i.e. about 0.1 
C yr-1
), which are three orders
3
of magnitude smaller. In cases where ranges overlap not all averages over a given set of
actual observations agree on trends (see Section 3), throwing into doubt for this case what
"warmer" or "cooler" mean.
It is clear that there are many misconceptions about nonequilibrium temperatures fields.
This paper serves to expose and identify them with specific reference to the measurement
of climate change. They may be summarized by the following points, which are treated in
detail later in the paper:
1. Sums or averages over the individual temperatures in the field are not temperatures.
Neither are they proxies for internal energy.
2. Temperatures from a field (individually or averaged) neither drive dynamics nor ther-
modynamics. Instead dynamics are driven by gradients and differences, in tempera-
tures and other variables.
3. A global spatial average cannot be an index for local conditions, otherwise nonlocal de-
pendence (i.e "thermodynamics at a distance") for local conditions would be required.
4. The utility of any global spatial average of the temperature field as an index for global
conditions has been presumed but not demonstrated.
5. It is easily demonstrated that different spatial averaging rules over temperatures can
have contrary trends in time (i.e. some increase while others decrease in time) when
the two fields being compared have range-overlap, as they do in this context. This
is demonstrated here in a basic example and subsequently with actual atmospheric
temperature-field observations.
6. No ground has been provided for choosing any one such statistic over the rest as the
one proper index for global climate.
7. If there are no physical or pragmatic grounds for choosing one over another, and
one increases while the others decreases, there is no basis for concluding that the
atmosphere as a whole is either warming or cooling.
2 Averaging and Physical Temperature
2.1 Temperature versus Average of Temperature Data
Thermodynamic variables come in two varieties: extensive and intensive. Extensive variables
are proportional to the size of the system. They are additive. In this category we find volume,
mass, energy, entropy, particle number etc. We can combine two systems and the values of
extensive variables for the whole system will simply be the sum of the values from the two
components. Correspondingly a mean subsystem (loosely called the average) will have this
sum divided by the number of components. Such an average over a quantity like mass is
meaningful because the sum is meaningful. For example average mass is of importance to
4
airlines because it is helpful to estimate the total load of an aircraft without having to weigh
every passenger.
Intensive variables, by contrast, are independent of system size and represent a quality
of the system: temperature, pressure, chemical potential etc. In this case combining two
systems will not yield an overall intensive quantity equal to the sum of its components. For
example two identical subsystems do not have a total temperature or pressure twice those
of its components. A sum over intensive variables carries no physical meaning. Dividing
meaningless totals by the number of components cannot reverse this outcome. In special
circumstances averaging might approximate the equilibrium temperature after mixing, but
this is irrelevant to the analysis of an out-of-equilibrium case like the Earth's climate.
Setting mixing aside, consider two disjoint isolated equilibrium systems, a and b, with
functions of state Ua
= fa
(Xa
1 , Xa
2 , . . . , Xa
n) and Ub
= fb
(Xb
1, Xb
2, . . . , Xb
n) respectively, where
Ua
and Ub
are the respective internal energies. Xa
j and Xb
j represent the corresponding ex-
tensive variables in systems a and b respectively. Obviously, given fa
, the extensive variables
for system a completely define the thermodynamic state of system a. Similarly for system b.
The partial derivatives, Ua
/Xa
j and Ub
/Xb
j are the jth intensive variables for the
respective systems. The temperature of each system is, of course, the particular partial
derivative with respect to the system's entropy,
T = U/S. (1)
Together, the intensive variables form the tangent spaces of the respective functions of state.
As such they are local properties of the state space and are thus independent of the scale of
the system.
Let us propose that an average over temperatures from both systems is required to be a
temperature. This proposition produces a contradiction. The state, and the temperature,
of system a, say, is completely determined by the variables {Xa
i } and does not change in
response to a change only in {Xb
i }. But any average is a function of both temperatures.
Thus, while each temperature is a function of the extensive variables in its own system only,
the average must depend explicitly on both sets of extensive variables, {Xa
i } and {Xb
k}.
That is it must depend on both states and it can change as a result of a change in either one.
Then the average cannot be a temperature for system a, because system a is mathematically
and thermodynamically independent of system b by assumption. Similarly, it cannot be a
temperature for system b. Consequently the average is not a temperature anywhere in the
system, which contradicts the proposition that the average is a temperature.
While it is thus simple, obvious, and unavoidable that there is no one physically defined
temperature for the combined system, the example illustrates the contradiction that arises
in requiring an average over a local equilibrium temperature field to be itself a temperature
of anything.
With reference to global temperature statistics, consider a projection of the full tempera-
ture field, T(r, t) onto a single real value, (t) at time t, usually the present. For concreteness
5
consider an average of temperatures:
(tM ) =
1
N
N
i=1
T(ri, tM ) -
1
M
M
j=1
T(ri, tj) . (2)
Outside of small details, such as mappings of actual measurements onto a uniform grid on
the surface,  is a common statistic actually used in discussions of global warming, intended
to quantify temperature change over the entire globe at time tM , by deviations from a sliding
average in the past. This statistic is also known as the global temperature anomaly. The
summation over i estimates a spatial integration of the field over the surface of the Earth,
where T(ri, tM ) is the air temperature at location ri and time tM , and the summation over
j accounts for a customary interval of time from the past to establish a local time mean (30
years is traditional).
 is explicitly a function of T(rk, t) for all rk
(i.e.  = f(T(r1, t1), T(r2, t1), . . . T(r1, t2), T(r2, t2), . . . T(rN , tM ) )). In contrast, local
equilibrium states in a field are defined at a particular location, r. Other locations not in
an infinitesimal neighborhood are independent, just as systems a and b are independent of
each other. The statistic , on the other hand, exhibits the same property as an average
of temperatures between the systems a and b: it depends on the states of all the local
subsystems, not just the state at any one location.
Thus the same paradox arises as for systems a and b. If  is supposed to be important at r,
it also depends explicitly on states at all remote locations {rk}. However, chemical, physical
or biological processes governed by temperature at r are not functions of temperature other
than at r. So if one insists on  defining processes like the melting of glaciers at some
location, say, the melting is forced into being a function of temperatures elsewhere on the
planet too, which is physically untenable. Claiming otherwise is tantamount to temperature
at a distance! While forces at a distance are considered an acceptable concept for classical
gravity and electromagnetism, they are not suitable for thermodynamic processes.
It is worth noting in passing that  suffers from the additional problem that it also
depends explicitly on temperatures from the past. So in the picture implied by them, not
only are the fate of glaciers today explicitly dependent on temperatures in a remote desert,
say, but also on temperatures in that desert from twenty years ago.
On top of this problem, even if we let the two systems a and b come to equilibrium so that
the temperature is the same everywhere and all possible definitions of average are equivalent
(see Sect. 3), this common temperature will not be uniquely specified by the original states
of a and b. It will also depend on the path taken by the process. Suppose two thermal
systems a and b have equal heat capacities but initially different temperatures Ta
and Tb
.
After putting them in thermal contact they will equilibrate at the common temperature
(Ta
+ Tb
)/2. If on the other hand they equilibrate reversibly, i.e. while producing work,
their common final temperature will be

TaTb.
Even in this simplest case "averaging" between these two systems is not unique.
6
2.2 Average Over Temperature Field as Index or Proxy
The argument is often heard that the global temperature, however it is calculated, while not
being really what is physically driving the climate, is a good index or proxy for whatever
does drive it. Science and engineering are used to such indirect measurements. Tempera-
ture itself, for example, is almost never measured according to its thermodynamic definition,
T = (U/S)V,N1,N2,N3..., but by measuring a volume, bending, electrical conductance, eigen-
frequency of a crystal, radiation spectrum etc. All such measurements rely on different as-
sumptions being met. Physicists and chemists are well aware of these restrictions. However,
no such physical arguments have been made for using a statistic as an index for climate driv-
ing force. Not even a statistical correlation has been forwarded for such a connection. ISO
standards for measuring and calculating such an index were optimistically promised after the
Johannesburg climate meeting [8], and standards have been published regarding measure-
ment of greenhouse gases, but the ISO has been conspicuously quiet on the measurement of
a global temperature.
If temperature field averages are not temperatures, can they function as an index or as a
proxy for climate dynamics? While an index need not be a physical variable explicitly, there
are nonetheless unavoidable requirements before a statistic can be regarded as an index. An
index is a statistic produced by a consistently applied rule that characterizes some, usually
complex, process usefully, even though the precise connections between it and the process
may not be fully known or understood. As an index the statistic usefully encapsulates a
possibly large set of data into one dimension for a specific purpose.
Proposing a statistic to be an index can be risky. What is regarded as "useful" may
prove to be quite imprecise and subjective. Therefore the burden to demonstrate the utility
of a statistic as an index must lay with those who propose it as one. Similarly the burden
should never be to prove a statistic is not an index.
However a peculiar circumstance exists for the case of averages over the temperature
field. False, but common, impressions that such averages are actual physical variables gen-
erally, and temperatures in particular, have left the task of demonstrating the utility of such
statistics undone. The apparent physical origins have led to the most uncritical thinking
about what trends in such statistics might imply.
While it is impossible to prove that such averages are not useful in any manner whatso-
ever, it can be shown that temperature field averages cannot be useful in certain specific roles
in which they are cited. For example, upward trends in  are said, uncritically, to indicate
trends in underlying dynamics, such as changes in the numbers or severity of storms, and
trends in rainfall, not to mention melting of glaciers or many other local physical dynamical
processes.
What connection exists between these dynamical processes and such averages? No phys-
ically precise reasoning has been proposed as the basis of such a connection. Indeed it is
unlikely that there is any such connection because intensive variables, like temperature, do
not generally drive dynamics, and in particular do not drive the dynamics of the atmosphere
and oceans, whose dynamics are caused by gradients, or differences in thermodynamic in-
tensities, rather than the intensities themselves.
7
To be more explicit, consider a free thermodynamic mixing process between two subsys-
tems a and b, isolated from everything except each other. In the absence of sources and
sinks due to chemical reactions, extensive quantities summed over subsystems a and b, ex-
cept entropy, will not change. The change in entropy is equivalent to the existence of the
process:
dS =
1
Ta
-
1
Tb
dUa
+
Pa
Ta
-
Pb
Tb
dV a
-
a
1
Ta
-
b
1
Tb
dNa
1 +    , (3)
where T is the temperature, P the pressure, and i is the chemical potential of species i.
Superscripts refer to the subsystems a and b. When the subsystems are in equilibrium with
each other, all intensities are the same in both systems (Ta
= Tb
, Pa
= Pb
, etc.). In that
case dS vanishes and nothing happens. Nowhere in Eq. (3) do intensities enter in isolation.
Differences cause process.
It is true that phase changes and rates of chemical and biological processes depend para-
metrically on (local) absolute temperature. However, if there are no differences in intensities,
like differences in chemical potential for chemical reactions, thermodynamic processes would
not happen in the first place. Even the sensation of warming or cooling is not a consequence
of a single temperature, but rather because of the thermodynamic forces that are set up
between the intensities in our bodies and those of our surroundings. Thus two people can
disagree on whether a given room is cold or hot.
If absolute intensities don't drive climate dynamics, there is no reason to expect that av-
erages over them will. Moreover, at least for small changes, an average over the temperature
field can be insensitive to climate change. Because of the local independence of functions
from their gradients, an average can change in time without any appreciable changes in
dynamics, while great climate changes can be envisioned under a constant average.
For example, simply rotating thermodynamic intensity fields with respect to the Earth's
surface leaves field averages invariant but can imply significant climate change. In the special
case of , invariance only requires changes that leave the sums in Eq. (2) unchanged. There
is an infinite range of possibilities of such changes with  invariant that would constitute
significant climate change. Indeed an overlooked type of climate modeling experiment is
climate change under fixed temperature field averages.
Ultimately, it can be no surprise that local states are not well represented by a globally
based statistic, casting further doubt on the role of  as an index. It is also worth remem-
bering that an invariant  would not generally imply invariance for other temperature field
statistics, from different averaging rules. As there are an infinite number of such statistics
extractable from a field, there is also the question of whether the different statistics convey
the same message or not, if any is conveyed at all. It is difficult to claim a statistic is a useful
index when many very similar statistics seem to say something different.
Thus the proponents of  as an index have been able to avoid two crucial tasks:
i) They have not been required to justify or demonstrate the physical usefulness of a
statistic extracted from the temperature field.
ii) They have not answered why  is the statistic of choice and not some other extract
of the temperature field with potentially very different behavior.
8
The very different behaviors of other statistics are discussed in Section 3.
2.3 Field Average As Energy Proxy
If a field average is not a temperature, and it is not an index, can it be simply a direct
proxy for internal energy instead? This calls for statistics like  to be given up as drivers of
climate or as indicators of local dynamics. Instead temperature is to be viewed as equivalent
to energy, and thus the numerator of the right side of Eq. (1) is to represent changes in the
global internal energy. Then any physical meaning is born by changes in energy instead of
temperature.
This strategy has two problems. First, substituting temperature for internal energy
amounts to extensifying an intensity, and second, the role for a global averaged internal
energy in climate is unclear, especially given the few percent that we could expect it to vary.
Despite popular thinking otherwise, temperature and energy are not equivalent. Tem-
peratures can be very high at very low energies. While heat is a form of energy, temperature
is, fundamentally, a measure of how energy is spread over quantum states. For example,
radiation from a small laser powered by flashlight batteries can have temperatures peaking
as high as  1011
K. This is higher than many stellar interiors but one cannot even feel the
heat of the beam on one's hand ([6]).
The myth of the equivalence of temperature and energy comes from the proportionality
of internal energy U to temperature T, through a heat capacity. In the case of an ideal gas
with no internal molecular structure we have
U(N, T) =
3
2
NkT . (4)
While Eq. (4) is not an equation of state in extensive variables alone, it is easily derived
from one by differentiating it according to the definition Eq. (1). In the result, U is not only
proportional to T but to N as well, so, being independent, changes in T are not equivalent
to changes in U.
Introducing a local version of Eq. (4) by dividing through by volume, V ,
u(n, T) =
3
2
nkT , (5)
where u  U/V and n  N/V , we see the same issue holds locally: measuring changes in T
cannot determine changes in u in environments where the number density is also variable.
But are changes in n large enough to matter for the processes in this case? Using Boyle's
law and Eq. (5),
du
u
=
dP
P
=
dn
n
+
dT
T
. (6)
As dP/P is about 5% across the entire Earth's surface while dT/T would be closer to 20%,
variation of n cannot be ignored, globally speaking, in comparison to variation in T for the
processes actually in question. As T varies about four times as much as u over the Earth,
temperature cannot be a proxy for internal energy in climate. In fact these numbers suggest
9
that places of high T tend to correspond to low n, which agrees with the well-known tendency
of aircraft to have diminished takeoff performance at high air temperatures.
Even if trends in global U were measurable and known (they aren't), the question re-
mains what it would mean for climate dynamics. Like variations in , significance is simply
presumed without further consideration.
3 Averaging Without Physics
The notion of averaging exists independently of physics. Loosely speaking, it is nothing more
than holding one mathematical object to stand in for many. Usually we think of a single
number standing in for a set of numbers or standing in for a more complex mathematical
object. While having a single representative taken in place of many can be convenient, clearly
something is lost in it. The representative may not exhibit what is physically relevant, as
the selection rule need not be set by the physics.
Setting aside the statistical analysis of physical data, averages arise within physical theory
in two basic ways. They can emerge, for example as an interpretation of a previously existing
structure rather than as an intrinsic property. In the case of quantum mechanics the term
expectation value is used for an eigenvalue that characterizes a mathematical operator and
thus a quantum state. The expectation value has the character of, if not the name of, an
average, but it need not be interpreted as one. The interpretation is definitely after the fact.
In other physical applications, averages arise in a somewhat ad hoc manner. They are
simply sums over physical quantities that have been normalized. In principle, normalization
is a convenience. It is not essential, and so neither is the notion of averaged quantities
essential to the mathematics in such physical theories. What is essential is that the sums
have meaning in the first instance.
As we saw in the preceding, sums are not generally meaningful over all variables. Tem-
perature, pressure, and chemical potential are just a few of the exceptions. Nonetheless, it
may happen that a variable is not additive, while a transformation on that variable is. For
example, a variable defined by the square of internal energy is not additive, but its square
root is. The physics thus presents a coordinate system for that quantity that is preferred over
another. Similarly, normalization of the corresponding sums results in a preferred average.
However for thermodynamic intensities there is no preferred coordinate system. The sum
has little physical meaning, so all transformations are in play. Normalized sums are not in-
variant under all transformations. Thus more than one average is unavoidable. Statisticians
have long known this. Unlike physicists they work with data unconnected to underlying
physical theory and have uncovered classes of averages that reflect this. One is faced with
having an infinity of averages, having all possible values within the range of the data set.
The most well known of the class are the H¨older or power means [16]. These arise
from transforming the mean under simple powers. Alternatively they can be deduced by
varying geometric assumptions about what constitutes the length of a vector. This method
has pedagogical significance because it forces an enlightening conflict between conventional
biases: one cannot have both the classical mean and Euclidean distance. This class of average
10
is by no means exhaustive.
This section makes three contributions to a thermodynamic discussion of nonequilibrium
fields:
1. A simple thermodynamic example is shown where different averages over given tem-
peratures exhibit contradictory time trends. Thus the notions of warming and cooling
are problematic for nonequilibrium temperature fields.
2. Global conditions are deduced for when it is possible to say, unequivocally, that one
temperature field is warmer or cooler than another.
3. It is demonstrated that any value of a data set can represent the mean in some coordi-
nate system. If there is no preferred coordinate system given for the average because
the physics does not provide one, then any value in the data set can stand as an average
of the temperature field.
3.1 Warming and Cooling Simultaneously?
3.1.1 The Averaging Operator and the Origin of Contradictory Trends
For the case of a finite data set, an averaging operator, A, maps the set onto a real number
within the domain of the original data. Clearly a value that does not satisfy this requirement
would be a pathological representative of the set. In the case of temperature data, the domain
of A is the range of data from the original field.
Thus the resulting real number is within the interval induced by the original set. This
definition is still quite general. All of the well-known (standard) averages, such as the mean,
the mode, the median, the root mean square, the harmonic mean, the geometric mean, or
the infinity of nonstandard, unnamed others satisfy this requirement, without necessarily
being equivalent to each other. It also follows from this definition that a set made up
of identical data values can have only a single average value, independent of the type of
averaging operation. That single value corresponds to the single repeated data value.
The aim of these mappings is of course beyond the mathematics and beyond any sta-
tistical theory. Neither mathematics nor statistics suffice to determine whether any result
of such mappings achieve any aim. Moreover, the data and its underlying physics do not
depend in any way on A.
Based on the definition, A is not a unique operation unless some additional rules, R,
beyond the fundamental definition are specified (c.f. the importance of path followed as
discussed at the end of Sect. 2.1). Thus if we average over a set  to get x, we must specify
the operation, AR in terms of R,
AR() = x . (7)
If there are two different rules, R and R, and one set, , then in general the averages
are different,
AR () = x (8)
AR
() = x ,
11
where x = x. Furthermore, if the  is a function of t (i.e. (t)), then ˙x(t) and ˙x(t) can
have opposite signs. This idea is illustrated in the following physical example.
3.1.2 A Physical Example of Contradictory Trends
Let's consider a specific example involving temperature. A glass of ice water at 2 
C is sitting
beside a cup of coffee at 33 
C. The two remain isolated, but are allowed to relax to room
temperature, which is 20 
C, according to Newtonian cooling (heating). To complete the
example, a plausible relaxation time of eight minutes for each container was set for the sake
of this illustration, but the phenomenon we will find is not unique to this value. In this
manner the ice water is allowed to warm, while the coffee cools accordingly.
For this example, the two independent temperatures were averaged in four different ways.
They are not exhaustive by any means. Furthermore, examples with other temperature units
and other averages may be formulated, but these would not add materially to the value of
the example.
The averages used are:
AR-1 (T1, T2) =
T-1
1 + T-1
2
2
-1
(9)
AR1 (T1, T2) =
T1 + T2
2
AR2 (T1, T2) =
T2
1 + T2
2
2
1
2
AR4 (T1, T2) =
T4
1 + T4
2
2
1
4
R-1 is the harmonic mean. It is precedented, for example, in the case of temperature in
connection with minimum entropy production and the radiation field [17]. It also appears
in connection with average resistance in a parallel circuit and in average travel times on a
road network with varying speed limits.
R1 is the simple mean, often used over small temperature ranges and in connection with
simple Newtonian heat exchange.
R2 is the root mean square, which is well precedented in statistics and statistical me-
chanics. In the latter case it appears particularly in connection with kinetic energy. It can
also emerge in connection with the potential energy of a spring.
R4 would appear in connection with black body radiation.
The results of the Newtonian cooling calculations are shown in Figure 1. It is clear
that the starting "system temperature" varies widely depending on the averaging formula
chosen. Because of the property that all averages must be the same for a set composed of
12
0 5 10 15 20 25 30
5
10
15
20
25
Minutes
TemperatureC
Figure 1: Four averages over one thermodynamic system. The rules for each, denoted by
R-1, R1, R2 and R4 are defined in Eqs. (9).
equal values, all averages must approach the room temperature. All of these averages clearly
exhibit this proper behavior.
Clearly whether the system seems "warmer" or "cooler" at time t = 5, say, in relation
to t = 0 depends on the average chosen. But the data are independent of the averaging rule
used, therefore the sign of the derivatives are not intrinsic to the data, but a property of the
averaging rule selected.
If the physics does not prescribe one rule to be used over another, as it does not for
temperature, we may use any rule. If one interpreter of the data chooses one rule, while
another chooses a different rule, there is no way to settle a disagreement as to whether the
system is getting "warmer" or "cooler" with time.
Alternatively, if one has a change of opinion about what rule to use, the ranking of the
two systems may be instantaneously reversed. In a mathematical sense this would just be
a quantitative alteration in the time derivative. However, "warming" versus "cooling" is
a qualitative distinction for thermodynamics. Changing the sign of the derivative reverses
the ranking of the two states catastrophically, in the sense of a qualitative change that is
extrinsic to the system studied. That is, rank order is catastrophically reversed by simply
changing how the data are interpreted -- a rank order catastrophe.
There are only two options. Admit the possibility that nonequilibrium systems can
simultaneously warm and cool or take the position that these terms have no meaning in such
cases. We adopt the latter convention. In either case, the ice water is warming and the coffee
is cooling, and while that is happening they are not in thermodynamic equilibrium with one
13
another or with the room. There is no physically coherent way to reduce this dynamic to
changes in a single temperature number for the system as a whole.
Arbitrary averages can and do give contradictory behaviors in more complex cases too
as we shall see below. Averaging does not represent any means of avoiding the fact that
a system is not in global thermodynamic equilibrium. Thus, in the case of nonequilibrium
thermodynamics, temperature averages fail in the most basic role of an average, which is for
one value to represent many.
3.2 Range Overlap and Global Ranking of Fields
In sections 3.1.1 and 3.1.2 it was shown that objects out of equilibrium cannot necessarily
be compared to each other as being "hotter" or "cooler". It makes problematic the claim
that Earth's temperature field is warmer or cooler today than it was a hundred years ago,
or that one century is hotter than another century.
In contrast, it is obviously valid to make the observation that the temperature field
of the Sun is hotter than the temperature field of Pluto, yet each of these bodies have
nonequilibrium temperature fields. What makes this comparison different? The simple
answer is that there are no common values in the respective temperature fields. This was
not the case for the example in the preceding section, where the interval spanned by the
range of temperature at time t = 0 contained the interval spanned at later times.
In the case of a temperature field on Pluto, the Sun, or the Earth, we work with a subset
of the field which constitutes a finite number of values. This is a practical necessity even if
it is not a mathematical one. The range of values from the field induces the domain of the
averaging operation. This domain is an interval, which according to the definition in 3.1.1
limits what an average can be. Consider the interval operation, I, which defines the domain
of the data set . For two non-overlapping data sets 1 and 2
I(1)  I(2) =  . (10)
This condition is global in that it is entirely independent of specific locations in the original
field. It follows that
AR(1) - AR(2) (11)
has no zeros over all possible rules R satisfying the definition in Section 3.1.1. No zeros
in Eq. (11) means it never changes sign, so there cannot be a rank order catastrophe as R
is varied over all rules. If the global condition Eq. (10) holds then it is not ambiguous to
say that one set is warmer or cooler than the other. In the case of the Sun and Pluto it is
reasonable to expect that Eq. (11) holds so that we may say that the Sun is hotter than
Pluto without contradicting the preceding.
But if
I(1)  I(2) =  , (12)
we will say that the fields yielding 1 and 2 have range overlap. In the case of range overlap,
we cannot guarantee that there are no zeros in Eq. (11) and therefore there can be rank order
14
catastrophes, which means that the ranking of temperature fields as "warmer" or "cooler"
is fundamentally problematic. It is possible for Eq. (11) to have no zeros over all reasonable
R in cases where Eq. (12) holds if a local condition is applied such that every location in the
field is increasing or decreasing. This condition does not occur in realistic fields such as that
of the Earth, and it is not global.
In the case of the Earth's temperature field, the temperatures form a set of values, .
The field at time t1 is being compared to the field at time t2. In that case Eq. (12) holds.
In fact this is among the most extreme cases of range overlap because,
I((t1))  I((t2)) (13)
from which we conclude that ranking the Earth's temperature fields at two instants in time
is highly problematic. It simply is not comparable to the case of Pluto versus the Sun.
3.3 Coordinate Transformations and Means
A naive view suggests that the simple mean is always the appropriate scalar summary for
any set of data, including temperature data. But this is untenable for reasons more general
than those related to the thermodynamic issues discussed above.
Data have no intrinsic meaning. The context establishes the meaning. It also establishes
whether the numbers themselves are used or some other numbers derived from the raw data.
"Derived" data always raises the issue of coordinate transformations on the raw data. As
noted above, temperature data used in climate measurement are inherently derived, since
air temperature is never measured using the thermodynamic definition Eq. (1) but using
instrument-based proxies such as length, volume, electrical conductance, etc., not to mention
natural proxies such as tree ring widths and wood density, ice core layer isotope ratios, and
so forth.
Even in the absence of a distinction between raw and derived data, questions about the
natural form of an average emerge. A context for a data set may take the form of an auxiliary
structure on the data that induces a geometry. The geometry can imply the natural form of
an average which can differ from the mean. In the absence of a context, different norms on
vectors formed from data can induce a class of averages, which have the same mathematics
as a range of transformations between raw and derived data. The change from raw to derived
data inevitably involves mappings and coordinate transformations.
One physical quantity is measured, but another physical property is sought. The two
quantities are linked by a relationship which amounts to a coordinate transformation. Con-
sider a case where basic observational data must be transformed to acquire the quantity for
which an average value is sought. Suppose that we measure a raw set {yi} with N positive
elements from which we derive a set {xi} by a nonsingular monotone transformation.
In examples such as kinetic speed or radiation energy as raw values, an average over
derived kinetic energy or derived temperature would be expressed in terms of a power law.
The raw and derived data are related by y = xr
for positive r, then the mean of {yi} is
15
equivalent to the mean over {xr
i },
AR1 ({yi}) = AR1 ({xr
i }) =
1
N i
xr
i . (14)
But the mean over the derived data is,
AR1 ({xi}) =
1
N i
xi . (15)
Thus in either case there are two data sets and two simple means -- one for each of the raw
and derived sets. For the radiation example we essentially measure {T4
i }, ignoring constants
of the classical black body law. To get temperature we must transform the measurements
by taking the 1/4 power. That is r = 4. Similarly for the speed problem we measure {|vi|}
and take the square, which means that r = 1/2.
But what is not mandated by the naive view is which mean to work with. That is, do
we use the transformation to compute (AR1 ({yi}))
1
r or do we compute AR1 ({y
1
r
i }) instead?
Do we take the mean first and then do the transformation, or do we do the transformation
and then take the mean? Statistical theory does not provide an answer.
Outside of exceptional cases
(AR1 ({yi}))
1
r = AR1 ({y
1
r
i }) , (16)
i.e. the mean operation does not generally commute with the powers. As a consequence, the
mean is not a coordinate system invariant.
3.3.1 Vector Norms, Geometry, and Averaging
The most important exception to Eq. (16) is when all the elements of {yi} are identical to
each other. In this case all acceptable definitions of averages must yield the same value.
Accordingly, for the degenerate class of sets where all elements are identical, averages must
also be invariants under transformations. This degenerate class turns out to tell us about
the geometry of averages.
Suppose one aims to find the average length of streets in a city. Let us use {xi} to denote
the set of lengths of the city's streets. Form a vector from this set, x1, x2, . . . , xN . Vectors
can have lengths in their own right. Seek another vector with the same length but having
all elements being equal. The equicomponent vector that results is
AR1 ({xi}), AR1({xi}), . . ., AR1 ({xi}) , provided the vector's length is computed as the sum
of the (positive) elements (e.g. x1 +x2 +  +xN ), which is the sum of the individual lengths
of the streets. In this case the traditional mean emerges as the average, and the length of
the vector is nothing more than the summation of lengths of all of the city streets, which
seems to make particular sense for this case.
It is notable here that the naive view of the mean runs afoul of the naive view of length.
The latter would hold that the vector's length ought to be Euclidean (i.e. (x2
1 + x2
2 +    +
16
x2
N )
1
2 ). But an equicomponent vector of equal Euclidean length is
AR2 ({xi}), AR2({xi}), . . ., AR2 ({xi}) or (AR1 ({x2
i }))
1
2
, (AR1 ({x2
i }))
1
2
, . . ., (AR1 ({x2
i }))
1
2
,
which does not yield the mean. It yields the root mean square. One cannot normally have
the mean and a Euclidean geometry for data.
Let us suppose, without loss of generality, that some data set, {xi}, has positive elements.
Just as this data does not necessarily have a context, vectors like x1, x2, . . ., xN need not
possess a mathematical geometry (an inner product) and hence any particular length. In
choosing a length, one induces a geometry and an average. The vector lengths or norms,
discussed above, both belong to a class of norms known as -norms, r, which depend on a
parameter, r,
(r)r

i
xr
i = NAR1 ({xr
i }) . (17)
This class of norms is not exhaustive, but it is sufficient for our discussion. For a data set
{xi} the equicomponent vector arising from any such -norm is
r
N
1
r
,
r
N
1
r
, . . . ,
r
N
1
r
. (18)
A component of Eq. (18) is the H¨older mean. Note that for the city streets example the
length is 1, which is known, appropriately, as the "taxicab" or "city streets" norm. The
Euclidean case was 2. All other positive values of r induce different geometries.
From this we conclude
(AR1 ({yi}))
1
r =
r
N
1
r
= AR1 ({xi}), (19)
where yi = xr
i was substituted so that we could observe that Eq. (19) is the same equation as
Eq. (16). However in this section we have not presumed a distinction between raw and derived
data. The issue is the underlying geometry, which brings exactly the same transformation
group into play even if transformations are not introduced at the outset. Considering the
geometric context generates the same infinite family of distinct averages as emerged in the
previous section.
3.3.2 Averages in General
This infinite family of averages agrees with the basic definition from Section 3.1.1. If we
take the limit as r   then  = max{xi}  xmax which is known as the -norm or
max-norm. According to Eq. (18) the average becomes xmax too. Thus it is not only the
largest element of the set, but it follows that it is the largest of the family of averages as r
is varied over positive values:
r
N
1
r
= i xr
i
N
1
r

Nxr
max
N
1
r
= xmax . (20)
17
For r = 0 we find r/N1/r
becomes the geometric mean in the limit: exp [( i ln xi)/N] 
exp [(N ln xmax)/N] = xmax.
Negative r is easily treated by forming the set {wi} where wi  x-1
i . Then wmax will be
the largest average over {wi} for positive r. As min{xi} = 1/wmax, then xmin  min{xi} is
the smallest average for negative r. As it is also smaller than the geometric mean, because
exp [( i ln xi)/N]  exp [(N ln xmin)/N] = xmin, it is the smallest average for all r.
Thus any value from the entire interval, I({xi}), is admissible as "the" average, even
values not in the original set but within the interval. This agrees with the basic notion of
the average from Section 3.1.1. Any member of the original set can be representative of the
whole set -- very democratic.
Even though we have resurrected the entire data domain of the data set to be the range of
possible averages, we have not encompassed all averaging rules. Potential averaging rules do
not end with simple powers. With the harmonic mean as a clue, any non-singular monotone
function, z() will do as long as it has an inverse, z-1
(),
z-1
(AR1 ({z(xi)}))  Az({xi}) (21)
Even this cannot be said to be comprehensive in itself as it does not yet include weighted
averages, nor have we extended the notion to a continuum, which we must face for the
atmosphere, at least at a formal level. A little bit of imagination will widen the possibilities
substantially.
Nonetheless this class is already more than broad enough for our purposes. In fact we
will reduce it slightly to use families of averages that include the -norm averages, which
are sometimes called the "mean of order r" family, hereinafter "r-means" for convenience.
We can also introduce the "mean of order s" family, hereinafter "s-means" [18], where
these families of symmetric means are important in the theory underlying index numbers in
economics. The s-means are given by z() = exp (s()), or the resulting average is
1
s
ln (AR1 ({esxi
})) (22)
or
r-mean: ARr  1
N
(xr
1 +    + xr
N )
1/r
s-mean: ARs  1
s
ln 1
N
(esx1
+    + (esxN
) .
(23)
These means will be used in the next section.
4 Averages and Actual Atmospheric Data
As discussed previously, the temperatures of the Earth form a continuous field, T(r, t) which
varies in time. However observations can only be made at specific locations, {ri}, and
times {tj}. While the desideratum of the global averaging operation would seem to be a
component of an equicomponent vector that has the same length as the temperature vector
extracted from the field itself as per section 3.3.1, it is still an intensive field that provides
18
no definite length. One might naively expect something like AR1 (T(ri, tj)) to be the average
used. However what is actually done is much more arcane than any example of averaging
operation discussed in the preceding sections ([1], [2], [7], [19], [4]).
The averaging rule varies in time. For example, there has been a substantial drop in the
number of available weather stations since the 1970s, and especially since 1990, when the
number of sampling points around the world fell roughly in half over four years ([7], Fig. 2).
The resulting land-based average is then combined (in some cases) with a series constructed
using samples of seawater temperature collected during the twentieth century by ships, using
a combination of bucket-and-thermometer observations and automated engine cooling-water
observations.
Observations are not made at a common instant, tj, for the whole field, as one might
naively expect in the first instance. Instead they have a common day, d, and measurements
are made in terms of local time of day time producing a set,
{T,d
ri
} = Ot(T(r, t)) , (24)
where ri is place and  is time on day, d. The operation Ot is the time dependent operator
that takes the full field to the set.
Land-based meteorological stations around the world provide records of daily maximum
and daily minimum temperatures, denoted max(T,d
ri
) and min(T,d
ri
). The daily average is
defined as max(T,d
ri
) + min(T,d
ri
) /2. Most land-based stations only report this type of
average, although some provide the min/max.
From section 3.3.1 this average is the mean of the  and - H¨older means over daily
temperatures. This combination is a new type of average in terms of this paper, which
naturally can be expected to have distinctive behaviors. We introduce a new averaging
operator Ax
R which only averages over x holding other variables fixed, thus producing a set
and not just a real number as output. In this case A
 indicates the average is over  and
we denote the mean of the H¨older means by .
Clearly, based on Section 3 and particularly 3.1.2, it may be problematic to determine
whether one location is warmer or colder than another on any particular day with this
statistic. Thus the observational network actually provides
A
 Ot T(r, t) . (25)
At every ri the mean of the daily values is taken over a month. The resulting series
of monthly values is translated by subtracting the mean of the monthly means themselves
further averaged over a thirty year interval (nowadays 1961-1990 is used). These reduced
values, known as anomalies, are then subjected to ad hoc adjustments which speculatively
reduce the effects attributable to any land-use changes since before human settlement in the
area, effects due to changes in equipment or measurement site, missing stretches of data, etc.
The adjusted anomalies are then subjected to a mapping to yield estimates of what the
anomalies might be if the actual observation sites were distributed with a uniform density
over the surface of Earth. The average series are grouped into latitude-longitude boxes and
19
averaged up to a "gridcell" level, with weights applied so that the variance is not a function
of the number of individual stations in the gridcell. In this way, records from isolated stations
count more heavily than records in more densely-sampled locations.
These translations to "anomaly" coordinates and the other mappings that speculate
on what the values of observations might be at different locations not measured, or under
different conditions not known are denoted by, M,
M A
 Ot T(r, t) . (26)
Finally, after taking the simple mean, we arrive at the global temperature statistic  that
is actually in use,
 = AR1 M A
 Ot T(r, t) . (27)
This statistic is clearly more complex than the  introduced in Eq. (2). It is far removed
from AR1 (T(r, t)), which  is widely confused with. There is no evidence that trends in
AR1 (T(r, t)) need agree with that of , nor is there a physical reason to prefer one over the
other.
Debates about the difference between trends in  and , where  is the output of an
entirely different averaging map of microwave fluxes, F, received from satellites, are also
problematic. There is no physical reason why  must equal  or even have the same trends.
 and  are entirely different statistics in a mathematical sense, even if they each are mapped
from the same T(r, t) for the Earth.
So far we have shown that different averages exist, they are used, and that contradictory
trends can emerge between them. We have shown that the conditions exist in the atmosphere
where such paradoxical behavior can be expected to be found. In the following we show that
it does in fact appear in real atmospheric data.
4.1 Contradictory Trends in Global Temperature Averages
Ranking a particular type of field average computed over a sequence of times amounts to
determining a trend in that average. Here we show that the sign and size of such a trend
computed statistically is dependent on the choice of averaging rules, which will suffice to
demonstrate both that the "global temperature trend" is not a unique physical variable, and
that ranking this or that year as the "warmest of the millennium" is not possible, since other
averages will give other results with no grounds for choosing among them.
To illustrate with actual temperature data, we computed averages of temperatures over
twelve sites (see Table 1) and computed a linear trend in each case. The trends through the
1979­2000 period were computed with r-means and s-means. The raw data are themselves
averaged (simple monthly means: r = 1) smoothing out some variability, but this could
not be avoided, and does not affect the main results below. Stations were selected to give
reasonable geographic variation, but whether it is a "global" sample or not is secondary for
the purpose of the example. Stations had to be in continuous use during the 1979­2000
20
Name Name
Phoenix, Arizona Sioux Falls, South Dakota
Cartagena, Colombia Egedesminde, Greenland
Dublin, Ireland Salehard, Russia
Chiang Mai, Thailand Ceduna, Australia
Jan Smuts, South Africa Halley, Antarctica
Honolulu, Hawaii Souda, India
Table 1: Twelve weather station records used for Figures 2 and 3. The data are monthly
averages from the Goddard Institute of Space Studies (GISS) [3].
-1-.8-.6-.4-.20.2
K/decade
-125 -100 -75 -50 -25 0 25 50 75 100 125
r parameter
Vertical dashed line denotes linear mean
Trends through r-means
Figure 2: Trends (in K/decade) through r-means from 12 Station sample
interval. Missing months were interpolated linearly as long as there was no more than one
missing month in sequence, and it wasn't at the start or finish of the sample.
For each value of r, s (cf. Eq. (23)) the monthly r, s-means across the stations were
computed, then a linear trend was fitted using ordinary least squares after deleting rows
with missing data. The trend values are plotted in Figures 2 and 3. For the simple mean
(r = 1, s = 0) the decadal "warming " trend was 0.06 
C/decade. This turns out to be
the peak value of the trend: for most values of r and s the trends are negative, indicating
"cooling" across the 1979 to 2000 interval.
It might seem contradictory that the same data show "global warming" of about 0.02 
C/decade
for s = 0.04, but "global cooling" of -0.04 
C/decade for s = -0.04. But there is no con-
tradiction in the data: They do not show "global" anything. The data are local. The
interpretation of "global" warming or cooling is an artificial imposition on the data achieved
21
-1-.8-.6-.4-.20.2
K/decade
-2 -1.5 -1 -.5 0 .5 1 1.5 2
s parameter
Vertical dashed line denotes linear mean
Trends through s-means
Figure 3: Trends (in K/decade) through s-means from 12 Station sample
by attaching a label to, respectively, a positive or negative trend in one particular average.
5 Conclusion
There is no global temperature. The reasons lie in the properties of the equation of state
governing local thermodynamic equilibrium, and the implications cannot be avoided by sub-
stituting statistics for physics.
Since temperature is an intensive variable, the total temperature is meaningless in terms
of the system being measured, and hence any one simple average has no necessary meaning.
Neither does temperature have a constant proportional relationship with energy or other
extensive thermodynamic properties.
Averages of the Earth's temperature field are thus devoid of a physical context which
would indicate how they are to be interpreted, or what meaning can be attached to changes
in their levels, up or down. Statistics cannot stand in as a replacement for the missing physics
because data alone are context-free. Assuming a context only leads to paradoxes such as
simultaneous warming and cooling in the same system based on arbitrary choice in some
free parameter. Considering even a restrictive class of admissible coordinate transformations
yields families of averaging rules that likewise generate opposite trends in the same data,
and by implication indicating contradictory rankings of years in terms of warmth.
The physics provides no guidance as to which interpretation of the data is warranted.
Since arbitrary indexes are being used to measure a physically non-existent quantity, it is
not surprising that different formulae yield different results with no apparent way to select
22
among them.
The purpose of this paper was to explain the fundamental meaninglessness of so-called
global temperature data. The problem can be (and has been) happily ignored in the name of
the empirical study of climate. But nature is not obliged to respect our statistical conventions
and conceptual shortcuts. Debates over the levels and trends in so-called global temperatures
will continue interminably, as will disputes over the significance of these things for the human
experience of climate, until some physical basis is established for the meaningful measurement
of climate variables, if indeed that is even possible.
It may happen that one particular average will one day prove to stand out with some
special physical significance. However, that is not so today. The burden rests with those
who calculate these statistics to prove their logic and value in terms of the governing dy-
namical equations, let alone the wider, less technical, contexts in which they are commonly
encountered.
6 Acknowledgement
Thanks to Kirill Kondratyev for encouraging the publication of this work.
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