Calculated Mean
Global Temperatures 16102012
Introduction
This monograph is a clarification and further refinement of
Reference 10 (references are listed at the end of this paper) which also considers only average global temperature. It does not
discuss weather, which is a complex study of energy moving about the planet. It
does not even address local climate, which includes precipitation. It does,
however, consider the issue of Global Warming and the mistaken perception that
human activity has a significant influence on it.
The word ‘trend’ is used here for temperatures in two
different contexts. To differentiate, αtrend applies to averagingout the
uncertainties in reported average global temperature measurements to produce
the average global temperature oscillation resulting from the net ocean surface
oscillation. The term βtrend applies to the slower average temperature change of the
planet which is associated with change to the temperature of the bulk volume of
the material (mostly water) involved.
The first paper to suggest the hypothesis that the sunspot
number timeintegral is a proxy for a substantial driver of average global
temperature change was made public 6/1/2009. The discovery started with
application of the first law of thermodynamics, conservation of energy, and the
hypothesis that the energy acquired, above or below breakeven (appropriately accounting
for energy radiated from the planet), is proportional to the timeintegral of
sunspot numbers. The derived equation revealed a rapid and sustained global energy
rise starting in about 1941. The true average global temperature anomaly change
βtrend is proportional to global energy change.
Measured temperature anomaly αtrends oscillate above and
below the temperature anomaly βtrend calculated using only the sunspot number anomaly timeintegral.
The existence of ocean oscillations, especially the Pacific Decadal Oscillation, led to the perception that there must be an effective net surface temperature oscillation for the planet with all named and unnamed ocean oscillations as participants. Plots of measured average global temperatures indicate that the net surface temperature oscillation has a period of 64 years with the most recent maximum in 2005.
The existence of ocean oscillations, especially the Pacific Decadal Oscillation, led to the perception that there must be an effective net surface temperature oscillation for the planet with all named and unnamed ocean oscillations as participants. Plots of measured average global temperatures indicate that the net surface temperature oscillation has a period of 64 years with the most recent maximum in 2005.
Combination of the effects results in the effect of the ocean
surface temperature oscillation (αtrend) decline 19411973 being slightly
stronger than the effect of the rapid rise from sunspots (βtrend) resulting in
a slight decline of the trend of reported average global temperatures. The steep
rise 19732005 occurred because the effects added. A high coefficient of
determination, R^{2}, demonstrates that the hypothesis is true.
Over the years, several refinements to this work (often resulting from other's comments which may or may not have been corroborative) slightly improved the
accuracy and led to the equations and figures in this paper.
Prior work
The law of conservation of energy is applied effectively the same as described in
Reference 2 in the development of a very similar equation that calculates
temperature anomalies. The difference is that the variation in energy ‘OUT’ has
been found to be adequately accounted for by variation of the sunspot number anomalies.
Thus the influence of the factor [T(i)/Tavg]^{4} is eliminated.
Change to the level of atmospheric carbon dioxide has no
significant effect on average global temperature. This was demonstrated in 2008
at Reference 6 and is corroborated at Reference 2
and again here.
As determined in Reference 3, reported average global
temperature anomaly measurements have a random uncertainty with equivalent
standard deviation ≈ 0.09 K. A substantial contributor to this variation
appears to be the apparent random variation in magnitude and period of el Nino.
Global Warming ended more than a decade ago as shown here,
and in Reference 4 and also Reference 2.
Average global temperature is very sensitive to cloud change
as shown in Reference 5.
The value used for average sunspot number was 43.97 (average
18501940) in Ref. 1, 42 (average 18951940) in Ref. 9, and 40 (average
16102012) in Ref. 10. It is set at 34 (average 16101940) in this paper. The
procession of values for average sunspot number produces slight but steady
improvement in R^{2} for the period of measured temperatures and
progressively greater credibility of average global temperature estimates for
the period prior to direct measurements becoming available.
Initial work is presented in several papers at http://climaterealists.com/index.php?tid=145&linkbox=true
The sunspot number anomaly timeintegral drives the temperature anomaly trend
It is axiomatic that change to the average temperature trend
of the planet is due to change to the net energy retained by the planet.
Table 1 in reference 2 shows the influence of atmospheric
carbon dioxide (CO_{2}) to be insignificant (tiny change in R^{2}
if considering CO_{2} or not) so it can be removed from the equation by
setting coefficient ‘C’ to zero. With ‘C’ set to zero, Equation 1 in Reference
2 calculates average global temperature anomalies (AGT) since 1895 with 89.82%
accuracy (R^{2} = 0.898220).
The current analysis determined that 34, the approximate
average of sunspot numbers from 16101940, provides a slightly better fit (in
fact, the best fit) to the measured temperature data than did 43.97 and other
values^{ 9,10}. The influence, of StephanBoltzmann radiation change
due to AGT change, on energy change is adequately accounted for by the sunspot
number anomaly timeintegral. With these refinements to Equation (1) in Reference 2 the
coefficients become A = 0.3588, B = 0.003461 and D = ‑ 0.4485. R^{2} increases slightly to 0.904906
and the calculated anomaly in 2005 is 0.5045 K. Also with these refinements the
equation calculates lower early temperature anomalies and projects slightly higher (0.3175 K vs. 0.269 K in 2020) future anomalies. The resulting equation for calculating the
AGT anomaly for any year, 1895 or later, is then:
Anom(y) = (0.3588,y)
+ 0.003461/17 Σ^{y}_{i=1895}
(s(i) – 34) – 0.4485
Where:
Anom(y) = calculated
temperature anomaly in year y, K
(0.3588,y)
= approximate contribution of ocean cycle effect to AGT in year y
s(i) =
average daily Brussels International sunspot number in year i
Measured temperature anomalies are from Figure 2 of Reference 3. The excellent
match of the up and down trends since before 1900 of calculated and measured temperature anomalies, shown here in Figure 1, demonstrates the usefulness and validity of the
calculations.
Projections until 2020 use the expected sunspot number trend for
the remainder of solar cycle 24 as provided^{ 11} by NASA. After 2020
the limiting cases are either assuming sunspots like from 1925 to 1941 or for
the case of no sunspots which is similar to the Maunder Minimum.
Some noteworthy volcanoes and the year they occurred are also shown
on Figure 1. No consistent AGT response is observed to be associated with
these. Any global temperature perturbation that might have been caused by
volcanoes of this size is lost in the temperature measurement uncertainty.
Much larger volcanoes can cause significant temporary global cooling from the added reflectivity of aerosols and airborne particulates. The Tambora eruption, which started on April 10, 1815 and continued to erupt for at least 6 months, was approximately ten times the magnitude of the next largest in recorded history and led to 1816 which has been referred to as ‘the year without a summer’. The cooling effect of that volcano exacerbated the already cool temperatures associated with the Dalton Minimum.
Much larger volcanoes can cause significant temporary global cooling from the added reflectivity of aerosols and airborne particulates. The Tambora eruption, which started on April 10, 1815 and continued to erupt for at least 6 months, was approximately ten times the magnitude of the next largest in recorded history and led to 1816 which has been referred to as ‘the year without a summer’. The cooling effect of that volcano exacerbated the already cool temperatures associated with the Dalton Minimum.
As discussed in Reference 2, ocean oscillations produce
oscillations of the ocean surface temperature with no significant change to the
average temperature of the bulk volume of water involved. The effect on AGT of
the full range of surface temperature oscillation is given by the coefficient
‘A’. (A, B, C, and D are the coefficients in Equation
1 of Reference 2)
The influence of ocean surface temperature oscillations can
be removed from the equation by setting ‘A’ to zero. To use all regularly
recorded sunspot numbers, the integration starts in 1610. The offset, ‘D’ must
be changed to 0.1993 to account for the different integration start point and
setting ‘A’ to zero. Setting ‘A’ to zero requires that the anomaly in 2005 be
0.5045  0.3588/2 = 0.3251 K. The result, Equation (1) here, then calculates
the trend 16102012 resulting from just the sunspot number anomaly timeintegral.
Trend3anom(y) = 0.003461/17 * Σ^{y}_{i = 1610}
[s(i)34] – 0.1993 (1)
Where:
Trend3anom(y) = calculated temperature anomaly βtrend in
year y, K degrees.
0.003461 = the proxy factor, B, W yr m^{2}.
17 = effective thermal capacitance of the planet, W Yr m^{2}
K^{1}
s(i) = average daily Brussels International sunspot number
in year i
34 ≈ average
sunspot number for 16101940.
0.1993 is
merely an offset that shifts the calculated trajectory vertically on the graph,
without changing its shape, so that the calculated temperature anomaly in 2005
is 0.3251 K which is the calculated anomaly for 2005 if the ocean oscillation
is not included.
Sunspot numbers back to 1610 are shown in Figure 2 of Reference
1.
Applying Equation (1) to the sunspot numbers of Figure 2 of Reference
1 produces the trace shown in Figure 2 below.
Figure 2: Anomaly
trend from just the sunspot number anomaly timeintegral using Equation (1).
Average global temperatures were not directly measured in
1610 (thermometers had not been invented yet). Recent estimates, using proxies,
are few. The temperature anomaly trend that Equation (1) calculates for that time is roughly
consistent with other estimates. The decline in the trace 16101700 on Figure 2
results from the low sunspot numbers for that period as shown on Figure 2 of
Reference 1.
How this phenomenon
could take place
Although the
connection between AGT and the sunspot number anomaly timeintegral is demonstrated,
the mechanism by which this takes place remains somewhat theoretical.
Various papers have been written
that indicate how the solar magnetic field associated with sunspots can influence
climate on earth. These papers posit that decreased sunspots are associated
with decreased solar magnetic field which decreases the deflection of and
therefore increases the flow of galactic cosmic rays on earth.
Henrik Svensmark, a Danish physicist, found that decreased
galactic cosmic rays caused decreased low level (<3 km) clouds and planet
warming. An abstract of his 2000 paper is at Reference 13. Marsden and
Lingenfelter also report this in the summary of their 2003 paper^{ 14} where they make the statement “…solar activity increases…providing more
shielding…less lowlevel cloud cover… increase surface air
temperature.” These findings have been
further corroborated by the cloud nucleation experiments^{ 15} at CERN.
These papers associated the increased lowlevel clouds with increased
albedo leading to lower temperatures. Increased low clouds would also result in
lower average cloud altitude and therefore higher average cloud temperature.
Although clouds are commonly acknowledged to increase albedo, they also radiate
energy to space so increasing their temperature increases radiation to space
which would cause the planet to cool. Increased albedo reduces the energy
received by the planet and increased radiation to space reduces the energy of
the planet. Thus the two effects work together to change the AGT of the planet.
Simple analyses^{ 5} indicate that either an increase of
approximately 186 meters in average cloud altitude or a decrease of average
albedo from 0.3 to the very slightly reduced value of 0.2928 would account for
all of the 20^{th} century increase in AGT of 0.74 °C. Because the
cloud effects work together and part of the temperature change is due to ocean
oscillation, substantially less cloud change is needed.
Combined Sunspot
Effect and Ocean Oscillation Effect
As a possibility, the period and amplitude of oscillations
attributed to ocean cycles demonstrated to be valid after 1895 are assumed to
maintain back to 1610. Equation (1) is modified as shown in Equation (2) to
account for including the effects of ocean oscillations. Since the expression
for the oscillations calculates values from zero to the full range but
oscillations must be centered on zero, it must be reduced by half the
oscillation range.
Trend4anom(y) = (0.3588,y) – 0.1794 + 0.003461/17 * Σ^{y}_{i
= 1610} [s(i)34] – 0.1993 (2)
The ocean oscillation factor, (0.3588,y) – 0.1794, is
applied to the period prior to the start of direct temperature measurements as a possibility. The
effective sea surface temperature anomaly, (A,y), is defined in Reference 2.
Applying Equation (2) to the sunspot numbers from Figure 2 of
Reference 1 produces the trend shown in Figure 3 next below. Available measured
average global temperatures from Figure 2 in Reference 3 are superimposed on the calculated
values.
Figure 3 shows that temperature anomalies calculated using
Equation (2) estimate possible trends since 1610 and actual trends of reported
temperatures since they have been accurately measured world wide. The match
from 1895 on has R^{2} = 0.9049 which means that 90.49% of average
global temperature anomaly measurements are explained. All factors not
explicitly considered (such as the 0.09 K s.d.
random uncertainty in reported annual measured temperature anomalies, aerosols, CO_{2,} other noncondensing ghg, volcanoes, ice change, etc.) must find room in that unexplained 9.51%. Note that a
coefficient of determination, R^{2} = 0.9049 means a correlation
coefficient of 0.95.
A survey^{ 12} of nontreering global temperature
estimates was conducted by Loehle including some for a period after 1610. A
simplification of the 95% limits found by Loehle are also shown on Figure 3.
The spread between the upper and lower 95% limits are fixed, but, since the
anomaly reference temperatures might be different, the limits are adjusted
vertically to approximately bracket the values calculated using the equations.
The fit appears reasonable considering the uncertainty in all values.
Calculated temperature anomalies look reasonable back to 1700 but indicate higher temperatures
prior to that than most proxy estimates. They are, however, consistent with the
low sunspot numbers in that period. They
qualitatively agree with Vostok, Antarctica ice core data but decidedly differ
from Sargasso Sea estimates during that time (see the graph for the last 1000
years in Reference 6). Credible worldwide assessments of average global
temperature that far back are sparse. Ocean oscillations might also have been
different from assumed.
Possible lower values
for average sunspot number
Possible lower assumed values for average sunspot number, with
coefficients adjusted to maximize R^{2}, result in noticeably lower
estimates of early (prior to direct measurement) temperatures with only a tiny
decrease in R^{2}. Calculated temperature anomalies resulting from using an average
sunspot number value of 26 are shown in Figure 4. The projected temperature anomaly trend
decline is slightly less steep (0.018 K warmer in 2020) than was shown in
Figure 1.
Figure 4: Calculated temperature anomalies from the sunspot number anomaly timeintegral plus ocean oscillation using 26
as the average sunspot number with superimposed available measured data from
Reference 3 and range estimates determined by Loehle.
Carbon dioxide change
has no significant influence
The influence that CO_{2} has on AGT can be
calculated by including ‘C’ in Equation (1) of Reference 2 as a coefficient to
be determined. The tiny increase in R^{2} demonstrates that
consideration of change to the CO_{2} level has no significant
influence on AGT. The coefficients and resulting R^{2} are given in
Table 1.
Table 1: A, B, C, D,
refer to coefficients in Equation 1 in Reference 2
Average daily SSN

ocean oscillation A

sunspots B

CO_{2} C

Offset
D

Coefficient of determination R^{2}

% cause of
19092005 AGT change


Sunspots

Ocean oscillation

CO_{2 }change


26

0.3416

0.002787

0

0.4746

0.903488

63.8

36.2

0

32

0.3537

0.003265

0

0.4562

0.904779

62.7

37.3

0

34

0.3588

0.003461

0

0.4485

0.904906

62.2

37.8

0

36

0.3642

0.003680

0

0.4395

0.904765

61.7

38.3

0

34

0.3368

0.002898

0.214

0.4393

0.906070

52.3

35.6

12.1

Science explains why CO_{2} change has no
significant effect on climate. (Added 10/2/14, revised
11/21/14, 11/29/14)
1) Firmly acknowledge the established fact
that gas molecules can absorb/emit photons only at specific discreet
wavelengths (which might be broadened from pressure, etc.). Full spectrum
(Plank’s law) StephanBoltzmann (SB) radiation applies to liquids and solids,
not to gases.
2) From gas kinetics, the time between
atmospheric molecule collisions is extremely short (The Hyperphysics calculator
calculates approximately 0.0001 microsecond at sea level pressure and
temperature).
3) The elapsed time between absorption and
emission of a photon by a CO_{2} gas molecule must be greater than zero
or there would be no evidence that absorptionemission had occurred. One
calculation is approximately 10 microseconds from the time that energy is
increased until a photon is emitted. http://rabett.blogspot.com/2013/04/thisiswhereelicamein.html
4) Thus, at sea level conditions, some or
all of the photon energy that is absorbed by a ghg molecule is immediately
transferred to other molecules by collision. The process of absorbing a photon
and transferring (thermal conduction in the gas) the added energy to other
molecules is thermalization. A common observation of thermalization by way of
water vapor is that nights cool faster when absolute water vapor content is
lower.
5) The reduced radiation flux on both
sides of the 15 micron CO_{2} absorption line, as observed in most TOA
measurements^{ 18} results from EMR energy absorbed by CO_{2}
that becomes thermalized. The spike at 15 microns results from
reversethermalization at very high altitude.
6) Thermalized energy carries no identity
of the molecule that absorbed it.
7) Jostling between the molecules
sometimes causes reversethermalization. Because, for terrestrial radiation wavelengths
(nearly all in the range 6100 microns), there are hundreds of absorption lines
for each water vapor molecule compared to the single one for a CO_{2} molecule,
at low to medium altitudes, EMR emission stimulated by reversethermalization
is essentially all by way of water vapor.
8) The thermalized radiation warms the
air, reducing its density, causing updrafts which are exploited by soaring
birds, sailplanes, and occasionally hail. Updrafts are matched by downdrafts
elsewhere, usually spread out but sometimes recognized by pilots and passengers
as ‘air pockets’ and micro bursts.
9) The population gradient of ghg
molecules, (especially water vapor above about 3 km, declining with increasing
altitude) favors radiation to space. Ghg molecules that emit a photon are
‘recharged’ by reversethermalization.
10) Clouds (average emissivity about 0.5)
consist of solid and/or liquid water particles (each containing millions of
molecules) that radiate according to SB law. Low amount of water vapor above
clouds and widening molecule spacing allows substantial radiation directly to
space.
11) The tiny increase in ghg from
increased CO_{2} causes absorption/thermalization to occur at slightly
lower altitude which slightly increases the convection rate.
12) In the wavelength range of significant
terrestrial radiation, water vapor at approximately 15,000 ppmv has more than
400 absorption lines (absorption opportunities) per molecule in the range 18100
microns compared to only one at 15 microns for CO_{2.} The tiny
increase in absorption lines due to a 100 ppmv CO_{2} increase, about 1
in 60,000, has no significant effect on climate.
Corroboration that CO_{2} has no
significant effect on AGT (added 1/29/15 updated 2/27/15)
A forcing adds
energy to earth analogous to speed adding distance on a trip. Both need to
operate for a duration to accumulate any energy change to the planet or
distance on the trip. If the forcing is constant, the energy change is simply
the forcing times the duration. If the forcing varies (or not), the energy
change is the timeintegral of the forcing. On the planet, the energy added
divided by the effective thermal capacitance (consistent units) gives the
average global temperature (AGT) change. Therefore the AGT change is equal to
an appropriate scale factor times the timeintegral of the forcing.
Atmospheric
carbon dioxide (CO_{2}), has been considered to be a forcing. If so,
the AGT change attributable to CO_{2} change must equal an appropriate
scale factor times the timeintegral of the CO_{2} level. It has been
demonstrated here that CO_{2} change has no
significant effect on AGT 18952012. Thus if CO_{2} is a
forcing the temperature could only increase (unless compensated for by an
asyetundiscovered forcing which magically disappeared as soon as credible
average global temperature measurements became available). Observation that the
AGT trend ever decreases to an earlier AGT is evidence that the scale factor is
zero, CO_{2} is not a significant forcing, and something else is
causing the temperature change.
According to
widely available data from Vostok, Antarctica ice cores, during previous
glaciations and interglacials, CO_{2} and AGT went up and down approximately
in lockstep (as so dramatically displayed in An Inconvenient Truth). This demonstrates that, at least up to
about 300 ppmv, CO_{2} has no significant effect on AGT.
Application of
this analysis methodology to CO_{2} levels for the entire Phanerozoic
eon (about 542 million years) (Berner, 2001) proves that CO_{2} levels
up to at least 6 times the present will have no significant effect on AGT.
Some might
assert that there must be a ‘break even’ CO_{2} level. Above this level
CO_{2} is a positive forcing and below the breakeven level the AGT
trajectory would indicate it was a negative forcing.
Pick any two end
temperatures. A ‘breakeven’ CO_{2} level and offset can be determined
such that the calculation will produce the two end temperatures. However, pick
two different end temperatures and a different ‘breakeven’ CO_{2}
level would be calculated. Since the possibility of many different ‘breakeven
levels is ludicrous, the conclusion that CO_{2} has no significant effect
on AGT prevails and something else is causing the temperature change.
Because CO_{2}
is only a trace gas in the atmosphere, if CO_{2} change does not cause
temperature change, it cannot cause climate change. THUS THE CO_{2}
CHANGE FROM BURNING FOSSIL FUELS HAS NO SIGNIFICANT EFFECT ON CLIMATE and
climate sensitivity (the effect on AGT of doubling CO_{2} is not
significantly different from zero.
The two factors
which explain the last 300+ years of climate change are also identified in a
peer reviewed paper published in Energy and Environment, vol. 25, No. 8,
14551471.
Further discussion of ocean cycles (Added 6/23/14)
The temperature contribution to AGT of ocean cycles is
approximated by a function that has a sawtooth trajectory profile. It is
represented in Equation (1) of Reference 2 by (A,y) where A is the total amplitude and y is
the year. The uptrends and down trends are each determined to be 32 years long
for a total period of 64 years. The total amplitude resulting from ocean oscillations
was found here to be 0.3588 K (case highlighted in Table 1).
Thus, for an ocean cycle surface temperature uptrend, the contribution of ocean oscillations
to AGT is approximated by adding (to the value calculated from the sunspot number anomaly timeintegral) 0.3588 multiplied by the fraction of the 32 year period that has
elapsed since a low. For an ocean cycle surface temperature down trend, the contribution is calculated by adding
0.3588 minus 0.3588 multiplied by the fraction of the 32 year period that has
elapsed since a high. The lows were found to be in 1909 and 1973 and the highs
in 1941 and 2005. The resulting trajectory, offset by half the amplitude, is shown
as ‘approximation’ in Figure 5.
Temperature data is available for three named cycles: PDO,
ENSO 3.4 and AMO. Successful accounting for oscillations is achieved for PDO
and ENSO when considering these as forcings (with appropriate proxy factors)
instead of direct measurements. As forcings, their influence accumulates with
time. The proxy factors must be determined separately for each forcing. The
measurements are available since 1900 for PDO^{ 16} and ENSO3.4^{ 17}.
This PDO data set has the PDO temperature measurements reduced by the average
SST measurements for the planet.
The contribution of PDO and ENSO3.4 to AGT is calculated by:
PDO_NINO = Σ^{y}_{i=1900} (0.017*PDO(i) + 0.009
* ENSO34(i)) (3)
Where:
PDO(i) =
PDO index^{ 16} in year i
ENSO34(i) =
ENSO 3.4 index^{ 17} in year i
How this calculation compares to the idealized approximation
used in Equation (2) is shown in Figure 5. The high coefficient of determination in Table 1 and the comparison in Figure 5 corroborate the assumption that the sawtooth profile provides an adequate approximation of the influence of all named and unnamed ocean cycles in the calculated AGT anomalies.
Figure 5: Comparison
of idealized approximation of ocean cycle effect and the calculated effect from
PDO and ENSO.
Conclusions
Others that have looked at only amplitude or only duration factors for solar cycles got poor correlations with average global temperature.
The good correlation comes by combining the two, which is what the
timeintegral of sunspot number anomalies does. As shown in Figure 2, the temperature anomaly trend
determined using the sunspot number anomaly timeintegral has experienced substantial
change over the recorded period. Prediction of future sunspot numbers more than
a decade or so into the future has not yet been confidently done although
assessments using planetary synodic periods appear to be relevant^{ 7,8}.
As displayed in Figure 2, the timeintegral of sunspot
number anomalies alone appears to show the estimated true average global temperature
trend (the net average global energy trend) during the planet warm up from the
depths of the Little Ice Age.
The net effect of ocean oscillations is to cause the surface
temperature trend to oscillate above and below the trend calculated using only
the sunspot number anomaly timeintegral. Equation (2) accounts for both and also,
because it matches measurements so well, shows that rational change to the level
of atmospheric carbon dioxide can have no significant influence.
Long term prediction of average global temperatures depends
primarily on long term prediction of sunspot numbers.
References:
11. Graphical
sunspot number prediction for the remainder of solar cycle 24 http://solarscience.msfc.nasa.gov/predict.shtml
13. Svensmark
paper, Phys. Rev. Lett. 85, 5004–5007
(2000) http://prl.aps.org/abstract/PRL/v85/i23/p5004_1
14. Marsden
& Lingenfelter 2003, Journal of the
Atmospheric Sciences 60: 626636 http://www.co2science.org/articles/V6/N16/C1.php
15. CLOUD experiment at CERN http://indico.cern.ch/event/197799/session/9/contribution/42/material/slides/0.pdf
(Linked from http://www.cgd.ucar.edu/cas/catalog/climind/TNI_N34/
)
18. Barrett, ‘Greenhouse
molecules, their spectra and function in the atmosphere’, Energy & Environment, Vol. 16, No. 6, 2005. http://www.warwickhughes.com/papers/barrett_ee05.pdf