For a spherical shape of the Earth, we have r E = r = c o n s t . and, hence, ψ ( r , θ , φ ) = ψ ( θ , φ ) . Usually, r E is represented by the volumetric mean radius of r E , V ≅ 6371.0   km . Thus, Equation (1) can be written as [10,42,44,46]

〈 ψ 〉 = r E 2 ∫ 0 2 π ∫ 0 π ψ ( θ , φ ) sin θ d θ d φ r E 2 ∫ 0 2 π ∫ 0 π sin θ d θ d φ = 1 4 π ∫ 0 2 π ∫ 0 π ψ ( θ , φ ) sin θ d θ d φ (2)

Since the average along a parallel of latitude, i.e., the zonal average, is defined by [44,46 - 49]

ψ ¯ ( θ ) = 1 2 π ∫ 0 2 π ψ ( θ , φ ) d φ (3)

Equation (2) can be written as

〈 ψ 〉 = 1 4 π ∫ 0 2 π ∫ 0 π ψ ( θ , φ ) sin θ d θ d φ = 2 π 4 π ∫ 0 π sin θ d θ ( 1 2 π ∫ 0 2 π ψ ( θ , φ ) d φ ) = 1 2 ∫ 0 π ψ ¯ ( θ ) sin θ d θ (4)

Inserting the averages for both the northern and the southern hemispheres given by [10]

〈 ψ 〉 N H = 1 2 π ∫ 0 2 π ∫ 0 π / 2 ψ ( θ , φ ) sin θ d θ d φ = ∫ 0 π / 2 ψ ¯ ( θ ) sin θ d θ (5)

and

〈 ψ 〉 S H = 1 2 π ∫ 0 2 π ∫ π / 2 π ψ ( θ , φ ) sin θ d θ d φ = ∫ π / 2 π ψ ¯ ( θ ) sin θ d θ (6)

leads to

〈 ψ 〉 = 1 2 ( 〈 ψ 〉 N H + 〈 ψ 〉 S H ) (7)

When we set, for instance, ψ ( θ , φ ) = T ( θ , φ ) ,we obtain the globally averaged near-surface air temperature

〈 T 〉 = 1 2 ∫ 0 π T ¯ ( θ ) sin θ d θ = 1 2 ( 〈 T 〉 N H + 〈 T 〉 S H ) (8)

Since the angle of the latitude, ϕ ,and the zenith angle, θ ,are related to each other by ϕ = π / 2 − θ ,Equation (8) can be written as

〈 T 〉 = 1 2 ∫ − π / 2 π / 2 T ¯ ( ϕ ) cos ϕ d ϕ = 1 2 ( 〈 T 〉 N H + 〈 T 〉 S H ) (9)

with

〈 T 〉 N H = ∫ 0 π / 2 T ¯ ( ϕ ) cos ϕ d ϕ (10)

and

〈 T 〉 S H = ∫ − π / 2 0 T ¯ ( ϕ ) cos ϕ d ϕ (11)

The Earth is, of course, not a sphere, but in a first approximation an oblate ellipsoid of revolution, i.e. an oblate spheroid, flattened at the poles and bulging at the Equator due to centrifugal forces. This oblate spheroid serves as a surface of reference for the mathematical reduction of geodetic and cartographic data [50]. Its parameters are the semi-major axis a ≅ 6378.1   km ,the semi-minor axis c ≅ 6356.8   km and the flattening f = ( a − c ) / a = 0 .003353 according to the World Geodetic System 1984 (WGS84) [50]. With respect to a Cartesian frame with its origin in the center of mass and its vertical axis congruent with Earth’s rotation axis, we have

x 2 a 2 + z 2 c 2 = 1 (12)

The coordinates are given by x = a cos ( π 2 − θ ) = a sin θ and z = c sin ( π 2 − θ ) = c cos θ . The ellipsoidal radius is

r ( θ ) = r E , V + Δ r ( θ ) = r E , V ( 1 + Δ r ( θ ) r E , V ) (13)

where Δ r ( θ ) is the difference between r ( θ ) and r E , V of the Earth. The ratio δ = Δ r ( θ ) / r E , V ranges from δ = − 0.00223 at the Poles to δ = 0 .00111 at the Equator (Figure 7). Using Equation (13) leads to

A E = ∫ 0 2 π ∫ 0 π r 2 ( θ , φ ) sin θ d θ d φ = 2 π r E , V 2 ∫ 0 π ( 1 + δ ) 2 sin θ d θ (14)

Since the surface of an oblate spheroid is also given by [51]

A E = 2 π a ( a + c 2 a 2 − c 2 arsinh ( a 2 − c 2 c ) ) (15)

or in the re-arranged form

A E = 2 π a 2 + π c 2 e ln ( 1 + e 1 − e ) (16)

with e = 1 − ( c / a ) 2 ,we have

∫ 0 π ( 1 + δ ) 2 sin θ d θ = 2 (17)

The equivalent spherical radius is r E , A ≅ r E , V ≅ 6371.0   km . In the case of a sphere, the rule A 3 = 36 π V 2 is exactly fulfilled, where A and V are the surface and the volume, respectively. Since of all surfaces of the same area, the sphere has the greatest volume, we have A 3 ≥ 36 π V 2 [52]. Consequently, r E , A is marginally larger than r E , V ,but this difference is negligible. Thus, we may write

〈 ψ 〉 = 1 4 π ∫ 0 2 π ∫ 0 π ψ ( δ , θ , φ ) ( 1 + δ ) 2 sin θ d θ d φ (18)

Using again the zonal average defined by Equation (3) leads to

〈 ψ 〉 = 1 2 ∫ 0 π ψ ¯ ( θ ) ( 1 + δ ) 2 sin θ d θ (19)

Inserting the averages for both the Northern and Southern Hemispheres defined by

〈 ψ 〉 N H = ∫ 0 π / 2 ψ ¯ ( θ ) ( 1 + δ ) 2 sin θ d θ (20)

and

〈 ψ 〉 S H = ∫ π / 2 π ψ ¯ ( θ ) ( 1 + δ ) 2 sin θ d θ (21)

leads to

〈 ψ 〉 = 1 2 ( 〈 ψ 〉 N H + 〈 ψ 〉 S H ) (22)

Thus, the globally averaged near-surface air temperature is given by

〈 T 〉 = 1 2 ∫ 0 π T ¯ ( θ ) ( 1 + δ ) 2 sin θ d θ = 1 2 ( 〈 T 〉 N H + 〈 T 〉 S H ) (23)

As in the case of spherical averaging, θ can be substituted by ϕ .

Hansen and Lebedeff [53] already discussed the problem of the incomplete spatial coverage of the Earth’s surface by the network of meteorological stations. They correctly stated:

“The principal limitation of this data set for global or hemispheric analysis is the incomplete spatial coverage, illustrated in Figure 1 [here, Figure 8] for four dates. Although the number and geographical extent of recording stations on land areas increased strongly between 1870 and 1900, there were still large areas in Africa and South America, and all of Antarctica, without coverage in 1900. Substantial station data for Antarctica begins in the 1950s. Large ocean areas remain without fixed meteorological stations at all times.”

However, as expressed by Equations (8) and (9), the uncertainty regarding observations in the south-polar region is of minor importance in determining the globally averaged near-surface temperature. Furthermore, von Hann and his fellow climate researchers not only gathered data from meteorological stations, but also eyewitness descriptions of weather and climate. Over land, the near-surface dry- and wet-bulb temperatures as well as the maximum and minimum temperatures were measured in thermometer boxes (Stevenson’s screens) at a height of 2 m or so above the ground using a standardized thermometer equipment as illustrated in Figure 9. The standardization of land stations goes back to the First Meteorological Congress held in Vienna in September 1873. Such an equipment suitable for sea-going conditions was also used aboard the H.M.S. Challenger during her 1873 to 1876 voyage [27], where either hourly or two-hourly meteorological observations were performed.

Measurements of the sea-surface temperature have been performed since the mid-18^th century; mainly wooden and canvas buckets served to take water probes. Krümmel [54] already discussed the accuracy of methods to measure the sea-surface and deep-sea temperatures. During the voyage of H.M.S. Challenger, the sea-surface temperature was observed every two hours [27].

Nonetheless, even today, network density and design affect the regional averages over landscapes. In 2009, PaiMazumber and Mölders [55] used the results of their WRF simulations performed over Russia for July and December 2005, 2006, and 2007 to create a ‘‘dataset’’ for assessing this impact theoretically. Based on the values at all WRF grid points, they calculated regional averages for various quantities for 2.8˚ × 2.8˚ areas as the ‘‘reference.’’ Then, regional averages determined based on 40 artificial networks and the actual 411 ‘‘sites’’ locations of a real network were compared with the reference regional averages. The 40 networks encompassed 10 networks of 500, 400, 200, or 100 different randomly taken WRF grid points as sites. The results showed that the real network’s site distribution misrepresents the landscape. Such misrepresentation leads to differences in regional averages that show geographical and temporal trends for most quantities: Errors are lower over shores of large lakes than coasts and lowest over flatland followed by low and high mountain ranges; offsets in timing occur during frontal passages when several sites are passed at nearly the same time. Generally, the real network underestimated regional averages of sea-level pressure, wind speed, and precipitation over Russia up to 4.8 hPa (4.8 hPa), 0.7 m∙s⁻¹ (0.5 m∙s⁻¹), and 0.2 mm day⁻¹ and overestimates regional averages of 2 m temperature, downward shortwave radiation, and soil temperature up to 1.9 K (1.4 K), 19 W∙m⁻² (14 W∙m⁻²), and 1.5 K (1.8 K) in July (December) [55].

Alexander von Humboldt was the first who drew annual isotherms on a map of the Earth in 1817 [56]. He used the temperature averages of 58 locations available around the world [57]. Based on the course of these isotherms, von Humboldt deduced the simplest laws of the heat distribution on the Earth’s surface. In his blog contribution “The first isothermal world maps”, Mike Klein, Senior Cartographic Specialist in the Geography and Map Division at the Library of Congress, stated:

“Humboldt withheld publishing his idea in the form of a generalized world map while he waited for data from weather stations around the world. Thus, the isotherm concept remained virtually unknown outside of the broader scientific community until 1838, when it was given wider recognition on the map below [here Figure 10]. Titled Alexander von Humboldt’s System der Isotherm-Kurven in Merkator’s Projection, it was published in Germany by Heinrich Berghaus in his Physikalischer Atlas, the first comprehensive physical world atlas.”

Figure 10 shows this isothermal chart published by Heinrich Berghaus in the 1845 Edition of his Physikalischer Atlas. (Note that the correct spelling is Mercator, but not Merkator as used in von Humboldt’s isothermal chart.)

Batchelder [11] reviewed earlier observations and the use of isothermal charts. By quoting von Hann [18], he underlined the importance to distinguish between water and land masses:

“Durch den Umstand, dass ein Parallelkreis teils über Land, teils über Wasser verläuft, entstehen klimatische Unterschiede zwischen West und Ost, oder Verschiedenheiten des Klimas nach den Meridianen, welche im solaren Klima nicht vorhanden wären. Neben der ungleichen Erwärmung und Erkaltung von Wasser und Land werden ausserdem durch das Vorhandensein des Landes gewisse konstante Luft- und Meeresströmungen erzeugt, welche ebenfalls eine Verschiedenheit des Klimas unter verschiedenen Meridianen desselben Parallels bedingen.”

The translation from German into English given by Batchelder [11] reads:

“Owing to the fact that a circle of latitude passes partly over land and partly over water, there arise climatic differences between West and East, or variations of the climate according to the longitudes, that in the solar climate would not exist. Following the dissimilar warming and cooling of water and land there are, moreover, set up, owing to the presence of the land, certain constant air and sea currents which also make possible a difference of climate on different meridians of the same parallel.”

The notion “solar climate” is discussed in Subsection 3.3.

Von Hann [18] and Spitaler [10] also discussed Forbes’ [23] formula for determining the average (or normal) temperature of a parallel of latitude derived for the Northern Hemisphere

T ( ϕ ) = A + B cos m ( ϕ ) ︸ waterglobe + C n cos ( 2 ϕ ) ︸ landeffect (24)

where n is the fraction of land compared to the circumference of the respective parallel. The coefficients A, B, C, and the exponent m must be derived from observations. Forbes [23] derived (here expressed with respect to the Celsius scale): A = −10.8˚C, B = 32.9˚C, C = 21.2˚C, and m = 1.25. Forbes also discussed the temperature difference between an aqueous globe (n = 0) and a rocky one (n = 1). Figure 11 shows the meridional distributions of the zonal averages of temperature, T ¯ ( ϕ ) ,between 50˚N and 50˚S. Forbes found for the latitude of 45˚ a surface temperature of 51.0 ˚ F ≅ 10.6 ˚ C for both the aqueous globe and the rocky one. For the Equator, he obtained 71.7 ˚ F ≅ 22.1 ˚ C for n = 0 and 109.8 ˚ F ≅ 43.2 ˚ C for n = 1. However, it is unlikely that in both cases the effect of the atmosphere would be the same. With respect to this result, von Hann [16] stated:

“Between latitude 40˚ and latitude 50˚ is the parallel on which both water and land hemispheres have the same temperature. In higher latitudes, the water hemisphere is warmer than the land hemisphere. Even Dove’s isothermal charts of the northern hemisphere show that the transition takes place between latitudes 40˚ and 45˚, and the graphic representation, above referred to, shows more precisely that this transition occurs at latitude 42.5˚. Beyond this latitude, as far as the pole, a water hemisphere is warmer than a hemisphere wholly covered by land.”

In his paper on von Hann’s contribution to modern climatology, Kahlig [58] reported that von Hann used a three-term moving average (running mean) procedure with unequal weights (1/4, 1/2, 1/4), called “hanning” (e.g., [59]), to smooth the raw data of land fraction listed in von Hann’s [15,16] table of the “Mean temperatures of the Parallels of Latitude” in his Handbook of Climatology (pp. 199-200). The raw data and the smoothed ones are listed in columns a and b in von Hann’s table (Figure 4). Thus, von Hann was able to handle Forbes’ [23] formula (Equation (24)) in a sophisticated manner.

Isothermal charts as developed by Dove [19], Wild [24], von Hann [25] and Buchan [27] played an important role in deriving the normal temperatures for numerous parallels of latitude. Schoch [20] used Dove’s [19] isothermal charts, Spitaler [10] used those of Wild [24] and von Hann [25] (eventually published in Berghaus’ Physikalischer Atlas), Batchelder [11], Arrhenius [14], and Hopfner [33] used Buchan’s [27] isothermal charts. The isothermal charts shown in Figures 12-16 were published by von Hann

[25] in 1887, Buchan [27,60] in 1889 and 1899, von Hann [57] in 1906, and Börnstein [31,35] in 1906/ 1913, respectively. Dove [19] used monthly means, seasonal means and annual means determined from the longest so far accessible series of observations at 900 stations. As reported by von Hann [25], his isothermal chart is based on all information available to him in January 1884. This information also included Wild’s [24] isothermal charts for the Russian Empire. Buchan’s [27] isothermal chart not only included the observations of the Challenger Expedition, but also the observations performed at 1620 stations during the fifteen-year period 1870-1884, except in the United States, where the observations started in October 1871, when the Signal Service of the War Department took charge of the Meteorological System of the United States. In his report “Summary of International Meteorological Observations”, Dunwoody [61] stated, for instance (Figure 17):

DeCourthy Ward [62] assessed von Hann’s [25] and Buchan’s [27] isothermal charts together with those prepared by Bartholomew and Herbertson and edited by Buchan [60] for the comprehensive work “Atlas of Meteorology: a series of over four hundred maps”. Figure 14 shows the isothermal chart of annual means. It is a reproduction of Buchan’s isothermal chart that appeared in the Challenger reports, but revised by him [60]. Figure 14 also shows Supan’s [63] temperature zones and climate provinces, and Dunwoody’s [61] map of annual simultaneous observations. The isothermal charts of von Hann [57] (Figure 15) and Börnstein [31,35] (Figure 16) slightly differ from each other. Unfortunately, the isotherms were drawn at different intervals. Only the 0˚C - and the 20˚C-isotherms are directly comparable.

More than 85 years later, Levitus [64] published the Climatological Atlas of the World Ocean. Figure 18 shows his global distribution of the annual mean potential temperature at the sea surface. Figure 19 shows the corresponding figure based on the Ocean Atlas 2018. These figures may be used to assess the distribution of the sea surface in the historical isothermal charts.

Spitaler [10] described his procedure to determine the normal temperature along the parallels of latitude as follows:

“Die Grundlage für die vorliegende Untersuchung lieferten die neuen Isothermenkarten von Wild und Prof. Hann, welche nach dem sämtlichen bis jetzt vorliegenden Beobachtungsmaterial der Erde gezeichnet wurden. Ich habe für jeden 10. Breitengrad von 5 zu 5 Längengraden, für die dazwischenliegenden Breitengrade aber nur für jeden 10. Längengrad die Temperatur des Jahresmittels bestimmt, sowie die mittleren Temperaturen der beiden extremen Monate Januar and Juli graphisch interpoliert und auf diese Weise einerseits aus je 72, andererseits aus je 36 äquidistanten Temperaturwerten die normale Temperatur der Breitenkreise bestimmt.”

The translation reads:

“The basis for the present study was the new isothermal charts of Wild and Prof. Hann, which were drawn based on all the Earth’s observation material available so far. For each 10th degree of latitude and for 5 to 5 degrees of longitude, but for the intermediate latitudes only for each 10th degree of longitude, I determined the annual-average temperature and interpolated the mean temperatures of the two extreme months of January and July graphically. In doing so, I determined the normal temperature of the parallels of latitude based on 72 and 36 equidistant temperature values, respectively.”

Batchelder [11] determined the normal temperature of each 10^th degree of latitude, from 80˚N to 50˚S, by averaging the observed temperatures at 36 equidistant points or stations on the circle, i.e., at every 10^thmeridian. The temperatures at these 36 points were determined, when an isotherm did not happen to fall directly over the point, by interpolation between the nearest isotherms. Referring to Forbes [23], he stated:

“Although one authority considers that ‘It is by no means an easy matter to deduce the mean temperature of a given parallel correctly from an isothermal chart,’ yet as Buchan constructed his isotherms at intervals of 5˚F, interpolation was usually simple and direct, and the limit of error rarely rose above one degree. The most unsatisfactory cases of interpolation were those succession of isotherms) as in July, 130˚W, 50˚N. Still, even here, 1˚F seems usually the error. A peculiar case occurs in January over the northern Amazon basin; as the observer moves north ward over South America he finds the temperature fall from the central part until above the equator, where occurs an isolated area of +85˚ over the Isthmus of Panama. The neighborhood of this area is very difficult to interpolate for; to the north the nearest isotherm is that of 75˚, a ten-degree skip, to the east and west are no neighboring isotherms, and far away to the south is the isotherm of 80˚ on a falling gradient. In the case of the parallel of 80˚N on the Annual Chart, too, the limit of error between 100˚W and 130˚W may rise as high as 4˚F. On the equator, owing to the slight gradient, the general limit of error is about 2˚F.”

Hopfner [33] also determined this normal temperature of each 10^th degree of latitude, but from 80˚N to 60˚S by averaging the observed temperatures at 36 equidistant points or stations on the respective parallel of latitude. He also discussed the accuracy of the use of isothermal charts in detail.

As argued by Köppen [37], notable variation exists in the annual mean temperatures of the lower atmosphere, reduced to sea level, for numerous parallels of latitude (cf. Table 3).

As listed in Table 3, the temperature difference between the Equator and 80˚N is 38˚C (maximum) and 44˚C (minimum). Whereas the difference between the Equator and 80˚S is 42˚C (maximum) and 48˚C (minimum) [37]. Thus, when computing the near-surface temperature of the Earth, we must address possible uncertainty related to such variations in annual averages.

The reduction of temperatures to sea level seems to be a problem. Even though the reduction of air pressure to sea level is widely accepted, this is not the case of near-surface air temperature. In his textbook “Lehrbuch der Meteorologie“ published in 1906, von Hann [57] stated:

“Die Isothermen stellen die Temperaturverteilung auf der Erde so dar, wie wenn alle Orte, nach deren Temperaturaufzeichnungen sie entworfen worden sind, im Meeresniveau liegen würden. Die Temperaturmittel müssen daher, bevor man sie auf der Karte einträgt, auf das Meeresniveau reduziert werden, wozu die früher angeführten Erfahrungen über die Wärmeabnahme mit der Höhe benützt werden.”

The translation reads:

“The isotherms represent the temperature distribution on Earth as if all the locations, which contributed to them with their temperature records, were at sea level. The temperature averages must, therefore, be reduced to sea level before entering the chart to which the previously mentioned experience about the heat decrease with height are used.”

Buchan [60] argued (his temperature notation has been updated):

“The mean temperature falls about 5.5˚C for every 1000 metres of ascent (1˚F per 330 feet or 16˚F per mile); but, in the Northern Hemisphere, between 30˚N and 70˚N, the mean temperature decreases only 0.75˚C (1.35˚F) for every increase of a degree of latitude; i.e. 0.0068˚C for every kilometre nearer the pole (0.0197˚F per mile). The temperature therefore falls 800 times more in the same distance in a vertical direction than in a horizontal one. This may be expressed by saying that the vertical temperature gradient is 800 times steeper than the horizontal one. This contrast can be seen in the sections on Plate 1, where the vertical distances are much exaggerated when compared with the horizontal ones, and hence the sections greatly minimise the contrast.”

Note that von Hann [66] argued in a similar manner. Based on the CRU dataset, Feulner et al. [67] recently illustrated the effect of station’s elevation. They argued that the effect of land (and ice sheet) elevation combined with the lapse rate on the temperature distribution is apparent, making Antarctica, Greenland, and the large mountain ranges markedly cooler than their surroundings.

Sellers [39] did not apply such reduction to sea level in determining his mean annual surface temperatures. He stated:

“Temperatures given in [his] Table 1 are estimated mean annual surface temperatures and are lower than the sea level values often quoted in the meteorological literature (Haurwitz and Austin, 1944 [40]; Trewartha, 1954 [68]), especially between 20˚N and 50˚N and south of 60˚S. The average global surface temperature is about 286˚K, compared with 288˚K for the average global sea level temperature.”

We considered Arrhenius’ [14,26] data re-corrected to mean terrain height above sea level and Sellers’ [39] data to illustrate the difference in the global average of the near-surface air temperature caused by zonal averages of temperatures at site elevation and those reduced to sea level.

Sellers considered the physical and climatological data of 10˚-latitude zones. Thus, we assigned his temperature data to the center of the respective latitude zones. Figure 20 reveals that Sellers’zonal temperature means are remarkably lower than those of the other authors. At latitudes beyond 60˚S, they are even lower than those in Köppen’s [37] distribution of zonal mean minimum temperatures. Beyond 30˚S, Ferrel’s [12] zonal temperature means slightly exceed Köppen’s.

The normal temperatures determined by the authors considered in this study are not completely independent. As aforementioned, subsequent authors improved them using the rapidly increased number of weather observations in the 2^nd half of the 19^th century (Figure 8). The differences between Dove’s [19] and Forbes’ [23] values were still marginal, but Wild [24], von Hann [25] and Buchan [27,60] used augmented weather information for editing their isothermal charts.

Graphic solutions have a long tradition in science and technology. Cremona-Maxwell diagrams, for instance, were used in statics with sufficient accuracy long before Konrad Zuse developed the first programmable computer in the late 1930s.

Forbes’ [23] formula (see Equation (24)) of the asymmetric water and land distribution may be considered as the first attempt to explain the difference between the isotherms of the Earth’s real climate and those of its so-called solar climate. According to von Hann [16,18], the solar climate, sometimes also called the mathematical climate, refers to the thought model of an Earth in the absence of its atmosphere. Von Hann [16] stated

“If the surface of the earth were occupied altogether by land, and if there were no surrounding atmosphere, the condition of our planet would be somewhat similar to that of the moon at the present time. Under these conditions, the distribution of temperature over the earth would depend solely upon the amount of heat received from the sun at any given place, and upon the loss of heat by radiation at that place. As these two factors would necessarily be the same at all points along the same parallel of latitude, the zones of equal temperature would coincide with the parallels of latitude. Even the presence of a vapourless atmosphere would interfere but little with this

distribution of temperature, for only the absolute amounts of heat received at, and radiated from, the surface of the earth would thereby be affected. It is true that convectional currents would be produced under these conditions; but as there would be no reason for the more frequent occurrence of warm or cold air currents along some meridians than others, the distribution of temperature in zones bounded by the parallels of latitude would not thereby be interfered with.”

The solar climate depends on the astronomic aspects like the distance between the Sun’s center and the Earth (Earth-Moon barycenter), the obliquity of Earth’s rotation axis with respect to the normal of the ecliptic plane, the angular velocity of the rotation, and the total solar irradiance (TSI) for 1 AU. Only the absorbed solar radiation and the thermal effect related to the regolith should be considered. With respect to geological time scales, also changes in the precession of Earth’s rotation axis, and long-term variations of the eccentricity, obliquity, and precession of the Perihelion caused by the Sun, Moon and planets of our solar system must be considered [69 - 75].

Figure 21 compares the meridional distributions of the zonal temperature averages of the real climate as given by Spitaler [10] (Sp1885), Arrhenius [26] (A1896-1), von Bezold [29] (vB1906), as well as von Hann-Süring [38] (vHS1939) and the solar climate predicted by Kramm et al. [44] for the year 2010. Kramm et al. [44] used the following local energy budget equation for a thin slab of 2 cm thickness adjacent to the surface of either the Earth’s Moon or the Earth in the absence of its atmosphere:

ϑ d d t 〈 c ρ T s l a b 〉 V = ( 1 − α ( Θ 0 , θ , φ ) ) F cos Θ 0 − ε ( θ , φ ) σ T s l a b 4 ( θ , φ ) − H s l ( θ , φ ) ,(25)

where 〈 … 〉 V is the volume average.Here, t is time, T s l a b , ρ ,and c, are the temperature, bulk density, and specific heat of this slab of soil, respectively. Furthermore, ( 1 − α ( Θ 0 , θ , φ ) ) F cos Θ 0 is the solar radiation absorbed by the slab, where F is the solar irradiance reaching the surface, α ( Θ 0 , θ , φ ) is the integral albedo inthe solar range, and Θ 0 is the local zenith distance of the Sun’s center. Kramm et al. [44] calculated α ( Θ 0 , θ , φ ) by Keihm’s [76] empirical formula rearranged to

α ( Θ 0 ) = α 0 + ( Θ 0 45 ∘ ) 3 ( a + b ( Θ 0 45 ∘ ) 5 ) (26)

where α 0 = 0.10 is the normal albedo, and a = 0.045 and b = 5.47 × 10 −   4 are empirical values. With exception of Keihm’s value for b, all other values are based on observations from the Lunar Reconnaissance Orbiter Diviner Lunar Radiometer Experiment [77,78]. In the case of the Earth, cos Θ 0 was determined using the rules of spherical trigonometry (e.g., [74,75,79,80])

cos Θ 0 = sin ϕ sin δ S u n + cos ϕ cos δ S u n cos h = cos θ sin δ S u n + sin θ cos δ S u n cos h (27)

Here, δ S u n is the declination of the Sun, ϕ is latitude, and h is the hour angle from the local meridian. The solar irradiance, F, is given by

F = ( r S u n r ) 2 F S u n (28)

where F S u n is the solar emittance [74,75,81,82], r S u n ≅ 6.963 × 10 5 km [83] is the radius of the Sun, and r is the actual distance between the Sun’s and Earth’s centers. Inserting the mean distance, r 0 (nearly 1 AU), into Equation (28) provides the solar constant for the Earth

S = ( r S u n r 0 ) 2 F S u n (29)

Combining formulae (28) and (29) yields

F = ( r 0 r ) 2 S (30)

Furthermore, F I R = ε ( θ , φ ) σ T s l a b 4 ( θ , φ ) is the emitted radiation according to the power law of Stefan

[84] and Boltzmann [85], where ε ( θ , φ ) is the integral relative emissivity, and H s l ( θ , φ ) is the vertical component of the soil heat flux provided by a multi-layer soil model. The direction of H s l ( θ , φ ) is governed by the difference between the absorbed solar radiation and the emitted infrared radiation. Moreover, the planetary and lunar ephemeris DE430 of the Jet Propulsion Laboratory (JPL) of the California Institute of Technology [86,87] to compute r and δ S u n . A TSI of 1361 W∙m⁻² at 1 AU was taken from Kopp and Lean [88] and Kopp et al. [89]. According to the TSI reconstruction by Kopp et al. [90], the TSI only changed by ±1 W∙m⁻² during 1850 to 2015. Thus, the solar climate derived by Kramm et al. [44] may be compared with the real climates represented by the datasets Sp1885 [10], A1896-1 [26], vB1906 [29], and vHS1939 [38].

Figure 21 also shows Krümmel’s [54] meridional distribution of the zonal mean sea-surface temperatures (Figure 18 and Figure 19 may be considered to assess Krümmel’s data.). Obviously, between 45˚N and 45˚S, the meridional distributions of normal temperatures mainly follow those of the zonal mean sea surface temperatures. Already Forbes’s formula (Equation (24)) illustrated in Figure 11 documents this fact. Krümmel’s sea-surface data would provide 〈 T s e a 〉 = 16.7 ˚ C for an aqueous globe. For comparison: Forbes’

formula (Equation (24)) would provide 〈 T s e a 〉 = 13.7 ˚ C for an aqueous globe and 〈 T l a n d 〉 = 20.7 ˚ C for a

rocky one.

The slab-temperature distribution predicted by Kramm et al. [44] fulfills the condition of the global radiation balance. Based on the globally averaged solar radiation of 〈 F 〉 = 340.2   W ⋅ m − 2 and the globally absorbed solar radiation is 〈 Q 〉 = 279 .7   W ⋅ m − 2 ,the global albedo in the solar range is about α E = 0.178 . The globally emitted infrared radiation is 〈 F I R 〉 = 279 .6   W ⋅ m − 2 . Thus, the radiative imbalance is 〈 Q 〉 − 〈 F I R 〉 ≅ 0.1   W ⋅ m − 2 . It is compensated by the globally averaged soil heat flux. As listed in Table 4, the

global average of this slab-temperature distribution is 〈 T s l a b 〉 = − 52.4 ˚ C = 220.7   K . Since 〈 T s l a b 4 ( θ , φ ) 〉 1 / 4 = 266.4   K ,one obtains 〈 T s l a b 〉 = 0.828 〈 T s l a b 4 ( θ , φ ) 〉 1 / 4 for the obliquely rotating Earth in the absence of its atmosphere.

Figure 21(a) shows the meridional distributions of zonal averages of the solar irradiance, F ¯ ( θ ) ,the absorbed solar irradiance, Q ¯ ( θ ) ,and various normal temperatures related to them for the Earth in the absence of its atmosphere. In contrast to Kramm et al. [44], Defant and Obst [36] used a solar constant of S = 2   cal ⋅ cm − 2 ⋅ min ≅ 1394.6   W ⋅ m − 2 . Replacing their too high solar constant with the current one would provide a meridional distribution of F ¯ ( θ ) that substantially agrees with that of Kramm et al. [44]. Based on Equation (4), integration of F ¯ ( θ ) from θ = 0 to θ = π provides for Defant and Obst’s original data 〈 F 〉 = 348.2   W ⋅ m − 2 and for the corrected data 〈 F 〉 = 340 .2   W ⋅ m − 2 ,respectively. The integration of

Q ¯ ( θ ) from θ = 0 to θ = π yields, of course, 〈 Q 〉 = 279 .7   W ⋅ m − 2 ,mentioned before.

Defant and Obst [36] used the power law of Stefan [84] and Boltzmann [85] to derive the zonal average of a so-called radiation temperature, T F ( θ ) ,from F ¯ ( θ ) . They assumed that the emitted radiation is equal to the completely absorbed solar irradiance. They compared the meridional distribution of this radiation temperature with that of the real zonal temperature means, T ¯ ( θ ) ,and interpreted the difference T ¯ ( θ ) − T F ( θ ) as the heat protection due to the atmosphere. The meridional distributions of both T ¯ (θ)

and T F ( θ ) are illustrated in Figure 21(b). Based on Equation (8), the integration of T F ¯ ( θ ) and T ¯ ( θ ) from θ = 0 to θ = π provides 〈 T F 〉 = 10.0 ˚ C and 〈 T 〉 = 13.9 ˚ C ,respectively.

This procedure, however, is incorrect from physical and mathematical points of view. Since the local radiation balance is given by [91]

Q ( θ ,   φ ) = ( 1 − α ( Θ 0 , θ ,   φ ) ) F   cos   Θ 0 =   σ   T s 4 ( θ ,   φ ) (31)

where Q ( θ , φ ) is the absorbed solar irradiance, and T s is the local surface temperature, zonal averaging would provide (see Equation (3))

Q ¯ ( θ ) = 1 2 π ∫ 0 2 π ( 1 − α ( Θ 0 , θ , φ ) ) F cos Θ 0 d φ = σ 2 π ∫ 0 2 π T s 4 ( θ , φ ) d φ (32)

and, hence,

Q ¯ ( θ ) = σ T s 4 ¯ ( θ ) (33)

In accord with the general inequality of Gerlich and Tscheuschner [91],

T s ¯ = ∫ X T s d W ≤ ∫ X T s 4 d W 4 = T s 4 ¯ 4 (34)

for a non-negative measurable function T s and a probability measure W, the zonal average of the surface temperature, fulfills the inequality

T s ¯ ( θ ) ≤ T Q ( θ ) = Q ¯ ( θ ) / σ 4 = T s 4 ¯ ( θ ) 4 (35)

In the case of completely absorbed solar irradiance, i.e., α ( Θ 0 , θ , φ ) = 0 ,we would have Q ¯ ( θ ) = F ¯ ( θ ) and, hence, T Q ( θ ) = T F ( θ ) . Thus, as illustrated in Figure 22(b), any zonal average of the surface temperature, T s ¯ ( θ ) based on Equation (31) is usually much smaller than the corresponding

radiation temperature T F ( θ ) . Based on Equation (4), the integration of T s ¯ ( θ ) from θ = 0 to θ = π

provided 〈 T s 〉 = 157.2   K . This global mean surface temperature substantially agrees with result of Gerlich and Tscheuschner [91] (cf. their Table 12), even though we considered an obliquely rotating Earth in the absence of its atmosphere. In the case α ( Θ 0 , θ , φ ) > 0 ,we would have T s ¯ ( θ ) ≤ T Q ( θ ) < T F ( θ ) . Consequently, one must conclude that the solar climate as derived by Defant and Obst [36] has to be discarded.

As illustrated by Figure 23, for any parallel of latitude, the slab temperature that represents the Earth’s surface temperature in the absence of the atmosphere varies with time due to the Earth’s daily rotation and annual orbit around the Sun. However, this slab temperature is based on a local energy balance equation (25) that also included the soil heat flux density [44]. The meridional distribution of the normal temperature along a parallel of latitude based on a local radiation balance would be the same as proposed

by Gerlich and Tscheuschner [91] because during nighttime the surface temperature derived from the local radiation balance would be zero Kelvin (see Equation (31)). Therefore, as illustrated in Figure 22(b), this meridional distribution of normal temperature is, by far, the lowest one.

Obviously, the existence of the atmosphere, the nonuniformity of the Earth’s surface, the uneven ocean-land distribution, the airflows and ocean currents cause the transition from the solar climate to the real climate [92]. The difference between the solar climate and the real one is usually called the atmospheric effect. Thus, this atmospheric effect would vary with the parallels of latitude.

The atmosphere is a complex thermo-fluid dynamic system with various degrees of freedom. It consists of various layers (e.g., troposphere, stratosphere, mesosphere, thermosphere) separated by conceptual partitions called pauses [48]. It exhibits an impressive amount of detail and huge spatial and temporal variability of its properties [48]. More than 99% of the atmospheric mass of about 5.15 × 10¹⁸ kg is below the altitude of 30 km above sea level. The atmospheric response time to an imposed change is of the order of days or weeks owing to its notable compressibility and its low density and specific heat [48]. Consequently, the response time corresponds to weather periods, rather than climate periods of, at least, 30 years. According to Fortak [93] and Peixoto and Oort [48], the total energy of the atmosphere is about 1.31 × 10²⁴ J, i.e., less than 0.1% of the total energy of the oceans.

Based on recent observations, the solar insolation at the top of the atmosphere (TOA) corresponds to a globally averaged value of about 〈 F 〉 T O A = 340.3   W ⋅ m − 2 (100 units). As sketched in Figure 24, only about 〈 Q 〉 = 238 .2   W ⋅ m − 2 is absorbed by the entire Earth-atmosphere system (EAS). Compared with 〈 Q 〉 = 279 .7   W ⋅ m − 2 of the solar climate of Kramm et al. [44], the solar radiation absorbed by the EAS is notably lower. The absorbed solar radiation of 〈 Q 〉 = 238 .2   W ⋅ m − 2 is approximately balanced, on global average, by the infrared radiation, 〈 F I R 〉 T O A emitted to space by the entire EAS (Figure 24). Note that the TOA may be interpreted as a height of the intervening atmospheric layer. Above this height, neither solar radiation nor infrared radiation is remarkably affected by atmospheric constituents.

As sketched in Figure 24, a notable portion of the solar insolation penetrating into the atmosphere is absorbed in the ultraviolet and visible ranges as well as in the near infrared range by various gaseous and particulate atmospheric constituents (20 units or 68.1 W∙m⁻²) and clouds (4 units or 13.6 W∙m⁻²). Especially the absorption of solar radiation by molecular oxygen (O₂) and ozone (O₃) heats the atmosphere directly [75,94]. Water vapor (H₂O) and O₂ are also active in the visible and near infrared range; nitrogen dioxide (NO₂) is active in the visible range, too. Furthermore, a considerable portion of the solar radiation is back-scattered by molecules (Rayleigh scattering), cloud and aerosol particles (Lorenz-Mie scattering). A notable amount of solar radiation reaching the Earth’s surface is reflected, either by the soil-vegetation and water systems on land or by the oceans. These processes contribute to a planetary albedo of about 30 units (102.1 W∙m⁻²), on global average (Figure 24). This means that the remaining 70 units (238.2 W∙m⁻²) of solar radiation, on global average, feed the EAS with energy. However, only about 46 units (156.5 W∙m⁻²) are absorbed by water (including ice) and land masses (including vegetation) in the close vicinity of the Earth’s surface (Figure 24). Note that a fraction of the incident solar radiation may penetrate into the water or snow to considerable depths without notable absorption.

Atmospheric motions are stochastic to a certain extent. However, organized patterns like Rossby waves, mountain-induced gravity and inertial-gravity waves, cyclones and anticyclones, jet streams, and circulation pattern of different sizes like Hadley, Ferrel, and polar cells, monsoonal circulation, small-scale land-sea breezes, convection roles, etc. can be identified. Variations of these flow patterns may affect the energy conversion at the Earth’s surface on local and regional scales. Turbulent motion can mainly be observed in the atmospheric boundary layer (ABL) and along jet streams. The exchange of sensible and latent heat between the land or water masses adjacent to the Earth’s surface and the atmosphere is strongly controlled by molecular and turbulent transfer processes within the Prandtl layer (also called the atmospheric surface layer, ASL). Over canopies of tall vegetation, the outer edge of the Prandtl layer may be at 100 m height above ground. The climates of landscapes may be described by the Köppen-Geiger climate classification. An updated version was derived by Peel et al. [96] in 2007.

During cloud formation huge amounts of latent heat are released that heat the ambient air directly. Clouds of various horizontal and vertical extensions and compositions of hydrometeors mainly occur in the troposphere. They strongly interact with both solar and infrared radiation by absorption, scattering, and emission (only infrared radiation). Clouds also affect the energy conversion at the interface Earth- atmosphere via radiation and by precipitation thereby altering the surface properties of vegetation and soils. Precipitation contributes in small amounts to the conversion of potential energy into kinetic energy which is finally converted into heat.

Solar radiation absorbed by water and soil layers adjacent to the Earth’s surface is converted into heat. Hence, it contributes to warming these layers [75]. These layers also exchange energy with the ABL by flux densities (simply denoted as fluxes) of sensible (5 units or 17 W∙m⁻²) and latent heat (23 units or 78.3 W∙m⁻²). These fluxes, on global average, heat the atmosphere from below and cause convective transports of energy and mass into the upper troposphere. There, especially the release of latent heat during phase transition processes contributes to establish atmospheric circulation systems of different spatial and temporal scales [75,93].

As the absorption of solar radiation by atmospheric constituents and the exchange of energy between the soil and/or water layers at the Earth-atmosphere interface by the fluxes of sensible and latent heat already heated the atmosphere (about 177.0 W∙m⁻² of the energetically relevant solar radiation, on global average), we have to expect that gaseous atmospheric constituents able to emit and absorb infrared (IR) radiation in finite spectral ranges, will emit energy in the IR range in all directions. The amount of this IR radiation depends on the local temperature of the mixture of these constituents. Therefore, it is indispensable to consider the down-welling IR radiation reaching the Earth’s surface, where most of it is absorbed. The same is true for hydrometeors.

The water and soil layers adjacent to the Earth-atmosphere interface, of course, also emit IR radiation depending on their local temperatures. The net emission in the IR range (emitted radiation minus absorbed down-welling radiation) is about 18 units (61.3 W∙m⁻²), on global average. A notable portion of this IR net emission is absorbed by atmospheric constituents and emitted in all directions, too. A small fraction propagates through the atmosphere (about 6 units or 20.4 W∙m⁻²) with marginal extinction by intervening constituents. Such a spectral region is the so-called atmospheric window ranging from 8.3 μm to 12.5 μm (e.g., [74,75,82,97,98]). It only contains the 9.6 μm-band of ozone. Satellite-borne radiometers use the atmospheric window between 10 μm and 12.5 μm to measure radiation up-welling from the Earth’s surface [97].

Gases like H₂O, carbon dioxide (CO₂), and O₃ and hydrometeors also emit IR radiation to space. As shown in Figure 24, on global average, the net emission by these gases is about 38 units (129.3 W∙m⁻²), and the emission by clouds to space is 26 units (88.5 W∙m⁻²).

Besides the solar climate, the oceanic circulation affects the meridional distribution of normal temperatures [99]. As Wunsch pointed out, the mass fluxes in the upper hundred meters of the ocean are primarily wind-driven yielding major features like the Gulf Stream and the Antarctic Circumpolar Current; and secondarily driven by tidal forces.

The role of the thermohaline circulation is assessed somewhat controversy. Wunsch [99] argued “that the “thermohaline circulation” should be reserved for the separate circulations of heat and salt, and not conflated into one vague circulation with unknown or impossible energetics. No shortcut exists for determining property fluxes from the mass circulation without knowledge of the corresponding property distribution.”On the contrary, Rahmstorf [100] stated that “the ocean’s density distribution, which determines pressure gradients and thus circulation, is itself affected by currents and mixing. Thermohaline and wind-driven currents therefore interact in non-linear ways and cannot be separated by oceanographic measurements.”

To calculate the global and the hemispheric averages of the near-surface air temperature, we used Equation (8) and the datasets of the historical climatological and annual averages of temperature along numerous parallels of latitude(i.e., the zonal averages of temperature or normal temperatures) published between 1852 (Dove [19]) and 1913 (Börnstein [35]). The results are listed in Table 4. For comparison, also the results derived from the meridional distributions of the normal temperatures published by Defant and Obst [36], von Hann-Süring [38] (eventually adopted by Haurwitz and Austin [40] and Blüthgen [41]) and Sellers [39] as well as for the solar climate predicted by Kramm et al. [44] (but not explicitly published) are listed in Table 4. In some cases, we used Equation (23) to calculate the respective spheroidal mean near-surface temperature to assess the accuracy of spherical averaging as compared to spheroidal averaging.

Figure 25 shows results of the integrand T ¯ ( θ ) sin θ ,as requested by Equation (8) for a spherical shape of the Earth, where Köppen’s [37] maximum and minimum temperatures for numerous parallels of latitude (Table 3) were considered. Using the polygons related to the points of this integrand yields a globally averaged minimum temperature of 〈 T min 〉 = 8.4 ˚ C and a globally averaged maximum temperature of 〈 T max 〉 = 15.4 ˚ C ,respectively. For fitting this small number of points of these polygons, we used the asymmetric double sigmoidal peak function,

I ( θ ) = I 0 + A 1 + exp ( − θ − θ c + B / 2 C ) ( 1 − 1 1 + exp ( − θ − θ c − B / 2 D ) ) (36)

because the temperature distributions illustrated by Figure 20 require an asymmetric peak function. The parameters A, B, C, and D are listed in Table 5. For Köppen’s [37] dataset, this fit function provided 〈 T min 〉 = 10.9 ˚ C and 〈 T max 〉 = 18.0 ˚ C ,respectively.

*)Re-corrected to mean terrain height above sea level.

*) Re-corrected to mean terrain height above sea level.

With respect to the polygons, Köppen’s [37] distributions of the zonal averages of minimum and maximum temperatures provided the highest and the lowest global and hemispheric averages, respectively. Beside the latter, only the datasets Do1853 [19] and Fo1859 [23] provided global averages remarkably lower than 〈 T 〉 = 14 ˚ C ,namely 〈 T 〉 = 13.3 ˚ C and 〈 T 〉 = 13.4 ˚ C ,respectively. The results derived from the other historical distributions of normal temperatures published between 1877 and 1913 are ranging from 〈 T 〉 = 14.0 ˚ C (Ba1894 [11]) to 〈 T 〉 = 15.1 ˚ C (Fe1877 [12]), with 〈 T 〉 = 14.5 ˚ C (vB1906 [29] and Bö1913 [35]) falling in between.

Obviously, the poor coverage of the Southern Hemisphere by observations during that time indicated by the normal temperatures of Dove [19] and Forbes [23] (no zonal temperature averages beyond 40˚S) led to a hemispheric average of 〈 T 〉 S H = 11.8 ˚ C . This value is remarkably lower than those derived from the meridional distributions of the historical normal temperatures published by the other authors. The respective averages for the Southern Hemisphere are ranging from 〈 T 〉 S H = 13.2 ˚ C (Ho1906-1 [33]) to

〈 T 〉 S H = 15.4 ˚ C (Fe1877 [12]). The datasets vB1906 and Bö1913 provided 〈 T 〉 S H = 13.7 ˚ C .

On the contrary, the average for the Northern Hemisphere based on Dove [19] and Forbes [23] is 〈 T 〉 N H = 14.9 ˚ C . Thus, it substantially agrees with the results derived from the datasets of the other authors. Values are ranging from 〈 T 〉 N H = 14.7 ˚ C (Ba1894 [11]) to 〈 T 〉 N H = 15.5 ˚ C (vB1906 [29], Bö1913 [35]). Our results also confirmed that the average temperature for the Northern Hemisphere slightly exceeds that for the Southern Hemisphere. This is also true in case of the solar climate, but not in the case of Ferrel’s [12] dataset Fe1877 (Table 4).

For comparison: The datasets of Defant and Obst [36] and von Hann-Süring [38] yielded: 〈 T 〉 = 13.9 ˚ C , 〈 T 〉 S H = 13.1 ˚ C ,and 〈 T 〉 N H = 14.7 ˚ C for De1923 and 〈 T 〉 = 14.2 ˚ C , 〈 T 〉 S H = 13.2 ˚ C ,and

〈 T 〉 N H = 15.1 ˚ C for vHS1939, respectively. Sellers’ [39] distribution of zonal averages of temperature, which were not reduced to sea level, provided 〈 T 〉 = 11.9 ˚ C , 〈 T 〉 S H = 11.1 ˚ C , 〈 T 〉 N H = 12.7 ˚ C . Note that Seller’s distribution prohibits to use Equation (8) in an exact manner to calculate hemispheric averages of temperature because it omitted a zonal average of temperature for the Equator. Thus, we determined the zonal average for the Equator by interpolation. The same was also done for Arrhenius’ datasets A1896-1 and A1896-2. Obviously, the results based on Sellers’ dataset are remarkably lower than 〈 T 〉 = 13.3 ˚ C , 〈 T 〉 S H = 13.0 ˚ C ,and 〈 T 〉 N H = 13.7 ˚ C derived from the dataset A1896-2 that represents the normal temperatures of various parallels of latitude, re-corrected by Arrhenius to mean terrain height above sea level. For completeness, the differences between the hemispheric temperature averages expressed by Δ ( N H − S H ) = 〈 T 〉 N H − 〈 T 〉 S H are listed in Table 4 as well.

As illustrated in Figure 25, the accuracy of the asymmetric double sigmoidal peak function (Equation (36)) seems to be acceptable, but the global averages of the minimum and maximum temperatures derived from Köppen’s [37] data (Table 3) notably differ from those provided by the polygons. However, as expected from a mathematical perspective, for higher numbers of points, the difference between polygons and Equation (36) should decrease remarkably, as illustrated in Figure 26 for Spitaler’s [10] complete dataset Sp1885 (Table 2) and an artificially reduced one based on it. Using Equation (8), the polygon and the Equation (36), provided for the complete dataset 〈 T 〉 = 14.9 ˚ C (Table 4). Whereas for the reduced dataset, the integration of the polygon yielded 〈 T 〉 = 14.4 ˚ C and that of Equation (36) provided 〈 T 〉 = 15.0 ˚ C . As aforementioned, the dataset of von Hann-Süring [38] (vHS1939) was adopted by Haurwitz and Austin [40] as well as Blüthgen [41]. However, Haurwitz and Austin [40] only considered the values at each 10^th degree of latitude, while Blüthgen [41] adopted the complete dataset. The dataset of Haurwitz and Austin [40] (Figure 27(a)) provided 〈 T 〉 = 13.6 ˚ C for the polygon and 〈 T 〉 = 14.1 ˚ C for Equation (36). The higher number of points as shown in Figure 27(b) yields 〈 T 〉 = 14.2 ˚ C for the polygon and 〈 T 〉 = 14.0 ˚ C based on Equation (36). The results derived from the reduced datasets are also listed in Table 4, but not further discussed.

By using Equation (36), we fitted the meridional distributions of the points of the integrand T ¯ ( θ ) sin θ related to the historical distributions of normal temperatures. The parameters of our fitting procedure are listed in Table 5; the calculated global and hemispheric averages are listed in Table 4. Global averages are ranging from 〈 T 〉 = 14.3 ˚ C (vB1906 [29], Bö1913 [35]) to 〈 T 〉 = 15.5 ˚ C (Fe1877 [12]). The hemispheric averages are ranging from 〈 T 〉 S H = 13.6 ˚ C (Ho1906-1 [33], vB1906 [29], Bö1913 [35]) to 〈 T 〉 S H = 15.9 ˚ C (Fe1877 [12]) for the Southern Hemisphere (i.e., the results reveal that the poor coverage of the Southern Hemisphere as indicated by the normal temperatures of Dove [19] and Forbes [23] plays only a minor role) and 〈 T 〉 N H = 15.0 ˚ C (vB1901-1 [7]) to 〈 T 〉 N H = 15.5 ˚ C (Ho1906-2 [33]) for the Northern Hemisphere.

For comparison: Based on Equation (36), the datasets of Defant and Obst [36] and von Hann-Süring [38] yielded: 〈 T 〉 = 14.5 ˚ C , 〈 T 〉 S H = 13.7 ˚ C ,and 〈 T 〉 N H = 15.2 ˚ C and 〈 T 〉 = 14.0 ˚ C , 〈 T 〉 S H = 13.2 ˚ C ,and 〈 T 〉 N H = 14.9 ˚ C ,respectively. Sellers’ [39] distribution of zonal averages of temperature (which are not reduced to sea level) provided 〈 T 〉 = 12.5 ˚ C , 〈 T 〉 S H = 11.8 ˚ C ,and 〈 T 〉 N H = 13.1 ˚ C . These results are remarkably lower than 〈 T 〉 = 13.5 ˚ C , 〈 T 〉 S H = 13.2 ˚ C , 〈 T 〉 N H = 13.8 ˚ C derived from A1896-2 that repre- sents the normal temperatures of various parallels of latitude, re-corrected by Arrhenius to mean terrain height above sea level.

To estimate the uncertainty of our results, the numerical solution was repeated 50 times to create an ensemble of 50 realizations, where for each solution the zonal averages of the temperature for all parallels of latitude were randomly modified by adding temperature values, Δ R T ,that are normally distributed within a standard deviation of σ = ± 2   K . No seed was presupposed. This procedure was applied to each of the datasets of the zonal averages of temperature. Figure 28 illustrates examples of these randomly added temperatures. Based on the 16 ensembles of 50 realizations each, we calculated the uncertainty of the global and hemispheric averages and the differences between the hemispheres. The results are listed in Table 6. As expected, the uncertainty depends on the number of zonal averages. The smallest values of uncertainty were obtained for vB1906 (Figure 3) and Bö1913 (Figure 5) followed by vB1901-1 (Figure 2) and vHS1939 (Table 2). The largest uncertainty occurred for Do1852, Fo1859, Fe1877, A1896-1, and A1896-2.

By using Equation (8), the integration of the polygon provided, for instance, for vB1906 and Bö1913 〈 T 〉 = 14.5 ˚ C , 〈 T 〉 S H = 13.7 ˚ C , 〈 T 〉 N H = 15.5 ˚ C ,and Δ ( N H − S H ) = 1.8 ˚ C (Table 4). Our uncertainty ana-

lysis yielded for these datasets 〈 T 〉 = ( 14.5 ± 0.3 ) ˚ C , 〈 T 〉 S H = ( 13.6 ± 0.4 ) ˚ C , 〈 T 〉 N H = ( 15.3 ± 0.4 ) ˚ C ,and

Δ ( N H − S H ) = ( 1.7 ± 0.6 ) ˚ C (Table 6). For comparison: vHS1939 yielded 〈 T 〉 = 14.2 ˚ C ,

〈 T 〉 S H = 13.2 ˚ C , 〈 T 〉 N H = 15.1 ˚ C ,and Δ ( N H − S H ) = 1.9 ˚ C (Table 4). Whereas the respective ensemble

*) Re-corrected to mean terrain height above sea level.

provided 〈 T 〉 = ( 14.1 ± 0.3 ) ˚ C , 〈 T 〉 S H = ( 13.1 ± 0.5 ) ˚ C , 〈 T 〉 N H = ( 15.0 ± 0.4 ) ˚ C ,and Δ ( N H − S H ) = ( 1.9 ± 0.7 ) ˚ C (Table 6).

Presupposing an oblate spheroid, the numerical solution of Equation (23) provided for vB1906 and Bö1913 〈 T 〉 = 14.6 ˚ C , 〈 T 〉 S H = 13.7 ˚ C , 〈 T 〉 N H = 15.5 ˚ C ,and Δ ( N H − S H ) = 1.8 ˚ C . One of these values is slightly higher than that provided by Equation (8); however, the increase is mainly a rounding effect due to a marginal change in the second decimal place. Based on our uncertainty analysis, we obtained for these datasets 〈 T 〉 = ( 14.5 ± 0.3 ) ˚ C , 〈 T 〉 S H = ( 13.7 ± 0.4 ) ˚ C , 〈 T 〉 N H = ( 15.4 ± 0.4 ) ˚ C ,and Δ ( N H − S H ) = ( 1.7 ± 0.6 ) ˚ C . Again, any increase is mainly a rounding effect due to a marginal change in the second decimal place. For comparison: vHS1939 provided 〈 T 〉 = ( 14.1 ± 0.3 ) ˚ C , 〈 T 〉 S H = ( 13.2 ± 0.5 ) ˚ C , 〈 T 〉 N H = ( 15.1 ± 0.4 ) ˚ C ,and Δ ( N H − S H ) = ( 1.9 ± 0.7 ) ˚ C .

The normal temperatures of Kramm et al. [44] were also used to assess the accuracy of the spherical averaging procedure (Equation (8)) in comparison with spheroidal averaging procedure (Equation (23). The results confirmed that Equation (8) provides results with an acceptable accuracy (±0.04˚C) even in the case of an oblate spheroid.