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METRONNo. 23, December 2007 26

Mehr ber PyrometrieMehr ber PyrometrieUta Stockmann, Budiman & AlexUta Stockmann, Budiman & Alex

The graph below appears in the famous book The Vis-ual Display of Quantitative Information by EdwardTufte. The graph is from a book by Johann Heinrich

Lambert (1779) called Pyrometrie.

Tufte wrote (p. 29): It was not until the late 1700sthat time series charts began to appear in scientificwritings. This drawing of Johann Heinrich Lambert,one of a long series, shows the periodic variation insoil temperature in relation to the depth under thesurface. The greater the depth, the greater the time-lag in temperature responsiveness. Modern graphic

designs showing time-series periodicities differ littlefrom those of Lambert although the data bases are far

larger.

This example shows one of the earliest time series inthe scientific literature and is actually on soil data. Aswe wrote in the previous issue of Pedometron thevariation of temperature with time and depth has at-tracted scientists to work on them using physical prin-ciples and other modern numerical tools such as Fou-

rier transform and wavelet analysis.

And indeed analysing soil temperature attracted earlyscientists such as Saussure in 1785, Leslie in 1815 andJames Forbes in 1846. Early studies were mainly inter-ested in the theory of the conduction of solar heat toearth. See also the history of the inductive sciences

http://tinyurl.com/2kue65

However until recently (e.g. Kuzyakova and Stahr,2006), pedometricians have been more interested in

applying modern mathematical/ statistical methods tothe time series.

Johann Heinrich Lambert(1728 1777), was a Ger-man mathematician,

physicist and astronomer.He is well known for hiswork on light and optics(Beer-Lambert law), wasthe first to introduce hy-perbolic functions intotrigonometry, and in-vented the Lambert cylin-drical equal-area projec-tion. Lambert also had agreat interest in the the-ory of heat conduction.Pyrometrie was his last

book completed in May 1777. He died on September25, 1777 in Berlin at the age of only 49.

The Pyrometrie book can now be viewed at: http://

num-scd-ulp.u-strasbg.fr:8080/64/

The temperature data of Lamberts famous graphcame from measurements of differences in soil tem-perature with increasing soil depth. It was conductedby Mr. Ott, a salesman of Zurich, Switzerland begin-ning in 1762 for a period of 4 years. The method is

as follows:The thermometers, surrounded by pipe, were in-

serted into soil at appropriate depths.

The thermometers were filled withWeingeiste (alcohol) and not mercury, with ascale from the freezing point of water (-10.4

degree) to boiling point of water (Fig. 3)

Measurements were based on du Crests tempera-

ture scale (after Micheli du Crest)

The thermometer readings were at , , 1, 2, 3,

4, 6 Fu below soil surface (which is equiva-lent to the imperial foot, 1 Fu = 0.31 m)

The mean temperature for each month was calcu-

lated for the period of 4 yearsLambert used the mean temperature values to drawthe famous figure 1. The curve is using the following

equation:

v = a.f (b + x) + c.f (d + 2x) + &c.

where v = stored heat. (The full equation is much more

complicated than this.)

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METRONNo. 23, December 2007 27

PyrometriePyrometrie

We present the data in Celsius in Table 1. We con-

verted the temperature from the du Crest scale intoCelsius by:

Degree C = 1.2 du Crest + 10.4

We also obtained the air temperature at Zurich. Andwe fitted the comtemporary model (see last issue of

Pedometron):

The data and fitted curve are presented in the abovefigure. We obtained the following parameter values:average temperature Tav = -2.02

oC, amplitude A of10.5oC, phase constant t0 = 0.33 year, with a dampingdepth zd of 2.8 m and a thermal diffusivity of 24.43m2 year-1. The amplitude is much higher comparedwith Forbes data, but the damping depth is similar tothe sand in Forbes study. No information is given onthe soil, but we expect that the soil where Mr. Ottconducted his experiment in Zurich was of a sandy

nature.

Month Depth of thermometer below soil surface (m)

0*0.08 0.16 0.31 0.62 0.93 1.25 1.88

Jan.-1.3 8.8 -2.1 -1.2 -0.3 0.9 2.4 4.7

Feb.2.3 -3.6 -2.4 -1.8 -0.6 0.2 1.8 3.2

March4.3 5.6 2.1 2.6 2.0 2.0 2.7 3.0

April8.3 10.5 7.0 7.0 5.8 5.6 5.2 5.2

May12.9 13.3 12.0 11.7 10.3 9.7 9.1 7.6

June15.9 17.9 15.7 15.0 13.6 12.7 11.7 10.2

July18.5 18.2 16.4 16.1 14.8 14.8 13.9 12.7

Aug.17.1 16.2 16.1 15.7 15.1 15.4 14.8 13.9

Sept.13.0 13.6 13.3 13.6 13.8 14.4 14.2 14.2

Oct.7.7 8.2 7.6 8.0 8.9 10.2 10.9 12.1

Nov.4.1 2.7 3.0 3.6 5.5 6.8 8.0 10.0

Dec. 0.3 -0.9 -0.7 0.0 1.5 3.0 4.7 7.0

-15

-10

-5

0

5

10

Month

Temperature(degreeCelsius)

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Soil temperature at Zurich at the depth of 0.08, 0.16, 0.31, 0.62,

0.9, 1.25, and 1.88 m below soil surface.

( ), 0exp sin z t avd d

z zT T A t t

z z

= +

Table 1. Mean soil temperature in degree Celsius (for 4 years)

* mean air temperature at Zurich from 1881 to 1885 obtained from http://data.giss.nasa.gov/gistemp/