Conductivity, Heat Capacity, and
Electromagnetics! OH MY!
Eric Russell
4/9/2010
Agron 577: Soil Physics

• What is remote sensing?
– Microwave remote sensing
• Very basic electromagnetics
– Blackbody radiation, Wien’s law, Stefan-Boltzmann law, brightness temperature
• Soil thermal properties
• Combining the previous two (the OH MY! part)
• Figures

• Taking measurements from a place when not being in physical contact of that place.
• Satellites, MRI’s, IR thermometers, RADAR,
LiDAR, camera
– For this presentation: microwaves
• Utilizes the electromagnetic spectrum (EM)

• Maxwell’s equations
– Set of equations that relate the characteristics and propagation of magnetic and electrical fields

• Theoretical concept
– Perfect absorber and emitter
• Objects can exhibit blackbody-like characteristics at certain temperatures
– Preferentially emits at specific wavelength/frequency
• Can use as an approximation (usually pretty good)

• Temperature is defined as the average kinetic energy of molecules in a substance
• Anything that has a temperature radiates via the Stefan-
Boltzmann law:
J = εσT 4
, where ε = emissivity and σ = 5.67x10
-8 [W/m 2 K 4 ]
• Wien’s Displacement law: l max
 b
T l
= wavelength, b = 2.8977685(51)×10 −3 m·K
• a (absorbtivity) + r (reflectivity) + t (transmissivity) = 1
• Kirchoff’s Law: at thermal equilibrium, emissivity ( ε ) = a
• Higher the temperature, greater the radiation emitted

• Standard measurement for remote sensing signal
• More strictly correct is the spectral irradiance
I( l
,T) obtained via Plank’s Law:
I
 

2 h c
2 l
3

 e h l kT


1


1
(J·s -1 ·m -2
• But brightness temperature is easier:
·sr -1 ·Hz -1 )
T b
= εT where T b
= brightness temperature (K), T = temperature of material (K), and ε = emissivity

• Bypass Plank’s law: estimate T b using the spectral brightness B l
(T) from the Rayleigh-
Jean law:
B l
 

2 ckT b l
4 where k = Boltzmann constant,
c = speed of light, T b
= brightness temperature, and λ= wavelength.
• Then back out the brightness temperature

• Thermal conductivity k
: Heat transfer through a unit area of soil (J/s m K, or W/m K)
• Heat capacity c r b
: Change in unit volume’s heat content per unit change in temperature (J/m 3 K)
• Soil Thermal Inertia: P

• From remote sensing: P
 k sat c sat r b
2
D
G
D
T
 where
D
G = variation in surface heat flux,
D
T = T max
– T min
, and ω = 2 p
/86400s

• As discussed, thermal properties depend upon many factors
– Focus on soil moisture (because it’s awesome… and where my research lies)
• Can create relationships between θ and thermal inertia (can’t separate the individual properties through remote sensing)
• We are now done with big scary equations and models

• Can’t separate conductivity from capacity from just remote sensing
– Properties depend on too many variables
– Can estimate thermal inertia P using model shown
– Can estimate parameters in thermal inertia if know soil type/texture/moisture content, etc.
• Due to variable needs in approximation, need more than one measurement
– Can model heat flux through energy balance
– Diurnal temperature changes are easy to get

Left: Nighttime temperature over bare soil
Right: Daytime temperature over bare soil
Minacapilli and Blanda 2009

(a) Ground heat flux G ≡ Q(0, t) (W m −2 ), and (b) surface (skin) temperature T s
≡ T(0, t) (°C) measured at the Lucky Hill site in the
Walnut Gulch Watershed, 5 –16 June 2008.
Wang et al 2010

Left: Soil thermal inertia P as a function of θ
Right: Normalized soil thermal inertia K p as a function of degree of saturation (normalized q
)
Lu et al. (2009)

Idso et al 1976

Idso et al1976

Smits et al 2010

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Four Wettable and Four Water-repellent Soils." Soil Sci. Soc. Am. J. 65 (2001): 1675-679.
• Campbell, Gaylon S., and John M. Norman. Introduction to Environmental Biophysics. 2nd ed. New
York: Springer, 1998.
• Hillel, Daniel. Introduction to Environmental Soil Physics. Amsterdam: Elsevier Academic, 2004.
• Idso, Sherwood B., Ray D. Jackson, and Robert J. Reginato. "Compensating for Environmental
Variability in the Thermal Inertia Approach to Remote Sensing of Soil Moisture." Journal of
Applied Meteorology 15 (1976): 811-17.
• Lu, Sen, Zhaoqiang Ju, Tusheng Ren, and Robert Horton. "A General Approach to Estimate Soil
Water Content from Thermal Inertia." Agricultural and Forest Meteorology 149 (2009): 1693-698.
• Lu, Xinrui, Tusheng Ren, and Yuanshi Gong. "Experimental Inverstigation of Thermal Dispersion in
Saturated Soils with One-Dimensional Water Flow." Soil Sci. Soc. Am. J. 73 (2009): 1912-920.
• Minacapilli, M., M. Iovino, and F. Blanda. "High Resolution Remote Estimation of Soil Surface
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• Smits, Kathleen M., Toshihiro Sakaki, Anuchit Limsuwat, and Tissa H. Illangasekare. "Thermal
Conductivity of Sands under Varying Moisture and Porosity in Drainage-Wetting Cycles." Vadose
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