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Research Collection

Master Thesis

Inverse heat conduction in soilsa new approach towards recovering soil moisture fromtemperature records

Author(s): Fuhrer, Oliver

Publication Date: 2000

Permanent Link: https://doi.org/10.3929/ethz-a-004086114

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Inverse Heat Conduction In Soils

A New Approach Towards Recovering Soil

Moisture From Temperature Records

Diploma ThesisOliver Fuhrer

Thesis SupervisorsProf. Dr. Ch. Schar, Climate Research ETH, ZurichSonia Seneviratne, Climate Research ETH, Zurich

ETH Z urich, Dept. PhysicsMarch 2000

Corresponding author addressOliver FuhrerBulachstr. 11iCH-8057 [email protected]

Contents

Contents i

Acknowledgements iii

Abstract v

1 Introduction 1

2 Heat Transfer in Soils 32.1 Heat Conduction Equation. . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Analytical Solutions of the Heat Conduction Equation. . . . . . . . . . . 4

2.2.1 Constant Thermal Properties. . . . . . . . . . . . . . . . . . . . 42.2.2 Depth Dependent Thermal Properties. . . . . . . . . . . . . . . 5

2.3 Numerical Solution of the Heat Conduction Equation. . . . . . . . . . . 62.4 The Effect of Soil Moisture on the Thermal Properties of Soils. . . . . . 6

2.4.1 Heat Capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . 72.4.3 Thermal Diffusivity. . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Inverse Determination of Thermal Properties 113.1 Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

3.1.1 Direct Amplitude Methods (DAM1/DAM2). . . . . . . . . . . . 123.1.2 Direct Phase Methods (DPM1/DPM2). . . . . . . . . . . . . . . 133.1.3 Direct Numerical Method (DNM). . . . . . . . . . . . . . . . . 133.1.4 Indirect Harmonic Methods (IHAM/IHPM). . . . . . . . . . . . 143.1.5 Indirect Numerical Method (INM). . . . . . . . . . . . . . . . . 14

3.2 Validation of Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.1 Overview of Idealised Experiments. . . . . . . . . . . . . . . . 153.2.2 Sensitivity to Quality of Temperature Measurements. . . . . . . 173.2.3 Sensitivity to Boundary Condition. . . . . . . . . . . . . . . . . 203.2.4 Sensitivity to Discretisation Intervals. . . . . . . . . . . . . . . 213.2.5 Variation of Thermal Diffusivity. . . . . . . . . . . . . . . . . . 24

i

4 Inverse Determination of Soil Moisture 274.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .274.2 Accuracy of Soil Moisture Determination. . . . . . . . . . . . . . . . . 284.3 Validation of Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Results of Experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Application to Real Datasets 335.1 Foulum Dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Norunda Dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 Ticino Dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Discussion 416.1 Validity of Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 416.2 Inversion of Heat Conduction Equation. . . . . . . . . . . . . . . . . . 426.3 Determination of Soil Moisture. . . . . . . . . . . . . . . . . . . . . . . 43

7 Conclusion and Outlook 45

References 50

Nomenclature 51

A Indirect Numerical Method (INM) 53

B Datasets 56B.1 Foulum Dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56B.2 Norunda Dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56B.3 Ticino Dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

C Selected Experiment Results 58C.1 Noise (Experiment 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . 59C.2 Linear Moisture Increase (Experiment 6). . . . . . . . . . . . . . . . . . 60C.3 Idealised Rain Event (Experiment 7). . . . . . . . . . . . . . . . . . . . 61

ii

Acknowledgements

My diploma thesis would not have been possible without the contributions and supportof many friends and colleagues from the Institute for Climate Research ETH and otherinstitutions.

• I would first like to thankSonia Seneviratne, who supported this project in all regards. Herprofessional and friendly way of helping me organise my results saved me from disappear-ing in a huge pile of printouts. This project would not be what it is today, had it not beenfor her highly contagious optimism. I really enjoyed the laughs we had in 25 J 86!

• I’d like to thankProf. Christoph Schar who made this project possible in the first place. Hehas been wonderfully supportive of my ideas and introduced me to a great research group.

• A big thanks also goes toMassimiliano Zappa andAchim Gurtz from Climate ReasearchETH for their efforts in showing me the measurement site in Claro and supplying me withlots of support for the Ticino dataset.

• Thanks also toManfred Stahli from the Institute of Terrestrial Ecology ETH for showinggreat interest in this project and supplying me with the Norunda dataset. The data collectedat Norunda were part of the Northern Hemisphere Climate Processes Landsurface Experi-ment (NOPEX). Thanks goes to all the people who were involved, especiallyErich Keller,who was in charge of the data used.

• I would also like to express my gratitude toKirsten Schelde from the Danish Institute ofAgricultural Sciences who generously supplied me with the dataset from the Foulum Re-search Centre.

• I’d like to thank all members of the Institute for Climate Research ETH for making me feelat home, namely alsoManfred Schwarb, Christoph Frei, Olaf Albrecht, Pier-Luigi Vidale,Vladislav Nespor, andHendrik Huwald who repeatedly answered my questions concerningUNIX, Fortran, and LATEX.

• I am very grateful to the proofreaders,Raelene Sheppard andMilen Blagoev for their effortsin making this text a bit more readable.

• A big thanks goes to my parents for their love and trust in me during the past four years.And also to my big brother Andreas, who has always ploughed the path for me. I will getyou yet. . .

iii

iv

Abstract

Recent numerical research indicates that the modeling of the summer climate in Europeis very sensitive to soil moisture content. Long-term climate data often lacks accurate soilmoisture measurements. In this study, a new approach towards recovering soil moisturefrom time-series of vertically stacked soil temperature measurements is investigated. Thestudy includes a comprehensive review of the literature, the development of several in-version procedures, idealised experiments using artificial temperature data generated byanalytical or numerical solutions of the heat conduction equation, and application to threeextended datasets of soil temperature and moisture time-series.

In a first step, soil apparent thermal diffusivity is estimated from temperature measure-ments by solving a one dimensional, inverse heat conduction problem. Seven methods forthis purpose are presented. The explicit direct methods are easy to implement and requireonly a few temperature measurements. However, sensitivity experiments indicate thattheir results are at times erratic. The indirect methods require more temperature measure-ments to implicitly determine values of apparent thermal diffusivity. Even though theyare computationally intensive, their estimates of apparent thermal diffusivity are muchmore reliable. An application of all methods to three temperature series from Centraland Northern Europe reveals inconsistencies and unphysically large values of apparentthermal diffusivity.

In the second step, the De Vries model is inverted to calculate soil moisture contentfrom the determined apparent thermal diffusivities. A short test to ascertain the validityof this method is conducted. In general, ambiguities in the determination lead to twosolutions of soil moisture content. Problems may arise if the apparent thermal diffusivityis not sufficiently accurate.

The feasibility of the new approach for determining soil moisture content is neitherconclusively proven nor refuted. Uncertainties in the determination of the apparent ther-mal diffusivity can only be eliminated by the availability of a time-series of field mea-surements of this quantity. Nevertheless, the idealised studies show that, if an accuratedetermination of apparent thermal diffusivity is attained, the soil moisture content can besuccessfully determined. Further, experiments using artificial temperature data demon-strate that the thermal diffusivity can be obtained by inversion with notable accuracy.

v

Zusammenfassung

Kurzliche numerische Untersuchungen weisen darauf hin, dass das Sommerklima in Europa einehohe Sensitivitat auf den Bodenwassergehalt aufweist. Genaue Messungen von Bodenwasserge-halt fehlen oft in langfristigen Klimadatensatzen. In dieser Arbeit wird ein neues Verfahrenin zwei Schritten zur Bestimmung des Bodenwassergehalts aus Zeitreihen von Bodentempera-turemessungen in verschiedenen Tiefen untersucht. Diese Arbeit enthalt eine umfassende Lit-eraturbesprechung, die Entwicklung von mehreren Inversionsmethoden, idealisierte Experimentemit kunstlichen Temperturedaten, welche anhand einer analytischen oder numerischen Losung derWarmeleitungsgleichung berechnet wurden, und die Anwendung auf drei umfangreiche Zeitrei-hen von Bodenfeuchte- und Bodentemperaturmessungen.

In einem ersten Schritt wird die scheinbare thermische Diffusivitat aus Temperaturmessun-gen bestimmt. Zur Losung dieses eindimensionalen, inversen Warmeleitungsproblems wurdensieben Methoden entwickelt. Die Implementierung der expliziten direkten Methoden ist relativeinfach und benotigt nur wenige Temperaturmessungen. Sensitivitatsanalysen zeigen aber, dassdie berechneten scheinbaren thermischen Diffusivitaten manchmal grosse Schwankung aufweisen.Im Vergleich benotigen die indirekten Methoden mehr Temperaturmessungen fur eine impliziteBestimmung der scheinbaren thermischen Diffusivitat. Sie benotigen zwar mehr Rechenzeit, bes-timmen aber die scheinbare thermischen Diffusivitat mit einer wesentlich hoheren Genauigkeit.Berechnungen von scheinbaren thermischen Diffusivitaten fur drei verschiedene Temperaturdatensatzeaus Zentral- und Nordeuropa weisen teilweise widerspruchliche, oder sogar unphysikalische Werteauf.

Im zweiten Schritt wird das De Vries Modell invertiert, um Bodenwassergehalte aus denberechneten scheinbaren thermischen Diffusivitaten zu berechnen. Ein einfacher Test zur Vali-dierung dieses Verfahrens wird durchgefuhrt. Im Allgemeinen werden wegen Zweideutigkeitenin der Inversion zwei Bodenwassergehalte bestimmt. Falls die scheinbare thermische Diffusivitatnicht genugend genau bekannt ist, ist eine Bestimmung des Bodenwassergehaltes nicht moglich.

Die Durchfuhrbarkeit dieser neuen Methode zur Ermittlung des Bodenwassergehaltes wurdeim Rahmen dieser Studie nicht eindeutig bewiesen oder widerlegt. Unsicherheiten in der Berech-nung der scheinbaren thermischen Diffusivitat konnen nur durch Vergleich mit einer Feldmessungdieser Grosse eliminiert werden. Trotzdem weisen die idealisierten Untersuchungen darauf hin,dass falls die scheinbare thermische Diffusivitat mit genugen hoher Genauigkeit ermittelt wer-den kann, eine Bestimmung des Bodenwassergehalts moglich ist. Die Experimente mit ideal-isierten Temperaturdaten zeigen, dass die scheinbare thermische Diffusivitat mit nennenswerterGenauigkeit bestimmt werden kann.

vi

Chapter 1

Introduction

Cognizance of soil moisture is relevant to many domains of climate research and agri-culture. As pointed out byHollinger and Isard(1994), the scheduling of field efforts byfarmers and surface water management decisions rely critically on the knowledge of soilmoisture content. In climate research, several recent studies have emphasised the demandfor datasets of soil moisture.Huang et al.(1996) analysed the sensitivity of long rangeweather forecasts to model-calculated soil moistures and concluded that, compared to an-tecedent precipitation, soil moisture is a better predictor of future monthly temperature. Ina process study using a regional climate modelSchar et al.(1999) found that summertimeEuropean precipitation climate depends heavily on the soil moisture content.

Accurate and nondestructive measurements of soil moisture require the use of com-paratively expensive equipment such as a neutron probe or time domain reflectometry sys-tems. As a result of the expense and difficulty of obtaining soil moisture measurementsat useful temporal and spatial scales, there are very few long-term soil moisture datasetsavailable. Exceptions are a dataset of gravimetric soil moisture measurements throughoutthe Soviet Union and a ten year soil moisture dataset from Illinois (USA) using neutronprobes (Hollinger and Isard, 1994). Due to this lack, most soil moisture information isoutput from computer models that are based on a limited number of soil moisture mea-surements. Also, in a comparison of the Soviet Union dataset with soil moisture output ofclimatological simulations,Vinnikov and Yeserkepova(1991) found that the soil moistureregime is not well modeled.

In this study, a new approach to compute soil moisture content on a daily basis ispresented. In a first step, field-measured values of soil temperature are used to calculatethe apparent thermal diffusivity of the upper soil layer. Several approaches can be fol-lowed for this purpose. In this work, seven methods were analysed both in terms of thecalculated results, and of the quantity and quality of the data required to perform the cal-culations. In a second step, a semi-empirical (De Vries and Afgan, 1975) and an empirical(Kersten, 1949) relation expressing thermal diffusivity as a function of soil moisture areinverted.

Time-series of soil temperature measurements are much more readily available thansoil moisture measurements. Using this new approach, soil temperature measurements atdifferent depths could be used to reconstruct soil moisture from already existing climato-logical datasets or on an operational basis from daily soil temperature measurements.

1

The feasibility of the above approach is studied using both artificial and measureddata. In order to ascertain the validity of the calculated thermal diffusivity, many idealisedstudies are presented. Due to the limited scope of this study, the focus of the present workis mainly set on the thermal diffusivity estimation. A very primitive model of heat transferhas been used. Freezing and thawing have been completely neglected and processes otherthan heat conduction are not considered.

2

Chapter 2

Heat Transfer in Soils

For the theoretical description of the transfer of heat in an inhomogeneous, anisotropicmedium such as soil, a macroscopic view must be adopted. All physical quantities referto a spatial average over a unit cell of soil. The linear dimension of such a unit cellis large when compared to the grain and pore sizes, and small in comparison with thetypical length scales of the variations of temperature and composition.

The mechanisms of heat transfer in soils are, in order of importance (De Vries andAfgan, 1975): conduction, convection and radiation. Conduction occurs throughout thesoil, but the main flow of heat is through the solid and liquid parts. Convection in theusual sense is at most times negligible, with the exception of rapid infiltration of water.This does not hold for the transport of latent heat by water vapour, which may contributegreatly to the heat transfer in the gasfilled pores. Heat transfer due to radiation is onlyof importance in dry soils at high temperatures and within large pores. In some regions,freezing and thawing may play a dominant role for heat transfer. In this study, heat transferin soils is modeled as being solely due to conduction of heat.

2.1 Heat Conduction Equation

If we neglect the anisotropy of the soil and assume that horizontal gradients of all rele-vant physical quantities are small compared to the vertical gradients, the heat conductionproblem can be reduced to one dimension. By further neglecting radiation and all formsof convection, the vertical heat flux can be described by Fourier’s Law

qh(z, t) = −kh,app(z, t) ∂zT (z, t) (2.1)

whereT is the soil temperature,qh is the flux of thermal energy andkh,app = kh+kh,pseudois the apparent thermal conductivity consisting of the macroscopic thermal conductivityand a contribution to account for the transfer of latent heat by processes which are notpurely conductive, such as latent heat transfer. Substituting Fourier’s Law(2.1) into theequation of continuity for thermal energy yields

∂tu(z, t) = ∂z (kh,app(z, t) ∂zT (z, t)) (2.2)

3

whereu is internal energy per unit volume. Assumingu is only a function of temperatureand using the definition of the volumetric heat capacitych = ∂u

∂T, we can writeEq. (2.2)

as a parabolic partial differential equation of temperature

ch(z, t) ∂tTh(z, t) = ∂z (kh,app(z, t) ∂zT (z, t)) (2.3)

subject to appropriate boundary conditions. Sincekh may depend on the temperature,Eq. (2.3) is generally a non-linear equation. Assumingch andkh,app are independent ofdepth, time and temperature,Eq. (2.3) becomes

∂tTh(z, t) = Dh,app ∂2zT (z, t) (2.4)

whereDh,app =kh,appch

is the apparent thermal diffusivity.

2.2 Analytical Solutions of the Heat Conduction Equa-tion

Analytical solutions of the non-linear, parabolic partial differentialEq. (2.3) are hard tofind and making a numerical approach is very appealing. Nevertheless, analytical solu-tions may contain a significant amount of physical insight and can be used to verify morecomplex numerical models of soil heat flow.

In order to use Fourier theory, we assume that the heat capacitych and the thermalconductivity kh are independent of temperature and time.Equation (2.3) can then beFourier transformed with respect to time to give an ordinary differential equation for thenth harmonic, namely

ch(z)an(z)iωn− d

dz

(kh(z)

d

dzan(z)

)(2.5)

whereω = 2πP

is the fundamental frequency andan is the Fourier coefficient of thenthharmonic.P is the period of the surface temperature signal and is usually chosen equalto one day. The soil temperature is given by the Fourier series

T (z, t) =∞∑n=0

an(z)eiwnt (2.6)

The physical values are given by the real parts. The boundary conditions for thean’s aredetermined by the Fourier transform of the boundary conditions imposed onT (z, t).

2.2.1 Constant Thermal Properties

Assuming depth, time and temperature independent thermal properties throughout thesoil, and imposing the boundary conditionsT (0, t) = T0(t) andT (z → ∞, t) = T∞, thephysically relevant solution ofEq. (2.5) immediately follows and yields

T (z, t) = T∞ +∞∑n=1

e−zdn

(bn cos

(ωnt− z

dn

)+ cn sin

(ωnt− z

dn

))(2.7)

4

wherebn andcn are Fourier coefficients determined by

bn =2

P

∫ P

0T0(t) cos(ωnt)dt (2.8)

and

cn =2

P

∫ P

0T0(t) sin(ωnt)dt (2.9)

The damping depthdn is determined by

dn =

√2Dh,app

ωn(2.10)

with the thermal diffusivityDh,app =kh,appch

. Thus, the diurnal temperature wave of a typi-cal soil has a damping depth of approximately5 cm to 20 cm. For the annual temperaturevariations these values are roughly

√365 ≈ 19 times larger. Rapid temperature variations

at the soil surface, for instance due to temporary cloud cover, have a smaller dampingdepth.

2.2.2 Depth Dependent Thermal Properties

In natural environments, thermal properties are rarely homogeneous throughout the soil.Soil type, soil composition and moisture content may vary considerably with depth andlocation. Several analytical (Massman, 1993; Wiltshire, 1983, 1982; Shao et al., 1998)and approximate solutions (Thakur and Momoh, 1983; Nerpin and Chudnovskii, 1984)of the heat conduction equation with depth dependent coefficients have been studied. Allanalytical solutions either depend on additional assumptions about the periodicity of thesolution or assume a parametric model of the thermal properties as a function of depth.According to a study byMassman(1993), the most successful approximate solution is theone proposed byNerpin and Chudnovskii(1984). It is based on the WKB1 approximationwhich describes the propagation of waves in inhomogeneous media.

For the sake of brevity, only a very simplified situation will be considered here. Weassume that the functional form of the thermal properties is known to be linear in a certainsoil layer

ch(z) = c0(1 + αz) (2.11)

kh(z) = k0(1 + αz) (2.12)

whereα is the slope of thech andkh gradient in the soil. The local coordinate system ischosen so thatch = c0 andkh = k0 at the top of the layer (z = 0). Using the substitution

ξ =

√ωnc0

α2k0

(1 + αz) (2.13)

in Eq. (2.5) leads to a differential equation which is solved by Bessel functions (Abramowitzand Stegun, 1972). The solution for thenth complex amplitude is

an(z) = βnJ0(ξe34πi) + γnN0(ξe

14πi) (2.14)

1Wentzel, Brillouin and Kramers

5

whereJ0 andN0 are Bessel functions of the first and second kind respectively, andβn andγn are coefficients which are determined by the boundary condition.

2.3 Numerical Solution of the Heat Conduction Equation

Even though analytical solutions ofEq. (2.3) can be readily found in simplified cases, inpractice numerical methods are more often applicable. In this study, an algorithm of theNumerical Algorithms Group Fortran Library (NAG, 1990) is used to integrateEq. (2.3)between two depthsz1 andz2. The integration is forward in time fromta to tb subject tothe boundary conditions

T (z = z1 . . . z2, ta) = Ta(z) (2.15)

T (zm, t = ta . . . tb) = Tm(t) m = 1, 2. (2.16)

The algorithm approximates the parabolic partial differential equation by an ordinary dif-ferential equation in time, obtained by replacing the spatial derivatives by finite differ-ences on a regular mesh. The discretisation interval in the time direction is chosen by theroutine to maintain a local accuracy specified by the user.

2.4 The Effect of Soil Moisture on the Thermal Proper-ties of Soils

The temporal variability of the parameters governing heat conduction in soil,ch andkhis determined mainly by the soil moisture. This is due to the fact that water and air arethe only soil constituents which can vary considerably on a daily basis. In this section,analytical models forch andkh are presented to describe this dependence.

2.4.1 Heat Capacity

The macroscopic volumetric heat capacity of a soil can be calculated by summing overall constituents and phases multiplied by their respective volumetric fractionsθj:

ch =n∑j=1

θjch,j (2.17)

Note thatch increases linearly with increasing soil moisture contentθw. Table 2.1 liststhe volumetric heat capacities of some major soil constituents. Due to the small heatcapacity of air, the contribution of air towards the total heat capacity may be neglected toa good approximation.Figure 2.1 shows the heat capacity calculated for a sandy loamsoil at Research Centre Foulum, Denmark (Schelde et al., 1998). As for all materials,the heat capacity of soils is dependent of temperature. The effect is very small over thetemperature range of interest though, and can be neglected to a good approximation.

6

0.0 0.1 0.2 0.3 0.4 0.50

1.106

2.106

3.106

4.106

Volumetric Soil Moisture θw [m3 m-3]

Hea

t Cap

acity

ch

[J m

-3 K

-1]

Figure 2.1: Heat capacitych as a function of volumetric soil moistureθw for an upper soil layer (5 cm to15 cm) at Foulum, Denmark (§B.1). The solid line is calculated usingEq. (2.17) and the valuesη = 0.54for porosity,θq = 0.39 for volumetric quartz content, andθc = 0.042 for volumetric clay content of thesoil.

2.4.2 Thermal Conductivity

Many models and empirical formulae to calculate the macroscopic thermal conductivityof soils have been proposed (De Vries and Afgan, 1975; Kasubuchi, 1984; Sepaskhahand Boersma, 1979; Kersten, 1949; Nakshabandi and Kohnke, 1994). A good overviewincluding a detailed evaluation of their applicability is given byFarouki (1986).

De Vries Model

In analogy to a model developed by Maxwell2, De Vries(1952a) developed a model forthe macroscopic thermal conductivity of ellipsoidal soil particles in a continuous medium

2Maxwell developed a model for the electrical conductivity of a mixture of uniform spheres dispersedat random in a continuous fluid

Substance ρ [ kgm3 ] ch [ Jm3K ] kh [ WmK ]

Quartz 2.66 103 2.0 106 8.8Clay 2.65 103 2.0 106 2.9Organic matter 1.3 103 2.5 106 0.25Water 1.0 103 4.2 103 0.57Air (dry) 1.25 1.25 103 0.025

Table 2.1: Thermal properties and densities of soil materials, water and air at10 ◦C according toDe Vriesand Afgan(1975).

7

of water (or air)

kh =

∑nj=1 κjθjkh,j∑nj=1 κjθj

(2.18)

wherekh,j is the thermal conductivity andθj is the volume fraction of thejth constituent.κj is the ratio of the space average of the temperature gradient in the soil grains of kindjand the space average of the temperature gradient in the water (or air). Assuming a needlelike shape for soil particles,Nobre and Thomson(1993) found that

κj =2

3

[1 +

(kh,jkh,w

− 1

)gj

]−1

+1

3

[1 +

(kh,jkh,w

− 1

)(1− 2gj)

]−1

(2.19)

where thegj ’s are the shape factors given inTab. 2.2.

Constituent Subscriptj Shape Factorgj

Quartz q 0.125Clay c 0.125Organic matter o 0.500Water w -Air (dry) a Variable

Table 2.2: Subscripts and shape factors of soil constituent according toNobre and Thomson(1993)

For air enclosures, the shape factor is dependent on the volumetric water content andis deduced by linear interpolation between the value for spherical shape at saturation anda value of0.013 at dryness (Kimball et al., 1976). Thus,

ga = 0.013 +

(0.022

θw,wilt+

0.298

η

)θw (2.20)

whereθw,wilt is the wilting point moisture content, andη is the porosity of the soil. Thisapproximation is only valid forθw > θw,wilt. For very dry soils3, the De Vries model maybe applied using air as the continuous medium.De Vries(1952a) suggests that one shoulddiscontinue calculations with water as a continuous medium atθw < 0.03 for coarse soilsor at θw < 0.05 − 0.1 for fine soils. Farouki (1986) indicates that the de Vries modelgives values within±10% over the applicable range ofθw. Figure 2.2 shows the thermalconductivity calculated using the de Vries model for a sandy loam soil at Research CentreFoulum, Denmark (Schelde et al., 1998).

Below θw ≈ 0.1, the De Vries model overpredicts thermal conductivities, since watercan no longer be considered a continuous medium in the soil. At complete dryness, theheat flow mainly passes through the grains and has to bridge the air-filled gaps betweenthe grains around their contact points. As with the heat capacity, temperature dependenceof kh can be neglected to a first approximation.

3When the moisture content of the medium is below the moisture content at which liquid flow becomesnegligible.

8

0.0 0.1 0.2 0.3 0.4 0.50.0

0.5

1.0

1.5

2.0

Volumetric Soil Moisture θw [m3 m-3]

The

rmal

Con

duct

ivity

kh

[W m

-1 K

-1]

Foulum Data

De Vries Model

Kersten Model

Figure 2.2: Thermal conductivitykh as a function of volumetric soil moistureθw for an upper soil layer(5 cm to 15 cm) at Foulum, Denmark (§B.1). Points are field measurements using the needle probe method(Schelde et al., 1998), the solid line is calculated using the de Vries equation (η = 0.54, θq = 0.39, θc =0.042), and the dashed line is calculated using the Kersten equation (ρd = 1.20, a1 = 1.24, a2 = −0.11,a3 = 0.62). The soil thermal conductivity measurements were determined in the laboratory at successivelyhigher suction levels (1.0, 5.0, 10 and50 kPa) using a minimum of three soil replicates for all layers.

Kersten Equation

Kersten(1949) proposes a purely empirical formula for the calculation of the thermalconductivity based on measurements for five different soils

kh = 0.1442 (a1 log θw − a2) 10a3ρd (2.21)

where the thermal conductivitykh is given in[WmK

], the dry densityρd in

[gcm3

], anda1, a2

anda3 are dimensionless empirical constants. Values ofa1, a2 anda3 valid for unfrozen

9

0.0 0.1 0.2 0.3 0.4 0.54.10-7

5.10-7

6.10-7

7.10-7

8.10-7

Volumetric Soil Moisture θw [m3 m-3]

The

rmal

Con

duct

ivity

kh

[m2

s-1]

Foulum Data

De Vries Model

Kersten Model

Figure 2.3: Thermal diffusivitych as a function of volumetric soil moistureθw for an upper soil layer(5 cm to 15 cm) at Foulum, Denmark (§B.1). The solid and dashed lines are calculated using the De Vriesand Kersten model, respectively(Fig. 2.2). Points are calculated using field measurements ofkh (see thecaption toFig. 2.2).

sand soils are0.750, 0.400 and0.625, respectively. According toFarouki(1986) the equa-tion generally applies to soils with low silt-clay content (less than about 20 %). It shouldideally be applied to coarse soils with an intermediate quartz content of about 60 % of thesoil solids. Kersten’s equation does not apply to dry soils or to crushed rocks.Figure 2.2shows the thermal conductivity calculated using the Kersten equation as compared to thede Vries model and field measurements. At moistures below approx.0.05 m3

m3 both modelsare no longer applicable.

2.4.3 Thermal Diffusivity

The thermal diffusivity is defined byEq. (2.22). It governs the temperature response of asoil to thermal perturbations.

Dh =khch

(2.22)

Figure 2.3 shows the thermal diffusivity calculated using the De Vries and Kersten equa-tion for a sandy loam soil at Research Centre Foulum, Denmark (Schelde et al., 1998).For mineral and loam soils, the thermal diffusivity shows a maximum value at a relativelylow value ofθw.

10

Chapter 3

Inverse Determination of ThermalProperties

3.1 Methods

Recovering volumetric soil moisture from time-series of temperature measurements canbe divided into two steps. This chapter describes methods to calculate thermal propertiesof the soil from temperature measurements. The second step, calculating volumetric soilmoisture from thermal properties of the soil, is discussed inSection §4.1.

Recovering thermal parameters of a body from temperature measurements inside thebody belongs to the class of parameter estimation problems. Even if the describingEq. (2.3) is linear,i.e. the thermal parameters are not dependent on temperature, parame-ter estimation may lead to a nonlinear problem. Literature dealing with the subject fallsinto two categories: The first deals with the general problem of inverse determinationof (temperature dependent) thermal properties of a solid from measurements at internaltemperature probes (Lam and Yeung, 1995; Krukovsky, 1995; Tervola, 1989; Huang andYan, 1995; Huang andOzisik, 1991; Lesnic et al., 1996). A general introductory textinto inverse thermal problems of this type can be found inKurpisz and Nowak(1995).The second category specifically deals with recovering the apparent thermal diffusivity ofsoils (Chen and Kling, 1996; Horton et al., 1983; Zhang and Osterkamp, 1995; Lettau,1971).

In the following sections, two classes of methods for determining thermal proper-ties are presented: Direct methods, which use temperature measurements to explicitlycalculate thermal soil properties and indirect methods, which minimise the differencebetween measured and calculated temperatures by arbitrarily varying thermal properties.Some general restrictions apply to all methods for inversely recoveringch andkh,app usingEq. (2.3). The heat conduction equation is invariant under the simultaneous transforma-tions

ch → Λch and kh,app → Λkh,app, (3.1)

whereΛ is an arbitrary, constant parameter. Therefore,kh,app and ch can only be de-termined simultaneously up to a multiplicative factor. This is most obvious in the case

11

of depth-independent thermal properties where the thermal properties are reduced to oneparameter, the apparent thermal diffusivityDh,app.

3.1.1 Direct Amplitude Methods (DAM1/DAM2)

Considering only the first harmonic of the diurnal temperature fluctuation yields theboundary conditions

T (0, t) = T∞ + T0 sin(ωt) (3.2)

T (z →∞, t) = T∞ (3.3)

whereT∞ is the average soil temperature,T0 is the amplitude andw = 2π24h

is the fun-damental frequency of the surface temperature wave. Assuming constant soil thermalproperties with depth and time, only the first harmonic ofEq. (2.7) has to be considered,giving

T (z, t) = T∞ + T0e−z√

ω2Dh,app sin

(ωt− z

√ω

2Dh,app

)(3.4)

If measurements of soil temperature in two depthsz1 andz2 are available, the apparentthermal diffusivity can be solved explicitly fromEq. (3.4) as

Dh,app =ω

2

(z2 − z1

ln(A1/A2)

)2

(3.5)

whereAm is the amplitude of the temperature wave atzm for m = 1, 2. Equation (3.5)is referred to as the Direct Amplitude Method 1 (DAM1). In order to applyEq. (3.5),four temperature measurements, the maximum and minimum values at two depths, arerequired. Accurate measurements of the time of occurence are not necessary. The DAM1determines one apparent thermal diffusivity per day. It should be noted that the assump-tions leading toEq. (3.5) rarely occur in real soils and the determined apparent thermaldiffusivity can only serve as an estimate.

Following the development ofHorton et al.(1983), more accurate estimates ofDh,app

can be made considering higher harmonics. The soil temperature at the upper boundarycan be approximated by a finite Fourier Series

T (0, t) = T∞ +N∑n=1

(bn cos(nωt) + cn sin(nωt)) (3.6)

whereT∞ is the mean temperature of the time interval considered,N the number of har-monics, andbn andcn are the amplitudes. The amplitudes can be determined by fittingEq. (3.6) to measurements of soil temperature using standard least squares methods. Con-sidering harmonics up toN = 2, we get

Dh,app =ω

2

(z2 − z1

arctanX

)2

(3.7)

where

X =(T1 − T3)(T ′2 − T ′4)− (T2 − T4)(T ′1 − T ′3)

(T1 − T3)(T ′1 − T ′3) + (T2 − T4)(T ′2 − T ′4)(3.8)

12

The temperaturesTk andT ′k are recorded every6h at two depthsz1 andz2 respectively.Equation (3.7) is referred to as the Direct Amplitude Method 2 (DAM2).

More harmonics may be considered, leading to rather complicated expressions for theapparent thermal diffusivity, but they have not been used in this study.

3.1.2 Direct Phase Methods (DPM1/DPM2)

In analogy to the DAM method, the phases of the temperature waves at two differentdepths can be equated. Again, we can explicitly solveEq. (3.4) for the apparent thermaldiffusivity, giving

Dh,app =1

2ω

(z2 − z1

t2 − t1

)2

(3.9)

wheretm is the measured time of occurence of either maximum or minimum soil tem-perature at depthzm. The same restrictions concerning the feasibility forEq. (3.5) apply.Furthermore, frequent measurements are necessary to ensure accurate estimates oftm. Oncloudy days, several maxima ofT may occur, rendering a determination oftm difficult.

Again, second harmonics may be considered (Horton et al., 1983). Using the sameassumptions leading toEq. (3.6), the apparent thermal diffusivity is explicitly given by

Dh,app =1

2ω

(z2 − z11

2ωlnY

)2

(3.10)

where

Y =(T1 − T3)2 + (T2 − T4)2

(T ′1 − T ′3)2 + (T ′2 − T ′4)2(3.11)

where the temperaturesTm andT ′m are recorded each6h at two depthsz1 andz2, respec-tively.

3.1.3 Direct Numerical Method (DNM)

Equation (2.4) for homogenous soils can be approximated with finite differences. Usinga forward scheme in time, a centered scheme in space, and solving forDh,app gives

Dh,app =(∆z)2

∆t

T (z, t+ ∆t)− T (z, t)

T (z + ∆z, t)− 2T (z, t) + T (z −∆z, t)(3.12)

where∆t is the time step between observations of temperature and∆z is the verticaldistance between two temperature sensors. The numerical scheme is stable if

Dh,app∆t

(∆z)2<

1

2(3.13)

is satisfied. Given a time series of temperature observations at several depths, estimatesof Dh,app can be calculated using the Direct Numerical Method (DNM) determined byEq. (3.12). Compared to the DAM and DPM, the DNM considers all harmonics butrequires measurements at three depths. At an apparent diffusivity ofDh,app = 1 mm2

s

and a sensor separation of∆z = 0.1 m, stability of the numerical scheme is ensured if∆t < 1.4 h.

13

3.1.4 Indirect Harmonic Methods (IHAM/IHPM)

Another method, which we will refer to as the Inverse Harmonic Method (IHM) is sug-gested byHorton et al.(1983). As in Eq. (3.6), we can approximate the upper boundarycondition with a finite Fourier Series. The coefficientsbn andcn are determined either bya discrete Fourier transformation of measurements at the boundary or by fitting the ap-proximate boundary condition to the observed temperatures using least square techniques.The first approach is followed in this study. For constant thermal properties, the analyti-cal solution of the heat equation is given byEq. (2.7). If temperature measurements at anadditional depth are available, the apparent thermal diffusivityDh,app is determined im-plicitly from Eq. (2.7). The value ofDh,app is chosen to minimise the sum of the squareddifferences between the calculated and measured temperature values.

The IHM is employed in this work in a somewhat simplified form. The temperaturedata at two levels of one day are Fourier transformed to give the amplitudes and phases of acertain number of harmonics. Due to the strong dominance of the diurnal oscillation, onlythe coefficients of the first harmonic,i.e. Adiurnal andφdiurnal are used for the IHAM andIHPM, respectively. Estimates of apparent thermal diffusivity are determined explicitlyusingEq. (3.5) andEq. (3.9) with Am = Am,diurnal andtm = ωφm,diurnal wherem = 1, 2refers to the depth levels.

The assumption made byHorton et al.(1983), of depth-independent thermal proper-ties, is not required for the IHM. If an analytical solution of the heat conduction equa-tion (2.3) for time-independent (but depth-dependent) thermal propertiesch andkh,app isknown, the above method can still be applied to determine bothch andkh,app. Only ifthe thermal properties are time-dependent can the upper boundary condition no longer beFourier transformed, and thus the IHM no longer be applied.

3.1.5 Indirect Numerical Method (INM)

The direct numerical method as described inSection §2.3 can be used to integrate theheat conduction equation(2.3) between temperature measurements at an upper and lowerboundary condition. If an arbitrary value of apparent thermal diffusivity is chosen, tem-peratures at one or more intermediate levels can be calculated. A value ofDh,app can beimplicitly determined by minimising the sum of squared differences between calculatedand measured temperatures at the intermediate levels. The goodness of fit can be usedas an error-estimate for the determinedDh,app. In order to apply this Indirect NumericalMethod (INM), temperature measurements at a minimum of three levels are necessary.The top- and bottom most temperature measurements serve as boundary conditions forthe numerical integration.

In this study, the integration of the heat conduction equation is completed over a periodof 24 h, giving a time-averaged thermal diffusivity〈Dh,app〉t. The integration is startedfour times a day at12 am,6 am,12 pm and6 pm. As compared to the direct methods andthe indirect harmonic methods, the INM is computationally intensive, since it requiresrepeated integration of the heat conduction equation to a high accuracy.

The assumption of time and depth independence of the soil thermal properties is notnecessary. Any parametric form ofch(z, t) andkh,app(z, t) can be solved. The only re-

14

striction is that, for a least squares fit to be well defined, the number of data must begreater or equal to the number of determined parameters.

3.2 Validation of Methods

All methods proposed inSection §3.1 except the DAM1 are implemented as indepen-dentFORTRAN 90modules, running under aSolaris environment. The DAM1 is notimplemented due to technical reasons and its high similarity with the DPM1. Testing ofthe above methods using artificial temperature data with known soil thermal properties isutilised. This section gives an overview of the tests that have been performed in order toascertain the validity, and compare the performance of the methods.Section §3.2.1 givesa short account of the data used for each experiment. The remaining sections present theresults of the experiments.

3.2.1 Overview of Idealised Experiments

In order to test the methods for inverse determination of the soil apparent thermal dif-fusivity, idealised temperature data are generated for a period of 20 days. Using thesetemperatures as input data the DAM2, DPM1, DPM2, DNM, IHAM and IHPM calcu-late 24, and the INM calculates 4 values of soil apparent thermal diffusivity per day. Astatistical evaluation is applied to the results. The mean bias is calculated

rn =1

N

N∑n=1

(Dn,calc −Dh,app,n) (3.14)

wherern refers to thenth residual andDn,calc is thenth calculated apparent thermal diffu-sivity. Dh,app,n is the apparent thermal diffusivity used to generate to artificial temperaturedata for thenth time-step. Also, a measure for the degree of noise is calculated.

σ = σ(rn − rn) =

√√√√ 1

N

N∑n=1

(Dn,calc −Dh,app,n − rl)2 (3.15)

Note thatrl in Eq. (3.15) is independent ofn. If Dh is constant, dividingrn andσ by theapparent thermal diffusivity determines the relative bias and relative noise, respectively.

Noise (Experiment 1)

Soil temperatures are calculated every hour at depths of10, 20 and30 cm for 20 daysusingEq. (3.4) with Dh = 0.7 mm2

s, T0 = 10◦C, andT∞ = 20◦C. Measurement errors

are simulated by adding a pseudo-random real number taken from a Gaussian distributionwith a zero mean and standard deviationσT = 0, 0.0003, 0.001, 0.003, 0.01, 0.03, 0.1, 0.3,or 1 ◦C. The upper two levels of these generated temperature values are used to calculateapparent thermal diffusivities using methods DPM1, DAM2, DPM2, IHAM and IHPM.Methods DNM and INM use the temperature values at all three levels. The apparentthermal diffusivities determined by the DNM are averaged using floating averages over a24 h period.

15

Temperature Shift (Experiment 2)

Soil temperatures are generated as in Experiment 1 withσT = 0.001 ◦C1. An offset of0.2 ◦C is added to all temperature values at a depth of20 cm. This temperature dataset isprocessed as in Experiment 1.

Non-Sinusoidal Boundary Condition (Experiment 3)

Soil temperatures are generated every hour at depths of10, 20 and30 cm for 20 daysusingEq. (2.7) as a finite sum of the form

T (z, t) = T∞ +N∑n=1

bne− zdn sin(ωnt− z

dn+ φn) (3.16)

with N = 5, Dh = 0.7 mm2

s, T∞ = 20◦C, bn=1...5 = 10.0, 3.11, 0.20, 0.77, 0.34◦C and

φn=1...5 = 3.61, 0.14, 2.89, 2.53,−1.32 rad. These temperature values represent a typicalday with clear skies. No artificial noise was added. This temperature dataset is processedas in Experiment 1.

Variation of ∆t and ∆z (Experiment 4)

Soil temperatures are generated with4, 12, 24, 48 or 144 samples per day at three depths,which are separated vertically by∆z = 0.05, 0.1 or 0.2m. The middle depth is located atz = 0.2m. They are calculated for a period of20 days usingEq. (3.4) withDh = 0.7 mm2

s,

T0 = 10◦C andT∞ = 20◦C. Measurement errors are simulated by adding a pseudo-random real number taken from a Gaussian distribution with a zero mean and standarddeviationσT = 0.001 ◦C. This temperature dataset is processed as in Experiment 1.

Various Values of Thermal Diffusivity (Experiment 5)

Soil temperatures are generated every hour at depths of10, 20 and30 cm for 20 days usingEq. (3.4) with Dh = 0.05, 0.1, 0.2, 0.4, 0.8 and1.2 mm2

s, T0 = 10◦C, andT∞ = 20◦C.

Measurement errors are simulated by adding a pseudo-random real number taken froma Gaussian distribution with a zero mean and standard deviationσT = 0.001 ◦C. Thistemperature dataset is processed as in Experiment 1.

Linear Moisture Increase (Experiment 6)

Soil temperatures are generated every hour at depths of10, 20 and30 cm for 20 daysusing a numerical solution of the heat conduction equation as described inSection §2.3.The boundary conditions of the integration are chosen as inEq. (3.2) and (3.3) usingT0 = 10 ◦C andT∞ = 20 ◦C. Volumetric soil moisture is chosen to increase linearly overa period of20 days from0.05 m3

m3 to 0.5 m3

m3 . The corresponding soil thermal propertiesch1Some methods are exact up to machine precision even in the presence of temperature shifts and non-

sinusoidal boudary conditions. A small error in the temperatures is chosen in order to facilitate comparisonsusing relative bias and noise measure.

16

2 3 4 5 6 7 8 9 10 11 12 13 14 155.6.10-7

5.8.10-7

6.0.10-7

6.2.10-7

6.4.10-7

6.6.10-7

0.0

0.1

0.2

0.3

0.4

0.5

Moisture

Diffusivity

Vol

umet

ric

Moi

stur

e θ w

[m3

m-3

]

Time [d]

Therm

al Diffusivity D

h [m2 s -1]

Figure 3.1: Artificial volumetric soil moisture and apparent thermal diffusivity for Experiment 7 as afunction of time.

andkh are determined usingEq. (2.17) and(2.18), respectively. All other soil properties(η = 0.535 m3

m3 , θq = 0.393 m3

m3 , θc = 0.042 m3

m3 , θo = 0.029 m3

m3 ) are chosen accordingto a real dataset (Foulum,§B.1). Measurement errors are simulated by adding a pseudo-random real number taken from a Gaussian distribution with a zero mean and standarddeviationσT = 0.001 ◦C. This temperature dataset is processed as in Experiment 1.

Idealised Rain Event (Experiment 7)

Experiment 7 only differs from Experiment 6 in the choice volumetric soil moisture. Anidealised rain event is simulated using

θw(t) =

{θ0 t < trainθ0 + θrain exp( t−train

λ) t ≥ train

(3.17)

whereθ0 = 0.1 m3

m3 , train = 5 d, θrain = 0.35 m3

m3 andλ = 3 d. The depth-dependenttime-shift and decrease in amplitude of the water front, as well as changes in temperaturedue to infiltration are not simulated.Figure 3.1 shows the calculated soil moisture andapparent thermal diffusivities as a function of time. This temperature dataset is processedas in Experiment 1.

3.2.2 Sensitivity to Quality of Temperature Measurements

Noise (Experiment 1)

Table 3.1 summarises the results of Experiment 1 in the caseσT = 0 ◦C. The second orderdirect methods (DAM2, DPM2) as well as the inverse harmonic methods (IHAM, IHPM)are exact to within machine precision. With a relative bias from the reference value ofapprox.−2%, the DNM already shows considerable deviation from the reference valuefor this highly idealised test. This is due to the fact thatEq. (2.4) cannot be solved forDh

if ∂2zT is equal to zero. The relative bias of the DPM1 of approx.1% is due to the limited

accuracy in calculating the time of occurence of minima and maxima of the temperaturecurve. This is done using a smooth spline interpolation of the 24 temperature data perday. The INM underestimatesDh by approx.0.3%. Smaller biases can be accomplished

17

by increasing the accuracy of the heat conduction equation integration, resulting in time-consuming calculations.

Figure 3.2 illustrates the relative mean bias as a function of the standard deviation ofthe artificial noise added to the temperature data. A complete illustration of the resultsfor σT = 0.001 ◦C can be found in the Appendix (§C.1). Both the DPM1 and DNMshow a considerably larger bias at smallσT than the other methods. AtσT ≈ 0.03 ◦C,the biases of the DPM1 and DNM increase rapidly. The DPM1 is very sensitive to anexact determination of the daily extremal temperatures. At high noise levels, several localextrema may occur and ambiguities in the determination arise. The failure of the DNMcan be accounted for by numerical instabilities at certain data, which give a larger overallerror. AtσT ≈ 0.1 ◦C, the biases of the higher order and indirect methods DAM2, DPM2,IHAM, IHPM and INM also exhibit strong deviations from the reference diffusivity.

Figure 3.3 shows the relative noise measureσ of the calculated apparent thermal dif-fusivities as a function of the standard deviation of Gaussian noise. At standard deviationssmaller thanσT = 0.1 ◦C, all methods except DPM1 and INM show an approximatelystraight line in the log/log plot, thus implying a linear dependence of the noise measureonσT . The DPM1 and INM have asymptotic behaviours forσT → 0 ◦C, due to the lim-itations in accuracy mentioned above. At errors greater thanσT = 0.1 ◦C, σ exhibits afaster than linear increase for the DPM1, DAM1, DPM2, DNM. These methods no longerwork reliably due to problems in the determination of daily temperature extrema and thenumerical instabilities mentioned above.

Temperature Shift (Experiment 2)

The results of Experiment 2 are summarised inTab. 3.2. Only the DNM and INM areaffected by a systematic error in the temperature data. All other methods rely on theamplitude or phase of the daily thermal wave, which is not affected by an offset. The biasof the INM is within the expected range, whereas the large bias of the DNM is due tofailure of the method at certain times. Using these two methods, care should be taken toavoid systematic errors of temperature measurements.

ri σ

Method 1 mm2/s rel. Bias 1 mm2/s rel. NoiseDAM2 ε − ε −DPM1 6.96 10−3 1.0% 6.09 10−2 8.7%DPM2 ε − ε −DNM −1.48 10−2 −2.1% 1.82 10−16 2.6 10−13

IHAM ε − ε −IHPM ε − ε −INM −1.93 10−3 −0.28% 8.19 10−4 0.12%

Table 3.1: Overview of the results from Experiment 1 forσT = 0 ◦C. ri andσ are the mean bias and noisemeasure as described inSection §3.2.1. The values should be seen in the light of the reference diffusivity,namelyDh = 0.7 mm2

s for Experiment 1. A valueε indicates an error smaller than machine precision,approximately10−15.

18

1.0.10-4 1.0.10-3 1.0.10-2 1.0.10-1 1.0.100-0.10

-0.05

0.00

0.05

0.10

Standard Deviation of Gaussian Noise σT [°C]

Rel

ativ

e B

ias

[1]

DAM2

DPM1

DPM2

DNM

IHAM

IHPM

INM

Figure 3.2: Relative mean bias of apparent thermal diffusivities as a function of standard deviation oftemperature dataσT . See Experiment 1 for a complete description.

Experiment 1 and 2 both suggest that the DPM1 and DNM are not suitable for de-termination of apparent thermal diffusivities of the soil. Even with the highly idealisedtemperature data of the above experiments, they exhibit strong sensitivity to measurementerrors. Due to these limitations, they will not be considered in the following experiments,with the exception of Experiment 3.

19

1.0.10-4 1.0.10-3 1.0.10-2 1.0.10-1 1.0.10010-5

10-4

10-3

10-2

10-1

100

101

Standard Deviation of Gaussian Noise σT [°C]

Rel

ativ

e N

oise

Mea

sure

[1]

DAM2

DPM1

DPM2

DNM

IHAM

IHPM

INM

Figure 3.3: Relative noise measure of apparent thermal diffusivities as a function of standard deviation oftemperature dataσT . See Experiment 1 for a complete description.

3.2.3 Sensitivity to Boundary Condition

Non-Sinusoidal Boundary Condition (Experiment 3)

In Experiment 3, a non-sinusoidal boundary condition was chosen.Table 3.3 summarisesthe results for all methods. As compared to Experiment 1 (σT = 0 ◦C), only the inverseharmonic methods (IHAM, IHPM) are still exact to within machine precision. The perfor-mance of the DPM1 and DNM are not acceptable for non-sinusoidal boundary conditions,and therefore should not be used for real soil-data. The second order direct methods(DAM2, DPM2) seem to handle boundary conditions having more than two harmonicswell, since higher harmonics are damped strongly. The noise measure of the INM method

20

is approximately equal to Experiment 1, and is dominated by the precision of the heatconduction integration.

In natural soils, the diurnal temperature wave is almost never purely sinusoidal. Ex-periment 3 indicates that methods which assume sinusoidal boundary conditions, such asthe DPM1, fail to give a satisfactory approximation of apparent thermal diffusivity withnon-sinusoidal boundary conditions.

3.2.4 Sensitivity to Discretisation Intervals

Variation of ∆t and ∆z (Experiment 4)

Experiment 4 consists of two tests. First, the time discretisation step,i.e. the numberof temperature data available per day, is varied. In the second test, the vertical distancebetween the temperature data is varied.Figure 3.4 and3.5 show the results of the first test.Both the bias and noise measure of the DAM2 and DPM2 are not sensitive to a variationof the number of available temperature measurements, since they always use four data.For a relative bias of less than5%, the INM requires more than four samples per day. If24 or more measurements per day are available, all tested methods determine the apparentthermal diffusivity with a bias of less than2%.

ri σ

Method 1 mm2/s ri/ri,noshift 1 mm2/s σ/σnoshiftDAM2 8.37 10−5 1.0 6.19 10−4 1.0DPM1 6.59 10−3 1.0 6.15 10−2 1.0DPM2 −8.49 10−5 1.0 6.33 10−4 1.0DNM −2.51 1.7 10+2 9.93 10−1 1.2 10+3

IHAM −8.70 10−5 1.0 2.24 10−4 1.0IHPM 8.27 10−5 1.0 2.68 10−4 1.0INM −3.80 10−3 1.9 2.47 10−2 27

Table 3.2: Overview of the results from Experiment 2. For a description ofri andσ seeTab. 3.1. The thirdand fifth columns compare the obtainedri andσ to the results of Experiment 1.

ri σ

Method 1 mm2/s rel. Bias 1 mm2/s rel. NoiseDAM2 2.16 10−5 3.08 10−5 1.79 10−2 2.6%DPM1 7.38 10−1 105% 1.24 10−1 18%DPM2 1.42 10−3 0.20% 3.36 10−2 0.48%DNM −2.84 10−1 −41% 1.59 10−15 2.27 10−15

IHAM ε − ε −IHPM ε − ε −INM −8.00 10−4 −0.11% 1.21 10−3 0.17%

Table 3.3: Overview of the results form Experiment 3. All tabulated quantities are explained in the captionto Tab. 3.1.)

21

1.0.101 1.0.102-0.02

0.00

0.02

0.04

0.06

Number of Samples per Day [1]

Rel

ativ

e B

ias

[1]

DAM2

DPM2

IHAM

IHPM

INM

Figure 3.4: Relative mean bias of apparent thermal conductivityri as a function of the number of temper-ature data per day. For a description see Experiment 4.

1.0.101 1.0.102

10-4

10-3

10-2

10-1

Number of Samples per Day [1]

Rel

ativ

e N

oise

Mea

sure

[1]

DAM2

DPM2

IHAM

IHPM

INM

Figure 3.5: Relative noise measure of apparent thermal conductivityσ as a function of the number oftemperature data per day. See Experiment 4 for a description.

For the IHAM and IHPM, the relative noise measure(Fig. 3.5) decreases steadilywith an increasing number of samples per day. The same is valid for the INM up to24samples per day. At higher sample rates, the noise measure of the INM is restricted bythe accuracy of the integration of the heat conduction equation. The DAM2 and DPM2determine the apparent thermal diffusivity with an approximately constant relative noiseof approximately0.1%. If highly time-resolved temperature measurements are available,these methods do not give an increase in accuracy.

The results of the second test of Experiment 4 are summarised inFig. 3.6 and3.7.Biases of the DAM2, DPM2, IHAM and IHPM are small compared to the bias of theINM. For the INM, the relative bias shows a strong sensitivity to the space discretisationinterval. The relative errors of the DAM2, DPM2 IHAM and IHPM decrease with in-

22

0.05 0.10 0.15 0.20

-0.02

-0.01

0.00

0.01

Discretisation Distance ∆z [m]

Rel

ativ

e B

ias

[1]

DAM2

DPM2

IHAM

IHPM

INM

Figure 3.6: Relative mean bias of apparent thermal conductivityri as a function of the discretisation step∆z, as described in Experiment 4.

0.05 0.10 0.15 0.2010-4

10-3

10-2

10-1

Discretisation Distance ∆z [m]

Rel

ativ

e E

rror

of D

iffu

sivi

ty [1

]

DAM2

DPM2

IHAM

IHPM

INM

Figure 3.7: Relative noise measure of apparent thermal conductivityσ as a function of the discretisationstep∆z. See Experiment 4 for a description..

creasing distance between temperature measurements. The apparent thermal diffusivityof the soil affects the amount of damping and phase shifting with depth of diurnal temper-ature waves. Hence, if the temperature data originate from levels far apart, the dampingand shifting is more pronounced and signal-to-noise is larger.

The results of Experiment 4 seem to indicate that, with the exception of the INM,large space discretisation intervals are more suitable for a determination of soil apparentthermal diffusivity. Nevertheless, care should be taken when using temperature measure-ments from a soil. The results are somewhat misleading because natural variation of soilthermal properties and water content with depth are not considered in this experiment.A distance longer than0.1 m can already lead to considerable error in the determinationof the diffusivity. Knowledge of the soil composition profile at the measurement site is

23

0.0 0.5 1.0 1.5-2.10-8

0

2.10-8

4.10-8

Apparent Thermal Diffusivity Dh [10-6 m2 s-1]

Bia

s [m

2 s-1

]

DAM2

DPM2

IHAM

IHPM

INM

Figure 3.8: Mean bias of apparent thermal conductivityri as a function of apparent thermal conductivityDh. See Experiment 5 for a description.

crucial for the ideal choice of∆z.

3.2.5 Variation of Thermal Diffusivity

Different Values of Thermal Diffusivity (Experiment 5)

Figure 3.8 and 3.9 illustrate the results of Experiment 5. Bias and noise measure arecalculated as a function of the apparent thermal diffusivity. As a consequence of thedifferent dependence of the damping and the phase-shifting on the apparent thermal dif-fusivity (3.4), a different behaviour of the amplitude-dependent (DAM2 and IHAM) andthe phase-dependent (DPM2 and IHPM) methods is expected. This can be observed inFig. 3.8. While the biases of the DPM2 and IHAM do not vary very strongly withDh, theDAM2, IHPM and INM show an increase in bias with increasingDh at values larger than0.1 mm2

s.

The noise measure(Fig. 3.9) is generally larger for the second order methods (DAM2,DPM2). Due to stronger damping of high frequency noise at low apparent thermal dif-fusivities, all methods show a steady decrease in noise with decreasing diffusivity forDh > 0.2 mm2

s. At very low diffusivities, the noise measures increase abruptly. Caution

in the application of the above methods should be exercised. In particular the accuracy ofthe second order methods (DAM2, DPM2) can be affected considerably by measurementserrors.

Linear Moisture Increase (Experiment 6)

In Experiment 6, the temperature data are generated by integrating the heat conductionequation subject to boundary conditions. Since this integration introduces additionalsources of errors, results cannot be compared to the previous experiments. Results are

24

0.0 0.5 1.0 1.5

10-9

10-8

10-7

Apparent Thermal Diffusivity Dh [10-6 m2 s-1]

Noi

se M

easu

re [m

2 s-1

]

DAM2

DPM2

IHAM

IHPM

INM

Figure 3.9: Noise measureσ of apparent thermal conductivity as a function of apparent thermal conduc-tivity Dh. See Experiment 5 for a description.

compared to a test run using a constant soil moisture ofθw = 0.3 m3

m3 . The complete re-sults of Experiment 6 are given in the Appendix(Fig. C.2). For the DAM2, DPM2, IHAMand IHPM a time-varying apparent thermal diffusivity introduces oscillations to the cal-culated diffusivities. Since these oscillations have a period of one day, averaging over24 h eliminates the oscillations. The DPM2 and IHAM overestimate (underestimate) theapparent thermal diffusivity ifdDh

dtis negative (positive), respectively. The DAM2, IHAM

and INM always underestimate the thermal diffusivity. For the DAM2 and IHAM, thisis also observed in the reference experiment, and is probably due to additional errorsintroduced by the direct integration of the heat conduction equation.

ri σ

Method 1 mm2/s ri/ri,ref 1 mm2/s σ/σrefDAM2 −3.04 10−3 0.97 2.32 10−3 3.8DPM2 1.96 10−4 −9.6 3.50 10−3 6.2IHAM 6.70 10−5 2.9 2.05 10−3 10IHPM −3.19 10−3 1.0 1.29 10−3 5.0INM −4.85 10−3 1.0 7.59 10−4 1.1

Table 3.4: Overview of the results from Experiment 6. For a description ofri andσ seeTab. 3.1. The thirdand fifth columns compare the obtainedri andσ to the results of a test run using a constant soil moistureof θw = 0.3 fracm3m3

Table 3.4 summarises the results of Experiment 6. The biases of the DPM2 and IHAMshow a strong change when compared to the reference run. Nevertheless, they still havesmaller biases than the other methods. The increase in noise measure for the DAM2,DPM2, IHAM and IHPM can be explained by the oscillations mentioned above.

The natural variation of the apparent thermal diffusivity due to a change in moisturerarely exceeds roughly10% of the absoluteDh value. In order to give reasonable es-timates of soil moisture, the methods for determining soil thermal diffusivities must be

25

accurate to within at least one or two percent. Even when using idealised temperaturedata, biases and noise measures of some methods are considerable. Averaging over a pe-riod of 24 h can help reduce the error of the calculated diffusivities, but biases are difficultto eliminate.

Idealised Rain Event (Experiment 7)

Experiment 7 evaluates the response of the methods to a fast change in the volumetricmoisture content of the soil. An illustration of the complete results of Experiment 7 isgiven in the Appendix(Fig. C.3). Since all methods use temperature data of one dayto recover the apparent thermal diffusivity, the determined diffusivities resemble a24 h-average of the reference diffusivity. As in Experiment 6, strong oscillations occur whenthe diffusivity varies with time for all methods except the INM. Even with the abruptchange of diffusivity att = 5 d, the INM exhibits a very stable behaviour.

ri σ

Method 1 mm2/s ri/ri,ref 1 mm2/s σ/σrefDAM2 −3.34 10−3 3.7 2.32 10−3 4.2DPM2 −4.04 10−5 2.0 3.50 10−3 8.5IHAM −3.55 10−5 1.6 2.05 10−3 15IHPM −3.34 10−3 1.1 1.29 10−3 12INM −4.84 10−3 1.0 7.59 10−4 2.0

Table 3.5: Overview of the results from Experiment 7. All tabulated quantities are explained in the captionof Tab. 3.1.

Bias and noise measure are given inTab. 3.5. The values reflect the description in thelast paragraph. The DAM2, IHPM and INM underestimate the diffusivity by a consider-able amount. As mentioned above, for the DAM2 and IHPM, this is artificially introducedby errors originating from the direct integration of the heat conduction equation.

26

Chapter 4

Inverse Determination of Soil Moisture

In a second step, the volumetric soil moistureθw is recovered from the soil thermal proper-ties. The following section describes the method used, andSection §4.2 gives an estimateof the error in the determination ofθw. In Section §4.3, a short test of the method isdiscussed.

4.1 Method

Due to the limitations mentioned inSection §3.1, simultaneously recoveringch(z, t) andkh,app(z, t) from temperature measurements is only possible up to a multiplicative facor.Thus, at a given depthza, only the apparent thermal diffusivity

Dh,app(za, t) =kh,app(za, t)

ch(za, t)(4.1)

can be determined unambiguously. Using the models forch andkh introduced inSec-tion §2.4, the apparent thermal diffusivity can be expressed as a function of soil moisturecontent

Dh,app(θw) =kh,app(θw)

ch(θw)(4.2)

wherech andkh are determined byEq. (2.17) and the De Vries equation(2.18), respec-tively. Given measurements or estimates of the volumetric composition of a soil (porosityη, quartz contentθq, clay contentθc and organic contentθo) at a certain depth, the De Vriesequation determines the apparent thermal conductivity as a function of the volumetric wa-ter contentθw.

Figure 4.1 shows the apparent thermal diffusivity calculated using the De Vries equa-tion as a function of volumetric soil moisture for mineral, clay and organic soils.Dh formineral and clay soils exhibits a maximum atθw ≈ 0.2 m3

m3 . For the organic soil,Dh

increases slowly and monotonically with increasingθw.It follows immediately that solvingEq. (4.2) for θw is problematic. Firstly, very small

variations of thermal diffusivity may lead to considerable errors in the inversion. Sec-ondly, a maximum in theDh(θw) function for mineral and clay soils renders a one toone inversion impossible. Thus, for mineral and clay soils, two values of volumetric soil

27

0.0 0.1 0.2 0.3 0.4 0.5 0.60

2.10-7

4.10-7

6.10-7

8.10-7

Volumetric Soil Moisture θw [m3 m-3]

The

rmal

Con

duct

ivity

Dh

[m2

s-1]

Mineral Soil

Clay Soil

Organic Soil

Figure 4.1: Thermal diffusivity ch as a function of volumetric soil moistureθw for mineral, clay andorganic soils as determined by the De Vries equation(2.18).

moisture may correspond to the same apparent thermal diffusivity. Choosing the correctsolution requires additional knowledge about the soil. Thirdly, the De Vries equationdoes not give reliable estimates ofDh at very low soil moistures. And lastly, a strongdependence ofDh on the soil type can be observed. If the soil composition is not accu-rately known or varies strongly with depth, a determination of soil moisture may be verydifficult.

In this study, solvingEq. (4.2) for θw is realised by splitting the range of volumetricsoil moistures into intervals whereDh(θw) is a monotonic function. This is done bysuccessively splitting intervals into two at a local extremum of the thermal diffusivitycurve. Normally, a maximum of two intervals is necessary. All possible solutions ofθw for a givenDh-value are then found by invertingEq. (4.2) on each interval using theC05ADFroutine of theNAG Fortran 77 Library. Due to the limited scope of this study,no algorithm for choosing the correct soil moisture is implemented.

4.2 Accuracy of Soil Moisture Determination

The apparent thermal diffusivityDh does not vary very strongly with soil moisture at highmoisture contents and near local maxima (seeFig. 4.1). This may lead to considerableerrors in the determination of soil moisture as described in the previous section. An esti-mation of the maximum tolerable error ofDh for three target accuracies of soil moistureis shown inFig. 4.3 as a function of soil moisture. The target standard deviations of volu-metric soil moisture areσθw = 0.01, 0.02 and0.05 m3

m3 for the solid, short dashed and longdashed lines, respectively. For the mineral and clay soils, maximum accuracy is requiredat the local maxima of apparent thermal diffusivity (compareFig. 4.1). For all three soiltypes, the maximum tolerable error ofDh increases with decreasingθw at θw < 0.2 m3

m3 .

28

For organic soils, the tolerable error inDh is generally smaller when compared to othersoil types.

It is important to note that organic soils have a smaller apparent thermal diffusivitythan the other soil types. Further study shows that the maximum tolerablerelative errorsas shown inFig. 4.4 have a similar order of magnitude for all three soil types with theexception ofθw values close to0.2 m3

m3 . Figure 4.4 shows that a very high degree ofaccuracy inDh is necessary in order to estimate the soil volumetric moisture. For a targeterror ofσθw = 0.05 m3

m3 over the whole range of soil moistures, the relative error inDh

should not exceed1%.

4.3 Validation of Method

The above method (§4.1) for determination of volumetric soil moisture is implementedas an independentFORTRAN 90module running under aSolaris environment. Thissection gives an account of the test that is performed in order to ascertain the validity ofthe method.

Overview of Idealised Experiment

Due to the simplicity of the method, only a very basic test is completed to ascertain thefunctionality of the algorithm for all values ofDh. In a similar fashion to the tests inSection §3.2, artificial apparent thermal diffusivities are computed and used as input forthe algorithm.

Linear Moisture Increase (Experiment 8)

Soil apparent thermal diffusivities are generated every hour at a depth of20 cm for 20 daysusingEq. (2.17) and(2.18). Volumetric soil moisture was chosen to increase linearly overa period of20 days from0.05 m3

m3 to 0.5 m3

m3 . All other soil properties (η = 0.535 m3

m3 ,θq = 0.393 m3

m3 , θc = 0.042 m3

m3 , θo = 0.029 m3

m3 ) are chosen to resemble the soil ofthe Foulum dataset (§B.1). No artificial noise was introduced. The generated apparentthermal diffusivities are used as input for the method described inSection §4.1 to recoverthe soil moisture.

4.4 Results of Experiment

Linear Moisture Increase (Experiment 8)

Figure 4.2 shows the results of Experiment 8. All possible solutions for soil moisturehave been graphed as a function of time. The linear increase in soil moisture as wellas a second solution are determined by the algorithm. A numerical comparison of thecorrect solution with the reference values shows that the inversion is accurate to machineprecision, if requested. An interesting feature is that the tendency of the soil moisture has

29

0 5 10 15 200.0

0.1

0.2

0.3

0.4

0.5

Reference

Solution 1

Solution 2

Time [d]

Moi

stur

e θ w

[m3

m-3

]

Figure 4.2: Results of Experiment 8: All solutions of volumetric soil moistureθw as a function of time.

opposite signs for the two solutions. This could be exploited in a method to choose whichsolution of soil moisture is correct.

30

0.0 0.1 0.2 0.3 0.4 0.50

1.10-8

2.10-8

3.10-8

4.10-8

5.10-8

Volumetric Soil Moisture θw [m3 m-3]

Max

imum

Tol

erab

le E

rror

in σ

Dh [m

2 s-1

] Mineral Soilσθw = 0.01 m3 m-3

σθw = 0.02 m3 m-3

σθw = 0.05 m3 m-3

0.0 0.1 0.2 0.3 0.4 0.50

1.10-8

2.10-8

3.10-8

4.10-8

5.10-8

Volumetric Soil Moisture θw [m3 m-3]

Max

imum

Tol

erab

le E

rror

in σ

Dh [m

2 s-1

]

σθw = 0.01 m3 m-3

σθw = 0.02 m3 m-3

σθw = 0.05 m3 m-3

Clay Soil

0.0 0.1 0.2 0.3 0.4 0.50

1.10-8

2.10-8

3.10-8

4.10-8

5.10-8

Volumetric Soil Moisture θw [m3 m-3]

Max

imum

Tol

erab

le E

rror

in σ

Dh [m

2 s-1

]

σθw = 0.01 m3 m-3

σθw = 0.02 m3 m-3

σθw = 0.05 m3 m-3

Organic Soil

Figure 4.3: Maximum tolerable absolute error inDh as a function of volumetric soil moisture for the targetaccuracies in soil moisture determination of0.01, 0.02 and0.05 m3

m3 .

31

0.0 0.1 0.2 0.3 0.4 0.51.0.10-4

1.0.10-3

1.0.10-2

1.0.10-1

Volumetric Soil Moisture θw [m3 m-3]

Max

imum

Tol

erab

le R

elat

ive

Err

or [1

]

σθw = 0.01 m3 m-3

σθw = 0.02 m3 m-3

σθw = 0.05 m3 m-3

Mineral Soil

0.0 0.1 0.2 0.3 0.4 0.51.0.10-4

1.0.10-3

1.0.10-2

1.0.10-1

Volumetric Soil Moisture θw [m3 m-3]

Max

imum

Tol

erab

le R

elat

ive

Err

or [1

] Clay Soil

σθw = 0.01 m3 m-3

σθw = 0.02 m3 m-3

σθw = 0.05 m3 m-3

0.0 0.1 0.2 0.3 0.4 0.51.0.10-4

1.0.10-3

1.0.10-2

1.0.10-1

Volumetric Soil Moisture θw [m3 m-3]

Max

imum

Tol

erab

le R

elat

ive

Err

or [1

] Organic Soil

σθw = 0.01 m3 m-3

σθw = 0.02 m3 m-3

σθw = 0.05 m3 m-3

Figure 4.4: Maximum tolerable relative errorσDh/Dh as a function of volumetric soil moisture for thetarget accuracies in soil moisture determination of0.01, 0.02 and0.05 m3 m−3

32

Chapter 5

Application to Real Datasets

Idealised studies (seeSection §3.2 and§4.3) can give significant insight concerning thevalidity and range of application of a model. However, only application to real measure-ments can show, whether the assumption of pure heat conduction in soils is acceptable.Three temperature records from field measurements were available for this study. Allsites are located in Central or Northern Europe. For a complete documentation of thethree measurement sites and details of the measurements done, see AppendixB.

In the following sections, the datasets and the determined apparent thermal diffusiv-ities are discussed. The second step of the inversion, the determination of soil moisturecontent from the calculated apparent thermal diffusivity, is not carried out. This is due tothe fact that the diffusivities are not satisfactorily determined. Instead, sources of errorsand problems of the model used are discussed.

5.1 Foulum Dataset

During a field campaign at the Research Centre Foulum in Denmark from 12th Septemberto 10th October 1991,Schelde et al.(1998) measured soil temperature and soil watercontent at several depths. A full description of the dataset and all quantities measured isgiven in the Appendix (§B.1).

Apparent thermal diffusivity is determined using the DPM2, DAM2, IHPM, IHAMand INM. The first four methods are run using temperature measurements at depths of0.05 m and0.15 m. The INM is run with an additional temperature series at a depth of0.25 m. Figure 5.1 shows a summary of the data used and results obtained. The temper-ature measurements clearly show the expected damping and phase-shifting with depth.At 5 cm, fluctuations of temperature due to cloud cover and synoptic events are present.During the observation period, the volumetric soil moisture exhibits a small increase fromapprox.0.24 m3

m3 to 0.3 m3

m3 as a consequence of precipitation events. In the second plotfrom the top, calculated thermal diffusivities are also shown. These will be referred toas reference diffusivities and are calculated using the De Vries equation. The requiredsoil composition (η, θq, θc andθo) was estimated from available measurements of thermalconductivity, porosity, density and grain size distribution at15 cm. Since soil moisture is

33

generally larger than0.2 m3

m3 , the reference diffusivity decreases with increasing moisture.Over the whole data range, it exhibits a maximum relative change of approx.2%.

The reference diffusivities as calculated by the De Vries model strongly depend on thevolumetric quartz content of the soilθq. Since no field measurements ofθq are available,the value ofθq used is only a best guess based on other soil properties. Also, the volumet-ric water content varies by approximately40% with depth from5 cm to 25 cm, makingthe assumption of vertical homogeneity used by all methods questionable. It follows thatthe reference diffusivity can at most be seen as a rough indication as to what apparentdiffusivities to expect in the soil.

The 24 h-averaged diffusivities obtained by the INM, IHPM, IHAM, DPM2 andDAM2 show little agreement with the reference diffusivity. The diffusivity as deter-mined by the INM is0.42 − 0.50 mm2

s, compared to the reference diffusivity of ap-

prox. 0.63 − 0.65 mm2

s. Further, instead of showing a decrease with time, the INM

diffusivity increases by relative change of approx.12%. Despite of these poor results,a comparison of the reference and INM diffusivity suggests a correlation between thetwo. Strong changes in INM diffusivity precede strong changes in the reference diffusiv-ity by approximately one day. The offset of one day can be explained by the length of theforward integration of the INM method of one day, the effect of taking24 h-averages ofthe results, as well as time shifts in the infiltration of the soil.

The performance of the IHAM and DAM2 is similar. Both determine values of dif-fusivities with a trend from approximately0.35 to 0.50 mm2

s. Considerable fluctuations,

especially for the DAM2, and complete failure to recover the diffusivity on day272 and273 are apparent. The IHPM and DAM2 exhibit a similar behaviour, though with strongerfluctuations and diffusivities which are on average0.2 mm2

slarger. Even though averaging

of the results of the IHPM, IHAM, DPM2 and DAM2 eliminates the strong fluctuationspresent, it is speculation to assume that the averaged results correctly estimate the appar-ent thermal diffusivity. Generally, the performance of the latter methods is not satisfac-tory.

Measurements of thermal conductivity for the Foulum soil are available and havebeen plotted inFig. 2.2 on page9. This puts the results of the De Vries equation, whichdescribes the measurements rather well, on a somewhat better footing for the Foulumdataset. Sadly, only four single measurements which were made under laboratory con-ditions and not as a time-series, are available. This renders a thorough verification orfalsification of the determined diffusivities difficult.

The fact that the trend of the reference diffusivity is exactly opposite to the trendexhibited by the determined diffusivities is very dissatisfying. An explanation may bethat the volumetric water content used for the determination of the reference diffusivity istoo large. As mentioned above, the water content increases by40% over a vertical distanceof 20 cm. It is not clear which of these soil moistures should be taken as representativefor a comparison with the methods determining an average value of diffusivity over thislayer. For this study, an average depth of15 cm has been chosen. Even if this mightpartly explain the discrepancy, no full explanation can be given and further research isnecessary.

34

5.2 Norunda Dataset

As part of the NOPEX1 project, Stahli et al. (1995) conducted soil temperature andmoisture measurements in a spruce-pine forest30 km north of Uppsala (Sweden) from20th August to 31st October 1994. A full description of the dataset and all quantitiesmeasured is given in the Appendix (§B.2).

Apparent thermal diffusivity is determined using the DPM2, DAM2, IHPM, IHAMand INM. The first four are run using temperature measurements in depths of0.1 mand0.2 m. The INM is run with an additional temperature series at a depth of0.3 m.Figure 5.2 shows a summary of the data used and results obtained. The temperaturesshow a seasonal trend with superimposed diurnal fluctuations. Volumetric soil moistureat 0.2 m exhibits a slow rise from0.05 to 0.19 m3

m3 in the first half of the data range anddoes not vary greatly in the second half. As explained for the Foulum dataset (§5.1), abest guess of the volumetric composition of the soil served as a basis of thermal diffusivitycalculations using the De Vries equation. The result of this calculation is shown in thesame plot as soil moisture and will be referred to as the reference diffusivity.

Starting around the 1. October (day 273), the groundwater level rises from a depth of3 m up to approximately1 m due to extensive rainfall typical for the fall season. Thisrise causes the soil moisture to stay more or less constant over the second half of the datarange. Also, the diurnal temperature variations are not very strong during this period.Two cooling events starting on day277 and289 are probably due to snowfall.

In the case of the Norunda dataset, no measurements of thermal properties were avail-able. Since the quartz content has not been measured directly, a best guess of volumetricquartz contentθq is made, based on standard values from literature (Kersten, 1949; DeVries, 1952a). Since absolute values of the diffusivity are very sensitive to quartz content,the values of the reference diffusivities cannot be expected to be correct. Nevertheless,variation of diffusivity due to volumetric water content is not very sensitive to quartz con-tent. Thus, the qualitative trend of the reference diffusivity can be used to compare againstthe diffusivities in the lower three charts ofFig. 5.2.

For the first half of the data range, the INM produces very stable results (compareFig. 5.2 on page38). A trend in diffusivity from 2 mm2

sto approximately3.5 mm2

son

day 265 is observable. Since such high values of apparent thermal diffusivity have notbeen found anywhere in literature, it is assumed that the determined diffusivities are anoverestimation. The source of error is unclear and further investigation is needed. Inspite of this failure, the increasing trend of the reference diffusivity (and soil moisture)can be discerned in the results of the INM. In the second half of the data range, the INMcompletely fails to give reliable estimate of diffusivity due to the lack of a clear diurnaltemperature variation.

The IHPM, IHAM, DPM2 and DAM2 do not give satisfactory results. Even though atrend similar to the INM results can be made out in the first half of the data range, strongfluctuations dominate even the24 h-averaged results.

1Northern Hemisphere Climate Processes Land-Surface Experiment

35

5.3 Ticino Dataset

During a field-measurement campaign of the Institute for Climate Research ETH as partof the EU-MAP-Project RAPHAEL2, soil temperatures and moistures were measured ona soccer field in Claro, Ticino (Switzerland). The measuring period was from 24th Julyto 11th November 1999. A full description of the dataset and all quantities measured isgiven in the Appendix (§B.3).

Apparent thermal diffusivity is determined using the DPM2, DAM2, IHPM, IHAMand INM. The first four are run using temperature measurements in depths of0.05 m and0.2 m. The INM is run with an additional temperature series at a depth of0.5 m. Fig-ure 5.3 on page39 shows a summary of the data used and results obtained. The diurnaltemperature variations have an amplitude of approximately3 ◦C. A strong infiltrationevent due to extensive precipitation on day 291 causes an abrupt soil temperature changeon all three levels. After a sharp increase from0.13 to 0.35 m3

m3 on day 219, volumetricsoil moisture in35 cm depth continually rises and drops at an average moisture of ap-proximately0.3 m3

m3 . No measurements of thermal conductivity are available. A referencediffusivity has been determined using the De Vries equation using a rough estimate of soilproperties.

The composition of a soil is extremely variable with depth. In the case of the Ticino,all temperature probes are located within different layers of the soil and are separated by adistance of at least15 cm. If these facts are seen in the light of the idealised experimentsin Section §3.2, the results of an application of the methods to the Ticino dataset havevery little significance. Only a very brief discussion of the results is presented here.

As for the Norunda dataset, apparent thermal diffusivities determined by the INMoverestimate the soil thermal diffusivity. Since no clear trend is present in the volumet-ric soil moisture, and consequently in the reference diffusivity, a correlation between thedetermined and reference diffusivity is not clearly visible. Strong fluctuations in INMdiffusivity often occur during precipitation events. All methods for determining soil ther-mal diffusivity neglect temperature changes caused by infiltration, and thus results duringsuch periods are expected to be incorrect. During the heavy precipitation event on day291, the failure of the INM to give a reasonable estimate of apparent thermal diffusivityis most apparent. To speculate, a slight increasing trend from day 225 to day 255 may beattributed to the trend also visible in the reference diffusivity.

For the Ticino dataset, the results of the phase-based methods (IHPM, DPM2) arequite different from the amplitude based methods (IHAM, DAM2). The IHPM and DPM2exhibit a complete failure to give reasonable values of apparent thermal diffusivity andstrong fluctuations dominate their output. The IHAM and DAM2 methods on the otherhand, show a clear connection to the INM results.

2Runoff and Atmospheric Processes for Flood Hazard Forecasting and Control

36

4.0.10-7

4.2.10-7

4.4.10-7

4.6.10-7

4.8.10-7

5.0.10-7

5.2.10-7

INM

2.10-7

4.10-7

6.10-7

8.10-7

1.10-6

IHPMIHAM

255 260 265 270 2752.0.10-7

4.0.10-7

6.0.10-7

8.0.10-7

1.0.10-6

DPM2DAM2

0.20

0.22

0.24

0.26

0.28

0.30

0.32

6.3.10-7

6.4.10-7

6.4.10-7

6.5.10-7

6.5.10-7

MoistureDiffusivity

Temperature

255.0 260.0 265.0 270.0 275.0

0.0

5.0

10.0

15.0

20.0

25.0

30.0

0.05 m

0.15 m

0.25 m

Tem

pera

ture

[°C

]

App

. The

rmal

Dif

fusi

vity

[m

2 s-

1 ]A

pp. T

herm

al D

iffu

sivi

ty [

m2

s-1 ]

App

. The

rmal

Dif

fusi

vity

[m

2 s-

1 ]

Moi

stur

e [m

3 m

-3]

App. T

hermal D

iffusivity [m2 s -1]

Figure 5.1: Overview of the Foulum dataset and the apparent thermal diffusivities as determined by theINM, IHPM, IHAM, DPM2 and DAM2. From top to bottom: Soil temperature (◦C) measurements at5 cm, 15 cm and25 cm, volumetric soil moisture (m3/m3) measurements and derived,24 h-averagedsoil apparent thermal diffusivities (m2/s) using the De Vries equation (scale on the right), and calculated,24 h-averaged diffusivities as determined by the different methods. Unit on the abscissa is days in year.Period shown corresponds to 12. Septermber 1991 to 10. October 1991.

37

230 240 250 260 270 280 290 300 3100.0.100

2.0.10-6

4.0.10-6

6.0.10-6

8.0.10-6

1.0.10-5

DPM2DAM2

0.0.100

1.0.10-6

2.0.10-6

3.0.10-6

4.0.10-6

5.0.10-6

6.0.10-6

7.0.10-6

8.0.10-6

INM

0

2.10-6

4.10-6

6.10-6

8.10-6

1.10-5

IHPMIHAM

0.00

0.05

0.10

0.15

0.20

0.25

0.30

7.0.10-7

7.5.10-7

8.0.10-7

8.5.10-7

9.0.10-7

9.5.10-7

1.0.10-6

MoistureDiffusivity

Temperature

230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0 310.0

0.0

5.0

10.0

15.0

10 cm

20 cm

30 cm

Tem

pera

ture

[°C

]

App

. The

rmal

Dif

fusi

vity

[m

2 s-

1 ]A

pp. T

herm

al D

iffu

sivi

ty [

m2

s-1 ]

App

. The

rmal

Dif

fusi

vity

[m

2 s-

1 ]

Moi

stur

e [m

3 m

-3]

App. T

hermal D

iffusivity [m2 s -1]

Figure 5.2: Overview of the Norunda dataset and the apparent thermal diffusivities as determined by theINM, IHPM, IHAM, DPM2 and DAM2. From top to bottom: Soil temperature (◦C) measurements at10 cm, 20 cm and30 cm, volumetric soil moisture (m3/m3) measurements and derived,24 h-averagedsoil apparent thermal diffusivities (m2/s) using the De Vries equation (scale on the right), and calculated,24 h-averaged diffusivities as determined by the different methods. Unit on the abscissa is days in year.Period shown corresponds to 20. August 1994 to 31. October 1994.

38

Temperature

200.0 210.0 220.0 230.0 240.0 250.0 260.0 270.0 280.0 290.0 300.0 310.0 320.0

0.0

10.0

20.0

30.0

0.05 cm

0.20 cm

0.50 cm

Tem

pera

ture

[°C

]

0.0.100

1.0.10-6

2.0.10-6

3.0.10-6

4.0.10-6

5.0.10-6

INM

App

. The

rmal

Dif

fusi

vity

[m

2 s-

1 ]

0

2.10-6

4.10-6

6.10-6

8.10-6

1.10-5

IHPMIHAM

App

. The

rmal

Dif

fusi

vity

[m

2 s-

1 ]

200 210 220 230 240 250 260 270 280 290 300 310 3200.0.100

4.0.10-6

8.0.10-6 DPM2DAM2

App

. The

rmal

Dif

fusi

vity

[m

2 s-

1 ]

0.00

0.10

0.20

0.30

0.40

0.50

5.0.10-7

5.5.10-7

6.0.10-7

6.5.10-7MoistureDiffusivity

Moi

stur

e [m

3 m

-3]

App. T

hermal D

iffusivity [m2 s -1]

Figure 5.3: Overview of the Ticino dataset and the apparent thermal diffusivities as determined by theINM, IHPM, IHAM, DPM2 and DAM2. From top to bottom: Soil temperature (◦C) measurements at5 cm, 20 cm and50 cm, volumetric soil moisture (m3/m3) measurements and derived,24 h-averagedsoil apparent thermal diffusivities (m2/s) using the De Vries equation (scale on the right), and calculated,24 h-averaged diffusivities as determined by the different methods. Unit on the abscissa is days in year.Period shown corresponds to 24. July 1999 to 11. November 1999.

39

40

Chapter 6

Discussion

This chapter gives an overview of the results and discusses the main issues concerningeach step of the inversion. Probable causes of the apparent discrepancies between theresults of the idealised experiments inSection §3.2 and application of the methods to fieldmeasurements of temperatures inChapter §5 are discussed.

6.1 Validity of Assumptions

The description of heat transfer in soils as well as the methods developed in this study arebased on several assumptions.

Dominance of Heat Conduction This entire study is based on the assumption that soiltemperatures can be modeled by the heat conduction equation(2.3). This impliesthat only heat transfer processes which produce a net heat flux proportional to thelocal temperature gradient are considered. Hence, the assumption of pure heat con-duction is not valid during strong infiltration due to precipitation or irrigation. Asmall study (which is not documented here) shows that temperatures of all threedatasets can be modeled to a satisfactory degree of accuracy using a direct integra-tion of the heat conduction equation. This is due to the fact that the direct integra-tion of temperatures at an intermediate level between two boundary conditions isnot very sensitive to the correct choice of apparent thermal diffusivitiy. It is impor-tant to note that even if a direct integration is successful, problems may arise in theinverse determination of the thermal properties.

Vertical Homogeneity of Soil All methods for a determination of the soil apparent ther-mal diffusivity rely on the assumption that thermal properties do not vary over agiven layer of soil. Field measurements show that soil composition and moisturemay vary considerably over distances of a fewcm’s. SolvingEq. (2.3) for the soilthermal diffusivity while retaining the depth dependent term yields

Dh,app =∂tT

∂2zT− ∂zkh,app · ∂zT

ch · ∂2zT

(6.1)

41

where the second term on the right is due to a depth dependence of the soil thermalproperties. This term may lead to daily oscillations inDh,app if vertical homogeneityis assumed for an inhomogeneous soil. In order to avoid this, care must be taken inthe choice of an appropriate soil layer, or other models should be used.

Time Independence of Soil MoistureThe DPM2, DAM2, IHAM, IHPM and INM as-sume constant soil thermal properties over the period of one day. The DPM1 andDNM assume the same for the sampling interval∆t. Soil moistures of all thedatasets indicate that these are good approximations over most of the data range.Only during strong precipitation or irrigation events does the soil moisture exhibitfast changes. Experiments 6 and 7 show that the IHAM, IHPM and INM methodsdetermine slow changes of soil moisture well.

Sinusoidal Boundary Conditions The DAM1 and DPM1 methods assume purely sinu-soidal boundary conditions. Experiment 3 indicates that this is not an accurate ap-proximation and may lead to failure of the methods if there are several temperaturemaxima per day.

6.2 Inversion of Heat Conduction Equation

Idealised experiments (§3.2) show that the inverse harmonic (IHAM/IHPM) and inversenumeric (INM) methods are suitable for the determination of apparent thermal diffusiv-ities under the assumptions of the heat conduction equation and vertical homogeneity.Even with artificial noise of up toσT = 0.1 ◦C, the mentioned methods still give accurateestimates of thermal diffusivity. Slow diffusivity changes over a period of several daysare well determined by both methods. Variations with a characteristic time shorter thanone day, such as changes of apparent thermal diffusivity due to infiltration, cannot beaccurately determined.

Large discrepancies between theoretical calculations of apparent thermal diffusivityand numerical determination using the IHAM, IHPM and INM methods are observedwhen real temperature measurements are used. The values determined by the methodsoverestimate the apparent thermal diffusivity of the soil. Due to a lack of measurementsseries of apparent thermal diffusivities, a verification of neither the reference diffusivitynor the diffusivities determined by the methods can be completed.

Application to real measurements indicates that the IHAM and IHPM are not verysuitable for a daily determination of soil apparent thermal diffusivity. Large fluctuationsmay occur if vertical inhomogeneities as well as other effects than thermal conduction arepresent.

The most stable performance is obtained using the INM method. However, it is alsoa very computationally intensive method. The determination of apparent thermal diffu-sivities to a target accuracy of10−3 mm2

sfor the Norunda dataset takes roughly12 h on

a Sun Ultra-60 workstation. Further, idealised experiments indicate that the INM un-derestimates apparent thermal diffusivities, which contrasts to the fact that it determinesunphysically large diffusivities in the case of real temperature measurements.

42

Results of this study indicate that simple methods relying on assumptions of sinusoidalboundary conditions (DPM1) or methods which do not take advantage of the availabilityof more than four temperature measurements per day (DPM2/DAM2) should not be usedin daily determination of soil apparent thermal diffusivity. Considerable inaccuraciesoccur in the idealised experiments ofSection §3.2.

6.3 Determination of Soil Moisture

Two models of thermal conductivity, the Kersten and De Vries models, are described inthis study. The De Vries model has been chosen for application to the data, as well asinverse determination of the volumetric soil moisture, due to its large range of applica-bility as well as its physical motivation. The large discrepancy between apparent thermaldiffusivities determined by the methods when compared to the ones calculated using theDe Vries equation also renders results of the latter questionable.

Due to the problems in the determination of apparent thermal diffusivity, only a verybasic method of soil moisture determination has been implemented. A test using diffu-sivities calculated using the De Vries equation indicates that the algorithm works withidealised data. Several problems and limitations of the method are apparent:

Model Character The De Vries equation uses a simplified physical model for describingthermal conductivity. It does not consider all processes contributing to the transferof heat in soils. Notably, the transfer of latent heat through the vapour phase maycontribute substantially at high soil temperatures. Field measurements indicate thatit describes the thermal conductivity to within10% accuracy for soil moistures rang-ing from 0.1 to 0.9 m3

m3 . Not all processes contributing to the apparent thermal con-ductivity are considered,e.g. the transport of latent heat through the vapour phaseis neglected.

Soil Parameters An exact knowledge of volumetric soil composition is necessary. Thediffusivity determined using the De Vries model is strongly dependent on the quartzcontent and porosity of the soil.

Low Moistures De Vries suggests that one should discontinue calculations using theDe Vries equation at volumetric moistureθw = 0.03 m3

m3 for coarse soils or atθw =

0.05 to 0.10 m3

m3 . Thus, the method cannot be used for very dry soils.

Two Solutions There exists no one to one mapping between apparent thermal diffusiv-ity and volumetric soil moisture. Except for very dry soil, two solutions of soilmoisture are obtained for a given apparent thermal diffusivity.

Small Sensitivity Even for strong changes in soil moisture, the apparent thermal diffu-sivity as determined by the De Vries equation rarely shows variations larger thanapprox.10%. Thus, a reliable estimation of volumetric soil moisture requires avery high skill in the determination of apparent thermal diffusivity. The idealisedstudies (§3.2) show that the INM, IHAM and IHPM are capable of determining the

43

apparent thermal diffusivity to an accuracy of more than10% even at high noiselevels. However, application to field-measured data does not yield results to withinthe required accuracy.

Quartz Content No measurements of volumetric quartz content is available for any ofthe three measurement sites. As mentioned before, the range of conductivities ob-tained using the De Vries equation largely depends on the quartz content of thesoil.

44

Chapter 7

Conclusion and Outlook

The feasibility of a determination of soil moisture from temperature measurements hasnot been conclusively proven in this study. The limited scope of this study did not allowfor a thorough investigation into many aspects of the determination of soil moisture fromtime-series of temperature measurements. Also, due to problems which arose in the de-termination of the soil apparent thermal diffusivity, none of the three available datasetswas used to determine soil moistures. Nevertheless, the idea of recovering soil moisturesolely from temperature measurements looks very promising. In particular idealised ex-periments with artificial temperature data demonstrate a high potential to quantitativelyestimate the apparent thermal diffusivity and to derive at least a qualitative soil moistureindicator. This chapter presents a survey of proposals which aim at a continuation of thisproject.

• One of the major handicaps was the unavailability of a time series of measurementsof thermal properties of the soil. Without such measurements, a rigorous verifi-cation of the determination of thermal diffusivity is difficult. Several companiesproduce equipment to measurech, kh andDh under field conditions. If a dedi-cated experiment for a validation of the methods presented in this study would beplanned, measurements of heat capacitych, thermal conductivitykh, temperatureT , heat fluxqh and soil moistureθw in a soil of known volumetric composition arenecessary.

• Both the inverse harmonic (IHAM/IHPM) and inverse numeric methods (INM) al-low a determination of depth dependent thermal properties. As has been shownin the discussion, assuming constancy with depth may lead to oscillations.Lettau(1971) has successfully determined depth dependent thermal conductivities usinga method similar to the inverse harmonic methods. Any further research effort onthis topic should try to quantify the importance of depth dependency and adapt themethods into this direction.

• There still exist several obvious lackings in the developed methods. The Fouriertransform of the temperature measurements used for the IHAM and IHPM methodsdoes not handle any temperature trends, which may lead to errors in the determi-nation of the amplitude and phase of the diurnal temperature wave. Further, as

45

described inSection §3.1.4, more harmonics can be used. For the INM, a consider-able acceleration could be obtained if a different algorithm for the time integrationis used.

• Apparent thermal diffusivity varies only by approx.10% for soil moistures above0.2 m3

m3 . As can be seen from illustrationsFig. 2.1 and2.2 on pages7 and9, respec-tively, bothch andkh show larger variability with soil moisture. Also, bothch andkh show a one to one mapping to soil moisture. A direct determination of any ofthese quantities from temperature measurements is not possible if the other is notknown. However, if for example heat flux measurements are available or one of thequantities is known for a given soil moisture, a determination ofch or kh is possible.This would facilitate the determination of soil moisture considerably.

46

References

Abramowitz, M., and I. E. Stegun,Handbook of mathematical functions, chap. 9, pp.359–389, U.S. Department of Commerce, National Bureau of Standards, 1972.5

Bear, J., and J. Bensabat, Heat and mass transfer in unsaturated porous media at a hotboundary: I. one-dimensional analytical model,Transport in Porous Media, 6, 281–298, 1991.

Carson, D. J., An introduction to the parametrization of land-surface processes. part ii.soil heat conduction and surface hydrology.,Meteorological Magazine, 116, 263–279,1987.

Chen, D., and J. Kling, Apparent thermal diffusivity in soil: Estimation from thermalrecords and suggestions for numerical modeling,Physical Geography, 17, 419–430,1996. 11

De Vries, D. A., Title unkown (referenced inde vries and afgan, 1975), in MededelingenLandbouwhogeschool, Wageningen, vol. 52, pp. 1–73, 1952a.7, 8, 35

De Vries, D. A., Title unknown (referenced inde vries and afgan, 1975), in Bulletin Inst.Int. du Froid, Annexe, vol. 1, pp. 115–131, 1952b.

De Vries, D. A., and N. H. Afgan,Heat and mass tranfer in the biosphere, John Wileyand Sons, 1975.1, 3, 7

Farouki, O. T., Thermal properties of soils,Series on Rock and Soil Mechanics, 11, 1–136,1986. 7, 8, 10

Fluhler, H., and K. Roth,Bodenphysik - Physik der ungesattigten Zone, EidgenossischeTechnische Hochschule Zurich, 1999.

Hillel, D., Introduction to soil physics, Academic Press, Inc., 1982.

Hollinger, S. E., and S. A. Isard, A soil moisture climatology of illinois,Journal of Cli-mate, 7, 822–833, 1994.1

Horton, R., P. J. Wierenga, and D. R. Nielsen, Evaluation of methods for determining theapparent thermal diffusivity of soil near surface,Soil Sci. Soc. Am. J., 47, 25–32, 1983.11, 12, 13, 14

47

Huang, C. H., and M. N.Ozisik, Direct integration approach for simultaneously esti-mating temperature dependent thermal conductivity and heat capacity,Numerical HeatTransfer, 20, 95–110, 1991.11

Huang, C.-H., and J.-Y. Yan, The function estimation in measuring temperature-dependent thermal conductivity in composite material,J. Appl. Phys., 78, 6949–6956,1995. 11

Huang, J., H. M. van den Dool, and K. P. Georgakakos, Analysis of model-calculated soilmoisture over the United States (1931-1993) and applications to long-range tempera-ture forecasts,Journal of Climate, 9, 1350–1362, 1996.1

Jansson, P.-E., Simulation model for soil water and heat conditions,Tech. rep., Departe-ment of Soil Sciences, Swedish University of Agricultural Sciences, 1998.

Kasubuchi, T., Heat conduction model of saturated soil and estimation of thermal con-ductivity of soil solid phase,Soil Science, 138, 240–247, 1984.7

Kersten, M. S., Thermal properties of soil,Bulletin of the University of Minnesota, Insti-tute of Technology, 52, 1–225, 1949.1, 7, 9, 35

Kimball, B. A., R. D. Jackson, R. J. Reginato, F. S. Nakazama, and S. B. Idso, Comparisonof field-measured and calculated soil-heat fluxes.,Proc. Soil Sci. Soc. Am., 40, 18–25,1976. 8

Krukovsky, P. G., A universal approach to solution of inverse heat transfer problems(method and software), in1995 National Heat Transfer Conference, vol. 10, pp. 107–112, 1995. 11

Kurpisz, K., and A. J. Nowak,Inverse thermal problems, Computational Mechanics Pub-lications, 1995.11

Lam, T. T., and W. K. Yeung, Inverse determination of thermal conductivity for one-dimensional problems,Journal of Thermophysics and Heat Transfer, 9, 335–344, 1995.11

Lesnic, D., L. Elliot, and D. B. Ingham, Identification of the thermal conductivity and heatcapactiy in unsteady nonlinear heat conduction problems using the boundary elementmethod,Journal of Computational Physics, 126, 410–420, 1996.11

Lettau, B., Determination of the thermal diffusivity in the upper layers of a natural groundcover,Soil Science, 112, 172–177, 1971.11, 45

Lettau, H., Improved models of thermal diffusion in the soil,Transactions of the AmericanGeophysical Union, 35, 121–132, 1954.

Leuenberger, D., Warmeleitung im boden,Tech. rep., Eidgenossische TechnischeHochschule Zurich, 1999.

48

Massman, W. J., Periodic temperature variations in an inhomogeneous soil: A comparisonof approximate and exact analytical expressions,Soil Science, 155, 331–338, 1993.5

Menzel, L., Modellierung der Evapotranspiration im System Boden-Pflanze Atmosphare,in Zurcher Geographische Schriften, vol. 67, pp. 37–39, 1997.

NAG, NAG Fortran Library Manual, Mark 14, National Algorithms Group Inc., 1990.6,53

Nakshabandi, G. A., and H. Kohnke, Thermal conductivity and diffusivity of soils asrelated to moisture tension and other physical properties,Agr. Meteorol., 2, 271–279,1994. 7

Narasimhan, T. N., Hydraulic characterization of aquifers, reservoir rocks, and soils: Ahistory of ideas,Water Resources Research, 34, 33–46, 1998.

Nerpin, S. V., and A. F. Chudnovskii,Heat and mass transfer in the plant-soil-air system.,pp. 33–50, Oxonian Press Pvt./ Ltd., New Delhi, 1984.5

Nobre, R. C. M., and N. R. Thomson, The effects of transient temperature gradients onsoil moisture dynamics,Journal of Hydrology, 157, 57–101, 1993.8

Prim, C., Rolle des Bodens bei der Vorhersagbarkeit der Wetterentwicklung, Master’sthesis, Eidgenossische Technische Hochschule Zurich, 1999.

Schar, C., D. Luthi, and U. Beyerle, The soil-precipitation feedback: A process study witha regional climate model,Journal of Climate, 12, 722–741, 1999.1

Schelde, K., A. Thomsen, T. Heidmann, and P. Schjønning, Diurnal fluctuations of waterand heat flows in a bare soil,Water Resources Research, 34, 2919–2929, 1998.6, 8, 9,10, 33, 56

Sepaskhah, A. R., and L. Boersma, Thermal conductivity of soils as a function of temper-ature and water content,Soil Sci. Soc. Am. J., 43, 439–444, 1979.7

Shao, M., R. Horton, and D. B. Jaynes, Analytical solution for one-dimensional heatconduction-convection equation,Soil Sci. Soc. Am. J., 62, 123–128, 1998.5

Sivia, D. S.,Data analysis: A Bayesian tutorial, Oxford Science Publications, 1996.

Stahli, M., K. Hessel, J. Eriksson, and A. Lindahl, Physical and chemical description ofthe soil at the NOPEX central tower site,Tech. Rep. 16, Department of Soil Sciences,Swedish University of Agricultural Sciences, Uppsala, Sweden, 1995.35, 56

Sundberg, J., Thermal properties of soils and rocks, Ph.D. thesis, Swedish GeotechnicalInstitute, 1988.

Tervola, P., A method to determine the thermal conductivity from measured temperatureprofiles,Int. J. Heat Mass Transfer, 32, 1425–1430, 1989.11

49

Thakur, A. K. S., and M. M. Momoh, Temperature variation in upper earth crust due toperiodic nature of solar insolation,Energy Conver. Manage., 23, 131–134, 1983.5

Vinnikov, K. Y., and A. Robock, Scales of temporal and spatial variability of midlatitudesoil moisture,Journal of Geophysical Research, 101, 7163–7174, 1996.

Vinnikov, K. Y., and I. B. Yeserkepova, Soil moisture: Empirical data and model results,Journal of Climate, 4, 66–79, 1991.1

Wiltshire, R. J., Solutions of the heat conduction equation in a non-uniform soil,EarthSurface Processes and Landforms, 7, 241–252, 1982.5

Wiltshire, R. J., Periodic heat conduction in a non-uniform soil,Earth Surface Processesand Landforms, 8, 547–555, 1983.5

Zabaras, N., K. A. Woodbury, and M. Raynaud (Eds.),Inverse problems in engineering,The American Society of Mechanical Engineers, 1993.

Zhang, T., and T. E. Osterkamp, Considerations in determining thermal diffusivity fromtemperature time series using finite difference methods,Cold Regions Science andTechnology, 23, 333–341, 1995.11

50

Nomenclature

Latin Symbols

Symbol Unit Descriptiona1, a2, a3 1 empirical constants for Kersten equationan

◦C complex Fourier coefficientA,Am,Adiurnal ◦C amplitude of temperature wave (mth harmonic)bn

◦C real Fourier coefficientch, c0 J m−3 K−1 volumetric heat capacity of soilcn

◦C real Fourier coefficientd, dn m damping depth (ofnth harmonic)Dh m2 s−1 thermal diffusivity of soilDh,app,Dn,Dcalc m2 s−1 apparent thermal diffusivity of soilgj 1 shape factor (De Vries equation)J0 1 Bessel function of first kindkh, k0, kh,j W m−1 K−1 thermal conductivity of soilkh,app W m−1 K−1 apparent thermal conductivity of soilkh,vap W m−1 K−1 kh due to latent heat transfer by water vapourN 1 number of harmonicsN0 1 Bessel function of second kindP s period of temperature signal, usuallyP = 1 dqh W m−2 heat fluxrn m2 s−1 nth residualt, ta, tb s timeT , T0, T∞, Tk, T ′ ◦C soil temperatureu J m−3 volumetric (thermal) internal energyX 1 placeholder for a complicated expressionY 1 placeholder for a complicated expressionz, zm m vertical distance, increasing downwards

Greek Symbols

Symbol Unit Descriptionα m−1 slope of linearly increasingch andkhβn

◦C coefficient (§2.2.2)∆t s time discretisation step∆z m depth discretisation stepε 1 machine precision, approx.10−15

51

γn◦C coefficient (§2.2.2)

η m3 m−3 soil porosityθ, θj , θw,wilt, θrain m3 m−3 volumetric contentκj 1 weight factor (De Vries equation)λ 1 decay time of moisture signalΛ 1 arbitrary constantξ 1 scaled depthρb kg m−3 dry soil densityσT

◦ standard deviation of temperatureσ m2 s−1 noise measure of apparent thermal diffusivityφn, φdiurnal rad phase of temperature wave (nth harmonic)ω s−1 fundamental frequency, usually2π/24 h

Subscripts

Subscript Values Descriptiona airc clayj q, c, o, w, a soil componentsk 1, . . . ,4 measurement index (DPM2/DAM2)l 1, . . . ,∞ time indexm 1, 2 boundary conditionsn 1, . . . ,∞ Fourier coefficients or time indexo organicq quartzv water vapourw water

Mathematical Symbols

Symbol Description∂x . . . partial derivative with respect tox¯. . . average over all samples〈. . .〉24 h time average over24 h

52

Appendix A

Indirect Numerical Method (INM)

In this section, the application of the INM is illustrated using the first day of the artificial tempera-tures of Experiment 3 described inSection §3.2.1. As described inSection §3.1.5 on page14, theINM involves repeated integration of the heat conduction equation for different values ofDh,app.This integration is done using the routineD03PBFof the Numerical Algorithms Group FortranLibrary (NAG, 1990). Figure A.1 shows the input and output of the routine. A total number of21 equidistant gridpoints has been used in thez-direction. The boundary conditions at0.1 and0.3 m (straight crosses,+) are given directly by the artificial data. The initial values att = 0 dare calculated by linear interpolation of the artificial temperatures at0.1, 0.2 and0.3 m onto thegridpoints. The mesh (solid lines,−) is the output of the forward integration for one day usinga value ofDh,app = 0.6 mm2

s . The artificial data at0.2 m do not exactly match the integrated

temperatures at the same depth, since they have been generated usingDh,app = 0.7 mm2

s . TheRMS1 difference between the integrated and artificial temperature data is approximately0.14 ◦C.

Figure A.2 illustrates the RMS difference between integrated and artificial temperature data asa function of the apparent thermal diffusivity used for the integration. A clear minimum is visibleat a apparent thermal diffusivity of approximately0.7 mm2

s . The INM determines the location ofthis minium using the routineE04ABFof theNumerical Algorithms Group Fortran Library. It isdetermined asDh,app = 0.6997 mm2

s at an RMS of0.026 ◦C.

1root mean square

53

Numerical Solution

Boundary Condition

Initial Condition

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.08 0.1

0.12 0.14

0.16 0.18 0.2

0.22 0.24

0.26 0.28 0.3

14

16

18

20

22

24

26

Tem

pera

ture

[¡C

]

Time [d]

Dep

th [m

]

Figure A.1: Illustration of the integration of the heat conduction equation using artificial data as boundarycondition. Explanation of quantities is given in text.

54

0.0 1.0 2.0 3.00.0

0.5

1.0

1.5

2.0

Apparent Thermal Conductivity kh,app [W m-1 K-1]

RM

S [°C

]

Figure A.2: Root mean square (RMS) of the differences between the integrated and artificial temperaturesat a depth of0.2 m as a function of the apparent thermal diffusivityDh,app used for the integration.

55

Appendix B

Datasets

B.1 Foulum Dataset

Extensive measurements of soil water tension, soil water content, soil temperature and soil thermalconductivity were made during an experiment on a25m2 bare soil plot (Schelde et al., 1998) atthe Research Centre Foulum (Denmark). The texture of the topsoil was classified as60% sandand fine sand,29% silt, 8% clay and3% organic material.Table B.1 summarises the conductedmeasurements.

Variable DescriptionTime 12. September, 1991 to 10. October 1991 (30 days)Resolution 30min.Soil Characteristics Porosity, density, grain size distribution, water retention

characteristicsTemperature Thermocouple probes at0 cm, 2.5 cm, 5 cm, 7.5 cm,

10 cm, 12.5 cm, 15 cm, 20 cm und25 cmThermal Conductivity Needle probe method (no time series, for different mois-

ture contents) at0 − 10 cm, 5 − 15 cm, 15 − 25 cm,25 − 35 cm, 35 − 45 cm, 45 − 55 cm, 55 − 65 cm,75− 85 cm

Volumetric Moisture TDR probed at2.5 cm, 5 cm, 7.5 cm, 10 cm, 12.5 cm,15 cm, 20 cm und25 cm

Moisture Tension Tensiometers at5 cm, 10 cm, 15 cm, 20 cm und25 cm

Table B.1: Summary of measurements conducted at the Research Centre Foulum, Denmark

B.2 Norunda Dataset

This dataset was measured during the NOPEX (Northern Hemisphere Climate Processes Land-Surface Experiment) project. All measurements were taken at the NOPEX Central Tower Site(Stahli et al., 1995), which is located 30 km north of Uppsala (Sweden) in an extended spruce-pine forest. The soil at the measurement site is a sandy loam soil containing many stones of up to80 cm diameter. It consists of a thin organic layer (0−9 cm), a reddish B-horizon (11−46 cm) and

56

a yellowish-gray C-horizon (46− 81 cm). Below about70 cm, the soil is very compact. Very fewroots were found in the compacted subsoil.Table B.2 summarises the conducted measurements.

Variable DescriptionTime 20. August, 1994 to 31. October, 1994 (72 days)Resolution 10min.Soil Characteristics Porosity, grain size distribution, water retention charac-

teristics, soil chemical analysisTemperature Thermocouple probes at10 cm, 20 cm, 30 cm, 50 cm and

80 cmHeat Flux Heatflux probe at7 cmVolumetric Moisture TDR probes at10 cm, 20 cm, 30 cm, 50 cmOther Ground water level

Table B.2: Summary of measurements conducted at Site A2, NOPEX Central Tower Site, Sweden

B.3 Ticino Dataset

In the summer of 1999, the Institute of Climate Research ETH conducted a field experiment aspart of the EU-MAP-Project RAPHAEL (Runoff and Atmospheric Processes for Flood HazardForecasting and Control). The principal research field was in Claro, Ticino (Switzerland). Thedata used in this study was measured at a site located next to a soccer field with flat topography. Aprofile of the soil was classified as humus at0− 10 cm, humus with roots at10− 40 cm and sandat40− 120 cm. Table B.3 summarises the conducted measurements.

Variable DescriptionTime 24. July, 1999 to 11. November, 1999 (102 days)Resolution 60min.Soil Characteristics PorosityTemperature Thermocouple probes at5 cm, 20 cm and50 cmHeat Flux HPC01SC Hukseflux probe at5 cmVolumetric Moisture TDR probes at5 cm, 15 cm, 35 cm, 50 cm, 80 cm and

105 cmOther Precipitation, Leaf Humidity, Atmospheric parameters

including turbulent fluxes

Table B.3: Summary of measurements conducted at the soccer field location in Claro, Switzerland

57

Appendix C

Selected Experiment Results

58

C.1 Noise (Experiment 1)

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0

6.95.10-7

7.00.10-7

7.05.10-7

Reference

Calculated

24h Average

DAM2

DPM2

Reference

Calculated

24h Average6.95.10-7

7.00.10-7

7.05.10-7

IHAM

Reference

Calculated

24h Average6.95.10-7

7.00.10-7

7.05.10-7

IHPM

Reference

Calculated

24h Average6.95.10-7

7.00.10-7

7.05.10-7

INM

Reference

Calculated

24h Average

2 3 4 5 6 7 8 9 10 11 12 13 14 15

6.95.10-7

7.00.10-7

7.05.10-7

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

Figure C.1: Results of Experiment 1: Apparent thermal diffusivity as determined by all methods as afunction of time. Also shown are a24 h average and the reference value of the diffusivity.

59

C.2 Linear Moisture Increase (Experiment 6)

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0

6.0.10-7

6.2.10-7

6.4.10-7

6.6.10-7

Reference

Calculated

24h Average

DAM2

6.0.10-7

6.2.10-7

6.4.10-7

6.6.10-7

DPM2 Reference

Calculated

24h Average

IHAM

6.0.10-7

6.2.10-7

6.4.10-7

6.6.10-7

Reference

Calculated

24h Average

6.0.10-7

6.2.10-7

6.4.10-7

6.6.10-7

IHPM Reference

Calculated

24h Average

2 3 4 5 6 7 8 9 10 11 12 13 14 15

6.0.10-7

6.2.10-7

6.4.10-7

6.6.10-7

INM Reference

Calculated

24h Average

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

Figure C.2: Results of Experiment 6: Apparent thermal diffusivity as determined by all methods as afunction of time. Also shown are a24 h average and the reference value of the diffusivity.

60

C.3 Idealised Rain Event (Experiment 7)

2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0

6.0.10-7

6.2.10-7

6.4.10-7

6.6.10-7

Reference

Calculated

24h Average

DAM2

6.0.10-7

6.2.10-7

6.4.10-7

6.6.10-7

DPM2

Reference

Calculated

24h Average

6.0.10-7

6.2.10-7

6.4.10-7

6.6.10-7

IHAM

Reference

Calculated

24h Average

6.0.10-7

6.2.10-7

6.4.10-7

6.6.10-7

IHPM

Reference

Calculated

24h Average

2 3 4 5 6 7 8 9 10 11 12 13 14 15

6.0.10-7

6.2.10-7

6.4.10-7

6.6.10-7

INM

Reference

Calculated

24h Average

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

App

. The

rmal

Dif

fusi

vity

[m

2 s

-1]

Figure C.3: Results of Experiment 7: Apparent thermal diffusivity as determined by all methods as afunction of time. Also shown are a24 h average and the reference value of the diffusivity.

61

62