comparative study on methods for computing soil heat storage and energy balance in arid and...

308 JOURNAL OF METEOROLOGICAL RESEARCH VOL.28

Comparative Study on Methods for Computing Soil Heat Storageand Energy Balance in Arid and Semi-Arid Areas

LI Yuan (� �), LIU Shuhua∗ (��), WANG Shu (� �), MIAO Yucong (��),

and CHEN Bicheng (��)

Department of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing 100871

(Received August 8, 2013; in final form January 3, 2014)

ABSTRACT

Observations collected in the Badan Jaran desert hinterland and edge during 19–23 August 2009 and inthe Jinta Oasis during 12–16 June 2005 are used to assess three methods for calculating the heat storage of the5–20-cm soil layer. The methods evaluated include the harmonic method, the conduction-convection method,and the temperature integral method. Soil heat storage calculated using the harmonic method provides theclosest match with measured values. The conduction-convection method underestimates nighttime soil heatstorage. The temperature integral method best captures fluctuations in soil heat storage on sub-diurnaltimescales, but overestimates the amplitude and peak values of the diurnal cycle. The relative performanceof each method varies with the underlying land surface. The land surface energy balance is evaluated usingobservations of soil heat flux at 5-cm depth and estimates of ground heat flux adjusted to account for soil heatstorage. The energy balance closure rate increases and energy balance is improved when the ground heatflux is adjusted to account for soil heat storage. The results achieved using the harmonic and temperatureintegral methods are superior to those achieved using the conduction-convection method.

Key words: soil heat storage, harmonic method, conduction-convection method, temperature integral

method, surface energy balance

Citation: Li Yuan, Liu Shuhua, Wang Shu, et al., 2014: Comparative study on methods for computing soilheat storage and energy balance in arid and semi-arid areas. J. Meteor. Res., 28(2), 308–322,doi: 10.1007/s13351-014-3043-5.

1. Introduction

In ideal conditions, the energy received and re-leased by the earth’s surface is equal. This tenet isone of the most basic principles of energy balance inthe earth system. Land surface energy balance clo-sure is used as a criterion to judge the quality of ob-servational data (e.g., to assess the accuracy of eddycorrelation data), and is crucial for accurate estimatesof surface CO2 and evaporative fluxes based on theenergy balance equation. Improvement in scientificunderstanding of mass and energy exchange betweenland surface and the atmosphere is also a common ba-sis for improving regional and global climate models(Twine et al., 2000; Wilson et al., 2002; Cava et al.,2008). However, observational assessments of the land

surface energy budget often contain significant imbal-ances. These imbalances may arise due to the com-plexity and non-uniformity of the underlying surface,or to the limits of observational technology and instru-ment precision. The magnitude of the energy imbal-ance may be as large as 30% in some cases (Foken andOncley, 1995; Oncley et al., 2007). Problems in landsurface energy balance closure have been widely stud-ied in recent decades.

Land surface energy imbalances are generally at-tributed to the following reasons: instrumental andobservational errors, overestimates of available energy(i.e., the sum of net radiation and soil heat flux), un-derestimates of effective energy (i.e., the sum of sensi-ble and latent heat), neglect of heat storage and advec-tion, and surface non-uniformity and non-stationarity.

Supported by the National Science and Technology Support Program of China (2012BAH29B03) and National (Key) BasicResearch and Development (973) Program of China (2009CB421402).

∗Corresponding author: [email protected].

©The Chinese Meteorological Society and Springer-Verlag Berlin Heidelberg 2014

NO.2 LI Yuan, LIU Shuhua, WANG Shu, et al. 309

Recent improvements in the accuracy of observationalinstruments and quality control techniques for turbu-lence data (such as coordinate rotation, plane fitting,high frequency response revisal, footprint analysis,density effect correction, and angle of attack revisal)have reduced the magnitude of instrumental and ob-servational errors and have often increased estimatesof effective energy. Although these new developmentshave improved land surface energy balance closure inmany cases, the problem is not yet completely solved(Moore, 1986; Wilczak et al., 2001; Nakai et al., 2006;Wang et al., 2007; Foken, 2008). Many studies in-dicate that the main factors in land surface energyimbalance are thermal storage terms (Ochsner et al.,2007; Foken, 2008), including heat storage in vegeta-tion, air, and soil. Canopy heat storage is particularlyimportant for ecosystems with tall, lush vegetation.Vegetation photosynthesis also consumes energy, butcan typically be ignored because it has little effect onthe energy balance. The heat storage of the atmo-spheric surface layer is approximately zero under thequasi-steady assumption, and Sun et al. (1995) sug-gested that this heat storage term is small enough tobe ignored. The ground heat flux cannot be measureddirectly by using existing observational technology andis usually approximated by the shallow soil heat flux(even though significant differences may exist betweenthese two terms). Soil heat storage is generally thedominant heat storage term because the shallow soillayer above the heat flux plate often stores consider-able heat. Ignoring soil heat storage could lead toa considerable energy imbalance. Shallow soil heatstorage appears to be particularly important in aridand semi-arid regions due to strong surface heating(Heusinkveld et al., 2004; Finnigan, 2006).

The traditional soil equation only considers theeffects of heat conduction. Horton et al. (1983) evalu-ated several common methods for calculating soil ther-mal diffusivity, including the amplitude method, thephase method, the arc tangent method, the logarith-mic method, the numerical method, and the harmonicmethod. These different methods rely on different as-sumptions and therefore yield different results. Hor-ton et al. (1983) also assessed the reliability of each

method and reported that the numerical and harmonicmethods were the most reliable. Upon comparing fiveof the six methods (except the numerical method),Verhoef et al. (1996) also judged the harmonic methodto be most reliable. Mo et al. (2002) provided furthervalidation for these methods. Wang et al. (2009) usedthe harmonic method and a temperature prediction-correction method to calculate shallow soil heat stor-age observed during the “Heihe Comprehensive Ob-servation Experiment”. They reported that the en-ergy closure rate was improved after modification ofthe soil heat flux using these two methods. Zuo etal. (2010) compared the results of harmonic, temper-ature prediction-correction, and temperature integralmethods for calculating the ground heat flux in ob-servations from the SACOL (Semi-Arid Climate andEnvironment Observatory, Lanzhou University) sta-tion, and concluded that the harmonic and tempera-ture prediction-correction techniques provided betterperformance.

Soil water is one of the most important factorsin the land surface energy balance of arid and semi-arid ecosystems. Moisture movement generates ther-mal convection effects that influence the soil temper-ature. Although the soil water content of arid andsemi-arid areas is relatively low, these effects still ex-ist and neglecting the influences of moisture movementwill introduce imbalances in the surface energy budget.Gao et al. (2003) showed that changes in soil temper-ature were connected with both soil heat conductionand heat convection caused by the vertical movementof liquid water in the soil. They also derived a one-dimensional heat conduction convection equation bycoupling heat conduction and convection. Gao (2005)proposed a formula for calculating the soil heat fluxthat considers the effects of thermal convection pro-cesses.

Most conventional methods for estimating groundheat flux rely on knowledge of the temporal evolu-tion of soil temperature. Wang et al. (2012) pro-posed a novel method that requires no information onsoil temperatures to supplement flux plate measure-ments. Wang (2012) extended this method to enablethe estimation of soil heat storage from a single depth


measurement. The method is based on the fundamen-tal solution of the one-dimensional heat equation andDuhamel’s principle. The only necessary thermal pa-rameter is the soil thermal diffusivity, which can be as-sumed constant in the absence of measurements. Thismethod is robust and preserves the accuracy of heatflux estimates in the face of reduced input information.The primary improvement over conventional methodsis in ease of use and expense, rather than accuracy(Wang, 2012; Wang et al., 2012).

Arid and semi-arid areas account for 30%–45% ofthe total global land area, and are the main land sur-face types in northern China. Arid and semi-arid areasare particularly sensitive to global climate change dueto their geographic locations and the fragility of theirecosystems. Land-atmosphere interactions in these ar-eas have important impacts on global redistribution ofenergy and global climate change (Fu and Wen, 2002;Huenneke et al., 2002). Deserts and oases are bothcommon natural landscapes in arid and semi-arid ar-eas; however, the underlying surfaces in desert hin-terland, desert edge, and oasis regions are substan-tially different. Interactions between deserts and oasespresent special challenges for projecting and under-standing regional climate changes. A number of re-cent studies have investigated the land surface energybalance over single underlying surface; however, fewof these studies have considered the effects of differentunderlying surface types.

We calculate the soil heat storage of differ-ent arid and semi-arid surfaces using the har-monic, conduction-convection, and temperature inte-gral methods. The results reveal important differencesin soil heat storage processes and land surface energybalance closure over different underlying surfaces. Weanalyze the energy balance after adjusting the shallowsoil heat flux to account for differences in land surfacetype.

2. Data and instrumentation

A portion of the observational data was obtainedfrom the Badan Jaran desert hinterland (39◦46′N,102◦9′E) and edge (39◦28′N, 102◦22′E) during 19–23

August 2009. This dataset was collected under theNational Basic Research and Development (973) Pro-gram of China “Energy and Water Cycle Experimentof Northwest China Typical Arid and Semi-arid Ar-eas”. Another dataset was obtained from the JintaOasis (39◦59′N, 98◦56′E) in Gansu Province during12–16 June 2005 under the “Energy and Water Cy-cle Field Experiment of the Oasis System” project ofthe Cold and Arid Regions Environmental and En-gineering Institute (CAREERI), Chinese Academy ofSciences (CAS).

All of the instruments used in the experimentwere automated and calibrated before use. The obser-vational data collected in the Badan Jaran desert weresampled as 30-min mean values with outliers elimi-nated during data quality control (Hu, 2004; Zuo et al.,2009). The observational data collected at the JintaOasis were sampled as 10-min means, and then aver-aged into 30-min means to facilitate direct comparisonof the two datasets. This data processing only reducedthe variability of the results and did not change thedirection of the gradient or the accuracy of the calcu-lations. All of the observations were taken under fineweather conditions.

3. Methodology

Soil heat flux is one of the most important com-ponents of the surface energy balance, but direct mea-surement of this term is difficult. Calculation of soilheat storage is a necessary step in calculating soil heatflux. Intelligent adjustment of soil heat flux prior toits inclusion in the land surface energy balance equa-tion may help to improve closure of the surface energybudget.

3.1 Methods for computing soil heat storage

Three forms of heat transfer occur in soils: con-duction, convection, and radiation. Radiative heattransfer at the surface is generally separated intoshort-wave and long-wave components. Short-waveradiation does not penetrate into the soil layer, andis reflected or absorbed at the surface. Long-waveradiation depends only on the temperature of an“infinitesimal” layer at the land surface. Because the


effects of radiation do not penetrate into the soil layer,radiative heat transfer can be neglected in the soil heatbudget.3.1.1 Harmonic method (HM)

Soil heat transfer occurs mainly via moleculartransmission. Bhumralker (1975) presented the heatconduction equation

∂T

∂t= k

∂2T

∂z2, (1)

where T is the soil temperature, t is time, z is depth,

and k =λ

Cvis the soil thermal diffusivity (with λ the

soil thermal conductivity and Cv the soil thermal ca-pacity). The soil thermal diffusivity k is a weak func-tion of soil water content and can be approximatedas constant across the depth of the soil layer (Wang,2012; Wang et al., 2012).

Changes of soil temperature with time can be ex-pressed as the superposition of n harmonics. The ini-tial boundary condition is defined as

T (z, 0) = T0 − γz, z � 0, (2)

where T0 is the average temperature of the soil surface,γ is the lapse rate of soil temperature with depth, andz is the soil depth. Hillel (1982) parameterized the di-urnal forcing at the surface as a pure sinusoidal func-tion. The upper boundary condition in this case isT (t)|z=0 = T0 + A sin(ωt) when t > 0, where A is the

amplitude and ω =2π

pis the angular velocity of the

earth rotation (p = 24 h is the harmonic period of landsurface temperature). The diurnal forcing is not a truesine function, which can be accounted for by replacingthe upper boundary condition with the Fourier series

T (t)|z=0 = T0 +n∑

i=1

Ai sin(iωt + ϕi), t > 0, (3)

where Ai is the amplitude of harmonic i and ϕi isthe initial phase of harmonic i. This formula indi-cates that variations in the land surface temperatureconsist of two parts: the constant T0 and n superim-posed sine waves. The periods of these sine waves arep, p/2, . . ., p/n, respectively, with the correspondingamplitudes A1, A2, . . ., An. Here, we assume that theinitial phases ϕi and the soil temperature below 1-m

depth are all constants. We can then use the vari-able separation method to solve the heat conductionequation (Eq. (1)). This approach yields

T (z, t) = T0 − γz +n∑

i=1

Aiexp(−Biz)

· sin(iωt + ϕi − Biz), (4)

with Bi =√

iω/2k. The parameters in Eq. (4) arederived by using the least squares method to find anoptimal fit to soil temperature measurements at twodepths. We can then obtain the harmonic soil heatflux formula as

G(z, t) = kCvγ + kCv

n∑

i=1

Ai

√2Biexp(−Biz)

· sin(iωt + ϕi + π/4 − Biz). (5)

Miao et al. (2012) fitted harmonic models of differentorders and found that the accuracy of the second-orderharmonic model (n = 2) was already sufficient for mostpurposes. Accordingly, we use the second-order har-monic for simplicity and ease of computation.3.1.2 Conduction-convection method (CM)

Moisture movement in the soil produces heat con-vection, which in turn affects the soil temperature.The convective heating Qv caused by the verticalmovement of water through a unit area of soil per unittime can be expressed as

Qv = CwwθΔT, (6)

where w is the water permeability in unit of m s−1

(positive upward), Cw is the liquid water thermal ca-pacity, θ is the soil water content, and ΔT is the ver-tical gradient of water temperature. Using the secondlaw of thermodynamics, the soil heat balance can beexpressed as

∂T

∂t= ka

∂2T

∂z2+ W

∂T

∂z, (7)

where ka is the thermal diffusivity (including only heat

conduction) and W =Cw

Cvwθ can be understood as the

liquid water flux density (in m s−1). Equations (1) and(7) are equivalent when W = 0 (i.e., the conduction-convection equation reduces to the traditional heat


conduction equation in soils with low water content).Gao et al. (2003) derived a first-order harmonic

analytical solution to the conduction-convection equa-tion (Eq. (7)):

T (z, t) = T0 − γz + Aexp(−Mz)

· sin(ωt + ϕ − Nz), (8)

ka =(z1 − z2)2ω ln(A1/A2)

(ϕ2 − ϕ1)[(ϕ2 − ϕ1)1 + ln2(A1/A2)], (9)

W =ω(z1 − z2)ϕ2 − ϕ1

[ 2 ln2(A1/A2)(ϕ2 − ϕ1)2 + ln2(A1/A2)

], (10)

where the amplitude A =Tmax − Tmin

2, M =

W

2ka+

√2

4ka

√W 2 +

√W 4 + 16k2

aω2, and N =

√2ω√

W 2 +√

W 4 + 16k2aω

2

. A1 and A2 are the ampli-

tudes of soil temperature variations at the depths z1

and z2, and ϕ1 and ϕ2 are the phases of soil temper-ature variations at depths z1 and z2. In this work, wetake z1 = 5 cm and z2 = 20 cm.

Fan and Tang (1994) calculated the heat flux us-ing a correlation form of the conduction-convectionmethod:

Qd = −Cvka∂T

∂z, (11)

Qw = −CvWaΔT, (12)

Qt = Qd + Qw = −Cvka∂T

∂z− CvWaΔT, (13)

where Qt is the total heat flux, Qd is the conduc-tion heat flux, Qw is the convective heat flux, and

Wa =θ(z) − θ(z + Δz)

θ1 − θ2

× W . Using Eq. (8),

Qd = −Cmk[−γ − MAexp(−Mz) × sin(ωt

+ϕ − Nz) − NAexp(−Mz)

× cos(ωt + ϕ − Nz)], (14)

and

ΔT = T (z2, t) − T (z1, t) = −γz2

+Aexp(−Mz2) × sin(ωt + ϕ − Nz2) + γz1

−Aexp(−Mz1) × sin(ωt + ϕ − Nz1). (15)

The soil heat storage calculated using eitherthe harmonic method or the conduction-convection

method is the difference between the soil heat fluxesat two different layers:

S = G1 − G2, (HM), (16)

S = Qt1 − Qt2, (CM). (17)

3.1.3 Temperature integral method (TIM)Based on the first law of thermodynamics, the in-

tegral form of one-dimensional heat conduction equa-tion is

G0 = Gz + Cv

0∫

z

∂Tz

∂tdz, (18)

where G0 is the ground heat flux, Gz is the soil heatflux observed at a heat flux plate at depth z, Tz is thetemperature profile in the soil between z and the landsurface, and t is time. The soil heat storage can thenbe calculated as

S = Cv

∫ 0

z

∂T

∂tdz, (19)

where∂T

∂tcan be approximated using a finite differ-

ence scheme. The soil heat storage can be written as

S =Cv

2Δt

0∑

z

[T (zi, t + Δt) − T (zi, t − Δt)]Δz. (20)

3.2 Expression of energy balance closure

We assume that the land surface is horizontal anduniform and the atmosphere is in a steady state. Us-ing the energy conservation and conversion laws andconsidering soil heat storage, the land surface energybalance equation can be expressed as

Rn = H + LE + G + S, (21)

where Rn is the net land surface radiation flux, H isthe land surface sensible heat flux, LE is the land sur-face latent heat flux, G is the ground heat flux, andS is the soil heat storage between the heat flux plateand the surface. H and LE can be approximated usingthe sensible and latent heat fluxes observed near thesurface. Sensible and latent heat fluxes were observedduring the campaign in the Badan Jaran desert hinter-land and edge, but were not directly observed during


the campaign at the Jinta Oasis. We use the aerody-namic method (Liu et al., 2009; Liu et al., 2010) tocalculate the sensible heat flux and latent heat fluxfrom the gradients of observations that were taken atthe Jinta Oasis. Rn is calculated from the radiationbalance equation as

Rn = (Rsd − Rsu) + (Rld − Rlu), (22)

where Rsd is the total downward short-wave radia-tion reaching the surface, Rsu is the amount of short-wave radiation reflected by the land surface, Rld is thedownward flux of long-wave radiation from the atmo-sphere, and Rlu is the upward flux of long-wave radi-ation from the land surface. Rsd, Rsu, Rld, and Rlu

have been directly observed.Several approaches may be used to assess the im-

balance in the land surface energy budget. These dif-ferent approaches may yield different results even forthe same dataset. We use an ordinary linear regres-sion (OLR) method to analyze the imbalances in theenergy budgets calculated for the Badan Jaran deserthinterland, Badan Jaran desert edge, and Jinta Oa-sis. In this method, the slope of the linear regressionbetween the quantities (H+LE) and (Rn − G − S) isused to represent the energy balance closure rate. En-ergy closure is accomplished if the slope of the linearregression is 1 and the intercept is 0.

4. Results and discussion

The three methods introduced above (HM, CM,and TIM) were tested and the results were validatedagainst observational data. The shallow soil heat fluxwas adjusted to the land surface energy balance andthe energy balance closure rate was analyzed. Thedetailed results of this analysis are presented and dis-cussed in the following section.

4.1 Soil heat storage

4.1.1 Analysis of observational dataThe soil water content at 5-cm depth was small-

est in the Badan Jaran desert hinterland and largest atthe Jinta Oasis, with the Badan Jaran desert edge inbetween (Fig. 1). The surface type in the Badan Jarandesert hinterland was sand, with a surface albedo of

approximately 0.33. The surface type at the BadanJaran desert edge was sparse desert reeds, with a veg-etation height of about 0.6 m and a surface albedo of0.23 (Ma et al., 2012). The most common soil typesat Jinta Oasis were irrigated silty soil, moist meadowsoil, and aeolian sandy soil. The surface was engagedas farmland. The typical crop in this area is wheat,but in mid June 2005, the primary crop was an initialgrowth of cotton. The surface albedo at Jinta Oasiswas approximately 0.19 (Chen et al., 2006; Ao et al.,2008). Soil moisture is an important factor in deter-mining surface albedo, as the presence of water arounda soil particle increases the absorption path of solar ra-diation. Greater soil moisture typically correlates withsmaller albedo. The surface albedo at Jinta Oasis waslower than the albedo at the other two locations be-cause the soil water content was highest. The differ-ences in albedo indicate that the underlying surfacesat the three sites have different physical characteris-tics.

Figure 2 shows the soil heat fluxes observed at5- and 20-cm depths at the three sites. The observedfluxes had obvious diurnal variations. The soil heatfluxes at 5-cm depth varied substantially from day today, while the soil heat fluxes at 20-cm depth variedmuch more smoothly. The diurnal variations of soilheat fluxes at 20-cm depth had smaller peaks, smalleramplitudes, and larger phase lags than the diurnalvariations at 5-cm depth. These differences illustratethat the amplitude of soil heat flux decays with depth,while the phase becomes delayed. The soil heat flux

Fig. 1. Time series of soil water content at 5-cm depth.


Fig. 2. Time series of soil heat fluxes measured at 5-

cm depth (G5) and 20-cm depth (G20) at the (a) Badan

Jaran desert hinterland, (b) Badan Jaran desert edge, and

(c) Jinta Oasis.

at 5 cm cannot be used as a direct measure of theground heat flux, but must be supplemented by thesoil heat storage terms in the land surface energy bal-

ance equation.4.1.2 Analysis of model results

Soil thermal capacity is needed to calculate soilthermal storage regardless of the numerical methodused. The soil thermal capacity at each measure-ment site can be obtained using the heat conduc-tion equation. These heat capacities are 0.69×10−6

J (m3 K)−1 for the Badan Jaran desert hinterland,0.80×10−6 J (m3 K)−1 for the Badan Jaran desertedge, and 0.92×10−6 J (m3 K)−1 for the Jinta Oa-sis. The HM and CM also require the soil thermaldiffusion, which is calculated according to the detailedmethod provided by Miao et al. (2012). The mainparameters for each method are listed in Tables 1–6.

Figure 3 shows soil heat storage at each site calcu-lated using the HM, CM, and TIM methods, as well asthe measured value of soil heat storage between 5- and20-cm depths. The soil heat storage varies substan-tially on the diurnal timescale. The daily variationsof soil heat storage calculated using the three meth-ods show good agreement with the variations in themeasured values. The HM method provides the clos-est fit because the iterative method of calculating soilthermal diffusivity makes the fullest use of the mea-surements. The CM method underestimates the soilheat storage at night. This underestimate may be at-tributable to the error inherent in assuming the soilmoisture flux density W to be constant throughoutthe day (Dai et al., 2009). During daytime, especiallyunder fine weather conditions, soil moisture moves up-ward because of surface evaporation (i.e., W > 0).Surface evaporation abates with the decrease of sur-face temperature at night, so soil moisture moves fromshallow layers to deeper layers (i.e., W < 0). Ignoringthese variations in soil water flux density induceserrors in the calculated soil heat storage. The valuesof soil heat storage calculated by using the HM and

Table 1. Amplitudes for the harmonic method

DayBadan Jaran desert hinterland Badan Jaran desert edge Jinta Oasis

A1 A2 A1 A2 A1 A2

1 –11.40 1.82 –9.82 1.36 –6.51 0.95

2 –10.36 1.82 –6.33 1.09 –6.97 1.61

3 –8.83 1.63 –7.03 1.01 –6.89 1.19

4 –8.82 2.07 –8.17 1.39 –6.40 1.27

5 –8.05 1.92 –6.38 1.36 –6.73 1.59


Table 2. Phases for the harmonic method


ϕ1 ϕ2 ϕ1 ϕ2 ϕ1 ϕ2

1 0.10π –0.08π 1.90π –0.30π 0.09π –0.10π

2 0.09π –0.02π 1.92π –0.17π 0.08π –0.07π

3 0.08π –0.10π 1.92π –0.23π 0.08π –0.05π

4 0.08π –0.10π 1.93π –0.18π 0.07π –0.02π

5 0.09π –0.05π 1.91π –0.15π 0.10π 0.05π

Table 3. Thermal diffusivity for the harmonic method


k (10−7 m2 s−1) k (10−7 m2 s−1) k (10−7 m2 s−1)

1 7.88 4.70 5.91

2 7.67 4.47 6.02

3 7.64 4.96 5.92

4 7.32 3.88 5.58

5 7.11 4.01 5.76

Table 4. Amplitudes for the conduction-convection method


A1 A2 A1 A2 A1 A2

1 11.44 4.02 9.56 2.12 6.67 2.04

2 10.66 3.60 6.65 1.45 7.77 2.17

3 9.15 3.10 7.40 1.32 7.61 2.11

4 9.27 3.02 8.44 1.74 7.15 2.00

5 8.99 2.51 6.87 1.46 7.52 2.19

Table 5. Phases for the conduction-convection method


ϕ1 ϕ2 ϕ1 ϕ2 ϕ1 ϕ2

1 1.10π 0.82π 1.00π 0.71π 1.09π 0.76π

2 1.00π 0.77π 0.92π 0.48π 1.08π 0.73π

3 1.08π 0.77π 0.92π 0.51π 1.08π 0.74π

4 1.08π 0.77π 0.93π 0.59π 1.07π 0.75π

5 1.07π 0.73π 0.91π 0.54π 1.10π 0.79π

Table 6. Thermal diffusivity and liquid water flux density for the conduction-convection method


ka (10−7 m2 s−1) W (10−6 m s−1) ka (10−7 m2 s−1) W (10−6 m s−1) ka (10−7 m2 s−1) W (10−6 m s−1)

1 10.20 1.95 7.07 2.95 7.24 1.43

2 9.29 1.54 5.02 0.90 6.61 1.40

3 9.54 1.75 5.62 1.37 7.11 1.90

4 10.35 2.37 5.80 3.04 7.69 2.34

5 10.44 1.43 5.56 2.41 8.04 2.30

CM are closer to the observed values; however, neitherof these methods is able to capture fluctuations in soilheat storage on smaller timescales. The diurnal ampli-tude of the soil heat storage calculated using the TIMmethod is larger, and the maxima are significantly

higher, but the time series captures these small-scalefluctuations. A comprehensive and objective compar-ison of the performance of these three methods in thiscase requires a more quantitative analysis.

Figure 4 shows scatter plots of calculated and


Fig. 3. The measured and calculated values of soil heat

storage between 5- and 20-cm depths at the (a) Badan

Jaran desert hinterland, (b) Badan Jaran desert edge,

and (c) Jinta Oasis measurement sites. Calculations have

been performed using the harmonic (HM), conduction-

convection (CM), and temperature integral (TIM) meth-

ods.

measured values of soil heat storage for each of thethree measurement sites. At the Badan Jaran deserthinterland site, the fitting coefficient is closest to 1 forthe HM results (0.9409), while the correlation coeffi-cient is highest for the TIM results (0.9297). At theBadan Jaran desert edge site, the fitting coefficientsare closer to 1 for HM (0.9004) and CM (1.0354), while

the correlation coefficients are highest for HM (0.9080)and TIM (0.8934). At the Jinta Oasis site, the fittingcoefficient is closest to 1 for CM (0.9072), while thecorrelation coefficient is highest for TIM (0.9141). Thefitting coefficient of the HM results gradually departsfrom 1 with increasing soil water content, while the fit-ting coefficient of the CM result gradually approaches1 under the same conditions. This difference arises be-cause CM considers the heat convection effects of soilwater, while the HM does not. TIM, which is based onenergy conservation, correlates well with the measuredvalues at all three sites; however, the fitting coefficientfor TIM is greater than 1 and the magnitude of theerror is larger. Whereas HM uses a second-order har-monic model, CM uses a true sine function to estimatesoil heat storage. This limits the accuracy of the CMcalculation, so the correlation coefficients for CM areonly 0.48–0.70.

Three statistics are used to further illustrate thedifferences between observations and calculations ofsoil heat storage using the three methods: the averagedeviation (Biss), the standard deviation (SEE), andthe relative standard deviation (NSEE). These statis-tics are defined as:

Biss =

n∑i=1

|S − S0|n

, (23)

SEE =

√√√√√n∑

i=1

(S − S0)2

n − 2, (24)

NSEE =

√√√√√√√

n∑i=1

(S − S0)2

n∑i=1

(S0)2, (25)

where n is the total number of samples and S and S0

are the calculated and measured values of soil heatstorage, respectively. The results are listed in Table 7.

The errors are smallest for calculations using HMfor the Badan Jaran desert hinterland and edge sites,while TIM performs better than CM. The HM calcula-tion is also most accurate for the Jinta Oasis site, butCM performs better than TIM in this case. Based onthese statistics, the calculation using HM is superior


Fig. 4. Scatter plots of calculated and measured values of soil heat storage at the (a1–a3) Badan Jaran desert hinterland,

(b1–b3) Badan Jaran desert edge, and (c1–c3) Jinta Oasis measurement sites. (a1–c1) HM, (a2–c2) CM, and (a3–c3)

TIM.

Table 7. Average deviation, standard deviation, and relative standard deviation for values of soil heat storage

calculated at each site

StatisticsBadan Jaran desert hinterland Badan Jaran desert edge Jinta Oasis

HM CM TIM HM CM TIM HM CM TIM

Biss 9.60 16.54 13.63 10.90 22.07 19.24 14.20 19.89 20.53

SEE 12.34 19.83 18.89 13.07 29.70 25.57 17.02 23.46 28.60

NSEE (%) 52.09 83.72 79.73 30.34 68.91 59.33 61.76 85.13 96.06

regardless of the underlying surface.

4.2 Energy balance

4.2.1 Analysis of observational dataThe definitions for Rn, H, LE, and G as terms in

the land surface energy balance equation (Eq. (21))have been introduced in Section 3.2. The followingsection presents an analysis of these four fluxes as ob-served at the measurement sites.

The accuracy of radiation observations has grea-


tly improved over the past 10–15 years with the adventof the global surface radiation reference station net-work (BSRN). The four-component radiometer usedto measure the net surface radiation Rn at these ex-periment sites is accurate to within ±5%. The errorin the observations is random. Under normal main-tenance conditions, the net radiation observations arethe most accurate among the four energy budget com-ponents.

Fig. 5. Time series of the net radiation flux at the sur-

face (Rn), the latent (LE) and sensible (H) heat fluxes, and

the soil heat flux (G5) measured at the (a) Badan Jaran

desert hinterland, (b) Badan Jaran desert edge, and (c)

Jinta Oasis sites.

The turbulent fluxes H and LE were collected us-ing eddy correlation methods. The sampling frequencywas 10–20 Hz, and the observations were then aver-aged into 30-min means. The instrument performedconsistently well during these experiments.

The soil heat flux G was observed by using a soilheat flux plate at a depth of 5 cm (denoted by G5).Heat flux measurements using this instrument are gen-

Fig. 6. Unadjusted and adjusted soil heat fluxes at the

(a) Badan Jaran desert hinterland, (b) Badan Jaran desert

edge, and (c) Jinta Oasis sites. The adjusted values (G0)

rely on calculations using the harmonic (HM), conduction-

convection (CM), and temperature integral (TIM) meth-

ods. The unadjusted values (G5) are raw observations

taken at 5-cm depth.


erally accurate to within ±3%. Errors in these obser-vations are not a primary reason for imbalance in theobserved energy budget.

Figure 5 shows measurements of Rn, H, LE, andG5 over the 5-day measurement periods at the BadanJaran desert hinterland and desert edge (19–23 Au-gust 2009) and at Jinta Oasis (12–16 June 2005). Allfour terms have obvious diurnal variations, with posi-tive anomalies during daytime and negative anomaliesat night. The phase of diurnal variations in G5 lagsslightly behind the phase of diurnal variations in Rn,H, and LE; in other words, the energy fluxes are notsynchronous. We therefore need to consider the role ofsoil heat storage between the soil heat flux plate andthe land surface to close the surface energy budget.4.2.2 Analysis of model results

Figure 6 shows the observed soil heat fluxes at 5-cm depth (G5) and ground heat fluxes (G0) adjustedbased on soil heat storage calculated by using the threemethods described in Section 3.1. G0 (HM) is calcu-lated by using Eq. (5) and G0 (CM) is calculated bydirectly using Eqs. (11)–(15) with the parameter val-ues listed in Tables 3–8. For G0 (TIM), the soil heatstorage from 5-cm depth to the land surface is calcu-lated by using Eq. (20) and the ground heat flux iscalculated by Eq. (18). The amplitudes of the calcu-lated ground heat fluxes are larger than the measuredsoil heat fluxes at 5-cm depth with a slight phase lead.These results are in line with the universal law men-tioned in Section 4.1. The diurnal variations of thecalculated values match well with the observed heatfluxes. The results indicate that these three methodsprovide a reliable means of calculating soil heat storageand adjusting soil heat flux for energy balance closure.

Figure 7 shows linear regression fits of (H+LE)and (Rn − G) using the unadjusted soil heat fluxes at5-cm depth and the ground heat fluxes adjusted usingcalculations of soil heat storage. This plot illustratesthe discrepencies from energy balance closure over thedifferent underlying surfaces. The regression parame-ters and correlation coefficients are listed in Table 8.

The slope of the linear regression between(H+LE) and (Rn − G) is closer to unity after adjust-ing the soil heat flux regardless of location. The en-

ergy closure rate for the Badan Jaran desert hinterlandsite increases by 3.83% when HM is used to adjustG, by 2.94% when CM is used, and by 3.83% whenTIM is used. The correlation coefficient also increasesby 0.0373, 0.0084, and 0.0337, respectively. HM andTIM provide a greater improvement than CM. The en-ergy closure rate for the Badan Jaran desert edge siteincreases by 3.19%, 3.11%, and 5.97%, respectively,while the correlation coefficient changes by 0.0042,–0.0145, and 0.0335, respectively. For this site, TIMprovides the largest improvement. The energy clo-sure rate for the Jinta Oasis site increases by 1.39%,1.49%, and 1.74%, respectively, while the correlationcoefficient changes by 0.007, 0.006, and –0.011, respec-tively. The differences in the results among the threemethods for this site are not significant.

Overall, the energy closure rate after adjustmentis better than before adjustment. The soil heat fluxesadjusted using HM and TIM provide better closurerates than those adjusted using CM. The diurnal vari-ations of soil heat flux do not follow a pure sine curve.HM uses a second-order harmonic model to simulatediurnal variations in heat flux, and is therefore moreaccurate than CM (which uses first-order harmonicmodel).

The results summarized in Fig. 7 and Table 8illustrate that errors in the energy balance have dif-ferent magnitudes over different underlying surfaces.The energy balance closure rate at the Badan Jarandesert hinterland site is approximately 80%, while theclosure rates at the Badan Jaran desert edge and JintaOasis sites are only about 55%. Even after accountingfor soil heat storage, the energy budgets at the lattertwo sites are still far from being closed. This suggeststhat the reasons for energy imbalance may vary sub-stantially for different underlying surfaces. In furtherresearch on energy balance closure, we must considerother factors in addition to soil heat storage. Eachunderlying surface type may have unique features thatcontribute to energy imbalance.

5. Conclusions

Three methods for estimating soil heat storage


Fig. 7. Linear regressions between the (H+LE) and (Rn − G) terms of the energy balance equation over the (a1–a4)

Badan Jaran desert hinterland, (b1–b4) Badan Jaran desert edge, and (c1–c4) Jinta Oasis measurement sites. (a1–c1)

The unadjusted observations, (a2–c2) HM, (a3–c3) CM, and (a4–c4) TIM.

Table 8. Energy balance closure metrics after adjusting for soil heat storage

Badan Jaran desert hinterland Badan Jaran desert edge Jinta Oasis

Closure rate R2 Closure rate R2 Closure rate R2

Unadjusted 0.7927 0.8744 0.5067 0.8434 0.5346 0.8272

HM 0.8310 0.9117 0.5386 0.8476 0.5485 0.8342

CM 0.8221 0.8828 0.5378 0.8289 0.5495 0.8332

TIM 0.8331 0.9081 0.5664 0.8769 0.5520 0.8162

have been evaluated for revealing their potential toimprove energy balance closure over arid and semi-arid surface types. The analysis is based on obser-vations collected at measurement sites in the BadanJaran desert hinterland, the Badan Jaran desert edge,and Jinta Oasis. Soil heat storage between 5- and 20-cm depths has been calculated using the HM, CM, andTIM methods. The soil heat flux at 5-cm depth hasbeen adjusted based on the results of these calcula-tions, and the resulting land surface energy balancehas been analyzed. The conclusions are as follows.

(1) The soil heat storage calculated using HM isclosest to the measured value because this methodmakes the fullest use of measurements in estimatingsoil thermal diffusivity. HM also benefits from the useof a second-order harmonic model of daily variationsin soil heat flux. CM couples heat conduction andconvection; theoretically, this method should providea more accurate representation of heat transfer pro-cesses in the soil than the other methods. CM appliedin this work neglects diurnal variations in soil waterflux density, and therefore underestimates nighttime


soil heat storage. CM also simulates the diurnal cy-cle of soil heat storage using a sine function, whichlimits the accuracy of the calculation. TIM calculatessoil heat storage based entirely on energy conserva-tion laws, so the calculation is consistent with observedfluctuations; however, the diurnal amplitude is largerthan observed and the peak values are substantiallyhigher.

(2) The relative performance of each methodvaries according to the soil water content of the under-lying surface. The minimum errors are achieved usingHM. Moreover, the HM calculations are the most ac-curate for the three surface types considered here. Theresults using TIM are better than those using CM forthe Badan Jaran desert sites, but the results usingCM are superior to those using TIM for the Jinta Oa-sis site.

(3) Accounting for soil heat storage in estimates ofground heat flux improves energy balance closure ratesat all three surface sites relative to unadjusted mea-surements collected at 5-cm depth. However, these im-provements are an order of magnitude smaller than thetotal land surface energy imbalance (0.01–0.05 com-pared to 0.2–0.5). The energy closure rate is especiallylow for the Badan Jaran desert edge and Jinta Oasismeasurement sites. This result indicates that soil heatstorage is just one among many factors that cause im-balance in observations of the energy budgets of aridand semi-arid surfaces. Future work on this topic mustconsider other factors in addition to soil heat storage,such as vertical thermal advection within the surfacelayer. Due to data limitations, we have not discussedthese aspects in this study. We will treat them indepth as observational data becomes more abundantin the future.

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