Urban Climate 3 (2013) 56–66

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Urban Climate

journal homepage: www.elsevier .com/locate/ucl im

Validation of two soil heat flux estimationtechniques against observations made in anengineered urban green space

2212-0955/$ - see front matter � 2012 Published by Elsevier B.V.http://dx.doi.org/10.1016/j.uclim.2012.11.001

⇑ Corresponding author. Tel.: +1 215 895 1385.E-mail address: fmontalto@coe.drexel.edu (F.A. Montalto).

L. Smalls-Mantey, K. DiGiovanni, M. Olson, F.A. Montalto ⇑Department of Civil, Architectural and Environmental Engineering, Drexel University, 3141 Chestnut Street,Philadelphia, PA 19104, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 August 2012Received in revised form 23 October 2012Accepted 5 November 2012

Keywords:Soil heat fluxGreen infrastructureHalf-order time derivativeGreen’s function

In highly developed landscapes, the soil heat flux that results fromchanges in land cover determines, in part, the extent to whichurban climatic conditions can be modulated (e.g. retrofitting citieswith new, engineered green spaces to reduce urban heat islandeffect). In this paper, temperature-dependent (TD) and tempera-ture-independent (TI) soil heat flux models are validated against6 days of measurements made in an engineered urban green space.The results suggest that the TI model (heat flux plate method) rep-resents the observations better than the temperature-dependentone, which consistently overestimates soil heat flux at night andduring the dusk-dawn period. Moreover, a sensitivity analysisreveals that the TD model is more sensitive to the selection of ther-mal conductivity and heat capacity values than the TI model. It canbe concluded that the TI model is a more robust predictive tool,and especially in urban applications where soil properties maybe highly uncertain.

� 2012 Published by Elsevier B.V.

Introduction

Soil (ground) heat flux (G) is the amount of thermal energy that moves through an area of soil overa unit of time, and is a key determinant of the temperature of the ground surface. Soil heat flux (SHF) is

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important in micrometeorology because it couples the energy transfer process at the surface (surfaceenergy balance (SEB)) with the energy transfer processes at the soil (soil thermal regime) (Sauer andHorton, 2005). By helping to regulate soil temperature, SHF is also related to a range of important,temperature-dependent ecohydrologic processes, such as plant growth, biomass production, wateruptake, evapotranspiration, and carbon assimilation.

The energy balance at the ground surface can be described through the following equation:

Rn � G0 ¼ LEþ H ð1Þ

where Rn is the net radiation, G0 is the ground (soil) heat flux, LE is the latent heat flux and H is thesensible heat flux. The left and right sides of the equation are often referred to as the available energyand turbulent fluxes, respectively.

Though it has been at times wholly omitted from some representations of the SEB, coarsely param-eterized from basic meteorological parameters (e.g. as a fixed percentage of the net radiation), andmeasured with rather simple techniques (e.g. by using uncorrected heat flux plate measurements)(Liebethal et al., 2005), SHF can be the most important term in the energy balance equation (e.g. overdry bare soil). Ogee et al. (2001) discovered that within a forest, SHF could represent 30–50% of the netradiation occurring over the understory. Similar orders of magnitude have been computed for othertypes of vegetated land cover (Idso et al., 1975; Santanello, 2003). To our knowledge, no efforts toquantify SHF in urban green spaces have been performed, nor have any previously developed predic-tive models been validated in this context.

The goal of this study was to compare the ability of two previously-developed models of SHF- onetemperature-dependent (Wang and Bras, 1999), and one temperature-independent (Wang andBou-Zeid, 2012) to replicate SHF measurements made in a newly, engineered, urban green space.The study area was located in Queens, NY, USA and equipped with a SHF plate and soil sensors thatmeasure both volumetric moisture content and soil temperature. Though field measurements at thissite have been ongoing for more than a year, this study analyzed data for a six-day period during thesummer of 2011. Two models and the experimental field site are described, followed by a statisticalanalysis of the goodness of fit and a sensitivity analysis of the two models.

Methodology

Two models were used to simulate data from the experimental field site. For each model goodnessof fit was assessed from a series of statistical tests.

Half-order time derivative method (temperature dependent (TD) method)

Wang and Bras (1999) developed a half-order time derivative method to compute SHF from the 1Dheat diffusion equation. Compared to previously proposed methods which require at least two layersof temperature data (Kimball and Jackson, 1975; Stull, 1988; Malek, 1993) only one layer of a soil tem-perature time series is necessary with the half-order time derivative method.

As stated above the half-order time derivative equation can be derived by combining the 1D heatdiffusion equation for heat transport and Fourier’s law in 1D, which states that for a homogenous en-tity, heat flux is proportional to the temperature gradient. With depth z increasing downward into thesoil, heat transfer in a vertically homogeneous soil can be represented as by the following equation(Sellers, 1965):

@T@t¼ j

@2T@z2 ð2Þ

where T is the soil temperature (K), z is a particular depth (m), t is time (s) and j is the thermaldiffusivity (m2 s�1).

By assuming a uniform initial soil temperature profile and a constant fixed temperature at infinitedepth, Eq. (2) can be solved using the following initial and boundary conditions:

T ¼ T0 for t ¼ 0; z > 0; ð3Þ

58 L. Smalls-Mantey et al. / Urban Climate 3 (2013) 56–66

T ¼ T0 for t > 0; z!1 ð4Þ

To meet these boundary conditions, the simulation must begin at a time when the SHF is at or nearzero throughout the entire soil profile. The solution of Eq. (2) subject to Eqs. (3) and (4) yields the SHFat any depth, z, and time, t. This is represented in the following half-order time derivative equation:

Gðz; tÞ ¼ffiffiffiffiffiffiffiffiffiffiffikqscs

p

r Z t

0

dTðz; sÞffiffiffiffiffiffiffiffiffiffit � sp ð5Þ

where s is a dummy integration variable, and k (W m�1 K�1) is the thermal conductivity, qscs is the soilvolumetric heat capacity. Following Wang and Bras (1999), we note that j = k/qscs, and the soil volu-metric heat capacity can be computed analytically as the sum of heat capacities of the dry soil and soilwater (Hsieh et al., 2009),

qscs ¼ ð1�uÞqmcm þ hqwcw ð6Þ

where / is the fractional soil porosity, h (m3 m�3) is the volumetric water content, qm is the densityof the dry mineral soil (kg m�3), qw (kg m�3) is the density of water and cm (J kg�1 K�1) and cw

(J kg�1 K�1) are the specific heats of dry and wet mineral soil, respectively.As previously stated, advantages of this method include the necessity of only one temperature time

series. However, the equation does require that the soil moisture content be known, and will, ofcourse, yield faulty SHF predictions if the soil temperature measurements are inaccurate, or if thethermisters are improperly placed in the soil.

Temperature independent numerical method (TI method)

Because accurate SHF predictions using a temperature-dependent (TD) model is highly dependentupon reliable measurements of the soil moisture content and temperature (Liebethal et al., 2005),Wang and Bou-Zeid (2012) recently developed a new method also based on the 1D heat diffusionequation that does not rely on measured soil temperature. This method utilizes the Green’s functionsolution (Cole et al., 2011) and more realistically represents boundary conditions at the soil surfaceusing Duhamel’s principle. Accordingly, the method presented by Wang and Bou-Zeid (2012) solvesEq. (2) subject to the following:

�k@TzðtÞ@z

����z¼0¼ f ðtÞ ð7Þ

�k@TzðtÞ@z

����z!1¼ 0 ð8Þ

where Tzðt ¼ 0Þ ¼ Ti

The implicit form of the authors’ general solution for SHF at any depth is:

GzðtÞ ¼ G0ðt � sÞderfcðz=ð2ffiffiffiffiffiffijsp

ÞÞ ð9Þ

where derfc(.) is the double precision complimentary error function.Solving for the SHF at the surface (z = 0) for a time continuum starting at t = 0 yields

G0ðnÞ ¼2GzðnÞ � Jn�1ðG0;DFzÞ

DFzð1Þð10Þ

where FzðjÞ ¼ erfcðz=ð2ffiffiffiffiffiffiffijtj

pÞÞ for discrete time series t ¼ ftj j j ¼ 0;1;2; . . . ;ng; ð11Þ

DFzðjÞ ¼ FzðjÞ � Fzðj� 1Þ and ð12Þ

Jn�1ðG0;DFzÞ ¼ G0ðn� 1ÞDFzð1Þ þXn�1

j¼2

½G0ðn� jþ 1Þ þ G0ðn� jÞ�DFzðjÞ ð13Þ

L. Smalls-Mantey et al. / Urban Climate 3 (2013) 56–66 59

Erfc(.) is the complimentary error function. To be numerically evaluated, Eq. (10) requires only thatthe soil thermal diffusivity be known and that a time series of G-values, obtainable using a heat fluxplate. Note, too, that by replacing the ‘‘0’’ subscript with other values, Gz, Eqs. (10)–(13) can also beused to calculate SHF at any depth. A more detailed version of the derivation is available by Wangand Bou-Zeid (2012).

Experimental field site

Field measurements were made at a curbside bioretention ‘‘Greenstreet’’ in Queens, NY, USA,owned by the New York City Department of Parks and Recreation (NYCDPR), and monitored by a teamled by Drexel University. The site is known as ‘‘Colfax’’ to the monitoring team. The 100 m2 bioreten-tion area receives direct precipitation only (unlike other such facilities which, as integral componentsof New York City’s green infrastructure program, are increasingly designed to also receive street run-off). Although the bioretention area is designed as a depression, its vertical soil profile is everywhereuniform and consists of: <10 cm mulch, placed on top of 0.6 m of imported ‘‘topsoil,’’ installed accord-ing to NYCDPR’s very strict specifications (Table 1). At this particular site, the underlying native soil iswell-drained and sandy. The site was planted with a variety of native trees, shrubs and grasses (Fig. 1).

Although a range of sensors and monitoring apparatus are installed at this site (Fig. 1), listed hereare only those instruments used in the SHF experiment presented. A self-calibrating SHF plate (Hukes-flux HFP01SC-50) was installed at a depth of 8 cm. A probe that simultaneously measures soil mois-ture, electrical conductivity and soil temperature (Decagon 5TE) was installed at a depth of 5 cm. Netradiation was measured with a net radiometer (Hukesflux NR01) installed on a pole at of height 4 mover the soil surface.

Data collection

Measurements of SHF, soil temperature, and net radiation over the Greenstreet were recorded atfive-minute intervals for a period of 6 days beginning August 31, 2011 and ending on September 5,2011. Note this collection period began 3 days after the end of Hurricane Irene that deposited a record17 cm of rain over New York City.

The heat flux at the soil surface was computed from the sensor measurements using the followingequation:

Table 1Topsoil

Gene

Cond

Orga

USD

pH

Solu

Gðz ¼ 0; tÞ ¼ Gðz; tÞ þ qscs@T@t

Dz; ð14Þ

where on the right side of the equation, the first term, G (z,t), represents the transient heat flux as mea-sured by the SHF plate and the second term represents the heat transferred through the overlying soillayer (of thickness, Dz and with soil temperature, T). Other soil properties required in the computation

specifications for Colfax Greenstreet.

ral classification USDA natural loam (40–65% sand)

ition Uniform quality, free from hard clods, stiff clay, hardpan, sods, partially disintegratedstone, coal lime, cement ashes, slag, concrete, tar residues, tarred paper, boards, chips,sticks or any other undesirable material

nic content 9–12% determined by the dry combustion method for total carbon and organic carbon

A soil gradation Rocks & stones 0%Gravel <5%Sand 40–65%Silt 10–35%Clay 15% maximum

6.0–7.0

ble salt value 0.45 mmhos cm�1

Fig. 1. Colfax Bioretention Greenstreet (Queens, NY). (A) Street view of data logger (mounted on pole) and climate station(including net radiometer). (B) Photo depicting mix of native vegetation across the site.

60 L. Smalls-Mantey et al. / Urban Climate 3 (2013) 56–66

were estimated based on soil testing performed previously on other NYCDPR’s Greenstreet topsoilmixes, and by consulting relevant literature. The soil’s thermal conductivity was estimated basedon research by van Duin (1963). The following are the dry soil properties used in the model simula-tion: thermal conductivity = 1.2 W m�1 K�1, soil porosity = 0.6, density of dry soil = 900 kg m�3, heatcapacity of dry soil = 600 J kg�1 K�1, thermal diffusivity = 1.02 � 106 m2 s�1. Note that although itwas not measured directly, the surface (z = 0) SHF was computed using Eq. (14) from the sensor read-ings and is hereafter referred to as the ‘‘observed’’ value.

Statistical measures of model goodness of fit

The heat flux predictions made using the TD and TI models were compared to its correspondingobserved value. Model goodness of fit was assessed using a variety of statistical parameters.

The predicted and observed time series were compared by plotting the residuals (e.g. observed–predicted values) at each time step and by computing the standard error of the estimate (SEE), andthe mean absolute error (MAE). Measures of overall model goodness of fit were calculated usingthe following equations:

SEE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN

i¼1

ðGðz ¼ 0; tiÞ � Gpðz ¼ 0; tiÞÞ2

ðN � 2Þ

vuuuut ð15Þ

MAE ¼PN

i¼1jGpðz ¼ 0; tiÞ � Gðz ¼ 0; tiÞjN

where i ¼ 1;2;3; . . . ;N ð16Þ

where N is the number of observations for which a comparison between the predictions and observa-tions could be made (e.g. 5 min sampling interval over 6 days), G(z = 0,t) is the observed surface soilheat flux (SSHF) per Eq. (14) and Gp (z = 0,t) is the predicted SSHF for the TD and TI models. Missingvalues were excluded from the statistical analysis and were therefore not included in the time inter-val. Values were also excluded from analysis due to the stipulations stated in the initial and boundaryconditions (Eqs. (3) and (4)); the simulation must begin at a time where there is a uniform soil tem-perature profile (SHF is at or near zero). In this study the models simulations started at 07:45 h, local

L. Smalls-Mantey et al. / Urban Climate 3 (2013) 56–66 61

time. Note that the units on SEE and MAE are identical to G (W m�2) and that smaller SEE and MAEvalues indicate better model predictions.

Scatter plots depicting the predicted and observed SSHF values were also generated, and the coef-ficient of determination (R2), slope (a), and y-intercept (b) of the trend lines were computed. Also com-puted was the Nash–Sutcliffe model coefficient of efficiency (E), which ranges from �1 to 1, and isuseful in evaluating whether differences between the observed and predicted data are as large asthe variability in the predicted data (Yoder et al., 2005).

Fig. 2

E ¼ 1�PN

i¼1ðGðz ¼ 0; tiÞ � Gpreðz ¼ 0; tiÞÞ2PNi¼1ðGðz ¼ 0; tiÞ � �Gðz ¼ 0ÞÞ2

; where i ¼ 1;2;3; . . . ;N ð17Þ

Values of E close to 1 indicate a good agreement between the observed and predicted data. A valueof E < 0 indicates that the observed mean is a better predictor than the model. The coefficient of effi-ciency represents an improvement over r2 for model evaluation purposes in that it is sensitive to dif-ferences in the observed and predicted means and variances, and will therefore always have a lowervalue (Legates and McCabe, 1999). The coefficient of determination is more sensitive to outliers thanto observations near the mean (Legates and Davis, 1997).

Results and discussion

Comparison of surface soil heat flux models to the observed values

Fig. 2 depicts the three SHF time series data (e.g. the observed values and the TD and TI modelpredictions) for the 6 day study period. In general, the TI model appears to predict the observedfield data better than the TD model. A plot of the residuals (observed value–predicted value)further validates this finding (Fig. 3). Positive errors indicate a model under-prediction whilenegative errors a model over-prediction. The TI model generally has smaller residuals than theTD model. In data analyzed before the presented study period (not shown), similar patternsregarding the residuals during various parts of the day held true, thus the pattern is not specificto the study period.

. Time series comparison of the observed SSHF and the TD and TI model predictions over the six-day study period.

12:00 24:00 12:00 24:00 12:00 24:00 12:00 24:00 12:00 24:00 12:00 24:00−20

−15

−10

−5

0

5

10

15

20

25

TimeDate (month/day/year)

Res

idua

ls fo

r the

sur

face

soi

l hea

t flu

x (W

m−2

)

TD ModelTI Model

8/31/11 9/2/11 9/6/119/1/11 9/1/11 9/2/11 9/3/11 9/4/11 9/5/119/4/11 9/5/119/3/11

Fig. 3. Time series of the residuals of the surface soil heat flux of the TD and TI model for the six-day study period.

−60 −40 −20 0 20 40 60 80 100 120 140−60

−40

−20

0

20

40

60

80

100

120

140

Observed Surface Soil Heat Flux (W m−2)

Pred

icte

d Su

rfac

e So

il H

eat F

lux

(W m

−2)

TI Model y=1.1x −0.24, R2=0.99TD Model y=0.89x + 6.9, R2=0.94

Fig. 4. Scatter plot depicting the predicted versus observed SSHF values over the six-day study period.

62 L. Smalls-Mantey et al. / Urban Climate 3 (2013) 56–66

Table 2Correlation factors between the predicted and observed surface soil heat flux values over a 6 day study period.

Model N MAE (W m�2) a b (W m�2) R2 E SEE (W m�2)

Observations 1545TD 1545 11 0.89 6.9 0.94 0.92 12TI 1545 2.9 1.1 �0.24 0.99 0.99 3.6

N: number of observations; MAE: mean absolute error; a: slope; b: intercept; R2: coefficient of determination; E: coefficient ofefficiency; SEE: standard error of the estimate.

L. Smalls-Mantey et al. / Urban Climate 3 (2013) 56–66 63

The regression statistics represented graphically in Fig. 4 and numerically in Table 2 also indicatethat the TI model outperforms the TD model. The coefficient of determination of the TI model isslightly higher than that of the TD model. The E values for the TD and TI models are 0.92 and 0.99,respectively, indicating that while both models fit the data fairly well, the TI model outperformsthe TD model. The SEE values, 12 W m�2 for the TD model and 3.6 W m�2 for the TI model, also con-firm this finding. The MAE (11 W m�2) of the TD model is almost 4 times larger than that of the TImodel (2.9 W m�2).

The relationship between surface soil heat flux and net radiation

The relationship between net radiation and SSHF is represented in Fig. 5. The linear regression be-tween the two reveals a regression line with the slope of 0.15 and 0.18 and R2 of 0.82 and 0.91 for theTD and TI models, respectively. As depicted in Fig. 5, on average 16% of the net radiation was

−100 0 100 200 300 400 500 600 700 800−60

−40

−20

0

20

40

60

80

100

120

140

Net Radiation (Rn) (W m−2)

Surf

ace

Soil

Hea

t Flu

x (G

) (W

m−2

)

TD Model (night)TI Model (night)TD Model (day)TI Model (day) TD Model Regression Line y=0.18x−23, R2=0.90 TI Model Regression Line y=0.15x−11, R2=0.82

Fig. 5. Relationship between the net radiation and SSHF for the TD and TI models during the 6-day study period. ( ) TD model(night), ( ) TD model (day), ( ) TI model (night), ( ) TI model (night).

64 L. Smalls-Mantey et al. / Urban Climate 3 (2013) 56–66

distributed as SHF. During the day SHF is derived from the net radiation, which accounts for the great-er magnitude of the SHF values compared to the night where SHF is derived from the ground. Day isconsidered as the time period between 06:00–18:00 h, local time, whereas night includes 18:00–06:00 h, local time. Respective slope values during the day and night for the TD model and TI modelare as follows, respectively: 0.15, 0.14, 0.16, and 0.16. Day and night spatial variability between thepercentages of SHF as a result of net radiation are small; however as previously discussed in the intro-duction section SHF does represent a significant portion of the energy budget equation represented inEq. (1). Neglecting to account for SHF could lead to an over estimation of the available energy (Rn�G),which is partitioned to latent and sensible heat flux.

Sensitivity analysis

A sensitivity analysis varying the thermal conductivity and volumetric water content (thus thethermal capacity through Eq. (6)) was conducted for both models to assess whether our finding thatthe TI model better represented the observations could be dependent upon parameter selection. Forthe sensitivity analysis, each of the models was run 62,500 times, varying the thermal conductivityfrom 0.01 W m�1 K�1 to 2.5 W m�1 K�1 at intervals of 0.01, and the volumetric moisture content from0.0020 (m3/m3) to 0.50 (m3/m3) at intervals of 0.0020. The SEE for each simulated run was computedand is graphically represented in contour plots (Figs. 6 and 7).

Larger errors were associated at the two extreme ends of the thermal conductivity spectrum,evaluated for the TD model (Fig. 6). Heat capacity had a greater influence on error at higher ther-mal conductivities than at lower values. High errors in the latter case were only seen at corre-lated high thermal conductivity and high thermal heat capacity. Extremely high errors wererecorded at the lower end of the conductivity spectrum with the TI model (Fig. 7-right). Realis-tically, values of thermal conductivity would not be this small. At values greater than 0.020(W m�1 K�1), at higher values of conductivity, the model is not as sensitive to errors with varyingthermal capacity.

Thermal Conductivity (W m−1 K−1)

Volu

met

ric T

herm

al C

apac

ity (W

m−3

K−1

)

2624

2220

18

16

14

14

1618

2022

2426

0.5 1 1.5 2 2.5

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

x 106

Fig. 6. Standard error of estimation (W m�2) of TD model SSHF based on varied values of volumetric thermal capacity andthermal conductivity.

Fig. 7. Standard error of estimation (W m�2) of TI model SSHF based on varied values of volumetric thermal capacity andthermal conductivity; left-higher values of thermal conductivity, right-lower values of thermal conductivity.

L. Smalls-Mantey et al. / Urban Climate 3 (2013) 56–66 65

Conclusion

In this study SSHF estimates from two numerical models were compared to observations made atan engineered urban green space. The performance of two models was examined along with their sen-sitivity to various parameters used in the calculation of the models. Examination of time series andregression statistics indicates that the TI model better predicts the field-estimated data for the Green-street soil type studied. The TD model consistently overestimates the ground heat flux during thenight (dusk-dawn). The difference between the air temperature and soil (max = 4.8 K) is not to the ex-treme where it would cause the temperature sensor to incur large errors. However based on the data itcan be inferred that the TD model consistently has large periods of overestimation from dusk tilldawn. The sensitivity analysis also revealed that the TD model is more sensitive to changes in thermalconductivity and heat capacity. Thus if the parameters are estimated or measured incorrectly thismodel becomes inaccurate. The TI model is not as sensitive to changes in these two parameters andmay be stated as a more robust model for estimating SHF. The TI model better characterized and accu-rately predicted heat flux values for the bioretention mix soil type.

Acknowledgements

The authors would like to acknowledge the New York City Department of Parks and Recreation fortheir assistance in accessing the field site, maintaining sensors, and supporting our research. Wewould also like to thank Jingfeng Wag and Zhihua Wang for their helpful comments and suggestions,Drexel University SWRE lab members, specifically Nathan Rostad and Stephen White, for their supportand assistance in the onsite sensor installation and skills in the fabrication of the monitoring systemcomponents and Dave Adams of Adams Environmental Systems Inc., for programming and systemsupport. This work was supported by an NSF RAPID Grant (CBET 1010131)

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