Directional radiometric temperature pro®les within a grass canopy

Russell J. Qualls a,*, David N. Yates b

a Biological and Agricultural Engineering, University of Idaho, P.O. Box 440904, Moscow, ID 83844-0904, USAb Civil, Environmental, and Architectural Engineering, University of Colorado at Boulder, Campus Box 428, Boulder, CO 80309-0428, USA

Received 4 January 1999; received in revised form 2 June 2000; accepted 18 June 2000

Abstract

The scalar roughness for sensible heat ¯ux, z0h, which appears in Monin and Obukhov similarity theory has been shown to

exhibit substantial variability. Brutsaert and Sugita (cf. W. Brutsaert and M. Sugita, J. Atmos. Sci. 53 (1996) 209±16) derived an

expression for z0h to account for variability in z0h as a result anisothermal vegetation skin temperature, which assumes an expo-

nential pro®le of canopy skin temperature. We examined this assumption through directional radiometric measurements of tem-

perature pro®les in a relatively uniform grass canopy. The temperature gradient steepened monotonely throughout the morning and

was modeled successfully with an exponential pro®le. The root-mean-square error (RMSE) between measured and modeled pro®les

was small and ranged between 0.038°C and 0.530°C. One of the most signi®cant ®ndings of this research is that the decay coe�cient,

b, of the exponential pro®le model was approximately constant with time for this canopy. This may provide a useful simpli®cation in

applying Brutsaert and Sugita's (loc. cit.) parameterization for z0h. Ó 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Radiometric surface temperature; Remote sensing; Scalar roughness; Vegetation; Sensible heat ¯ux

1. Introduction

The exchange of mass and energy between land andatmosphere has signi®cant e�ects on the hydrology andweather of a region. These ¯uxes can be modeled withvarying degrees of complexity with sets of energy bal-ance and continuity equations commonly referred to asland surface models (LSMs) or surface-vegetation±atmosphere transfer schemes (SVATs).

It is desirable to use directional radiometric surfacetemperatures, hs, in these models because of the spatialinformation that they contain. A directional radiometrictemperature (DRT), is a radiometric surface tempera-ture measurement made along a speci®c sensor zenithview angle (i.e., the angle measured with respect to avertically downward view). It is derived from a radiativeenergy balance of the surface using a radiancemeasurement from a single direction with a narrow ®eldof view, and provides the best approximation of thethermodynamic temperature based on a measurement ofradiance [14]. However, vegetated surfaces impose dif-®culties on the modeling process. In general, hs values

for a vegetated surface are not unique due to the com-plicated structure and the vertical and horizontal dis-tribution of the foliage skin temperatures. A markedvertical distribution of canopy surface temperature re-sults under conditions of strong shortwave radiationand moderately sparse leaf area density.

Complex multilayer canopy models have been de-veloped to accommodate this vertical complexity[13,20,26]. However, these complex models introduce asigni®cant computational burden, and stringent re-quirements in terms of initialization and updating thatare undesirable especially for operational weathermodels, which cover subcontinental to continentalscales.

Other models take a lumped approach and treat thesurface as a single ``big leaf''. That is, they consider thecanopy to be an isothermal entity [15,19]. Many of thesemodels employ a bulk aerodynamic relationship forsurface sensible (H) heat ¯ux. An example is given byChen et al. [3] as

H � qcpChua�hsfc ÿ h� � qcp�hsfc ÿ h�=rh; �1�where q is the air density; cp the coe�cient of speci®cheat for air at constant pressure; Ch the bulk exchangecoe�cient for heat; ua the wind speed; hsfc the bulksurface temperature; h the temperature of the air; and rh

www.elsevier.com/locate/advwatres

Advances in Water Resources 24 (2001) 145±155

* Corresponding author. Tel.: +1-208-885-6184; fax: +1-208-885-

7908.

E-mail address: rqualls@uidaho.edu (R.J. Qualls).

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 4 0 - 3

is the aerodynamic resistance to heat transfer,rÿ1

h � Chua.Ch may be formulated directly from similarity theory

in terms of atmospheric stability correction functions wm

and wh for momentum and heat, respectively, and sur-face roughness lengths. When distinct roughness lengthsz0m and z0h for momentum and heat are retained, Ch canbe expressed as

Ch � k2=R�

lnz

z0m

� ����ÿ wm

zL

� �� wm

z0m

L

� ��ln

zz0h

� ��ÿ wh

zL

� �� wh

z0h

L

� ���; �2�

where z is the measurement height for wind speed andair temperature, k the von K�arm�an's constant, L theObukhov length, and R, estimated at 1.0, is the ratio ofthe momentum and heat exchange coe�cients in theneutral limit.

The scalar roughness z0h which appears in (2) is animportant quantity. Verhoef et al. [27] notes that allmeteorological weather models currently include aparameterization for z0h, usually in the form of ln�z0m=z0h�. Furthermore, recent tests of the Pan andMahrt [15] LSM by Chen et al. [3] show these bulkequations to be highly sensitive to the parameterizationof z0h.

Numerous parameterizations of z0h appear in theliterature. Garratt and Francey [6] proposed a constantratio z0m=z0h � e2. Chen et al. [3] employed the formu-lation of Zilitinkevich [29], z0m=z0h � expfkC

pRe�g, in

the National Center for Environmental Prediction(NCEP/NOAA) Eta model, which is an operationalmesoscale model. Here C is an empirical constant; andRe� the roughness Reynolds number which describesturbulence intensity. Using FIFE data, Qualls andHopson [18] employed z0m=z0h � expfb1 � b2�R�e=LAI�b � b3Ug, where LAI is leaf area index, the bis andb were empirically determined constants and U was afunction of daily maximum solar zenith angle. Althoughthe constants were determined empirically, this rela-tionship is mathematically consistent with the expres-sion for z0h derived by Brutsaert and Sugita [2]. It doesnot account for diurnal variation of solar elevation nordoes it account for sensor view angle e�ects. Stewartet al. [22] analyzed values of z0m=z0h obtained from eightdi�erent ®eld experiments. Kustas et al. [10] tested anumber of empirical expressions including z0m=z0h �expfbua�hs ÿ h�g where hs was a DRT of the surface.Other examples exist (e.g., Qualls and Brutsaert [17]: z0h

is a function of LAI; Sun and Mahrt [25]: z0h is afunction of �hs ÿ h�=H ; Sugita and Brutsaert [23] andSugita and Kubota [24]: z0h is a function of solar ele-vation; Matsushima and Kondo [13]: the aerodynamic

conductance, which is related to the bulk transfer coef-®cient, Ch and z0h, is taken as a function of LAI, windspeed and atmospheric stability).

Many of these studies show that z0h is neither con-stant nor proportional to the momentum roughnesslength [3,10,13,17,18,25,27]. In order to accommodatethis variability of z0h, Chen et al. [3] modi®ed the LSMof Pan and Mahrt [15] by parameterizing z0h as afunction of roughness Reynolds number. The revisedLSM has been implemented into the NCEP operationalmesoscale Eta model.

Although the revised LSM simulated spatially aver-aged directional radiometric temperatures (DRTs) fromFIFE reasonably well, there are instances where thesimulated and measured DRTs di�ered by as much as 2±6°C (see Fig. 5(d), panel C in Chen et al., [3]). Whendi�erences occur in the comparisons provided, themodel tends to under-predict mid-day DRTs, and over-predict nighttime DRTs. Due to its sensitivity to z0h,when the model is used in a mode in which DRTs serveas input to calculate sensible heat ¯uxes, these temper-ature di�erences would produce signi®cant errors in themodeled sensible heat ¯uxes. Studies by Qualls andHopson [18] and Verhoef et al. [27] indicate thatparameterization of ln �z0m=z0h� by means of theroughness Reynolds number alone leaves some unex-plained variability.

Some of this variability may be explained by verticalskin temperature gradients within the canopy and adistinct ground surface temperature. These cause thevalue of a DRT measured over a given surface to dependon the sensor view angle [8,12,28].

Fig. 1 illustrates the e�ect that a vertical foliage skintemperature gradient has on scalar roughness. Bulkformulations such as (3), shown later, operate under theassumption that the logarithmic air temperature pro®leha extends down into the canopy (see single-dashed linein Fig. 1), as if the atmospheric surface sublayer (SSL)extended all the way to the surface.

Fig. 1. Air and canopy temperature pro®les within the surface

sublayer (SSL) and canopy sublayer (CSL).

146 R.J. Qualls, D.N. Yates / Advances in Water Resources 24 (2001) 145±155

In reality, the canopy interferes with many of theprocesses so that we designate the region below the topof the vegetation as the canopy sublayer (CSL). No-tably, both the actual temperature pro®le of the airwithin the canopy (ha(f)) and the foliage skin tempera-ture (hf (f)) are disturbed from the ha pro®le, where f is anon-dimensional depth into the canopy whose value iszero at z� h, and one at z� 0. As a result, the ¯uxes atdepth f within the CSL are modulated by the�hf�f� ÿ ha�f�� di�erence.

When a DRT of the canopy is used for the surfacetemperature in the bulk formulation, the assumed``height'' z0h corresponding to this temperature must bethe height, where that value of hs would occur on theextrapolated logarithmic ha pro®le. This may be seen bythe dash-double-dotted line extending down from hs toha and then across to z0h in Fig. 1. As noted above, hs

depends on both the skin temperature pro®le throughthe canopy and on the view angle of the sensor, hencez0h must be variable.

Under sparse canopy conditions, the view angle e�ectmay be dominated by a combination of two ``sources'': asoil surface temperature and a single canopy tempera-ture. ``Dual-source'' models have been developed toaddress these conditions [5,11]. However, with moredense canopies, including some grasslands, vertical skintemperature variations through the canopy itself mayhave a signi®cant e�ect [13,20,21,26]. In order to addressproblems related to these vertical temperature gradients,canopy radiative transfer algorithms are often used.Based on the results of coupled canopy radiative trans-fer and climate models, Prevot et al. [16] found z0h todepend on sensor view angle. They concluded that theirmodel produced reasonable results but that ®eld re-search was needed to verify the model. Furthermore,coupling a canopy radiative transfer algorithm to anLSM introduces a substantial computational burden.

Many experimental data sets include DRTs measuredin situ, and from aircraft and satellite platforms. Someeven have data for which DRTs were measured over thesame point on the ground with sensors oriented at dif-ferent view angles [28]. However, these data sets requireone to infer the nature of the canopy temperature pro-®les or to generate the pro®les by modeling [4]; thepro®les cannot be validated by means of these data. Thisis not to say there have not been studies of the viewangle e�ects on DRT [8,11±13,21,26], however, only afew studies exist, where canopy temperature pro®leswere measured [9].

Despite studies that have explicitly addressed the ef-fects of view angle on DRT, there is no clear consensuson a method to overcome the problems associated withit. Some recent studies have shown that certain optimalview angles exist which minimize the view angle e�ects[2,13]. These typically range between 50° and 70° fromnadir. However, these optimal angles are of little prac-

tical value since surface temperature data from satelliteremote sensing images are measured across a range ofangles, usually less than 50°.

In order to receive the bene®t of spatial continuitywhich may be achieved through the use of spatially-distributed DRTs, while maintaining computational ef-®ciency of a single canopy layer in an LSM, one mustaccommodate the anisothermal canopy pro®le andsensor view angle e�ects. Recent theoretical advance-ments in embedding the CSL e�ects within a bulktransfer equation by Brutsaert and Sugita [2], whichhave been furthered by Qualls and Hopson [18] andCrago [4], have created the opportunity to accountfor these e�ects more rigorously than was previouslypossible. To aid these developments, experimentalstudies regarding the representation of the canopytemperature pro®le are crucial.

The purpose of this paper is to present and model®eld observations of plant canopy DRT pro®les and toshow some realistic values of temperature gradients andtheir corresponding semi-diurnal variability that mightbe expected. In addition, we will conduct a sensitivityanalysis of Brutsaert and Sugita's [2] analytical expres-sion for scalar roughness z0h, particularly with respect torealistic shapes and magnitudes of vertical skin tem-perature gradients within a plant canopy.

2. Methods

2.1. The anisothermal scalar roughness length

In practical applications, there are several possibleways to parameterize the heat transfer characteristics ofa land surface covered with vegetation. One of thesimpler methods is to use the bulk transfer approachgiven by Monin±Obukhov (M±O) similarity theory [1],which includes the scalar roughness length for sensibleheat, z0h, given as

hs ÿ h � Hahku�qcp

lnzÿ d0

z0h

� ��ÿ w

zÿ d0

L

� ��; �3�

where h is the potential temperature of the air at somereference height z within the surface sublayer (SSL); hs

an e�ective or bulk surface temperature intended here tobe supplied by DRT; aÿ1

h the turbulent Prandtl number(which is a constant close to unity); k von K�arm�an'sconstant; u� the friction velocity; q the air density; cp thecoe�cient of speci®c heat at constant pressure; d0 thezero plane displacement height; L the Obukhov length;and wh the stability correction function for the tem-perature pro®le in the SSL.

Unfortunately, z0h is neither constant, nor propor-tional to the momentum roughness length, z0m as notedearlier. In order to derive an expression for z0h for an-isothermal canopies, Brutsaert and Sugita [2] formulated

R.J. Qualls, D.N. Yates / Advances in Water Resources 24 (2001) 145±155 147

and solved a di�erential equation for within-canopyscalar transport, and matched it to the M±O similarity¯ux pro®le equation for the SSL. The main results oftheir work were expressions for the isothermal and an-isothermal roughness lengths, z0hi and z0h, respectively,

z0h � z0hi expahku�qcp�hfg ÿ hfh�

H

�� �r2 � bC2�

r2�b2 � baÿ C2��

� w��; �4�

z0hi � z0m exph

�hÿ d0�r2

�� ln

hÿ d0

z0m

� ��; �5�

where z0m is the momentum roughness; h the canopyheight; C2 � �2LAI � Ctfh�=�ahk�hÿ d0��; LAI the leafarea index; Ctf the bulk transfer coe�cient for foliageelements introduced in Brutsaert and Sugita [2];r2 � �aÿ �a2 � 4C2�1=2�=2; a the extinction coe�cient inthe exponential shear stress pro®le and in the eddy dif-fusivity pro®le, which are assumed equal and have val-ues between 2 and 4 for dense ¯exible elements (wheat,oats, immature corn); b is the temperature pro®le decaycoe�cient described below; H the sensible heat ¯ux; andw a temperature-weighting coe�cient which relates theview angle dependent DRT to the foliage temperature atthe top and bottom of the canopy, as discussed below.Note that z0hi only depends on the bulk canopy char-acteristics.

In their derivation, Brutsaert and Sugita [2] assume avertically uniform canopy structure for which the foliageskin temperature in the canopy can be represented withan exponential decay model

hf�f� � hfg � �hfh ÿ hfg�eÿbf; �6�where hf (f) is the foliage skin temperature at depth fwithin the canopy de®ned as f � 1ÿ z=h where z is theheight above the ground surface, and h is the canopyheight. f varies from 0 at the top of the canopy to 1 atthe ground. hfh is the foliage skin temperature at the topof the canopy and hfg is the asymptotic limit of the fo-liage temperature far below the bottom of the canopy asf becomes large.

2.2. Weighting coe�cient, w

One of the long-term goals of this work is to developa method to incorporate DRTs into LSM's. Severalstudies have determined ``optimal'' view angles, whichsatisfy various criteria for sensible heat ¯ux calculation[2,4,8,13,28]. One rarely has control over the view angleof the sensor. In fact, from scanning sensors a widerange of zenith view angles are presented in any givensurface temperature image. Therefore, the ability toaccommodate DRTs measured from arbitrary view an-gles is important. The relationship between a DRT and

a given canopy temperature pro®le depends on thesensor view angle, and on the density and structure ofthe vegetation. These factors may be combined throughthe temperature-weighting coe�cient, w, which appearsin (4) and is discussed below.

Brutsaert and Sugita [2] employ the approximationthat DRT can be represented as a linear combination ofthe foliage skin temperature at the top and bottom ofthe canopy

hs � whfh � �1ÿ w�hfg; �7�where hs is the DRT measured at a known view angle.The weighting coe�cient, w, depends on canopy density,architecture, and skin temperature pro®le shape (i.e. thedecay coe�cient, b), as well as on the view angle of thesensor.

Similarly in dual-source models [5,11], the DRT istaken to be a combination of the soil surface tempera-ture, hsoil, and of the canopy temperature, hveg, assumedto be isothermal, weighted by their respective fractionsfsoil and 1ÿ fsoil visible to the sensor,

hs � fsoilhsoil � �1ÿ fsoil�hveg: �8�More correctly, the radiances corresponding to thesetemperatures should be added, but the linearization is agood approximation provided hsoil and hveg are not toodi�erent.

For anisothermal canopies, (8) may be modi®ed byintegrating through the canopy,

hs � fsoilhsoil �Z 1

0

�ÿ df

df

�hf�f�df; �9�

where f is the fraction of the viewer ®eld not yet ob-scured by foliage at the depth f into the canopy; hf (f) thefoliage skin temperature given by (6).

For a vertically uniform canopy, the fraction f may begiven by Beer's law,

f �f� � exp�ÿ g0LAIf=cosm

; �10�

where g0 is the leaf angle distribution parameter; LAI theleaf area index; m the sensor zenith view angle. Friedl [5]estimated g0 using g0 � cos (ul), where ul is the mean leafinclination angle measured from a horizontal plane. Forgrasslands and grass-like crops such as wheat, ul will betaken as 65°. Values of ul for these and other types ofvegetation may be obtained from Gates [7]. The fractionfsoil, which appears in (9) may be obtained by setting fequal to 1 in (10).

For canopies whose foliage density is non-uniform inthe vertical direction, LAI*f may be replaced byR f

0a�f�df, where a(f) is a function which describes the

vertical distribution of canopy density as in Matsushimaand Kondo [13].

Crago [4] obtained an expression for the weightingcoe�cient w in (7). He did so by integrating (9) with

148 R.J. Qualls, D.N. Yates / Advances in Water Resources 24 (2001) 145±155

(10), and setting the result equal to the right-hand side of(7).

w � �1ÿ fsoileÿb� bcosm

g0LAI

�� 1

�ÿ1

: �11�

This requires the assumption that hsoil� hfg (i.e., thesoil surface temperature is equal to the foliage skintemperature deep within the canopy at the limit of theexponential canopy model (6), where f!1). This is areasonable assumption since (6) can even accommodatea canopy which is isothermal with a discontinuity be-tween the soil temperature and the foliage temperature,by setting b� 0, and letting hfh be the canopy tempera-ture and hfg be the soil temperature.

Eqs. (4), (5) and (11) provide the information re-quired for the sensitivity analysis of z0h. It is worthnoting that for the purpose of modeling z0h with (4), theactual values of hfg and hfh are unimportant, but onlytheir di�erence.

2.3. Pro®le ®tting procedure

In (4), hfh is the temperature at the top of the canopy,and hfg refers to the asymptotic limit of the exponentialpro®le as f (the normalized depth into the canopymeasured from the top downward) becomes large, i.e.,below the bottom of the canopy. hfh, hfg and b are un-knowns for each pro®le. We took hfh directly from DRTmeasurements at the ``top'' of the canopy, which wedescribe more fully later. We estimated hfg and b by®nding their values that minimized the sum of squareddi�erences between measured, hm(fi), and modeled,h(fi), temperature pro®lesXn

i�1

�hm�fi� ÿ h�fi��2: �12�

We accomplished this by the standard linear leastsquares procedure in which we took the partial deriva-tive of (12) with respect to hfg after substituting the right-hand side of (6) for h(fi), set the result equal to zero, andsolved for hfg

hfg �hfh

PX 2

i ÿP

Xi

� �ÿP hm�fi�Xi �P

hm�fi�PXi ÿ 2

PXi � n

;

�13�where Xi � exp�ÿbfi�, and n is the number ofmeasurement heights in each pro®le.

Following a similar procedure for b produces animplicit equation for b. Therefore, we linearized theproblem by rearranging (6) and taking the natural log-arithm.

Yi � lnhfg ÿ hm�fi�

hfg ÿ hfh

� �� ÿbfi: �14�

If we set the partial derivative with respect to b of thesum of the squared di�erences of the right- and left-

hand sides of (14), equal to zero, we can solve explicitlyfor b

b � ÿP

fiYiPf2

i

: �15�

Eq. (15) may be substituted into (13) to eliminate b.The resulting equation is an implicit equation for hfg

since b in (15) is a function of hfg through Yi

hfg � g�hfg�: �16�Next, we de®ned a function h�hfg� � hfg ÿ g�hfg� � 0

and found the values of hfg, which caused h(hfg) to bezero. The exponent b is then calculated with (15). Thesevalues of hfg and b are those which minimize the sum ofsquared di�erences between measured and modeledtemperatures for each pro®le.

In addition, we imposed the constraints: (1) b > 0;and (2) hfg�ti� > hfg�tj� when hfh�ti� > hfh�tj�, where ti

and tj are the times of two di�erent pro®les. The ®rst ofthese constraints is required in order for h(f) to ap-proach hfg as f becomes large. The second constraintwas imposed on the basis of observations, which showedthat pro®les tended not to overlap and that successivepro®les increase in curvature and steepness as tem-perature increases when averaged over a suitable timeperiod on clear days.

In the following sections, we describe the data setused, and present results of ®eld observations of direc-tional radiometric canopy temperature pro®les fromwhich we obtain values for hfg, hfh, and b, and a sensi-tivity analysis.

3. Data set

The data used here came from an experiment carriedout from June 19 to July 9, 1997 in a hay ®eld north ofBoulder, Colorado, located at latitude 40° 60 52.800 Nand longitude 105° 150 51.600 W. During this experiment,DRT measurements of canopy temperature pro®leswere collected. The data used here were from the com-pletely cloudless morning of July 1. The grass wascharacterized by a dense lower story 46 cm tall withsparse individual stocks extending up to 87 cm. The 15°®eld of view infrared thermometer was oriented hori-zontally, facing west, and moved manually up and downthrough the grass. Each pro®le consisted of measure-ments at 12 evenly spaced heights from 7 up to 84 cm.One measurement was collected each second, 5 at eachheight before moving the sensor to the next height, sothat a complete pro®le was measured in 1 min. Themeasurements at each height during each minute wereaveraged together to represent the DRT at that height.Data were collected continuously from 6:18 to 11:50a.m. mountain daylight savings time (MDST) with theexception of a 4-min and 20-min break in the data

R.J. Qualls, D.N. Yates / Advances in Water Resources 24 (2001) 145±155 149

starting at 7:00 and 9:10 a.m., respectively. From thesemeasurements, more than 18 500 values were consoli-dated into 30-min averages to which we refer in the textusing the ending time of the 30-min (e.g., we refer to theaverage from 7 to 7:30 a.m. as ``pro®le 7.5''). A few ofthe time periods were shorter than 30 min (6:18±6:30;7:04±7:30; 9:00±9:10; and 11:30±11:50). We still refer tothese by the ending time of the half-hour period. Table 1contains the values of these 30-min pro®les as a functionof time and height above the ground.

4. Results and discussion

4.1. General observations

The 30-min averaged DRT pro®les appear in Fig. 2.Several important features show up here. First, thepro®les begin nearly isothermal at 6:30 a.m., and themean pro®le temperature increases and the temperaturegradient steepens as the morning progresses. Second, the

sparse upper canopy remains nearly isothermalthroughout the morning, and most of the gradient isconcentrated in the dense lower canopy.

Since z0h given by (4) depends on the di�erence intemperatures between the bottom and top of the cano-py, �hfg ÿ hfh�, we have plotted �hf�7 cm� ÿ hfh� fromeach 1-min pro®le as a function of time in Fig. 3, to getan idea of how �hfg ÿ hfh� might behave. hfh is the av-erage of all DRTs measured at and above z� 42 cm, andhf (7 cm) is the temperature measured closest to theground, at z� 7 cm. Hereafter, hf (7 cm) will be denotedhf7. The vertically averaged temperature, hfh, was used torepresent the top of canopy temperature because theupper canopy was relatively sparse, approximately iso-thermal, and if the pro®les exhibit an exponential shape,this occurs in the lower part of the canopy at and below42 cm, as is clear in Fig. 2.

Table 1

DRT values of 30-min averaged pro®les in °C as functions of time and height above the ground surface

Height

(cm)Time (h)

6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12

84 12.7 14.4 17.9 18.7 19.3 20.0 20.2 21.7 22.1 22.7 23.0 23.6

77 12.3 14.2 17.5 18.3 19.0 19.8 20.1 21.6 21.9 22.6 22.8 23.5

70 11.9 13.9 17.2 18.0 18.8 19.7 20.1 21.7 21.9 22.6 22.7 23.5

63 11.6 13.7 17.1 17.9 18.7 19.7 20.2 21.9 22.0 22.7 22.9 23.8

56 11.2 13.6 16.9 17.9 18.7 19.7 20.3 22.0 22.3 23.0 23.0 23.9

49 10.8 13.5 16.8 17.8 18.6 19.6 20.2 22.2 22.5 23.4 23.6 24.2

42 10.6 13.1 16.6 17.5 18.5 19.5 20.2 22.3 22.7 23.8 24.2 25.1

35 10.6 12.8 16.7 18.1 19.1 20.3 20.7 23.1 24.0 25.0 25.9 27.2

28 10.5 12.6 17.1 18.7 19.7 20.9 21.5 24.1 25.6 27.1 28.0 29.4

21 10.5 12.5 16.8 18.9 20.2 21.8 22.4 25.4 26.9 28.5 29.7 31.3

14 10.9 12.8 16.6 19.1 20.8 22.3 23.2 26.6 28.0 29.6 31.1 33.1

7 11.4 13.2 16.4 19.1 21.4 22.8 23.9 27.4 28.8 30.5 32.2 34.0

Fig. 2. A 30-min averaged DRT pro®les. Numbers indicate range of

times (HHMM) included in each pro®le.

Fig. 3. Evolution of bottom to top canopy temperature di�erences,

hf7 ÿ hfh, for each one-minute pro®le, and for 30-min averages with

time. Also shown is the evolution of solar zenith angle for comparison

with temperature changes.

150 R.J. Qualls, D.N. Yates / Advances in Water Resources 24 (2001) 145±155

The solid line in Fig. 3 connects the values of�hf7 ÿ hfh� taken from the 30-min averaged data. Inaddition, since we are interested in the diurnal pro-gression of �hf7 ÿ hfh� and its causes, we have plotted thesolar zenith angle (dashed line) versus time in Fig. 3.This illustrates qualitatively, that as the sun rises higherin the sky (decreasing zenith angle), it penetrates moredeeply into the canopy and causes signi®cant warmingof the lower canopy. The upper canopy remains coolerdue to exposure to the wind.

The mean di�erence �hf7 ÿ hfh� is zero at around 7:30a.m. It then increases relatively steadily throughout themorning to a mean di�erence of about 10°C (�2°C) by11:30 a.m. Solar noon occurs at this location at ap-proximately 12:58 MDST. If �hf7 ÿ hfh� continued toincrease at the same rate shown in Fig. 3, one mightexpect �hf7 ÿ hfh� to increase by another 3°C to a valueof 13°C by solar noon at which time the solar zenithangle would be approximately 17°. This increase in�hf7 ÿ hfh� would correspond to a decrease in z0h

throughout the morning according to (4), other factorsremaining the same.

4.2. Fitted pro®les

The derivation of (4) in Brutsaert and Sugita [2] as-sumes an exponential temperature pro®le. To evaluatethe realism of this assumption, we ®tted exponentialcurves described by (6) to the 30-min average DRTpro®les, using the least squares procedure describedearlier. We used the average of the temperatures at andabove 42 cm for hfh for the reasons explained above.

hfg and b converged to solutions which satis®edh(hfg)� 0 and the imposed constraints for ®ve of thepro®les (#s 8, 10.5, 11, 11.5, and 12). The other casesfailed for the following reasons:1. Pro®les 7.5, 8.5, and 9.5 violated constraint 2. Due to

the small range in h(f), curvature in the pro®le is ob-scured by the scatter in the data. The optimizationprocedure tends toward a straight line for h(f) overthe applicable range of f. This requires b! 0 andhfg !1, which caused a divide by zero error in (13).

2. Pro®les 9 and 10 violated constraint 2. As in (1)above, the curvature is slight. The optimization pro-cedure found a solution but hfg was larger than thatfor later pro®les.

3. Pro®les 6.5 and 7 reveal internal minima within theirpro®les so that the least squares solution becomes avertical line through the average value of the lower®ve temperatures. This forces b to become excessivelylarge in order for h(0) to equal hfh.

In order to ®t an exponential pro®le to those pro®leswhich failed to converge and/or satisfy the constraints,we examined the temporal trend in �hfg ÿ hfh� for the ®vepro®les for which the solution converged and satis®edthe constraints, and compared the trend to that for

�hf7 ÿ hfh�. We illustrate the comparison in Fig. 4.Considering the solid squares and their correspondingdiamonds, it appears that �hfg ÿ hfh� is roughly propor-tional to �hf7 ÿ hfh�. This is physically realistic since (1)both �hfg ÿ hfh� and �hf7 ÿ hfh� should become 0 simul-taneously when the pro®le is isothermal, and (2) eachsuccessive pro®le appears to envelop the previous pro-®les and increase in steepness and curvature in Fig. 2.The average value of the ratio of �hfg ÿ hfh�=�hf7 ÿ hfh�was �1.46. We determined �hfg ÿ hfh� for the otherpro®les by multiplying their corresponding �hf7 ÿ hfh�value by this ratio. The results are shown by the hollowsquare symbols in Fig. 4.

For each pro®le, we used these values of �hfg ÿ hfh� todetermine hfg, and then determined b by means of (15).Doing so gives the value of b that minimizes the sum ofsquared errors associated with the linearized tempera-ture pro®le equation (14) for the speci®ed value of hfg.For each time period, Table 2 presents the values of hfg,b �hfg ÿ hfh�, and the root-mean-square error (RMSE)between measured and modeled temperature pro®les.The measured (symbols) and modeled (lines) pro®lesdetermined from (6) using the data in Table 2 are shownin Fig. 5. The resulting values of b are all very similarand range between about 1.1 and 1.4 with an averagevalue of 1.26, as shown in Fig. 6. The relatively constantnature of b is one of the important ®nding of this paper.

It is important to note that the RMSE did not in-crease substantially for those pro®les for which hfg and bwere estimated based on the ration method. In fact, theRMSE was relatively insensitive to hfg. We illustrate thisin Fig. 7. The solid line in panel (a) shows the RMSE asa function of hfg for pro®le 10. In decreasing the globaloptimum value of hfg� 66°C by 26±40°C, RMSE onlyincreases from about 0.11±14°C. Thereafter, the slope ofRMSE increases, but in going to hfg� 30.0°C (the value

Fig. 4. Comparison between the temporal trends in hf7 ÿ hfh (diamond

symbols) and hfg ÿ hfh (square symbols). The solid square symbols

denote hfg ÿ hfh values determined by the least squares pro®le ®tting

procedure. The hollow square symbols denote hfg ÿ hfh values deter-

mined by proportionality to hf7 ÿ hfh.

R.J. Qualls, D.N. Yates / Advances in Water Resources 24 (2001) 145±155 151

obtained by the ratio method), RMSE only increases to0.34°C. Although this is larger than the minimum valueby a factor of 3, it still represents only 3.75% of the

di�erence �hfg ÿ hfh�, which is certainly an acceptableerror. The signi®cance of this is that determination of hfg

by the ratio method did not substantially increase theerror in modeling the measured pro®le.

In contrast, the pro®les 8, 10.5, 11, 11.5 and 12,which satis®ed the constraints in our original optimi-zation run, exhibited much more signi®cant changes inRMSE for small changes in hfg. The solid line in panel bof Fig. 7 shows that an increase in RMSE for pro®le 12from the global minimum to a value of 0.3°C onlyallows a range in hfg from about 35.5°C to 44°C. This ismuch smaller than the range 30°C to +1 allowed forpro®le 10 for the same change in RMSE, and indicatesthat the certainty in the value of hfg for pro®les 8, 10.5,11, 11.5, and 12, which satis®ed the constraints in theoriginal optimization run is much higher than for theother pro®les.

4.3. Sensitivity analysis

To demonstrate the behavior of z0h, we plotted, on alogarithmic scale in Figs. 8(a±d), the ratio z0h=z0hi de-termined from (4) and (5) as a function of several vari-ables. Values of the variables for the base scenario andtheir ranges are given in columns 2 and 3 of Table 3.

Fig. 7. Value of root mean square error (RMSE) as a function of hfg.

The dashed line represents the locally optimum value of b corre-

sponding to hfg. The RMSE for any pair of hfg, b values is given by the

value of the RMSE line directly above the pair. Panel (a) corresponds

to pro®le 10; panel (b) corresponds to pro®le 12.

Fig. 5. Measured (symbols) and modeled (lines) temperature pro®les

for lower half of the canopy, where the exponential pro®le appears to

exist.

Fig. 6. Illustration of approximate constancy of the coe�cient, b, in

the exponential pro®le model (6).

Table 2

Values of variables used to model the canopy temperature pro®le by

means of (6), and the RMSE between measured and modeled values

Time (h) hfg (°C) b hfg ÿ hfh

(°C)

RMSE

(°C)

6.5 9.98 1.07 )1.61 0.530

7 11.9 1.18 )1.87 0.455

7.5 16.1 1.18 )1.04 0.163

8 19.7 1.44 1.63 0.137

8.5 22.5 1.19 3.74 0.255

9 24.2 1.29 4.48 0.208

9.5 25.6 1.24 5.39 0.322

10 30.0 1.29 8.03 0.339

10.5 32.9 1.16 10.7 0.050

11 34.3 1.31 11.4 0.099

11.5 36.3 1.39 13.2 0.038

12 38.3 1.46 14.4 0.099

152 R.J. Qualls, D.N. Yates / Advances in Water Resources 24 (2001) 145±155

Panel (a) of Fig. 8 shows that z0h=z0hi decreases lin-early on a logarithmic scale as a function of increasingtemperature di�erence �hfg ÿ hfh�. This is also clearbased on simple inspection of (4). Panel (a) also showsthe e�ects of increasing LAI, which are not as readilyapparent from (4). An eightfold increase in LAI from0.5 to 4 causes an increase in z0h=z0hi from about 0.024 to0.31 for the base scenario. The incremental increase inz0h=z0hi decreases for each additional unit increase inLAI (e.g., there is a bigger change in going from LAI� 1to 2, than from 2 to 3, etc.).

Although z0h is non-linearly related to u�, due to thepresence of u� in (4) in the numerator and in

C2 ��f �Ctf�Re��u�����, the direct in¯uence of u� in thenumerator dominates, so that z0h=z0hi is essentiallylinearly related to u� on a logarithmic scale. z0h=z0hi

decreases from about 0.66 to about 0.02 for a changein u� from 0.1 to 1 m/s for the base scenario (notshown).

The ratio z0h=z0hi is not very sensitive to b. Panel (b)shows that a change in b from 0.5 to 1.5 produces anegligible change even for �hfg ÿ hfh�� 15°C. This issigni®cant given the narrow range of b observed for thisdata set. Furthermore, even a large change in b from 1.5to 7 only increases z0h=z0hi from around 0.26 to 0.4 forthe base scenario. Crago [4] demonstrates that theweighting coe�cient, w, is very sensitive to b, however,apparently this sensitivity does not propagate its waythrough to z0h. Panel (c) shows that z0h=z0hi becomesmore sensitive to b as LAI decreases.

Finally, the e�ect of zenith view angle for a sensorlocated above the canopy is shown in panel (d). Viewangle a�ects z0h=z0hi by means of changes in theweighting coe�cient, w given by (11), which relatesDRT to hfg and hfh through (6). Note here that, unlikeLAI, a unit increase in view angle creates successivelylarger increases in z0h=z0hi as view angle increases. Forthe base scenario, a view angle of 71.39° causes z0h=z0hi

to equal 1 for all values of �hfg ÿ hfh�. Others havecommented on the fact that various models and obser-vations suggest that a large view angle causes z0h to

Fig. 8. Results of sensitivity analysis for the base scenario and variable parameter values indicated in Table 3.

Table 3

Values of variables used for base scenario of sensitivity analysis and

ranges of values allowed to vary in Figs. 8(a±d)

Variable Base scenario Range

hfg ÿ hfh 5°C 0±15°C

B 1 0.5±7

LAI 3 0.5±4

u� 0.4 m sÿ1 � � �m 20° 0±70°H 0.42 m � � �A 3 � � �z0m h/8 � � �d0 2/3h � � �Ctf 0.0715 � � �H 150 W mÿ2 � � �

R.J. Qualls, D.N. Yates / Advances in Water Resources 24 (2001) 145±155 153

behave as though the canopy were isothermal[2,4,8,13,28].

5. Summary and conclusions: signi®cance to modeling z0h

To summarize these results, it is useful to review someof the background on the scalar roughness, z0h. First, aparameterization for z0h appears in all meteorologicalweather models [27], therefore improvements to it willhave broad impacts. Secondly, many parameterizationsfor z0h have been proposed. Research indicates that z0h isneither constant nor is it proportional to the momentumroughness, z0m. Furthermore, the roughness Reynoldsnumber does not explain all of its diurnal variability.

Some of the diurnal variability may be due to verticalskin temperature gradients that form within vegetationas the sun rises higher in the sky and penetrates moredeeply into the canopy. Brutsaert and Sugita [2] derivedan expression to account for the in¯uence of tempera-ture gradients on z0h. Their derivation assumed an ex-ponential pro®le shape for the temperature gradientsgiven by (6). This assumption limits the applicability oftheir method to canopies with a relatively uniform ver-tical structure. In this paper, we present data to evaluatethis assumption and examine the semi-diurnal behaviorof several variables, b, hfg, and hfh which are importantparameters in the exponential temperature pro®le (6).The speci®c conclusions from this work are listed below:1. The exponential decay coe�cient, b, may be approx-

imately constant for a given canopy at a given growthstage, although it is likely to be a function of LAI(Fig. 6). This, together with the fact that z0h doesnot appear to be extremely sensitive to b (Fig. 8(b)),suggests that for a given canopy, b may not have asigni®cant in¯uence on z0h. Furthermore, b may beable to be estimated for a given canopy on the basisof a few pro®le measurements when solar zenith angleis small so that the curvature and range of h(f) in thepro®le is large.

2. �hfg ÿ hfh� appears to increase in a consistent fashionon clear days as solar zenith angle decreases (Figs. 3and 4). This behavior and the ®nding that b is rela-tively constant may provide useful constraints for es-timating �hfg ÿ hfh� and b from time series of DRTsmeasured from multiple view angles.

3. If we extrapolate the rate of increase in �hfg ÿ hfh� tosolar noon (�1 p.m. MDST), it would attain a valueof �20°C. Thus for this canopy/location/time of year,a rough estimate for the diurnal range of �hfg ÿ hfh� is0±20°C.

Although these conclusions do not provide enough in-formation to apply Brutsaert and Sugita's [2] par-ameterization for z0h operationally, they do providesome simpli®cations that will be helpful to evaluate thediurnal variability of z0h.

Acknowledgements

This material is based upon work partially supportedby the National Science Foundation under Grant No.EAR-9903117 and by the National Aeronautics andSpace Administration under Grant No. NAG5-8534.

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