Surface energy balance model of transpiration from variable canopy cover and evaporation from residue-covered or bare-soil systems

ORIGINAL PAPER

Surface energy balance model of transpiration from variablecanopy cover and evaporation from residue-covered or bare-soilsystems

Luis Octavio Lagos Æ Derrel L. Martin ÆShashi B. Verma Æ Andrew Suyker ÆSuat Irmak

Received: 16 October 2008 / Accepted: 2 June 2009 / Published online: 29 August 2009

� Springer-Verlag 2009

Abstract A surface energy balance model based on the

Shuttleworth and Wallace (Q J R Meteorol Soc 111:839–

855, 1985) and Choudhury and Monteith (Q J R Meteorol

Soc 114:373–398, 1988) methods was developed to esti-

mate evaporation from soil and crop residue, and transpi-

ration from crop canopies. The model describes the energy

balance and flux resistances for vegetated and residue-

covered surfaces. The model estimates latent, sensible and

soil heat fluxes to provide a method to partition evapo-

transpiration (ET) into soil/residue evaporation and plant

transpiration. This facilitates estimates of the effect of

residue on ET and consequently on water balance studies,

and allows for simulation of ET during periods of crop

dormancy. ET estimated with the model agreed favorably

with eddy covariance flux measurements from an irrigated

maize field and accurately simulated diurnal variations and

hourly amounts of ET during periods with a range of crop

canopy covers. For hourly estimations, the root mean

square error was 41.4 W m-2, the mean absolute error was

29.9 W m-2, the Nash–Sutcliffe coefficient was 0.92 and

the index of agreement was 0.97.

List of symbols

Cd Drag coefficient

Cp Specific heat of air (J kg-1 �C-1)

d Zero plane displacement (m)

Dv Water vapor diffusion coefficient (m2 s-1)

ea Vapor pressure of the air (mbar)

eb Vapor pressure of the air at the canopy level (mbar)

e�a Saturated vapor pressure of the air (mbar)

e�b Saturated vapor pressure at the canopy level (mbar)

e�L Saturated vapor pressure at the top of the wet layer

(mbar)

e�Lr Saturated vapor pressure at the top of the wet layer

for the residue-covered soil (mbar)

e�1 Saturated vapor pressure at the canopy (mbar)

kE Total latent heat flux (W m-2)

kEc Latent heat flux from the canopy (W m-2)

kEr Latent heat flux from the residue-covered soil

(W m-2)

kEs Latent heat flux from the soil (W m-2)

Gor Conduction flux from the residue-covered soil

surface (W m-2)

Gos Conduction flux from the soil surface (W m-2)

Gr Soil heat flux for residue-covered soil (W m-2)

Gs Soil heat flux for bare soil (W m-2)

h Vegetation height (m)

H Total sensible heat flux (W m-2)

Hc Sensible heat flux from the canopy (W m-2)

Hr Sensible heat flux from the residue-covered soil

(W m-2)

Hs Sensible heat flux from the soil (W m-2)

K von Karman constant

k1 Thermal diffusivity (m2 s-1)

Communicated by S. Ortega-Farias.

L. O. Lagos � D. L. Martin � S. Irmak

Department of Biological Systems Engineering,

University of Nebraska-Lincoln, Lincoln,

NE 68583-0726, USA

S. B. Verma � A. Suyker

School of Natural Resources, University of Nebraska-Lincoln,

Lincoln, NE 68583-0726, USA

Present Address:L. O. Lagos (&)

Departamento de Recursos Hıdricos,

Universidad de Concepcion, Chillan, Chile

e-mail: [email protected]

123

Irrig Sci (2009) 28:51–64

DOI 10.1007/s00271-009-0181-0

K Thermal conductivity of the soil, upper layer

(W m-1 �C-1)

K(z) Eddy diffusion coefficient (m2 s-1)

K0 Thermal conductivity of the soil, lower layer

(W m-1 �C-1)

Kr Thermal conductivity of the residue layer

(W m-1 �C-1)

Lm Lower layer depth (m)

Lr Thickness of the residue layer (m)

Lt Thickness of soil layer (m)

rah Aerodynamic resistance for heat transfer (s m-1)

ram Aerodynamic resistance for momentum transfer

(s m-1)

raw Aerodynamic resistance for water vapor (s m-1)

rb Boundary layer resistance (s m-1)

rbh Excess resistance term for heat transfer (s m-1)

rbw Excess resistance term for water vapor (s m-1)

rc Surface canopy resistance (s m-1)

rL Soil heat flux resistance for the lower layer (s m-1)

rr Residue resistance for water vapor flux (s m-1)

rrh Residue resistance for heat flux (s m-1)

rs Soil surface resistance for water vapor flux (s m-1)

rso Soil surface resistance to the vapor flux for a dry

layer (m s-1)

ru Soil heat flux resistance for the upper layer (s m-1)

r1 Aerodynamic resistance between the canopy and the

air at the canopy level (s m-1)

r2 Aerodynamic resistance between the soil and the air

at the canopy level (s m-1)

Rn Net radiation (W m-2)

Rnc Net radiation absorbed by the canopy (W m-2)

Rns Net radiation absorbed by the soil (W m-2)

Ta Air temperature (�C)

Tb Air temperature at canopy height (�C)

TL Soil temperature at the interface between the upper

and lower layers for bare soil (�C)

TLr Soil temperature at the interface between the upper

and lower layers for residue-covered soil (�C)

Tm Soil temperature at the bottom of the lower layer (�C)

T1 Canopy temperature (�C)

T2 Soil surface temperature (�C)

T2r Soil surface temperature below the residue (�C)

uh Wind speed at the top of the canopy (m s-1)

u2 Wind speed at 2 m above the surface (m s-1)

u* Friction velocity (m s-1)

w Mean leaf width (m)

z Reference height (m)

zo Surface roughness length (m)

z0o Roughness length of the soil surface (m)

Greek symbols

a Attenuation coefficient for eddy diffusion coefficient

within the canopy

b Fitting parameter

D Mean rate of change of saturated vapor pressure with

temperature between the canopy and the air at the

canopy level (mbar �C-1)

/ Soil porosity

/r Residue porosity

h Volumetric soil water content (m3 m-3)

hs Saturation water content of the soil (m3 m-3)

q Density of moist air (kg m-3)

s Soil tortuosity

sr Residue tortuosity

Introduction

Evapotranspiration (ET) is often equivalent to 80–90% of

the annual precipitation in semiarid and subhumid regions.

Altering land use practices can change ET and may affect

the regional water balance. Water managers and individual

producers are interested in the impact of water conserva-

tion measures, such as reduced tillage, on ET during crop

growing seasons and dormant periods. Crop residue affects

many of the processes that determine the evaporation rate,

including net radiation, soil heat flux, aerodynamic and

surface resistances to heat and water vapor fluxes (Steiner

1994). Residue generally increases infiltration and reduces

evaporation from the soil. Caprio et al. (1985) compared

evaporation from three mini-lysimeters for bare soil and

beneath wheat stubble that was 14, and 28 cm tall. Evap-

oration from lysimeters with stubble was 60% of the

evaporation from bare soil after a 9-day period. Todd et al.

(1991) used mini-lysimeters to show that crop canopy and

straw mulch both reduced evaporation compared to bare-

soil conditions. The effects of canopy and residue were

about equal for limited or full irrigation. Enz et al. (1988)

evaluated daily evaporation for bare and stubble-covered

soil. Evaporation was larger from bare soil initially; how-

ever, later in the process, evaporation from stubble-covered

soil exceeded that from bare soil because the residue-

covered soil was wetter.

Evaporation from soil surfaces has been described as

occurring in three stages. An initial energy-limited stage

occurs when enough soil water is available to satisfy the

potential evaporation rate. A falling-rate stage occurs when

soil water limits flow to the soil surface. The third stage

involves very low evaporation rates that are nearly constant

rate for very dry soil (Jalota and Prihar 1998). Steiner

(1989) evaluated the effect of residue (from cotton, sor-

ghum and wheat) on the initial, energy-limited, rate of

evaporation. The evaporation rate relative to bare-soil

evaporation was described by a logarithmic relationship.

Increasing the amount of residue on the soil reduced

evaporation during the initial stage. Bristow et al. (1986)

52 Irrig Sci (2009) 28:51–64

123

predicted soil heat and water budgets using a soil–residue–

atmosphere model. Model results indicated that surface

residue decreased evaporation by roughly 36% compared

with simulations from bare soil; however, the impact

decreased with the length of the drying period.

Process models of ET have progressed through phases

over time. Initially, Penman (1948) estimated ET by par-

titioning net radiation into sensible and latent heat fluxes

for a layer extending from a reference height to a uniform

surface. This concept has been applied to crops by

approximating the canopy as a single surface (i.e., single

big leaf) for the Penman–Monteith (P–M) method

(Monteith 1965). The P–M method is widely accepted for

predicting ET of reference crops (Jensen et al. 1990; Allen

et al. 1998; ASCE 2002). Crop coefficients for reference

crop ET methods have not been developed to account for

the effects of land use practices.

The P–M model has been used to estimate crop ET in a

one-step approach. Accuracy has varied but the model

generally performs best when the leaf area index (LAI) of

the crop exceeds 2. Ortega-Farias et al. (2004) evaluated

the P–M model for soybeans for varying soil water and

atmospheric conditions. Large disagreements were found

for hourly estimates of ET; however, performance on a

daily basis was more acceptable when LAI ranged from 0.3

to 4. Kjelgaard and Stockle (2001) evaluated three surface

resistance methods in the P–M model for maize and

potatoes by comparing ET estimates to daily Bowen ratio

energy balance system measurements. None of the surface

resistance methods were reliable for estimating ET for

maize; however, all methods performed well for potatoes.

Rana et al. (1997) estimated ET with the P–M model for

stressed soybeans and produced good results for hourly,

daily and seasonal time scales. Shuttleworth (2006)

presented a theoretical analysis to use the P–M model for

one-step estimation of crop water requirements. He used a

blending height in the atmospheric boundary layer where

meteorological conditions are independent of the underly-

ing crop. Aerodynamic resistance and vapor pressure

deficit (from climate variables at 2 m) were used to

estimate ET. Flores (2007) used the P–M method for maize

ET and characterized uncertainties when weather data are

measured above grass. The model worked well for a full-

crop cover under well-watered conditions. The uncertainty

introduced from measuring weather data over grass was

small.

The next phase of modeling recognized that sparse

vegetation and crops with partial canopy cover may not

satisfy the big leaf assumption and models were developed

to predict transpiration and evaporation separately.

Shuttleworth and Wallace (1985) combined one-dimensional

models of transpiration and evaporation using surface

resistances to regulate heat and mass transfer from plant

and soil surfaces, and aerodynamic resistance to regulate

fluxes to the atmosphere. Several studies evaluated the

Shuttleworth and Wallace (1985) (S–W) model. Farahani

and Bausch (1995) compared the P–M and S–W models for

irrigated maize and found that the P–M model performed

poorly when the LAI was less than 2 because soil evapo-

ration was neglected in calculating surface resistance. The

S–W model performed satisfactorily for the entire range of

canopy cover. Stannard (1993) compared the P–M, S–W

and Priestley–Taylor ET models for sparsely vegetated,

semiarid rangeland. The P–M model was not sufficiently

accurate while the S–W model performed significantly

better for hourly and daily data. Lafleur and Rouse (1990)

compared the S–W model with ET measured with a Bowen

ratio energy balance system for crop cover ranging from no

vegetation to full cover. The S–W model agreed with

hourly and daily measurements for all values of LAI.

Farahani and Ahuja (1996) extended the S–W model to

include the effects of crop residues on soil evaporation by

adding a partially covered soil area and partitioning

evaporation between bare and residue-covered areas. Iritz

et al. (2001) modified the S–W model to estimate ET for a

forest. The modification consisted of a two-layer soil

module which calculated soil surface resistance as a

function of the wetness of the top soil. They simulated

seasonal evaporation fairly well. Tourula and Heikinheimo

(1998) modified soil surface and aerodynamic resistances

in the S–W model to produce daily and hourly estimates

that agreed with measured ET. The S–W model compared

well to data from eddy covariance systems for a vineyard in

an arid environment (Ortega-Farias et al. 2007).

The Shuttleworth–Wallace approach requires soil heat

flux (G) to solve the energy balance. Commonly, G is

calculated as a percentage of net radiation (Rn). However,

although soil heat flux is related to net radiation, it is also

affected by others parameters (i.e., surface cover, soil

moisture content, and soil thermal conductivity).

Shuttleworth and Wallace (1985) estimated G as 20% of Rn.

Allen et al. (1998) estimated daily reference ET assuming

that G for a fully vegetated grass or alfalfa is small

compared to Rn (i.e., G = 0). For hourly simulation, G

was estimated as 10% of Rn during the day and half of

Rn during the night for a reference grass surface, and 4%

of Rn for the day and 20% of Rn during the night for an

alfalfa reference.

A more complete surface energy balance including the

estimation of soil heat flux was presented by Choudhury

and Monteith (1988), they developed a four-layer model

for the heat budget of homogeneous land surfaces using

explicit solutions for the conservation of heat and water

vapor in a uniform vegetation and soil system. An impor-

tant feature was the ability of the model to partition the

available energy into sensible heat, latent heat, and soil

Irrig Sci (2009) 28:51–64 53

123

heat flux for the canopy/soil system. This model offers the

possibility to include the effect of residue on total ET.

Therefore, our goal was to develop a multiple-layer

surface energy balance (SEB) model that accounts for the

effects of canopy and residue on ET. This paper describes

the development of the model and the procedures used to

compute parameters for use in the energy balance. Simu-

lated ET during the growing season for maize (Zea mays

L.) and before maize growth was compared to measure-

ments using an eddy covariance flux system (e.g., Suyker

and Verma 2009) to assess model performance.

Materials and methods

SEB model

Our model combines and extends previous ET models by

Shuttleworth and Wallace (1985) and Choudhury and

Monteith (1988). The model has four layers (Fig. 1). The

first layer extended from the reference height above the

vegetation to the sink for momentum within the canopy,

a second layer between the canopy and the soil surface, a

third layer corresponding to the top soil layer where

surface resistance can be calculated as a function of soil

water content and the fourth, a lower soil layer where the

soil atmosphere is nearly saturated with water vapor. The

daily soil temperature is held constant at the bottom of

the lower layer. The SEB model distributes net radiation

(Rn) into sensible heat (H), latent heat (kE), and soil heat

fluxes (G) through the soil–canopy system (Fig. 2). Total

latent heat (kE) is the sum of latent heat from the canopy

(kEc), latent heat from the soil (kEs) and latent heat from

the residue-covered soil (kEr). Similarly, sensible heat is

calculated as the sum of sensible heat from the canopy

(Hc), sensible heat from the soil (Hs) and sensible heat

from the residue-covered soil (Hr). Horizontal gradients

of the potentials and physical and biochemical energy

storage terms in the canopy/residue/soil system are

neglected.

The total net radiation is divided into that absorbed by

the canopy (Rnc) and the soil (Rns) and is given by

Rn = Rnc ? Rns. The net radiation absorbed by the can-

opy is divided into latent heat and sensible heat fluxes as

Rnc = kEc ? Hc. Similarly, for the soil, Rns = Gos ? Hs,

where Gos is a conduction term downwards from the soil

surface and is expressed as Gos = kEs ? Gs, where Gs is

the soil heat flux for bare soil. Similarly, for the residue-

covered soil, Rns = Gor ? Hr, where Gor is the conduction

downwards from the soil covered by residue. The con-

duction is given by Gor = kEr ? Gr, where Gr is the soil

heat flux for residue-covered soil.

The total latent heat flux from the canopy/residue/soil

system is given by

kE ¼ kEc þ ð1� frÞkEs þ frkEr ð1Þ

where fr is the fraction of the soil affected by residue. The

total sensible heat is given as

Fig. 1 Fluxes of the surface

energy balance (SEB) model

54 Irrig Sci (2009) 28:51–64

123

H ¼ Hc þ ð1� frÞHs þ frHr ð2Þ

The differences in vapor pressure and temperature

between levels can be expressed with an Ohm’s law

analogy using appropriate resistance and flux terms

(Fig. 2). The sensible and latent heat fluxes from the

canopy, from bare soil and soil covered by residue are

expressed by

Hc ¼qCpðT1 � TbÞ

r1

and kEc ¼qCpðe�1 � ebÞ

cðr1 þ rcÞð3Þ

Hs ¼qCpðT2 � TbÞ

r2

and kEs ¼qCpðe�L � ebÞ

cðr2 þ rsÞð4Þ

Hr ¼qCpðT2r � TbÞ

r2 þ rrh

and kEr ¼qCpðe�Lr � ebÞcðr2 þ rs þ rrÞ

ð5Þ

where q is the density of moist air, Cp is the specific heat

of air, c is the psychrometric constant, T1 is the mean

canopy temperature, T2 is the temperature at the soil

surface, Tb is the air temperature within the canopy, T2r is

the temperature of the soil covered by residue, r1 is an

aerodynamic resistance between the canopy and the air, rc

is the surface canopy resistance, r2 is the aerodynamic

resistance between the soil and the canopy, rs is the

resistance to the diffusion of water vapor at the top soil

layer, rrh is the residue resistance to transfer of heat, rr is

the residue resistance to transfer of vapor acting in series

with the soil resistance rs, eb is the vapor pressure of the

atmosphere at the canopy level, e�1 is the saturation vapor

pressure in the canopy, e�L is the saturation vapor pressure

at the top of the wet layer, and e�Lr is the saturation vapor

pressure at the top of the wet layer for the soil covered by

residue.

Conduction of heat for the bare-soil and residue-covered

surfaces are given by

Gos ¼qCpðT2 � TLÞ

ru

and Gs ¼qCpðTL � TmÞ

rL

ð6Þ

Gor ¼qCpðT2r � TLrÞ

ru

and Gr ¼qCpðTLr � TmÞ

rL

ð7Þ

where ru and rL are resistance to the transport of heat for

the upper and lower soil layers, respectively, TL and TLr are

the temperatures at the interface between the upper and

lower layers for the bare soil and the residue-covered soil,

and Tm is the temperature at the bottom of the lower layer

which was assumed to be constant on a daily basis.

Choudhury and Monteith (1988) expressed differences in

saturation vapor pressure between points in the system as

linear functions of the corresponding temperature differences.

They found that a single value of the slope of the saturation

vapor pressure, D, when evaluated at the air temperature

Ta gave acceptable results for the components of the heat

balance. The vapor pressure differences were given by

e�1 � e�b ¼ DðT1 � TbÞ; e�L � e�b ¼ DðTL � TbÞ;e�b � e�a ¼ DðTb � TaÞ; and e�Lr � e�b ¼ DðTLr � TbÞ

ð8Þ

The above equations were combined and solved to estimate

fluxes. Details are provided by Lagos (2008). The solution

gives the latent and sensible heat fluxes from the canopy as

kEc ¼Dr1Rnc þ qCpðe�b � ebÞ

Dr1 þ cðr1 þ rcÞand

Hc ¼cðr1 þ rcÞRnc � qCpðe�b � ebÞ

Dr1 þ cðr1 þ rcÞ

ð9Þ

Similarly, latent and sensible heat fluxes from bare-soil

surfaces are estimated by

kEs

¼ RnsDr2rL þ qCp½ðe�b � ebÞðru þ rL þ r2Þ þ ðTm � TbÞDðru þ r2Þ�cðr2 þ rsÞðru þ rL þ r2Þ þ DrLðru þ r2Þ

ð10Þ

Hs ¼RnsrLD�kEs½rLDþ cðr2þ rsÞ�þqCpðe�b� ebÞ�qCpDðTb�TmÞ

rLD

ð11Þ

The latent and sensible heat fluxes from the residue-

covered soil are simulated with

Values for Tb and eb are necessary to estimate latent heat

and sensible heat fluxes. The values of the parameters can

be expressed as

kEr ¼RnsDðr2 þ rrhÞrL þ qCp½ðe�b � ebÞðru þ rL þ r2 þ rrhÞ þ ðTm � TbÞDðru þ r2 þ rrÞ�

cðr2 þ rs þ rrÞðru þ rL þ r2 þ rrhÞ þ DrLðru þ r2 þ rrhÞð12Þ

Hr ¼RnsrLD� kEr½rLDþ cðr2 þ rs þ rrÞ� þ qCpðe�b � ebÞ � qCpDðTb � TmÞ

rLDð13Þ

Irrig Sci (2009) 28:51–64 55

123

Fig. 2 A schematic resistance

network of the surface energy

balance (SEB) model: a latent

heat flux and b sensible heat

flux

eb¼ TbðDA2�A3ÞþA1

qCp�DA2TaþA2e�aþTmA3þ

ea

craw

� �

� craw

1þA2craw

� �ð14Þ

Tb ¼B1

qCpþ Ta

1

rah

� DB2

� �þ ðe�a � ebÞB2 þ TmB3

� �

� rah

1� DB2rah þ B3rah

� �ð15Þ

56 Irrig Sci (2009) 28:51–64

123

where rah is the aerodynamic resistance for heat transport,

raw is the aerodynamic resistance for water vapor transport,

ea is the vapor pressure at the reference height, and e�a is the

saturated vapor pressure at the reference height. Six coef-

ficients (A1, A2, A3 and B1, B2 and B3) are involved in these

expressions. These coefficients depend on environmental

conditions and other parameters. The expressions to com-

pute the coefficients are given in ‘‘Appendix’’.

These relationships define the SEB model which is

applicable to conditions ranging from closed canopies to

surfaces with bare soil or those partially covered with

residue. Without residue the model is similar to that by

Choudhury and Monteith (1988).

Model parameters

We also developed procedures to compute parameter val-

ues for the model. This process is as important as the

formulation of the energy balance equations.

Aerodynamic resistances

Thom (1972) stated that heat and mass transfer encounters

greater aerodynamic resistance than the transfer of

momentum. Accordingly, aerodynamic resistances to heat

(rah) and water vapor transfer (raw) can be estimated as

rah ¼ ram þ rbh and raw ¼ ram þ rbw ð16Þ

where ram is the aerodynamic resistance to momentum

transfer, and rbh and rbw are excess resistance terms for heat

and water vapor transfer.

Shuttleworth and Gurney (1990) built on the work of

Choudhury and Monteith (1988) to estimate ram by

integrating the eddy diffusion coefficient over the sink

of momentum in the canopy to a reference height zr

above the canopy giving the following relationship for

ram:

ram ¼1

ku�ln

zr � d

h� d

� �þ h

aKh

exp a 1� zo þ d

h

� �� �� 1

� �

ð17Þ

where k is the von Karman constant, u* is the friction

velocity, zo is the surface roughness, d is the zero plane

displacement height, Kh is the value of eddy diffusion

coefficient at the top of the canopy, h is the height of

vegetation, and a is the attenuation coefficient. A value of

a = 2.5, which is typical for agricultural crops, was rec-

ommended by Shuttleworth and Wallace (1985) and

Shuttleworth and Gurney (1990).

Verma (1989) expressed the excess resistance for heat

transfer as

rbh ¼kB�1

ku�ð18Þ

where B-1 represents a dimensionless bulk parameter.

Thom (1972) suggests that the product kB-1 equal

approximately 2 for most arable crops.

Excess resistance was derived primarily from heat

transfer observations (Weseley and Hicks 1977). Aerody-

namic resistance to water vapor was modified by the ratio

of thermal and water vapor diffusivity:

rbw ¼kB�1

ku�k1

Dv

� �2=3

ð19Þ

where k1 is the thermal diffusivity and Dv is the molecular

diffusivity of water vapor in air.

Similarly, Shuttleworth and Gurney (1990) expressed

the aerodynamic resistance (r2) by integrating the eddy

diffusion coefficient between the soil surface and the sink

of momentum in the canopy to yield:

r2 ¼h expðaÞ

aKh

exp�az0o

h

� �� exp

�aðd þ zoÞh

� �� �ð20Þ

where z0o is the roughness length of the soil surface. Values

of surface roughness (zo) and displacement height (d) are

functions of LAI and can be estimated using the expres-

sions given by Shaw and Pereira (1982).

The diffusion coefficients between the soil surface and

the canopy, and therefore the resistance for momentum,

heat, and vapor transport are assumed equal although it is

recognized that this is a weakness in the use of the K theory

to describe through-canopy transfer (Shuttleworth and

Gurney 1990). Stability was not considered.

Canopy resistances

The mean boundary layer resistance of the canopy r1, for

latent and sensible heat flux, is influenced by the surface

area of vegetation (Shuttleworth and Wallace 1985):

r1 ¼rb

2LAIð21Þ

where rb is the resistance of the leaf boundary layer, which

is proportional to the temperature difference between the

leaf and surrounding air divided by the associated flux

(Choudhury and Monteith 1988). Shuttleworth and

Wallace (1985) noted that resistance rb exhibits some

dependence on in-canopy wind speed, with typical values

of 25 s m-1. Shuttleworth and Gurney (1990) represented

rb as

rb ¼100

aw

uh

� �1=2

1� exp�a2

� �� ��1

ð22Þ

Irrig Sci (2009) 28:51–64 57

123

where w is the representative leaf width and uh is the wind

speed at the top of the canopy. This resistance is only

significant when acting in combination with a much larger

canopy surface resistance and Shuttleworth and Gurney

(1990) suggest that r1 could be neglected for foliage

completely covering the ground. Using rb = 25 s m-1 with

an LAI = 4, the corresponding canopy boundary layer

resistance is r1 = 3 s m-1.

Canopy surface resistance, rc, can be calculated by

dividing the minimum surface resistance for a single leaf

(rL) by the effective canopy LAI. Five environmental

factors have been found to affect stomata resistance: solar

radiation, air temperature, humidity, CO2 concentration

and soil water potential (Yu et al. 2004). Several models

have been developed to estimate stomata conductance and

canopy resistance. Stannard (1993) estimated rc as a

function of vapor pressure deficit, LAI, and solar radiation

as

rc ¼ C1

LAI

LAImax

C2

C2 þ VPDa

RadðRadmax þ C3ÞRadmaxðRadþ C3Þ

� ��1

ð23Þ

where LAImax is the maximum value of LAI, VPDa is

vapor pressure deficit, Rad is solar radiation, Radmax is

maximum value of solar radiation (estimated at

1,000 W m-2) and C1, C2 and C3 are regression coeffi-

cients. The canopy resistance does not account for soil

water stress effects.

Soil resistances

Farahani and Bausch (1995), Anadranistakis et al. (2000)

and Lindburg (2002) found that soil resistance (rs)

can be related to volumetric soil water content in the top

soil layer. Farahani and Ahuja (1996) found that the ratio

of soil resistance when the surface layer is wet rela-

tive to its upper limit depends on the degree of satura-

tion (h/hs) and can be described by an exponential

function as

rs ¼ rso exp �bhhs

� �and rso ¼

Ltss

Dv/ð24Þ

where Lt is the thickness of the surface soil layer, ss is a soil

tortuosity factor, Dv is the water vapor diffusion coefficient

and / is soil porosity, h is the average volumetric water

content in the surface layer, hs is the saturation water

content, and b is a fitting parameter. Measurements of hfrom the top 0.05-m soil layer were more effective in

modeling rs than h for thinner layers.

Choudhury and Monteith (1988) expressed the soil

resistance for heat flux (rL) in the soil layer extending from

depth Lt to Lm as

rL ¼qCpðLm � LtÞ

Kð25Þ

where K is the thermal conductivity of the soil. Similarly,

the corresponding resistance for the upper layer (ru) of

depth Lt and conductivity K0 as

ru ¼qCpLt

K 0ð26Þ

Residue resistances

Surface residue is an integral part of many cropping sys-

tems. Bristow and Horton (1996) showed that partial sur-

face mulch cover can have dramatic effects on the soil

physical environment. The vapor conductance through

residue has been described as a linear function of wind

speed. Farahani and Ahuja (1996) used results from Tanner

and Shen (1990) to develop the resistance of surface resi-

due (rr) as

rr ¼Lrsr

Dv/r

ð1þ 0:7u2Þ�1 ð27Þ

where Lr is residue thickness, sr is residue tortuosity, Dv is

vapor diffusivity in still air, /r is residue porosity and u2 is

wind speed measured 2 m above the surface. Due to the

porous nature of field crop residue layers, the ratio sr//r is

about one (Farahani and Ahuja 1996).

Similar to the soil resistance, Bristow and Horton (1996)

and Horton et al. (1996) expressed the resistance of residue

for heat transfer, rrh, as

rrh ¼qCpLr

Kr

ð28Þ

where Kr is the thermal conductivity of the residue.

The fraction of the soil covered by residue (fr) can be

estimated using the amount and type of residue (Steiner

et al. 2000). The soil covered by residue and the residue

thickness were estimated using the expressions developed

by Gregory (1982).

Model inputs necessary to solve the surface energy

balance are as follows: net radiation, solar radiation, air

temperature, relative humidity, wind speed, LAI, crop

height, soil texture, soil temperature, soil water content,

residue type, and residue amount. All others parameters

can be calibrated or defined from literature accordingly to

canopy, soil and residue characteristics.

Net radiation

Similar to the Shuttleworth and Wallace (1985) and

Choudhury and Monteith (1988) models, measurements of

net radiation and estimations of net radiation absorbed by

the canopy are necessary for the SEB model. We used

58 Irrig Sci (2009) 28:51–64

123

Beer’s law to estimate the penetration of radiation through

the canopy and estimated the net radiation reaching the

surface (Rns) as

Rns ¼ Rn expð�CextLAIÞ ð29Þ

where Cext is the extinction coefficient of the crop for net

radiation. Consequently, net radiation absorbed by the

canopy (Rnc) can be estimated as Rnc = Rn - Rns.

Study site

An irrigated maize field site located at the University of

Nebraska Agricultural Research and Development Center

near Mead, NE (41�09053.500N, 96�28012.300W, elevation

362 m) was used for model evaluation. This site is a 49 ha

production field that provide sufficient upwind fetch of

uniform cover required for adequately measuring mass and

energy fluxes using eddy covariance systems. The climate

in this area is humid continental climate and the soil cor-

responds to a deep silty clay loam (Suyker and Verma

2009). The field has not been tilled since 2001. Detailed

information about planting densities and crop manage-

ments are provided by Verma et al. (2005) and Suyker and

Verma (2009).

Soil water content was measured continuously at four

depths (0.10, 0.25, 0.5 and 1.0 m) with Theta probes

(Delta-T Device, Cambridge, UK). Destructive green LAI

and biomass measurements were taken bi-monthly during

the growing season. The eddy covariance measurements

of latent heat, sensible heat, and momentum fluxes were

made using an omnidirectional three-dimensional sonic

anemometer (Model R3, Gill Instruments Ltd., Lyming-

ton, UK1) and an open-path infrared CO2/H2O gas ana-

lyzer system (Model LI7500, Li-Cor Inc., Lincoln, NE).

Fluxes were corrected for sensor frequency response and

variations in air density. More details of measurements

and calculations are given in Verma et al. (2005). Air

temperature and humidity were measured at 3 and 6 m

(Humitter 50Y, Vaisala, Helsinki, Finland), net radiation

at 5.5 m (CNR1, Kipp and Zonen, Delft, the Netherlands)

and soil heat flux at 0.06 m (Radiation and Energy Bal-

ance Systems Inc., Seattle, WA). Soil temperature was

measured at 0.06, 0.1, 0.2 and 0.5 m depths (Platinum

RTD, Omega Engineering, Stamford, CT). More details

are given in Verma et al. (2005) and Suyker and Verma

(2009).

Results and discussions

Sensitivity analysis

A sensitivity analysis of ET rates predicted with the

modified SEB model to parameter value changes was

evaluated. Calculations were based on typical midday

conditions during the growing season of maize in south-

eastern Nebraska. Results are expressed as the percent

difference between results for parameter changes relative

to results for the base parameter value (Table 1). Simulated

ET was most sensitive to changes in surface canopy

resistance for high LAI conditions. Simulated ET varied by

about ±8% of the base ET when the canopy resistance was

varied by ±30% for high LAI values. Soil surface resis-

tance and residue resistance were most significant for low

LAI conditions with variations in ET of approximately

±50% of base ET for soil resistance and ±15% for residue

resistance changes. The soil and residue resistances were

varied to represent very wet conditions (i.e., no resistance)

and dry conditions. The model was sensitive to the soil heat

flux resistance for the lower soil layer especially for small

LAI values with variations of about ±13% for an LAI of

0.1. For LAI values of 3 and 6, the effect of the soil heat

flux resistance of the lower layer was between (±3%). The

model was less sensitive to the extinction coefficient,

attenuation coefficient, crop height and soil heat flux of the

upper soil layer. More details are given by Lagos (2008).

Model evaluation

Evapotranspiration predictions from the SEB model were

compared with eddy covariance flux measurements during

2003 for an irrigated maize field. To evaluate the energy

balance closure of eddy covariance measurements, net

radiation was compared against the sum of latent heat,

sensible heat, soil heat flux and storage terms. Storage

terms include soil heat storage, canopy heat storage, and

energy used in photosynthesis. Storage terms were calcu-

lated by Suyker and Verma (2009) following Meyers and

Hollinger (2004). During these days, the regression slope

for energy balance closure was 0.89 with a correlation

coefficient of r2 = 0.98.

For model evaluation, 15 days under different LAI

conditions were selected to initially test the model; how-

ever, further work is needed to test the model for entire

growing seasons and during longer periods. Hourly data for

three 5-day periods with varying LAI conditions (LAI = 0,

1.5 and 5.4) were used to compare measured ET to model

predictions. Input data of the model included hourly values

for: net radiation, air temperature, relative humidity, soil

temperature at 50 cm, wind speed, solar radiation and soil

water content. During the first 5-day period, which was

1 Mention of product names is for information only and does not

imply endorsement by the authors or the University of Nebraska-

Lincoln.

Irrig Sci (2009) 28:51–64 59

123

prior to germination, the maximum net radiation ranged

from 240 to 720 W m-2, air temperature ranged from 10 to

30�C, soil temperature was fairly constant at 16�C, and

wind speed ranged from 1 to 9 m s-1 but was generally

less than 6 m s-1 (Fig. 3). Soil water content in the

evaporation zone averaged 0.34 m3 m-3 and the residue

density was 12.5 ton ha-1 on 6 June 2003. Precipitation

occurred on the second and fifth days totaling 17 mm.

ET estimated with the SEB model and measured using

the eddy covariance system is given in Fig. 4. ET fluxes

were the highest at midday on 6 June reaching approxi-

mately 350 W m-2. The lowest ET rates occurred on the

second day. Estimated ET tracked measured latent heat

fluxes reasonably well. Estimates were better for days

without precipitation than for days when rainfall occurred.

The effect of crop residue on evaporation from the soil is

shown in Fig. 4 for this period. Residue reduced cumula-

tive evaporation by approximately 17% during this 5-day

period. Evaporation estimated with the SEB model on 6

and 9 June was approximately 3.5 mm day-1, totaling

approximately half of the total evaporation for 5 days.

During the second 5-day period, when plants partially

shaded the soil surface (LAI = 1.5), the maximum net

radiation ranged from 350 to 720 W m-2 and air temper-

ature ranged from 10 to 33�C (Fig. 5). The soil temperature

was nearly constant at 20�C. Wind speed ranged from 0.3

to 8 m s-1 but was generally less than 6 m s-1. The soil

water content was about 0.31 m3 m-3 and the residue

density was 12.2 ton ha-1 on 24 June 2003. Precipitation

occurred on the fifth day totaling 3 mm. The predicted rate

of ET estimated with the SEB model was close to the

observed data (Fig. 6). Estimates were smaller than mea-

sured values for 24 June which was the hottest and windiest

of the period. The ability of the model to partition ET into

evaporation and transpiration for partial canopy conditions

is also illustrated in Fig. 6. Evaporation from the soil

represented the majority of the water use during the night,

and early or late in the day. During the middle of the day,

transpiration represented approximately half of the hourly

ET flux.

Table 1 Parameter values and relative difference of evapotranspiration estimate for sensitivity analysis

Parameter Parameter value Relative differences for leaf area indexes (%)

LAI = 0.1 LAI = 3 LAI = 6

Base Low High Low High Low High Low High

Canopy resistance, rc (s m-1) 65.8 46.1 85.5 -2.12 1.36 -8.61 7.01 -8.99 7.45

Soil resistance, rs (s m-1) 227 0 1,500 -45.7 54.3 -4.87 4.86 -2.02 1.69

Residue resistance, rr (s m-1) 400 0 1,000 -18.3 12.1 -1.70 1.01 -0.60 -0.02

Attenuation coefficient, a 2.5 1 3.5 -0.79 3.96 -0.52 0.70 -0.45 0.30

Crop height, h (m) 2.3 1.6 3 0.01 -0.01 -0.07 0.43 1.67 -1.42

Soil heat flux resistance, ru (s m-1) 63.5 44.4 82.6 -1.35 1.29 0.07 -0.06 0.03 -0.03

Extinction coefficient, Cext 0.6 0.4 0.8 0.15 -0.14 2.69 -1.48 1.11 -0.33

Soil heat flux resistance, rL (s m-1) 415 290 540 14.5 -10.5 2.91 -2.03 1.07 -0.73

Fig. 3 Environmental conditions during a 5-day period without

canopy cover for net radiation (Rn), air temperature (Ta), soil

temperature (Tm), precipitation (Prec.), vapor pressure deficit (VPD),

and wind speed (u)

60 Irrig Sci (2009) 28:51–64

123

The last period represents a fully developed maize

canopy that completely shaded the soil surface. The crop

height was 2.3 m and the LAI was 5.4. Environmental

conditions for the period are given in Fig. 7. The maximum

net radiation ranged from 700 to 740 W m-2 and air

temperature ranged from 15 to 36�C during the period. Soil

temperature was fairly constant during the 5 days at 21.5�C

and wind speed ranged from 0.3 to 4 m s-1. The soil water

content was about 0.25 m3 m-3 and the residue density

was 11.8 ton ha-1 on 16 July 2003. Precipitation occurred

on the third day totaling 29 mm. Observed and predicted

ET fluxes agreed for most days with some differences early

in the morning during the first day and during the middle of

several days (Fig. 8). Transpiration simulated with the SEB

model was nearly equal to the simulated ET for the period

as evaporation rates from the soil was very small.

Hourly measurements and SEB predictions for the three

5-day periods were combined to evaluate the overall per-

formance of the model (Fig. 9). Results show variation

about the 1:1 line; however, there is a strong correlation

and the data are reasonably well distributed about the line.

Modeled ET is less than measured for latent heat fluxes

above 450 W m-2. The model underestimates ET during

hours with high values of vapor pressure deficit (Figs. 6,

8), this suggests that the linear effect of vapor pressure

deficit in canopy resistance estimated with Eq. 23 pro-

duces a reduction on ET estimations. Further work is

required to evaluate and explore if different canopy

resistance models improve the performance of ET pre-

dictions under these conditions. Various statistical tech-

niques were used to evaluate the performance of the

model. The coefficient of determination, Nash–Sutcliffe

coefficient, index of agreement, root mean square error

and the mean absolute error were used for model evalua-

tion (Legates and McCabe 1999; Krause et al. 2005;

Moriasi et al. 2007; Coffey et al. 2004). The coefficient of

Fig. 5 Environmental conditions for a 5-day period with partial crop

cover for net radiation (Rn), air temperature (Ta), soil temperature

(Tm), precipitation (Prec.), vapor pressure deficit (VPD), and wind

speed (u)

Fig. 4 Evapotranspiration estimated by the surface energy balance

(SEB) model and measured by an eddy covariance system and

simulated cumulative evaporation from bare and residue-covered soil

for a period without plant canopy cover

Fig. 6 Evapotranspiration and transpiration estimated by the surface

energy balance (SEB) model and ET measured by an eddy covariance

system for a 5-day period with partial canopy cover

Irrig Sci (2009) 28:51–64 61

123

determination was 0.92 with a slope of 0.90 over the range

of hourly ET values. The root mean square error was

41.4 W m-2, the mean absolute error was 29.9 W m-2,

the Nash–Sutcliffe coefficient was 0.92 and the index of

agreement was 0.97. The statistical parameters show that

the model reasonably well represents field measurements.

Similar performance was obtained for daily ET estimations

(Table 2). Analysis is underway to evaluate the model for

more conditions and longer periods. Simulations reported

here relied on literature-reported parameter values. We are

also exploring calibration methods to improve model

performance.

Conclusions

A SEB model based on the Shuttleworth–Wallace and

Choudhury–Monteith models was developed to account for

the effect of residue, soil evaporation and canopy transpi-

ration on ET. The model describes the energy balance of

vegetated and residue-covered surfaces in terms of driving

potential and resistances to flux. Improvements in the SEB

model were the incorporation of residue in the energy

balance and modification in aerodynamic resistances for

heat and water transfer, canopy resistance for water flux,

residue resistances for heat and water flux, and soil resis-

tance for water transfer. The model requires hourly data for

net radiation, solar radiation, air temperature, relative

humidity, and wind speed. LAI and crop height plus soil

texture, temperature and water content as well as the type

and amount of crop residue are also required. An important

feature of the model is the ability to estimate latent, sen-

sible and soil heat fluxes. The model provides a method for

partitioning ET into soil/residue evaporation and plant

transpiration, and a tool to estimate the effect of residue ET

and consequently on water balance studies. Comparison

between estimated ET and measurements from an irrigated

maize field provides support for the validity of the SEB

Fig. 8 Evapotranspiration and transpiration estimated by the surface

energy balance (SEB) model and ET measured by an eddy covariance

system during a period with full canopy cover

Fig. 9 Measured versus modeled hourly latent heat fluxes

Fig. 7 Environmental conditions for 5-day period with full canopy

cover for net radiation (Rn), air temperature (Ta), soil temperature

(Tm), precipitation (Prec.), vapor pressure deficit (VPD) and wind

speed (u)

62 Irrig Sci (2009) 28:51–64

123

model. Further evaluation of the model is underway for

agricultural and natural ecosystems during growing seasons

and dormant periods. We are developing calibration pro-

cedures to refine parameters and improve model results.

Acknowledgments This project was partially supported by funding

from the US EPA, the University of Nebraska Program of Excellence

and the University of Nebraska-Lincoln Institute of Agriculture and

Natural Resources. Their support is gratefully recognized.

Appendix

A1¼Dr1Rnc

Dr1þ cðr1þ rcÞþð1� frÞ

RnsDr2rL

cðr2þ rsÞðruþ rLþ r2ÞþDrLðruþ r2Þ

�

þ frRnsDðr2þ rrhÞrL

cðr2þ rsþ rrÞðruþ rLþ r2þ rrhÞþDrLðruþ r2þ rrhÞ

�

A2 ¼1

Dr1þ cðr1þ rcÞþð1� frÞ

ðruþ rLþ r2Þcðr2þ rsÞðruþ rLþ r2ÞþDrLðruþ r2Þ

�

þ frðruþ rLþ r2þ rrhÞ

cðr2þ rsþ rrÞðruþ rLþ r2þ rrhÞþDrLðruþ r2þ rrhÞ

�

A3¼ ð1� frÞDðruþ r2Þ

cðr2þ rsÞðruþ rLþ r2ÞþDrLðruþ r2Þ

�

þ fr

Dðruþ r2þ rrhÞcðr2þ rsþ rrÞðruþ rLþ r2þ rrhÞþDrLðruþ r2þ rrhÞ

�

B1 ¼ Rnc

cðr1 þ rcÞDr1 þ cðr1 þ rcÞ

þ Rnsðð1� frÞð1� Dr2rLXsÞ�

þ frð1� Dðr2 þ rrhÞrLXrÞÞ�

B2 ¼�1

Dr1 þ cðr1 þ rcÞþ ð1� frÞ

1

rLD� ðru þ rL þ r2ÞXs

� ��

þ fr1

rLD� ðru þ rL þ r2 þ rrhÞXr

� ��

B3 ¼ ð1� frÞ1

rL

� Dðru þ r2ÞXs

� ��

þ fr1

rL

� Dðru þ r2 þ rrhÞXr

� ��

Xs ¼1

cðr2 þ rsÞðru þ rL þ r2Þ þ DrLðru þ r2Þ

� �

� ðrLDþ cðr2 þ rsÞÞrLD

� �

Xr¼1

cðr2þ rsþ rrÞðruþ rLþ r2þ rrhÞþDrLðruþ r2þ rrhÞ

� �

� ðrLDþcðr2þ rsþ rrÞÞrLD

� �

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