DRAFT REPORT ON POTENTIAL EVAPOTRANSPIRATION

MODELS

PREPARED BY:

Nicolette Taylor 1, John Annandale 1, Edossa Etissa 1 and Mark Gush 2 1University of Pretoria, 2CSIR

Deliverable submitted to the Water Research Commiss ion – Project K5/1770

March 2008

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CONTENTS……………………………………………………………………PAGE

1. INTRODUCTION ......................................................................................1

2. POTENTIAL EVAPOTRANSPIRATION MODELS ................ ..................3 2.1 Generic crop models .........................................................................3

2.1.1 FAO56 ........................................................................................3 2.1.2 SAPWAT and SAPWAT3 ...........................................................8 2.1.3 Soil Water Balance (SWB) .......................................................15 2.1.4 Penman-Monteith calculations of evapotranspiration ...............18 2.1.5 The Priestley-Taylor formula ....................................................21

2.2 Remote Sensing ..............................................................................22

2.2.1 Surface Energy Balance Algorithm for Land (SEBAL)..............22

3 POTENTIAL GROWTH MODELS............................ ..............................24 3.1 Generic Growth Models ...................................................................27

3.1.1 MAESTRA ................................................................................27 3.2 Specific crop growth models............................................................29

3.2.1 Pecan tree growth model..........................................................29 3.2.2 PEACH.....................................................................................32 3.2.3 ‘CITROS’ – A Dynamic Model of Citrus Productivity.................34

4. DISCUSSION AND CONCLUSIONS ......................... ............................35

5. REFERENCES .......................................................................................47

6. APPENDIX A - FLOW DIAGRAM INDICATING THE PROCEDURE FOR THE ESTIMATION OF CROP EVAPOTRANSPIRATION (ET C)...........55

7. APPENDIX B - FLOW CHART OF THE VARIOUS SUB-MODELS WITHIN THE SWB-2D MODEL............................ .................................56

8. APPENDIX C - FLOW CHART OF THE VARIOUS OBJECTS WITH IN THE PECAN TREE GROWTH MODEL (ANDALES ET AL. 2006).. .....60

1

1. INTRODUCTION

The need to improve irrigation scheduling and water use efficiency of fruit tree

crops in the summer and winter rainfall areas of South Africa, where water

stress is increasing, has been identified (Roux 2006). Existing models in

South Africa cannot confidently simulate water use of fruit trees for different

climate, soil, water and management conditions. This information is absolutely

essential when drawing up on-farm water management plans for fruit

production and for issuing licenses for water use. Many models have been

developed for field crops which have a one-dimensional canopy radiation

interception and water redistribution procedure. Fruit tree orchards are more

complex and this complexity needs to be taken into account when developing

evapotranspiration models for fruit trees. Process-based research on the

water use of fruit tree orchards is therefore required to provide accurate site

specific information, which can be extrapolated across regions, soils and

management practices in South Africa.

Most horticultural species have discontinuous canopies which results in

complex light interception dynamics, aerodynamics and gas exchange (Gary

et al. 1998). As a result of planting fruit trees in hedgerows, and allowing for

large spaces between rows for vehicle movement, distribution of energy is not

uniform across these widely spaced rows (Annandale et al. 2002). The

distribution of energy across rows will therefore depend on a number of

factors including, solar and row orientation, tree size and shape, as well as

slope and land aspect (Annandale et al. 2002). In addition, localised irrigation

(micro- or drip) used in orchards only wets a limited area under the canopy of

the trees, so evaporation from the surface is not uniform. The tree canopy

intercepts and channels precipitation down the stem, so even rainfall will not

be evenly distributed at the surface. All this makes traditional soil water

balance approaches to determining water use of fruit trees problematic. It is

hoped that with field measured sap flow rates, together with soil water content

(in at least a 2-D profile), total evaporation and the driving weather variables

2

for atmospheric demand, an ideal data set will be generated for calibrating

and verifying evapotranspiration models. Nicolas et al. (2005), Rana et al.

(2005) and Pereira et al. (2007b) have all used sap flow data to validate their

model.

It has been suggested that models fall within two broad categories: 1)

mechanistic models developed for scientific understanding of the processes in

nature and 2) functional or empirical models developed to solve management

problems (Passioura et al. 1996). Mechanistic models are based on

hypotheses, which may or may not be correct, on how plants grow. These

models are often difficult to parameterise, require large numbers of inputs and

as a result are not favoured by users due to their complexity. On the other

hand, models based on empirical relationships established in a given

environment are unlikely to apply outside that environment. These models

should only be used within the range over which they were calibrated. The

overall aim of a model should be accurate prediction on which to base sound

advice. As we would like to extrapolate to a range of management practices,

soil types and growing regions in South Africa, it is very important to use a

more mechanistic approach in order to gain an understanding of the system

and the driving variables for water use in the various fruit trees.

Prior to setting up a particular model the appropriateness of using the model

must be gauged. Numerous aspects need to be considered including the

availability of required input data, ability of the model to simulate the desired

processes and complexity of operation (ease of use / user support) among

others. We must be able to differentiate between water-supply- or

evaporative-demand-limited water uptake. A mechanistic model would

therefore be preferable, as empirical models often assume that water loss is

demand-limited. For real-time decisions the model should include a daily time

step, however, for planning purposes a monthly time step should be

adequate. The models proposed for evaluation in this project are the SWB

model, the FAO56 model and SAPWAT. These are general or generic crop

models that are applied to a wide range of crops. In addition to these, crop

specific growth models are available, e.g. pecan (Andales et al. 2006) and

3

peach (Grossman and Dejong 1994). It should be kept in mind that growth

models are often difficult to parameterise and a large number of variables will

have to be quantified as a result. More generic growth models are also

available such as MAESTRA (Medlyn 2004), which is a development of the

earlier MAESTRO model (Wang and Jarvis 1990) and is a three dimensional

model of forest canopy radiation absorption, photosynthesis and transpiration.

A different approach is the use of remote sensing to estimate the spatially

distributed surface energy balance in composite terrains, such as that found in

SEBAL (Surface Energy Balance Algorithm for Land) (Bastiaanssen et al.

1998). Possible collaboration with these authors could be invaluable when

considering the water use of fruit trees on a broader scale. The project team

are also aware of the drafting of new FAO33 yield response to water

guidelines for fruit trees and vines, which will deal particularly with how to

manage water during times of limited supply.

2. POTENTIAL EVAPOTRANSPIRATION MODELS

2.1 Generic crop models

2.1.1 FAO56

The FAO56 model (Allen et al. 1998) provides a means of calculating

reference and crop evapotranspiration from meteorological data and crop

coefficients. The effect of climate on crop water requirements is given by the

reference evapotranspiration (ETo), and the effect of the crop by the crop

coefficient Kc. Actual crop evapotranspiration (ETc) is calculated by multiplying

ETo by Kc :

ETc = Kc x ETo (1)

The calculation of ETo is based on the Penman-Monteith combination method,

and represents the evapotranspiration of a hypothetical reference crop (short

grass). The equation is as follows (Allen et al. 1998):

4

)34.01(

)(273

900)(408.0

2

2

u

eeuT

GRET

asn

o ++∆

−+

+−∆=

γ

γ (2)

where ETo reference evapotranspiration [mm day-1], Rn net radiation at the crop surface [MJ m-2 day-1], G soil heat flux density [MJ m-2 day-1], T mean daily air temperature at 2 m height [°C], u2 wind speed at 2 m height [m s-1], es saturation vapour pressure [kPa], ea actual vapour pressure [kPa], es - ea saturation vapour pressure deficit [kPa], ∆ slope of the vapour pressure curve [kPa °C -1], γ psychrometric constant [kPa °C -1].

The reference crop has the following unambiguous definition “A hypothetical

reference crop with an assumed height of 0.12 m, a fixed surface resistance

of 70 s m-1 and an albedo of 0.23, closely resembling the evaporation of an

extensive surface of green grass of uniform height, actively growing and

adequately watered.” The technique uses standard climatic data that can be

easily measured or derived from commonly collected weather station data (air

temperature, humidity, radiation and wind speed) and takes into account the

location (altitude above sea level and latitude).

Differences in the canopy and aerodynamic resistances of the crop being

simulated, relative to the reference crop, are accounted for within the crop

coefficient (Kc). Kc serves as an aggregation of the physical and physiological

differences between crops and takes into account canopy properties and the

aerodynamic resistance of the crop. Two calculation methods to derive ETc

from ETo are possible. The first approach integrates the relationships between

evapotranspiration of the crop and the reference surface into a single

coefficient (Kc). This is used for normal irrigation planning and management

purposes, for the development of basic irrigation schedules and for most

hydrological water balance studies. The second approach splits Kc into two

factors that separately describe the evaporation (Ke) and transpiration (Kcb)

components. This allows Kc to be calculated on a daily time step. There is an

attempt to simulate supply-limited water use by introducing a water stress co-

5

efficient (Ks) (Allen et al. 1998). The threshold point at which soil water

content is low enough to limit evapotranspiration takes into account the

atmospheric evaporative demand. In this case crop evapotranspiration will be

calculated as follows:

ETcadj = [KsKcb + Ke] x ETo (3)

A flow chart for the calculation of ETc is presented in Appendix A.

The FAO56 publication lists crop coefficients for numerous crops under

“standard conditions”, which implies that no limitations are placed on crop

growth or evapotranspiration from soil water and salinity stress, crop density,

pests and diseases, weed infestations or low fertility. The season is also split

into 4 distinct growth phases, viz. an initial phase, a development phase, a

mid-season phase and finally an end phase, each with its own associated Kc

value (Figure 1).

Figure 1: Crop coefficient curve for the four growth stages (Allen et al. 1998).

0.00

0.20

0.40

0.60

0.80

1.00

1.20

Time of season (days)

Kc

Kc ini

Kc mid

Kc end

initial mid-season late season

crop developmentt

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The list includes several deciduous and woody species such as certain fruit

and nut trees. Table 1 lists the crop factors for the selected species in this

study. Data for macadamia could not be found. Crop factors could be utilised

for the specific species they represent, or modified to better represent crop

development stages (and associated dates) for additional species not yet

represented. Multiplication of the reference evaporation by the basal crop

coefficient represents the upper envelope of crop transpiration where no

limitations are placed on plant growth or evapotranspiration. The option to

split the crop coefficient (Kc) into two factors that separately describe the

evaporation (Ke) and transpiration (Kcb) components is particularly suited to

this particular study because the field data that will be collected will yield

values of transpiration (excluding soil evaporation), and are consequently

directly comparable to the simulated values of Kcb. In addition, wetted areas

and frequency of wetting will have a big effect on water use. Another major

advantage of FAO56 is that all the necessary data to successfully run the

model will be collected at site and will be sufficient to use as input into the

model. This includes weather data, stage of growth and leaf area index.

A common occurrence with crop factors is that inaccuracies occur due to

assumptions made about the plant-soil system. Crop factor curves assume

that plant growth and development is dependent on calendar days, but

thermal time (degree days) and water supply play a more important role in

canopy development (Ritchie and NeSmith 1991). This limits the “universality”

of a particular crop factor. In addition, crop factors do not always take into

account processes that are taking place in the soil which impact on water

uptake by the plant. This includes infiltration, soil water potential and rooting

density, all of which vary with soil depth and distance from the tree row

(Annandale et al. 2000).

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Table 1: Crop coefficients, Kc, for non stressed, well-managed cops in

subhumid climates (RHmin≈ 45%, u2 ≈ 2 m s-1) for use with FAO Penman-

Monteith (Allen et al. 1998). Values for Macadamia could not be found in the

guidelines.

* The midseason value is lower than the initial and ending values due to the effects of stomatal closure during periods of peak ET (Allen et al. 1998). For humid and subhumid environments, where there is less control of stomates by citrus, the values can be increased by 0.1-0.2.

Fruit tree specie K c ini Kc mid Kc end Maximum crop height (m)

Apples

-no ground cover, killing frost

0.45 0.95 0.7 4

- no ground cover, no frost

0.60 0.95 0.75 4

- active ground cover, killing frost

0.50 1.20 0.95 4

-active ground cover, no frost 0.80 1.20 0.85 4

Peaches

-no ground cover, killing frost

0.45 0.90 0.65 3

- no ground cover, no frost 0.55 0.90 0.65 3

- active ground cover, killing frost 0.50 1.15 0.90 3

-active ground cover, no frost 0.80 1.15 0.85 3

Citrus , no ground cover

- 70% canopy 0.70 0.65* 0.70 4

- 50% canopy 0.65 0.60 0.65 3

- 20% canopy 0.50 0.45 0.50 2

Citrus , with active ground cover or weeds

- 70% canopy 0.75 0.70 0.75 4

- 50% canopy 0.80 0.80 0.80 3

- 20% canopy 0.85 0.85 0.85 2

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2.1.2 SAPWAT and SAPWAT3

SAPWAT is a user-friendly computer programme whose primary function is to

estimate crop evapotranspiration for planning purposes in a user-friendly

format (Crosby and Crosby 1999). In this way irrigation requirements for

application in planning, design and management can be deduced. It was not

designed as a real-time irrigation management tool. SAPWAT links to and is a

development of the FAO planning model CROPWAT, which has been derived

from several FAO irrigation and drainage papers. SAPWAT does not attempt

to model crop growth but is rather a management and planning aid that is

supported by extensive South African climate and crop databases. Some of

the major innovations incorporated into SAPWAT include (SAPWAT 2008):

1) the replacement of the American Class A evaporation pan with

reference evaporation from a short grass surface calculated from

climatic data by means of the standardised Penman-Monteith

procedure. The short grass surface has the characteristics of a

growing crop and this automatically compensates for regional

climatic differences

2) the use of FAO four stage methodology whereby crop factors

compatible with the short grass reference evaporation can be

developed and modified by applying simple and understandable

procedures

3) the facility to differentiate between soil evaporation and plant

transpiration, making it possible to cater for field crops, orchards

and vegetables in conjunction with a full range of irrigation methods

and strategies.

This is in accordance with FAO56.

For predictive purposes, SAPWAT uses average climatic data from weather

stations throughout South Africa (locally obtained weather data can be

inserted into the model) and crop factors which are specific for a geographic

region, planting date and time of year. According to the user manual, when

calculating crop factors and determining the crop factor curve for a particular

location provision is made for selecting the percentage cover at full growth.

9

Figures of less than 100% are usually found in the case of permanent crops,

such as orchards, where young trees will have a final cover of as low as 10%

in a given season and even when mature seldom reach 100%. In such an

instance, where a final cover of less than 100% is entered, a leaf area index is

calculated by the program. The percentage wetted area is also accounted for

when determining the crop factor curve, which is particularly important in

orchard crops where localised irrigation is used, in the form of micro- or drip

irrigation. From here the crop evapotranspiration is calculated, which allows

the estimation of crop water requirements and therefore irrigation. If

significant, rainfall can be included in the irrigation requirement calculations,

otherwise rainfall is ignored and irrigation requirements are calculated as if all

crop water requirements will be derived from irrigation alone. Irrigation

requirements are calculated according to the system used, the efficiency of

the irrigation, the target yield and the uniformity of distribution of the irrigation.

SAPWAT is a valuable tool for Water User Associations for registration and

licensing purposes, for macro-planning and for water demand management

strategies. The crop specific transpiration data generated by this particular

study will be very useful in validating the current selection of crop factors for

the selected fruit tree species as shown in Figures 2 to 5. We are confident

that this study will, at least, improve on the estimated crop factors for the

selected fruit tree species in South Africa.

10

Figure 2: Crop factors for early/short, medium and late/long apples in South

Africa. Data obtained from http://www.sapwat.org.za.

11

Figure 3: Crop factors for citrus in South Africa. Tables for “above average”,

“average” and “below average” citrus do not differ. Data obtained from

http://www.sapwat.org.za.

Figure 4: Crop factors for macadamia in South Africa Data obtained from

http://www.sapwat.org.za.

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Figure 5: Crop factors for short/early, medium and long/late peaches in South

Africa. Data obtained from http://www.sapwat.org.za.

The information contained in the above tables is interpreted as follows:

St.1 to St. 4 indicates the number of days per growth cycle, which is in

accordance with the four-stage crop factor curve. It should be noted that

SAPWAT uses a 360 day year. The initial stage in perennial crops starts at

13

the onset of full or semi-dormancy until re-growth has started. The second

phase is the crop development stage which lasts until effective full ground

cover is reached, during that particular season. The mid-season stage lasts

from reaching full effective ground cover, till the beginning of maturity, as

indicated by a change in leaf colour and leaf abscission. The final stage is the

late season stage and lasts from the end of the mid-season stage until full

maturity. In fruit trees harvesting takes place before the onset or at the

beginning of this stage. Kc is the maximum crop factor value expected and

therefore maximum transpiration. This usually varies between 1.1 and 1.2 for

crops at full cover. When determining crop factors, crops that do not give full

cover at maturity e.g. orchards, can be accounted for by adjusting the

percentage cover at full growth. Maximum Fv (foliage value) lies between 0.1

and 1.0 and accounts for those instances where transpiration is suppressed

(supply limited) e.g. in stressed or diseased plants. A value of 0.8 implies that

even though full canopy has been reached, crop transpiration is at an 80%

level of efficiency. Fv end (foliage value at harvest or at onset of dormancy) is

a value between 1 and 100. For crops that are harvested at full maturity this

would be 1, for those that are harvested at the end of stage 3, for example,

this value could be 100. Fv start (foliage value at the end of the dormancy

period for perennial crops) is a value between 1 and 100. This value is usually

1 for perennial crops that restart growth from bare branches and increases for

semi-dormant crops, such as subtropical fruit species. There is also provision

made for perennial crops. Within the months column, the onset of the fullest

level of dormancy is marked for perennial crops with a ‘Y’.

Water / irrigation requirements are estimated in SAPWAT by making use of

general approaches to root development, rooting depth and soil water holding

capacity. However, when using the management module, provision is made to

adjust these parameters according to the specific area and crop. In terms of

soil, characteristics can be changed by the user, this includes total available

water (total water holding capacity of the soil), maximum rain infiltration rate

(total amount of rain that the soil can absorb in the course of a day), initial soil

depletion (level which soil moisture is beneath field capacity at the beginning

of the cycle) and soil rooting depth (rooting depth of the crop or soil depth in

14

the case of an impermeable layer). Rainfall can be included or excluded in

water management calculations.

Crop characteristics for application by SAPWAT were mainly collected by

means of surveys of researchers, technicians and farmers and where possible

evaluated against existing published data (van Heerden 2008). It is therefore

unlikely that the majority of crop factors and growth stages derived for fruit

tree species would have been validated. There is therefore concern that the

currently available crop factors for fruit tree species in South Africa are not

robust enough to be applicable to all regions in South Africa where the fruit

trees are grown. In addition the four stage canopy growth descriptors may not

be sufficient to predict the changes that occur in evergreen canopies

throughout the year.

SAPWAT is currently being upgraded to SAPWAT3 (van Heerden et al.

2008). SAPWAT3 integrates an upgraded version of PLANWAT with the latest

SAPWAT crop irrigation requirement engine. This allows for data to be stored,

which was not possible for SAPWAT. SAPWAT3 also uses an internationally

accepted climate system, the Köppen-Geiger Climate System, in order to

standardize regions that form the background of the update of the crop factor

data. This system is based on a combination of temperature and precipitation

and is computed in terms of monthly or annual values. This makes SAPWAT

universally applicable. In addition, 50 years of calendar weather data,

including ETo has been incorporated into SAPWAT3 on a quaternary basis.

SAPWAT3 also makes provision for adjusting the slope of the crop factor

curve for the dominant third stage, as it is believed that a horizontal curve is

an over-simplification, particularly for fruit tree crops. This is achieved by

including a start and end entry for the 3rd growth phase when setting up the

growth factor curve in SAPWAT3. The dual crop coefficient approach is used

in SAPWAT3, where crop transpiration and evaporation from the soil surface

are calculated separately (Allen e al. 1998, van Heerden et al. 2008). Crop

water requirements are therefore calculated as follows

ETc = (Kcb + Ke)ETo (1)

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Although this is a more complex approach it gives a better understanding of

the role of soil evaporation as part of crop irrigation requirement estimation

and how different irrigation strategies can increase or reduce this loss (van

Heerden et al. 2008)

2.1.3 Soil Water Balance (SWB)

SWB is a mechanistic, real-time, generic crop growth / soil water balance

model, which was developed by Annandale et al., (1999), as an irrigation

management tool. From soil, weather and crop inputs it simulates crop growth

and provides insights into the one-dimensional biophysical links between the

atmosphere (environment), plant systems and the soil system. It has been

calibrated and tested under a range of conditions and crops and has proven

to be very helpful, with reliable correctness. Canopy growth and crop water

uptake is well simulated under both water-supply- and energy-demand-limited

conditions (Annandale et al. 2000). Subsequent to the development of the

original SWB model, a new version that performs energy interception and

water balance modelling in two dimensions, by trees planted in hedgerows

(SWB-2D), was developed and verified as an extension of the former model.

Two types of models are included in SWB-2D for hedgerow crops. Firstly, a

mechanistic two-dimensional energy interception and finite difference,

Richard’s equation based soil water balance model and secondly, an FAO-

based crop factor model, with a quasi-2D cascading soil water balance model

(Annandale et al. 2002). The first model calculates the two-dimensional

energy interception for hedgerow crops based on solar and row orientation,

tree size and shape, and leaf area density. A two-dimensional water

redistribution is also calculated with a finite difference solution. Some of the

input parameters for this model, such as leaf area density and soil saturated

hydraulic conductivities, are not always easy to obtain, which affects the ease

with which this model can be used. This prompted the development of the

second model which is based on the FAO crop factor approach and allows for

the prediction of crop water requirements with limited input data. This model,

16

includes a semi-empirical approach for partitioning above-ground energy, a

cascading soil redistribution that separates the wetted and non-wetted portion

of the ground and a prediction of crop yield according to the CROPWAT

model develop by the FAO (Smith 1992a). The two-dimensional model was

incorporated into the original SWB model (Annandale et al., 1999; Annandale

et al., 2002; Annandale et al., 2003; Annandale et al., 2004).

Input data to run the two dimensional canopy interception model are: day of

year, latitude, standard meridian, longitude, daily solar radiation, row width

and orientation, canopy height and width, stem height and distance to the

bottom of the canopy, extinction coefficient, absorptivity and leaf area density

(Annandale et al. 2002). A flow diagram for the energy interception model is

shown in Appendix B.

Spatial distribution of evaporation at the soil surface is calculated by the

model in two steps (Annandale et al. 2002):

1) potential evaporation at each node is estimated by applying the

Penman-Monteith equation (Allen et al. 1998), using radiation

estimated locally as an input, and

2) evaporation from the soil surface at each node is calculated as a

function of potential evaporation, air humidity and humidity of the soil

surface.

In order to simulate two dimensional water movement in the soil the following

inputs are required: altitude, rainfall and irrigation water amounts and

minimum and maximum daily temperature. The following soil measurements

are required for each soil layer (after establishing a grid of nodes): two points

on the water retention function (normally field capacity and permanent wilting

point), initial volumetric soil water content and bulk density. Row distance,

wetted diameter of micro-jets or drippers, fraction of roots in the wetted

volume of soil and distance of the nodes from the tree row are also required

as inputs. The flow diagram of the two-dimensional soil water balance for

hedgerow crops is shown in Appendix B.

17

Potential transpiration from the trees is calculated as follows (Annandale et al.

2002): potential evapotranspiration (PET) is partitioned between potential

evaporation from the soil and potential transpiration from the canopy. PET is

calculated using the Penman-Monteith equation (Allen et al. 1998) using

locally obtained weather data and the maximum crop factor after rainfall

occurs (Jovanovic and Annandale 1999). Local potential evaporation

calculated at each radiation node are weighted according to the surface and

are accumulated over the whole surface to calculate overall potential

evaporation (PE). Potential transpiration is then taken as the difference

between PET and PE. Root densities at different soil depths are accounted for

in the calculation of root water uptake using the approach of Campbell and

Diaz (1998). This allows for differentiation between atmospheric-limited and

supply-limited water demand. Root depth and the root fraction in the wetted

and non-wetted volume of the soil can also be entered by the user. Due to the

complexity of this model it has not been widely implemented as an irrigation

scheduling tool, but it may be a useful mechanistic approach to generating

orchard specific crop factors.

The FAO-based crop factor model does not attempt to grow the canopy

mechanistically and therefore the impact of water stress on canopy growth is

not simulated (Annandale et al. 2002). This model will, however, still perform

satisfactorily if the estimated canopy cover closely resembles that found in the

field. The following crop parameters need to be known in order to calculate

the crop factor: Kcb for the initial, mid- and late stages; crop growth periods in

days for initial, development, mid- and late stages; initial and maximum root

depth and initial crop height and maximum crop height. The following inputs

are required to run the FAO-type crop factor model: planting date (before

flowering for evergreen trees and bud burst for deciduous trees), latitude,

altitude and maximum and minimum air temperatures. In the absence of the

preferred measured data SWB estimates solar radiation, vapour pressure and

wind speed according to FAO recommendations (Smith 1992b, Smith et al.

1996). A flow diagram of the FAO-type crop factor model included in SWB is

included in Appendix B.

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The soil water balance under localised irrigation is also calculated in SWB

(crop factor approach) using a simplified procedure for the calculation of non-

uniform wetting of the soil surface, evaporation and transpiration (Annandale

et al. 2002). The data input required to run the two-dimensional cascading

model are rainfall and irrigation amounts, volumetric soil water content at field

capacity and permanent wilting point and initial soil water content for each

layer. Row spacing, wetted diameter, distance between micro-jets or drippers,

and the fraction of roots in the wetted volume are also required. A flow

diagram of the cascading soil water balance for fruit tree crops under localised

irrigation is included in Appendix B.

2.1.4 Penman-Monteith calculations of evapotranspiration

Evapotranspiration from a Clementine (Citrus reticulate Blanco) orchard in

Southern Italy, with a Mediterranean climate, was estimating using a modified

Penman-Monteith formula (Rana et al. 2005). The authors compared the sap

flow method for quantifying actual evapotranspiration with eddy covariance (a

micrometeorological method) and then examined the effectiveness of the crop

coefficient method to determine the water requirements of the orchard. Finally

a model of actual evapotranspiration on an hourly scale was developed

following the Penman-Monteith approach and simple standard meteorological

variables as inputs for determination of the canopy resistance. Good

correlation was found between sap flow and eddy covariance measurement of

transpiration, indicating that sap flow is a valid method for estimating field

scale evapotranspiration. Rana et al. (2005) found that the generic citrus crop

factor given by Allen et al. (1998) did not correspond well with the calculated

crop factor during a 105 day period during bud swelling and flowering. During

this time the calculated crop factor was considerably higher than the

prescribed factor given by Allen et al. (1998). This was probably due to it

being a period of active growth, with associated high stomatal conductance,

high wind speeds and high vapour pressure deficits (Rana et al. 2005). The

evaporation (E) modelling used a simple formulation based on the Penman-

Monteith model.

19

)/1(

/)(

ac

ap

rr

rDcAE

++∆+∆

=γ

ρλ (1)

where A = Rn – G is the available energy in W m-2, ρ is the air density in kg

m-3, ∆ is the slope of the saturation pressure deficit versus temperature

function in kPa °C -1, γ is the psychrometric constant in kPa °C -1, cp is the

specific heat of moist air in J kg-1 °C -1, D the vapour pressure deficit of the air

in kPa, rc is the bulk canopy resistance in s m-1 and ra is the aerodynamic

resistance in s m-1.

For irrigated crops the canopy resistance rc is not constant and varies

according to the available energy and the vapour pressure deficit. Katerji and

Perrier (1983) proposed to calculate rc as

br

ra

r

r

aa

c +=*

(2)

where a and b are empirical calibration coefficients which require

experimental determination. r* (s m-1) is given as:

A

Dcr pρ

γγ

∆+∆=* (3)

The resistance r* is linked to the isothermal resistance (ri = ρcpD/γH) (Monteith

1965) and can be considered as “climatic” resistance as it is dependent on

weather variables.

The canopy resistance (rc) was calculated from Eq. (1) by using E values

calculated from the eddy covariance method, together with the measured

values of vapour pressure deficit and available energy and the estimated

value for ra (Eq. (4)). These values of rc were combined with Eq. (2) to

estimate the parameters a and b. The model input variables are air

20

temperature and humidity, wind speed and direction, global radiation, rainfall

and mean height of the orchard.

*

)/()ln(

ku

dhdzr c

a

−−= (4)

where, z is a reference point sited in the boundary layer above the canopy, d

(m) is the zero plane displacement, hc is the mean height of the orchard, k is

the von Kármán constant and u* is the friction velocity (m s-1).

The canopy resistance of irrigated crops is therefore considered a function of

radiation and vapour pressure deficit (Eq. (3)). The final expression of the

model, on an hourly scale is:

)/(226.00042.0(

)/(*

a

ap

rr

rDcAE

++∆+∆

=γ

ρλ (5)

The model was tested for a year and compared with E measured by the sap

flow method. These two methods were found to agree closely. The model was

therefore shown to be preferable for calculating the water requirement of the

Clementine orchard, over an approach based on a crop coefficient which is

considered constant during the different growth stages.

Pereira et al. (2006) demonstrated that well irrigated canopy transpiration can

be accurately estimated using the conventional grass reference evaporation

and the Penman-Monteith formula described by FAO56 (Allen et al. 1998) in a

number of different species in New Zealand. These authors did not use the

recommended crop factors, but rather the computed reference evaporation

was corrected using the ratio between the canopy leaf area and the 2.88 m2

of the leaf area assumed for the hypothetical reference grass surface. A good

correlation was found between estimates using this method and sap flow

determined by the compensation heat-pulse technique (Pereira et al. 2006).

No other empirical correction or adjustment factor was required, but a reliable

21

method for estimating canopy leaf area is needed. This was tested in normal

apples, dwarf apples, olives, walnuts, grapevines and kiwifruit and it was

evident that under the same Eo conditions the sap flow of irrigated trees is a

positive function of the total canopy leaf area. In addition, when sap flow was

converted to per unit leaf area (L M-2 leaf d-1) an empirical function indicated

that sap flow corresponded, on average, to 35% of Eo. It is, however, unlikely

that this empirical relationship will be universally applicable.

These methods could prove very useful in improving estimation of water use

of fruit trees for growers, as they are relatively simple and take into account

local conditions and management practices. The major stumbling block,

however, remains the simple and reliable estimation of leaf area. If this can be

adequately addressed these types of models should prove to be extremely

useful. However, more mechanistic approaches, that include soil

measurements, need to be followed in order to ensure the accurate estimation

and then simulation of water use of fruit trees in various regions of South

Africa.

2.1.5 The Priestley-Taylor formula

Pereira et al. (2007b) describe a simple method for estimating daily sap flow

from well-irrigated orchards by using empirical relationships between radiation

absorbed by the canopy and daily sap flow measurements. This approach

was validated in an apple orchard consisting of large trees, an apple orchard

consisting of dwarf trees, an olive orchard and an isolated walnut tree. These

authors adapted the Priestley-Taylor formula to estimate daily sap flow of a

tree (T, L tree-1 day-1), using only net all-wave radiation, average air

temperature and tree leaf area. This is viewed as a simpler approach than the

traditional Penman-Monteith FAO56 model (Allen et al. 1998) which requires

local climate data, which is not always available from a standard climate

station.

22

The adapted Priestley-Taylor formula is as follows

λγα A

s

sT

+= (1)

where, s (kPa °C -1) is the slope of the saturation vapour pressure curve at

average daily temperature, γ is the psychrometric constant (= 0.066 kPa °C -1),

λ (=2.45 MJ L-1) is the latent heat of vaporization, A (MJ tree-1 day-1) is the

total amount of net (all-wave) radiation absorbed by the leaf canopy and α is

the Priestley-Taylor parameter. A was estimated from a measurement of the

net radiation over grass (RN, MJ m-2 day-1) and the trees total, one-sided, leaf

area (LA, m2 tree-1). According to Pereira et al. (2007a) the relationship is A =

0.303 (±0.032) RN LA. However using a fixed value of 0.32 for A, regardless of

the tree species, and α = 1.26, the authors were able to get a good estimate

of the daily sap flow recorded in the trunks of fully irrigated orchard trees

(Pereira et al. 2007b). The fixed value for A was determined previously for

citrus (Pereira et al. 2001). By determining leaf area, a more accurate

estimation can be obtained, but obtaining reliable leaf area measurements is

problematic and the authors are still trying to develop a rapid way of

assessing leaf area. Pereira et al. (2007b) found that empirically, sap flow per

unit tree leaf area is one quarter of the net radiation over a grass sward. This

is a very simple method for estimating the water use of orchards. The

applicability of this empirical relationship to different locations is questionable.

These two approaches were considered by the authors to be a simple working

alternative to the traditional crop-coefficient approach which relates crop water

use to potential evapotranspiration. Once again the lack of soil measurements

is a concern.

2.2 Remote Sensing

2.2.1 Surface Energy Balance Algorithm for Land (SEBAL)

SEBAL uses surface temperature To, hemispherical surface reflectance ro and

Normalized Difference Vegetation Index (NDVI), as well as their

23

interrelationships to infer surface fluxes for a wide spectrum of land types

(Bastiaanssen et al. 1998). Actual satellite data is inserted in the derivation of

the regression coefficients. Figure 6 illustrates the principal components of

SEBAL which converts remotely measured spectrally emitted and reflected

radiances into the surface energy balance and land wetness indicators. It

provides an opportunity to monitor energy balances from individual lands to

scheme level. Comparisons of predicted ET by SEBAL to lysimeter

measurements indicate a relatively good accuracy and show promise for

application in river basin planning and water rights management (Allen et al.

2001).

Figure 6: Principle components of SEBAL which converts remotely measured

spectrally emitted and reflected radiances into the surface energy balance

and land wetness indicators (Bastiaanssen et al. 1998).

Advantages of the SEBAL model include: minimal collateral data is required; it

is a physical concept and is therefore applicable for various climates; there is

no need for land use classification or the involvement of data demanding

hydrological models; evaporation is retrieved directly without the use of crop

factors; the method is suitable for all visible, near-infrared and thermal-

infrared radiometers, which implies it can be applied at different spatial and

temporal resolutions; for high resolution images, the results can be verified

with in-situ fluxes and soil water measurements; and it is a modular approach

(Bastiaanssen et al. 1998). The disadvantages include: the requirement for

Surface resistance

Visible

Near infrared

Thermal- infrared

Surface albedo

Vegetation

Surface temperature

Conversion

Net radiation flux density

Soil heat flux density

Sensible heat flux density

Latent heat flux density

Bowen-ratio

Evaporation fraction

SEBAL

Satellite radiances

Surface parameters

Land surface parameterization

Surface energy balance

Wetness indicator

24

cloud-free conditions; the presence of both drylands (Λ=0) and wetlands

(Λ=1); surface roughness is poorly described; it is only suitable for flat terrain

(Bastiaanssen et al. 1998) and it does not explain the causes, it only

measures the net effects of land surface processes (Stevens et al. 2005).

3 POTENTIAL GROWTH MODELS

Crop modelling is just one step in the process of designing better

management tools. Physical yield of a crop is determined by dry matter (DM)

production, DM distribution and DM content of the harvestable organ

(Marcelis et al. 1998). Dry matter production is largely driven by

photosynthesis, whilst photosynthesis is largely dependent on intercepted

radiation. Intercepted radiation in turn is determined by leaf area. Most crop

growth models are deterministic, which implies that in a given context, defined

by the set of variables, a unique output is calculated (Gary et al. 1998). In this

instance the variability in the system is ignored. In contrast, stochastic models

take uncertainty into account. In some instances these models are used to

reduce possible variability in the marketable product. These models are,

however, generally, not mechanistic and they are based on probability density

functions (Gary et al. 1998). Plant morphology is considered an important

input into growth models to simulate light interception and photosynthesis.

The dynamics of the spatial organisation of shoot and root systems has been

the main characteristic addressed in fruit trees to date (Gary et al. 1998). It is

also important to remember when modelling perennial fruit tree growth that

there will be an interaction between the current and subsequent growth cycle

(Monselise and Goldschmidt 1982).

Two basic frameworks have been adapted for modelling growth and

development. Firstly, those models which analyse growth, and secondly,

models which are photosynthesis-driven (Gary et al. 1998). Growth analysis

models attempt to represent key features of crop behaviour without a detailed

account of the respective mechanisms. In some instances the relative growth

rate is determined in terms of the changes in size and activity of the canopy

25

i.e. the leaf area ratio (m2 g-1) and the net assimilation rate (g m-2 d-1). These

models have proven to be highly effective in studying the reaction of the plant

to environmental conditions and comparisons between species. However,

they have proved ineffectual in modelling mutual shading of leaves (Marcelis

et al. 1998). Development of photosynthesis-based models has been the

dominant strategy in horticulture and agriculture in general. For a comparison

of empirical and mechanistic growth models see Figure 7. These models are

based on fairly detailed descriptions of the fate of carbon in the crop, from the

production to the storage and/or assimilation of photoassimilates. These

models allow for long term prediction of biomass accumulation and they can

also simulate short term (hourly) responses to the environment. They

calculate the interception of radiation by the leaf area in order to simulate the

production of photosynthates (Marcelis et al. 1998). The use of

photosynthates for respiration, conversion into structural dry matter and the

partitioning to different plant organs is calculated and finally fresh mass can

be estimated from dry mass. In order to simulate crop photosynthesis in detail

Figure 7: Comparison of A) empirical and B) mechanistic approaches to yield

prediction. The empirical model describes the basic processes which may

limit yield. In a mechanistic approach a more realistic description is possible,

provided the more numerous parameters can be estimated (Gary et al. 1998).

26

the calculation of total radiation interception and spatial variation in radiation

interception is required (Marcelis et al. 1998). These models also need to take

into account stomatal resistance, as this might limit photosynthesis especially

under high light conditions (Marcelis et al. 1998). This is achieved by

simulating a negative influence of stomatal resistance on the maximum rate of

leaf photosynthesis. Strong features of photosynthesis-based models are the

simulation of light interception and gross photosynthesis and weak features

are the simulation of leaf area development, maintenance respiration, organ

abortion, DM content and product quality (Marcelis et al. 1998).

Growth models must also model respiration, which is most often divided into

two components, viz. growth and maintenance (Amthor 1989). Maintenance

respiration is assumed to have priority over growth. The amount of assimilates

required for maintenance (Rm) are therefore subtracted from gross

photosynthesis (Pg) and the remaining assimilates are available for dry mass

increase (dw/dt)

dw/dt = Yg x (Pg - Rm) (1)

where, Yg is the growth conversion efficiency, which is the weight of dry mass

formed per unit weight of assimilates. The amount of assimilates used for

maintenance respiration as a fraction of the gross assimilate production has

been estimated to be 33% for peach (Grossman and DeJong 1994) and

depending on latitude and harvest date it varied between 30-80% for apple

(Wagenmakers 1995). It is important for fruit crops that assimilates partitioned

to the fruit should not be too high or an alternate bearing cycle will be initiated.

Thus thinning of fruits is often practiced in deciduous fruit orchards and on

occasion in citrus to avoid the entry into severe alternate bearing cycles. The

partitioning of this dry matter at the crop level is still poorly understood and

there seems to be great diversity in the manner in which crops partition

assimilates (Marcelis et al. 1998). As a result current available models are

fairly species specific.

27

The definition of morphological types and the adoption of some standards are

necessary in order to model the diversity of horticultural crops (Gary et al.

1998). The difficulty in modelling fruit tree species lies partly in the

heterogeneity found in orchard situations. Variability may be found in the

genetic make-up of the plants, the root and shoot environment and plant

management strategies (e.g. pruning). Attempts to include variability around

each parameter will result in unrealistic noise in the model (Gary et al. 1998)

and thus it is considered best to characterise variability in the environment,

e.g. the spatial variability in soil.

The modelling of sink strength has found numerous applications in

reproductive crops, including peach and citrus. Effects of environmental

conditions, pruning strategies and plant densities can be simulated through

their effects of sink strengths of individual organs, number of organs, or

indirectly through effects of source strength on organ formation (Marcelis et al.

1998).

By modelling photosynthesis and assimilate partitioning and therefore growth,

it is possible to mechanistically grow a canopy. Changes in canopy size

during a season can therefore be accurately estimated in a number of

different locations for use in evapotranspiration calculations.

3.1 Generic Growth Models

3.1.1 MAESTRA

The MAESTRA model (Medlyn 2004) is a development of the earlier

MAESTRO model (Wang and Jarvis 1990) and is a generic three dimensional

crop growth model that was initially developed for forestry. It predicts radiation

absorption, photosynthesis and transpiration of individual crowns of trees in a

stand and by the stand as a whole. It has proved useful in scaling up leaf

measurements to the whole canopy and whole stand. Each crown is divided

into 6 horizontal layers with each layer divided into 12 grid points of equal

volume. A number of physical and physiological properties are specified for

28

each layer, which includes radiation, temperature, leaf area index and leaf

nitrogen content. Environmental variables that drive model simulations are

radiation, air temperature, wind speed and atmospheric CO2 concentration

above the canopy. It is assumed that air humidity, temperature and CO2

concentration are uniform within the canopy.

The radiation penetrating to each point is calculated for three wavebands

(PAR, near infrared and thermal or longwave) and direct diffuse and scattered

radiation are considered separately (Wang and Jarvis 1990). This also takes

into account the hourly position of the sun in the sky and the daylength for a

specified date and latitude. The amount of absorbed radiation by the leaves

considers leaf inclination and the distribution of leaf ages.

Leaf boundary layer and stomatal conductance estimations require reference

height, wind speed, air temperature and water vapour saturation deficit at that

height, crown dimensions of all trees within the plot and PAR flux density

incident normal to the leaf surface. Photosynthesis and transpiration at each

grid point is calculated from absorbed radiation. Transpiration is calculated by

applying the Penman-Monteith equation (Jarvis and McNaughton 1986) to

each grid point and then summing over all grid points. The inputs are the net

radiation flux density absorbed by the leaf surface, the water vapour

saturation deficit of the ambient air, atmospheric pressure, leaf boundary layer

and stomatal conductances (Wang and Jarvis 1990).

Photosynthesis is estimated through a mechanistic C3 photosynthesis model

developed by Farquhar and Von Caemmerer (1982) and stomatal

conductance is estimated through a more empirical formulation, described by

Ball et al. (1987). Inputs for photosynthesis include leaf temperature, the leaf

boundary and stomatal and mesophyll conductances. Respiration is

calculated for leaves, branches and the stem/trunk.

Importantly MAESTRA does not attempt to simulate changes in soil water

content which is very important for micro-irrigated fruit tree orchards. In

addition, this model is unnecessarily complex for our purposes due to its

29

three-dimensional nature. Whilst young orchards are three-dimensional, as

they mature and form hedgerows the orchard becomes two-dimensional and,

as we will be quantifying water use in mature canopies, it will not be

necessary to use a three dimensional model.

3.2 Specific crop growth models

In a literature search conducted by Gary et al. (1998) they found that apple,

peach and kiwifruit represent half of the references on modelling of fruit crops.

Although difficult to parameterise these models are often very accurate due to

the species specific nature of carbon partitioning to various organs.

3.2.1 Pecan tree growth model

A simulation model of pecan growth and yield was developed by Andales et

al. (2006) based on an existing growth-irrigation scheduling model (GISM) (Al-

Jamal et al. 2002) as a tool for managing irrigation and the amount of pruning

required and the timing thereof. Modelling yield in fruit trees is complicated by

the alternate bearing habit of many fruit tree species, which is often a carry-

over from the previous season’s growth, as carbohydrate reserves from the

previous season impact flowering and yield in the current season. This model

simulates irrigation, alternate bearing, shoot biomass allocation and growth on

a daily basis. Figure 8 illustrates a flow chart of the model.

The irrigation object simulates pecan evapotranspiration, using the equation

developed by Samani and Pessarakli (1986) or Penman (Snyder and Pruitt

1992) and an adjusted crop coefficient, soil water balance and water stress.

The calculated non-stressed ET for a closed canopy is reduced by soil

moisture stress (Ks) and a canopy scale factor (Kca). The canopy scale factor

decreases ET from a closed canopy to an amount associated with the percent

cover at each simulation time interval (Andales et al. 2006). The soil moisture

stress function is dependent on the water holding capacity of the soil, the

30

rooting depth and the soil moisture at each time interval. The soil moisture

level at each particular interval takes into account precipitation.

Figure 8: Overall flow chart of the pecan model (Andales et al. 2006).

The biomass allocation object calculates the potential biomass allocation

ratios to different branches and the trunk. The model determines crown

distribution by separating the canopy into levels through successive

numbering of branching i.e. the trunk is level 1 and branches originating from

the trunk are level 2. The branches are distributed uniformly among north,

east, south, west and top side, i.e. 20% of the total number of branches is

located in each sector. The potential yearly growth is calculated according to

the original cross-section area of each branch and trunk, new growth cross-

section area, branch length, number and density. From this, the biomass

allocation ratio is determined.

The pruning object simulates removal of branches and affects the objects of

irrigation, biomass allocation ratios and alternate bearing. Shoot biomass

allocation ratios are adjusted according to pruning inputs. Nut yield and bud

growth rate are adjusted according to the calculated pruning coefficient (Kp).

31

The alternate bearing object simulates carbohydrate reserve amount in the

roots in order to determine fluctuations in yield from year-to-year. This object

also takes into account pruning. Finally, the growth object simulates tree and

nut growth and allocates the biomass components to the tree components

according to ET, WUE, water stress, allocation ratios, pruning effects and the

carbohydrate reserve amount. The allocation of aboveground dry matter is

allocated in the following order: leaves, nuts, reserve carbohydrate pools,

branches and the trunk. The leaf area per tree (m2 tree-1) is modelled by

multiplying total leaf biomass per tree by the specific leaf area (m2 kg-1).

Seasonal growth of each organ is controlled by critical growth stages

expressed as thermal time. Flow diagrams for the various objects included in

this model are shown in Appendix C.

Inputs to the model include daily maximum and minimum temperature and

humidity, solar radiation, wind speed, rainfall and soil temperature. In addition,

irrigation time and amount, pruning time, maximum diameter of branches to

be pruned and sides to be pruned are also included. The growth simulation

requires several initial conditions including tree trunk radius, root depth, root

carbohydrate reserve amount and soil water. To illustrate the difficulty in

parameterising these crop specific growth models a complete list of

parameters for the pecan growth model is as follows: wood density (kg m-3),

lateral outermost branch angle from the horizontal (degrees); maximum LAI

(m-2 m-2); specific leaf area (m2 kg-1); root:shoot biomass ratio; number of

trees per hectare; tree spacing (m); soil water holding capacity (m m-1); initial

trunk radius (m); initial root depth (m); initial carbohydrate reserve (kg tree-1);

initial soil water (m m-1); maximum root depth (m); optimum carbohydrate

reserve in mid winter (kg tree-1); reserve ratio (RR) lower limit; equilibrium RR,

RRe (0-1); maximum leaf growth rate (kg tree-1 GDD-1, where GDD is growing

degree days); maximum bud growth rate in spring (kg tree-1 GDD-1); maximum

shell growth rate (kg tree-1 GDD-1); maximum kernel growth rate (kg tree-1

GDD-1); maximum husk growth rate (kg tree-1 GDD-1); root growth rate (mm

GDD-1); growing degree days (GDD) to bud break; GDDs of leaf

photosynthesis beginning; GDDs to pollination; GDDs to shell hardening;

32

GDDs to leaf fall; GDDs to shuck split (Andales et al. 2006). These

parameters were either measured by the authors or taken from literature.

The growth component of this model is too complex for our purposes, whilst

the irrigation object, which estimates transpiration, is not mechanistic enough

for our purposes. Note should, however, be made of the manner in which

potential yearly cross section can be calculated from the original cross-section

area and how reserve starch levels can be used to estimate alternate bearing.

3.2.2 PEACH

PEACH describes the daily and seasonal carbon balance of peach trees

(Grossman and DeJong 1994a), which is driven by competition amongst

individual, semiautonomous plant organs based on the organ’s growth

potential. Carbon assimilation, maintenance respiration and the growth of

vegetative and reproductive organs of peach trees are all simulated by the

model. Photosynthetic carbon assimilation is simulated using daily minimum

and maximum temperatures and solar radiation as inputs. Carbohydrate

partitioning is simulated based on sink strength, proximity to sources, and the

quantity of carbohydrate available (Grossman and DeJong 1994a). The sink

strength of an organ is based on its genetically determined organ growth

potential, i.e. the maximum rate at which the organ can accumulate dry matter

per unit time. This is closely related to the ability to unload assimilates from

the phloem (DeJong and Grossman 1992). Carbon is partitioned first to

maintenance respiration, then to leaves, fruits, stems and branches and finally

the trunk (Figure 9). Residual carbohydrate from aboveground growth is used

to support root growth. Sink strength can be altered by suboptimal

environmental conditions and/or insufficient available resources e.g.

temperature and water availability.

PEACH is a state-variable model where fruit (number at full bloom, number at

thinning and weight), leaf weight, current-year stem weight, branch weight,

trunk weight and root weight are the state variables, and minimum and

33

maximum air and soil temperature, degree days and solar radiation are the

driving variables. The rate variables that characterise carbohydrate

assimilation and utilization were derived from previous in depth studies of

photosynthesis, respiration and growth potential in peach trees (DeJong and

Goudriaan 1989, DeJong et al. 1990, Grossman 1993, Grossman and

DeJong 1994b).

Figure 9: Flow diagram indicating the partitioning of carbon by the PEACH

model (Grossman and DeJong 1994). Boxes are state variables and valves

are rate variables. Solid lines represent carbon flows and dashed lines

represent information flows.

This brief description of the model illustrates the in-depth physiological studies

that are required in order to accurately parameterise the model. This includes

studies of photosynthesis, respiration and growth potential. These kinds of

34

studies are beyond the scope of this project. A detailed account of the

phenology of the species to be studied, together with accurate determinations

of leaf area should be sufficient to account for changes that occur in the

canopy during any season. However, as this model has been parameterised it

could be included in an evapotranspiration model in order to accurately

simulate canopy growth.

3.2.3 ‘CITROS’ – A Dynamic Model of Citrus Productivity

‘CITROS’ is based on a earlier citrus model by Goldschmidt and Monselise

(1977) who suggested that there are three stages in which “a decision should

be taken by the tree” in quantitative terms: how many flowers to produce, how

many fruit to set and how much will the fruit enlarge? Following this, a

mathematical model was developed based on empirical assumptions and

semi-quantitative information, which described alternate bearing and was able

to predict the impact of fruit thinning on final yield. ‘CITROS’ has been

developed to identify and quantify productivity problems, predict yields and

devise methods to optimise citrus production (Bustan et al. 1999). The model

is based on the assumption that when all the needs are optimally managed,

dry matter production and allocation become the fundamental processes that

limit productivity. Parameters that required determination include, the potential

growth rate of the fruit, relative growth rate of fruit under continuous optimum

conditions, conversion coefficient from fresh to dry matter (kDM) and a

conversion factor from dry matter to glucose (I/YG), coefficient of maintenance

respiration response to temperature (Q10), daily carbon exchange rate of the

canopy (estimated to be 340 g glucose by Syvertsen and Lloyd (1994)), and

technical coefficients for the estimation of fruit drop and sensitivity to

deficiencies (Bustan et al 1999).

Temperature was used to estimate physiological time. Effective Heat Units

(EHU) are used and are based on an optimum response curve of citrus to

temperature. This allows the negative effect of high temperature on growth to

be predicted and it accounts for the minimum temperature at which growth is

35

possible. The relative growth rate of the fruit (RGROPT), under continuous

optimal conditions, is related to temperature by dividing the potential relative

growth rate (RGRP, determined under sink-limited growth) by a temperature

coefficient (the ratio of actual accumulated EHU and the maximum EHU).

The potential growth rate function (GRPOT) can then be expressed as the

product of RGROPT, the current fresh weight of the fruit and the current

temperature coefficient (kT).

As with most growth models, ‘CITROS’ distinguishes between growth and

maintenance respiration and gives priority to maintenance respiration (Bustan

et al. 1999). In order to calculate maintenance respiration (rm), growth

respiration is subtracted from total respiration. Growth respiration is calculated

from dry weight composition (McDermitt and Loomis 1981) and the response

to temperature is assumed to be the same as that for growth rate. The

product of fruit dry weight (w), rm and kdTr (temperature response coefficient)

divided by 1.466 (to convert CO2 to glucose) gives the demand for glucose by

fruit maintenance respiration (DEMr). Priority is given to the fulfilment of DEMr

and only the remaining carbohydrates are available for growth. The total

amount of carbohydrates required daily is dependent on the number of fruits

on the tree, which varies greatly during the first few months of fruit

development. The abscission of fruit 80 days after anthesis (DAA) has also

been accounted for in the model, by including a mechanism of self-thinning

which responds to carbohydrate shortages. Photosynthetic rates were

adopted from a previous publication (Syvertsen and Lloyd 1994) and adapted

to local conditions using the EHU method (Bustan et al. 1999). Root demand

for carbohydrates has not been accounted for in this model.

4. DISCUSSION AND CONCLUSIONS

The increasing stress on water resources in South Africa has resulted in the

identification of the need to improve knowledge on water use of fruit tree

species. This will allow the more efficient scheduling of irrigation and the

issuing of water licenses that adequately reflect water requirements of fruit

36

tree orchards. As it is virtually impossible to measure water use under all the

possible combinations of climate, soils and management conditions in South

Africa, it is necessary to develop a model, or verify existing models, which can

accurately simulate water use of fruit trees across all conditions. Modelling

evapotranspiration in orchards is complex due to the discontinuous nature of

the canopy, the localised irrigation (micro- or drip) used in orchards and the

fact that this is a perennial crop, with carryover effects from the previous

season impacting the current season’s growth. It is therefore important that

the correct input variables are chosen and measured in order to accurately

parameterise the model.

As it is vital that the chosen model is able to extrapolate water use of fruit tree

species over a wide range of conditions and management practices it will be

necessary to use a mechanistic approach, where a thorough understanding of

the mechanisms driving transpiration is evident. This largely eliminates

models based on empirical relationships, which can seldom be used in

conditions outside the environment in which they were calibrated. It is also

important that the model should be able to differentiate between water-supply-

or atmospheric-demand-limited water uptake, which implies that

measurements should take into account the soil-plant-atmosphere continuum.

This review has covered both evapotranspiration models and growth models.

Whilst evapotranspiration models attempt to estimate water use of plants,

growth models are largely concerned with carbon assimilation and partitioning

in order to predict yield in many instances. These growth models can be

incorporated into evapotranspiration models in order to accurately “grow the

canopy” over a season, thereby enabling accurate estimations of water use as

the canopy dimensions change. These models can, however, be difficult to

parameterise and as a result the number of input variables needed to be

quantified is drastically increased. Tables 2 to 5 list the variables and

parameters required for each of the evapotranspiration and growth models

included in this review. The number of input variables and difficulty in

parameterising growth models is clearly illustrated in Table 5.

37

Table 2: Variables that must be quantified for the evapotranspiration models

SWB-2D Penman-Monteith FAO 56 SAPWAT Radiation

interception Crop factor Rana et al.

2005 Pereira et al. 2006

Priestley- Taylor

Net radiation Net radiation Day of year Max. and min. air temperature Net radiation Net radiation

Net radiation over a grass reference

Min., max. and mean air temperature

Min., max. and mean air temperature

Daily solar radiation Solar radiation Wind speed Wind speed Average air

temperature

Humidity (saturated vapour pressure deficit)

Humidity (saturated vapour pressure deficit)

Rainfall and irrigation amounts

Humidity Friction velocity Air temperature

Wind speed Wind speed Min. and max. daily temperatures

Wind speed Soil heat flux

Humidity Rainfall and irrigation amounts

Air temperature

Wind speed Vapour pressure deficit

Rainfall

38

Table 3: Parameters required for the evapotranspiration models

SWB-2D Penman-Monteith FAO 56 SAPWAT Radiation

interception Crop factor Rana et al.

2005 Pereira et al. 2006

Priestley- Taylor

Length of crop growth stages

Length of crop growth stages Longitude Length of growth

stages

Empirical calibration coefficients, a and b for the canopy resistance

Canopy leaf area Leaf area

Crop factors Crop factors Standard meridian Crop factors Mean orchard

height

Leaf area of the grass reference surface

Psychrometric coefficient

Threshold point at which soil water limits transpiration

Planting date Latitude Initial max. rooting depth

Psychrometric coefficient Altitude Altitude

Altitude Threshold point at which soil water limits transpiration

Row width & orientation

Initial max. crop height Altitude Atmospheric

pressure Atmospheric pressure

Atmospheric pressure Leaf area index Ellipse height and width Planting date

Atmospheric pressure

Latent heat of vaporization α = 1.26

Latent heat of vaporization Crop height Bare stem

height Altitude Psychrometric coefficient

Psychrometric coefficient Rooting depth Extinction

coefficient Latitude

Leaf resistances Absorptivity

Volumetric water content at field capacity and permanent wilting point

Irrigation type Leaf area density Row spacing

39

Irrigation frequency Bulk density of the soil

Wetted diameter of irrigation

Wetted diameter

Volumetric water content at field capacity and permanent wilting point

Distance between microjets/drippers

Altitude Initial volumetric soil water content

Fraction of roots in the wetted zone

Atmospheric pressure Wetted diameter of irrigation

Latent heat of vaporization Rooting depth

Psychrometric constant

Fraction of roots in the wetted volume of soil

40

Table 4: Variables that must be quantified for SEBAL and the various growth models

SEBAL MAESTRA Pecan PEACH ‘CITROS’

Satellite images Day of year Daily max. and min. temperature

Min. and max. air temperature Days after anthesis

Surface temperature Net radiation flux density absorbed by the leaves Humidity Soil temperature Air temperature

Shortwave atmospheric transmittance PAR flux density Solar radiation Degree-days No. of fruit per tree

Vegetation height Air and leaf temperature Wind speed Solar radiation Fruit weight

Leaf area index Rainfall

Wind speed Soil temperature

Atmospheric CO2 above the canopy Irrigation time and amount

Water vapour saturation deficit Pruning time

Atmospheric pressure Max. diameter of branches to be pruned

Soil surface temperature Side of the tree to be pruned

Tree trunk radius

Root depth

Root carbohydrate reserve amount

41

Table 5: Parameters required for SEBAL and the various growth models

SEBAL MAESTRA Pecan PEACH ‘CITROS’

Thermal infrared emissivity of the atmosphere Hemisphere Wood density Fruit number at bloom Maintenance respiration

Soil heat flux Latitude Lateral outermost branch angle from the horizontal

Fruit number at thinning (defruited, heavily thinned at bloom, thinned at two weeks, thinned at four weeks, thinned at eight weeks, unthinned)

Optimum and minimum temperature for growth

Surface roughness length Longitude Maximum LAI Individual fruit weight Coefficient of maintenance respiration response to temperature

Friction velocity Slope Specific leaf area Leaf weight Max. canopy photosynthesis

Monin-Obukhov length Plot dimensions Root:shoot biomass ratio Current-year stem weight Potential relative growth rate

Total number of trees Number of trees per ha Branch weight Dry weight composition

Transmittance and reflectance of PAR, NIR and thermal radiation of leaves

Tree spacing Trunk weight

Reflectance of PAR, NIR and thermal radiation at the soil surface

Soil water holding capacity Root weight

Inclination angle of leaves Initial trunk radius Rate variables for growth potential

Leaf area density Initial root depth Rate variables for photosynthesis

Leaf age distribution Optimum CHO reserve in mid-winter

Rate variables for respiration

42

Quantum yield Equilibrium reserve ratio (RR) Specific respiration rate

Maximum rate of photosynthesis RR lower limit

Maintenance respiration rate (leaf, current-year stem, branch, trunk, root and fruit)

Empirical approach: convexity of photosynthesis light response curve

Max. leaf growth rate Growth respiration coefficient

Mechanistic approach: max. rate of carboxylation and potential electron transport, effective Michaelis-Menton constant for CO2, initial slope and convexity of the light response curve of the potential electron transport

Max. bud growth rate in spring Relative growth rate

Temperature coefficient of the dark respiration rate Max. shell growth rate Net sink strength of growing

organs

dark respiration at leaf temperature of 0.0°C Max. kernel growth rate

coefficients defining the temperature response of the CO2 compensation point

Max. husk growth rate

Root growth rate

Growing degree days (GGD) to bud break

GDDs of leaf photosynthesis beginning

GDDs to pollination, shell hardening, leaf fall and shuck split

43

In terms of evapotranspiration models, FAO56 (Allen et al. 1998) is the most

widely used in fruit tree orchards due to its physiological and biological basis

(Pereira et al. 2006). The four-stage approach for the canopy size works well

in field crops and is largely applicable to deciduous fruit crops, however, it is

unlikely that such an approach will adequately accommodate evergreen

species, such as citrus and macadamia, which grow in flushes and at any one

stage will have leaves of vastly different ages on the tree, with different

associated resistances. Crop factors also assume that plant growth and

development is dependent on calendar time but thermal time (degree days)

and water supply are important influences on canopy development (Ritchie

and NeSmith 1991). A more mechanistic approach for converting ETo,

estimated using the Penman-Monteith equation, to crop transpiration is

therefore required. Both Rana et al. (2005) and Pereira et al. (2006)

attempted this. Rana et al. (2005) included estimates for canopy resistance,

climatic resistance and aerodynamic resistance in an adapted Penman-

Monteith equation and concluded that this was a more accurate manner to

estimate ET in a Clementine orchard, than a constant crop factor during the

different growth stages. In contrast, Pereira et al. (2006) used the ratio

between canopy leaf area and the leaf area of the reference grass surface to

adjust ETo to ETc in a number of different orchards. While these methods

provide a more accurate manner to convert ETo to ETc for a given orchard,

they are not mechanistic enough and exclude soil conditions. In addition, the

method of Pereira et al. (2006) is based on an empirical relationship which

may not be applicable to all situations.

SAPWAT (Crosby and Crosby 1999) is based on FAO56 and thus encounters

the same shortfalls when trying to estimate water use of fruit tree orchards

over different environmental and management conditions. Although

SAPWAT3 (van Heerden et al. 2008) has tried to improve the prediction of the

crop factor curve, e.g. by allowing the crop factor curve to be sloped during

phase 3 of growth, these improvements need to be validated with

experimental data. As SAPWAT is primarily used as a planning tool it will be

extremely valuable for Water Use Associations, governmental organisations

44

and growers in South Africa to have access to improved crop factors for fruit

trees in order to accurately estimate water use of orchards. This will be

achieved through the accurate estimation of the input variables and the

characterisation of the growth stages required to construct the crop factor

curve (see Table 3 for the required parameters). The need to improve crop

factors stems from the ease with which water use can be estimated using

FAO56 or SAPWAT, which is very attractive to growers. As the accuracy of

estimated transpiration from ETo hinges on the appropriateness of the crop

factor, a more mechanistic approach for crop factor determination should

result in more accurate estimations of crop water use, which should

subsequently improve the efficiency of irrigation scheduling in South Africa.

SWB-2D, on the other hand, allows a more mechanistic approach for the

estimation of water use in fruit tree crops, as it attempts to account for the

complexity associated with hedgerow crops, by modelling radiation

interception across the discontinuous canopy and soil water content in a two

dimensional profile (Annandale et al. 2002). It also provides insight into the

soil-plant-atmosphere continuum, which allows the differentiation between

supply- and demand-limited water uptake. The biggest drawback of this model

is the perceived complexity, but it may be a very useful mechanistic approach

to generating orchard specific crop factors.

The advent of the modern computer and computer programming prompted

the development of numerous crop growth models. Annual crops were the

first to be modelled, followed by the more complex horticultural crops. As

mentioned, these models are difficult to parameterise, largely due to the

required measurements of photosynthesis and respiration that are needed in

order to accurately simulate carbon assimilation and partitioning in the plant.

The incorporation of such growth models in an evapotranspiration model

should allow the accurate estimation of water use over a season, as the

canopy and root system enlarges. However, it has been noted that if soil

water is not limiting, the transpiration will largely be determined by the leaf

area (Pereira et al. 2006). This is reflected in many of the models which

require leaf area as an input variable (see Table 3 and 4). Thus, while

45

modelling tree growth will undoubtedly contribute to a more mechanistic

approach to the estimation of evapotranspiration, it may be more expedient to

develop a reliable and simple method for estimating leaf area. This will

improve the water use estimations considerably without having to quantify

photosynthesis and respiration parameters. Numerous methods have been

used to determine leaf area, which includes Lang and Xiang (1986), Villalobos

et al. (1995), Wünsche and Lakso (1998), Jovanovic and Annandale (1998),

Broadhead et al. (2003) and Jonckheere et al. 2004).

Data needed to calibrate and validate models will be obtained by determining

sap flow rates. In addition, total evaporation will be determined either through

the use of scintillometry or the eddy covariance method (depending on

orchard layout and topography). The scintillometry technique integrates

evaporation rates from the soil and vegetation represented along a bean path.

Together with supporting data on net radiation, soil heat fluxes, wind speed,

air temperature and air pressure, scintillometry allows the calculation of mean

surface sensible heat along the light path. The latent heat fraction that

equates to the evaporation rate is derived from an energy balance equation.

Using the additional sap flow data, the soil evaporation component may be

deduced by subtraction, thereby allowing the different components of

evaporation to be quantified which can be used to compare model estimates

against.

As water use of fruit tree species needs to be estimated over a wide range of

environmental, soil and management conditions, a generic, mechanistic

model of evapotranspiration is required. Of the models reviewed SWB-2D

(Annandale et al. 2002) fits these requirements the best. Importantly, it

includes either an hourly- or daily-time step and can therefore be used in real-

time. The minimum input variables that need to be quantified therefore include

(Table 3): longitude, latitude, standard meridian, row width and orientation,

ellipse height and width, bare stem height, extinction coefficient, absorptivity,

leaf area density, bulk density of the soil, volumetric water content at field

capacity and permanent wilting point, initial volumetric soil water content,

wetted diameter of irrigation, rooting depth and fraction of roots in the wetted

46

zone (Annandale et al. 2002). Variables needed to run the model include day

of year, daily solar radiation, rainfall and irrigation amounts, minimum and

maximum daily temperatures, humidity and wind speed. The inclusion of

rainfall and rooting volumes in the model are very important as it should allow

the grower to make more efficient use of rain and therefore reduce

dependency on available water resources. If a crop specific growth model

exists that has already been parameterised, e.g. PEACH or ‘CITROS’, then

this could be incorporated in SWB-2D in order to mechanistically “grow a

canopy”. However, if unavailable or unsuitable the modelling of leaf area

could prove a far better parameter to accurately estimate water use. The

problem encountered with evergreen crops, such a citrus and macadamia,

where there are leaves of vastly different ages on the tree at any one time

could be partly overcome by including parameters found in the MAESTRA

model which accounts for this variability.

47

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55

6. APPENDIX A - FLOW DIAGRAM INDICATING THE PROCEDU RE

FOR THE ESTIMATION OF CROP EVAPOTRANSPIRATION (ET C)

Calculate Reference ET o

Select stage lengths Verify and supplement locally

Kcb + Ke Dual crop coefficient

Select values for Kcb ini, Kcb mid & Kcb end

Kc

Single crop coefficient

Adjust Kcb min & Kcb end to local climatic conditions

Construct Kcb curve

Determine daily Ke values for surface evaporation

Kc = Kcb + Ke

Select values for Kc ini, Kc mid & Kc end

Adjust Kc ini to reflect wetting frequency

of soil surface

Adjust Kc mid & Kc end to local climatic conditions

Construct Kc curve

ETc = KcETo

56

7. APPENDIX B - FLOW CHART OF THE VARIOUS SUB-MODEL S

WITHIN THE SWB-2D MODEL

Figure 1: Flow diagram of the two-dimensional energy interception model for hedgerow tree crops

57

Figure 2: Flow diagram of the two-dimensional soil water balance model for hedgerow tree crops

58

Figure 3: Flow diagram of the FAO-type crop factor model

59

Figure 4: Flow diagram of the cascading soil water balance for tree crops under localised irrigation

60

8. APPENDIX C - FLOW CHART OF THE VARIOUS OBJECTS W ITHIN

THE PECAN TREE GROWTH MODEL (ANDALES ET AL. 2006)

Figure 1: Flow chart of the irrigation object (Andales et al. 2006).

61

Figure 2: Flow chart of the shoot biomass allocation ratio object (Andales et al. 2006).

62

Figure 3: Flow chart of the pruning object (Andales et al. 2006).

Figure 4: Flow chart of the alternate bearing object (Andales et al. 2006).